Mathematics - Questions from the 2022 JEE

Question 1

The total number of functions, f : {1, 2, 3, 4} → {1, 2, 3, 4, 5, 6} such that f(1) + f(2) = f(3), is equal to
(A) 60
(B) 90
(C) 108
(D) 126

Answer: (B) 90

Question 2

If α, β, γ, δ are the roots of the equation x^4 + x^3 + x^2 + x + 1 = 0, then α^2021 + β^2021 + γ^2021 + δ^2021 is equal to
(A) –4
(B) –1
(C) 1
(D) 4

Answer: (B) –1

Question 3

The number of q∈ (0, 4π) for which the system of linear equations
3(sin 3θ) x – y + z = 2
3(cos 2θ) x + 4y + 3z = 3
6x + 7y + 7z = 9
has no solution, is
(A) 6
(B) 7
(C) 8
(D) 9

Answer: (B) 7

Question 4

If \(\displaystyle \lim_{n\rightarrow\infty}\left(\sqrt{n^2-n-1}+n\alpha+\beta\right)=0 \) then 8(α + β) is equal to
(A) 4
(B) –8
(C) –4
(D) 8

Answer: (C) –4

Question 5

If the absolute maximum value of the function \(f(x) = (x^2 – 2x + 7) e^{(4x^3-12x^2-180x + 31)} \) in the interval [–3, 0] is f(α), then
(A) α = 0
(B) α = –3
(C) α∈ (–1, 0)
(D) α∈ (–3, –1]

Answer: (B) α = –3

Question 6

The curve y(x) = ax^3 + bx^2 + cx + 5 touches the x-axis at the point P(–2, 0) and cuts the y-axis at the point Q, where y′ is equal to 3. Then the local maximum value of y(x) is
(A) \(\frac{27}{4} \)
(B) \(\frac{29}{4} \)
(C) \(\frac{37}{4} \)
(D) \(\frac{9}{2} \)

Answer: (A) \(\frac{27}{4} \)

Question 7

The area of the region given by
\(A=\left\{\left(x,y\right);x^2\leq y \leq \text{min}\left\{x+2, 4-3x \right\} \right\}\) is
(A) \(\frac{31}{8}\)
(B) \(\frac{17}{6}\)
(C) \(\frac{19}{6}\)
(D) \(\frac{27}{8}\)

Answer: (B) \(\frac{17}{6}\)

Question 8

For any real number x, let [x] denote the largest integer less than equal to x. Let f be a real valued function defined on the interval [–10, 10] by \(f\left(x\right)=\left\{\begin{matrix}x-[x], \text{ if } [x] \text{ is odd} \\1+[x]-x, \text{if }[x]\text{ is even}\end{matrix}\right.\)
Then the value of \(\frac{\pi^2}{10}\displaystyle\int\limits_{-10}^{10}f\left(x\right)\cos\pi x\ dx\) is
(A) 4
(B) 2
(C) 1
(D) 0

Answer: (A) 4

Question 9

The slope of the tangent to a curve C : y = y(x) at any point (x, y) on it is \(\frac{2e^{2x}-6e^{-x}+9}{2+9e^{-2x}}\). If C passes through the points \(\left(0, \frac{1}{2}+\frac{\pi}{2\sqrt{2}}\right)\text{ and }\left(\alpha,\frac{1}{2}e^{2\alpha}\right),\) then e^α is equal to
(A) \(\frac{3+\sqrt{2}}{3-\sqrt{2}} \)
(B) \(\frac{3}{\sqrt{2}}\left(\frac{3+\sqrt{2}}{3-\sqrt{2}} \right)\)
(C) \(\frac{1}{\sqrt{2}}\left(\frac{\sqrt{2}+1}{\sqrt{2}-1} \right)\)
(D) \(\frac{\sqrt{2}+1}{\sqrt{2}-1}\)

Answer: (B) \(\frac{3}{\sqrt{2}}\left(\frac{3+\sqrt{2}}{3-\sqrt{2}} \right)\)

Question 10

The general solution of the differential equation (x – y^2)dx + y(5x + y^2)dy = 0 is :
(A) \(\left(y^2+x\right)^4=C\left|\left(y^2+2x\right)^3\right|\)
(B) \(\left(y^2+2x\right)^4=C\left|\left(y^2+x\right)^3\right|\)
(C) \(\left|\left(y^2+x\right)^3\right|=C\left(2y^2+x\right)^4\)
(D) \(\left|\left(y^2+2x\right)^3\right|=C\left(2y^2+x\right)^4\)

Answer: (A) \(\left(y^2+x\right)^4=C\left|\left(y^2+2x\right)^3\right|\)

Question 11

A line, with the slope greater than one, passes through the point A(4, 3) and intersects the line
x – y – 2 = 0 at the point B. If the length of the line segment AB is \(\frac{\sqrt{29}}{3}, \ \text{then B also lies on the line :}\) 
(A) 2x + y = 9
(B) 3x – 2y = 7
(C) x + 2y = 6
(D) 2x – 3y = 3

Answer: (C) x + 2y = 6

Question 12

Let the locus of the centre (α, β), β> 0, of the circle which touches the circle x^2 +(y – 1)^2 = 1 externally and also touches the x-axis be L. Then the area bounded by L and the line y = 4 is :
(A) \(\frac{32\sqrt{2}}{3}\)
(B) \(\frac{40\sqrt{2}}{3}\)
(C) \(\frac{64}{3}\)
(D) \(\frac{32}{3}\)

Answer: (C) \(\frac{64}{3}\)

Question 13

Let P be the plane containing the straight line \(\frac{x-3}{9}=\frac{y+4}{-1}=\frac{z-7}{-5}\) and perpendicular to the plane containing the straight lines \(\frac{x}{2}=\frac{y}{3}=\frac{z}{5} \) and \(\frac{x}{3}=\frac{y}{7}=\frac{z}{8}.\) If d is the distance P from the point (2, –5, 11), then d^2 is equal to :
(A) \(\frac{147}{2}\)
(B) 96
(C) \(\frac{32}{3}\)
(D) 54

Answer: (C) \(\frac{32}{3}\)

Question 14

Let ABC be a triangle such that \(\overrightarrow{BC}=\overrightarrow{a},\overrightarrow{CA}=\overrightarrow{b},\overrightarrow{AB}=\overrightarrow{c},\left|\overrightarrow{a}\right|=6\sqrt{2},\left|\overrightarrow{b}\right|=2\sqrt{3}\) and \(\overrightarrow{b}\cdot\overrightarrow{c}=12.\) Consider the statements :
\(\left(S1\right):\left|\left(\overrightarrow{a}\times\overrightarrow{b}\right)+\left(\overrightarrow{c}\times\overrightarrow{b}\right)\right|-\left|\overrightarrow{c}\right|=6\left(2\sqrt{2}-1\right)\)
\(\left(S2\right):\angle ACB=\cos^{-1}\left(\sqrt{\frac{2}{3}}\right) \)
Then
(A) Both (S1) and (S2) are true
(B) Only (S1) is true
(C) Only (S2) is true
(D) Both (S1) and (S2) are false

Answer: (C) Only (S2) is true

Question 15

If the sum and the product of mean and variance of a binomial distribution are 24 and 128 respectively, then the probability of one or two successes is :
(A) \(\frac{33}{2^{32}} \)
(B) \(\frac{33}{2^{29}} \)
(C) \(\frac{33}{2^{28}} \)
(D) \(\frac{33}{2^{27}} \)

Answer: (C) \(\frac{33}{2^{28}} \)

Question 16

If the numbers appeared on the two throws of a fair six faced die are α and β, then the probability that x^2 + αx + β> 0, for all x ∈ R, is :
(A) \(\frac{17}{36}\)
(B) \(\frac{4}{9}\)
(C) \(\frac{1}{2}\)
(D) \(\frac{19}{36}\)

Answer: (A) \(\frac{17}{36}\)

Question 17

The number of solutions of |cos x| = sinx, such that –4π ≤ x ≤ 4π is :
(A) 4
(B) 6
(C) 8
(D) 12

Answer: (C) 8

Question 18

A tower PQ stands on a horizontal ground with base Q on the ground. The point R divides the tower in two parts such that QR = 15 m. If from a point A on the ground the angle of elevation of R is 60° and the part PR of the tower subtends an angle of 15° at A, then the height of the tower is :
(A) \(5\left(2\sqrt{3}+3\right)\text{ m}\)
(B) \(5\left(\sqrt{3}+3\right)\text{ m}\)
(C) \(10\left(\sqrt{3}+1\right)\text{ m}\)
(D) \(10\left(2\sqrt{3}+1\right)\text{ m}\)

Answer: (A) \(5\left(2\sqrt{3}+3\right)\text{ m}\)

Question 19

Which of the following statements is a tautology?
(A) \(\left(\left(\sim p\right)\vee q \right)\Rightarrow p\)
(B) \(p\Rightarrow \left(\left(\sim p\right)\vee q\right)\)
(C) \(\left(\left(\sim p\right)\vee q\right)\Rightarrow q\)
(D) \(q\Rightarrow \left(\left(\sim p\right)\vee q\right)\)

Answer: (D) \(q\Rightarrow \left(\left(\sim p\right)\vee q\right)\)

Question 20

For \(z \in \mathbb{C}\ \text{if the minimum value of}\ \left(|z-3\sqrt{2}| + |z-p\sqrt{2}i|\right)\)  is 5√2, then a value of p is _________.
(A) 3
(B) \(\frac{7}{2}\)
(C) 4
(D) \(\frac{9}{2}\)

Answer: (C) 4

Question 21

The number of real values of λ, such that the system of linear equations
2x – 3y + 5z = 9
x + 3y – z = –18
3x – y + (λ^2 – | λ |)z = 16
has no solutions, is
(A) 0
(B) 1
(C) 2
(D) 4

Answer: (C) 2

Question 22

The number of bijective functions f : {1, 3, 5, 7, …, 99} → {2, 4, 6, 8, ….., 100} such that \(f(3)\ge f(9)\ge f(15)\ge f(21)\ge … \ge f(99)\) is_____.
(A) \(^{50}P_{17}\)
(B) \(^{50}P_{33}\)
(C) \(33!\times 17!\)
(D) \(\frac{50!}{2}\)

Answer: (B) \(^{50}P_{33}\)

Question 23

The remainder when (11)^1011 + (1011)^11 is divided by 9 is
(A) 1
(B) 4
(C) 6
(D) 8

Answer: (D) 8

Question 24

The sum \(\sum_{n=1}^{21}\frac{3}{(4n-1)(4n+3)}\) is equal to
(A) \(\frac{7}{87}\)
(B) \(\frac{7}{29}\)
(C) \(\frac{14}{87}\)
(D) \(\frac{21}{29}\)

Answer: (B) \(\frac{7}{29}\)

Question 25

\(\displaystyle \lim_{ x\to \frac{x}{4}}\frac{8\sqrt{2}-(\cos x + \sin x)^7}{\sqrt{2}-\sqrt{2}\sin 2x}\) is equal to
(A) 14
(B) 7
(C) \(14\sqrt{2}\)
(D) \(7\sqrt{2}\)

Answer: (A) 14

Question 26

\(\displaystyle \lim_{n \to \infty}\frac{1}{2^n}\left(\frac{1}{\sqrt{1-\frac{1}{2^n}}} + \frac{1}{\sqrt{1-\frac{2}{2^n}}} + \frac{1}{\sqrt{1-\frac{3}{2^n}}} + … + \frac{1}{\sqrt{1-\frac{2^n -1}{2^n}}}\right)\) is equal to
(A)
(B) 1
(C) 2
(D) –2

Answer: (C) 2

Question 27

If A and B are two events such that \(P(A)=\frac{1}{3}, P(B)=\frac{1}{5}\ \text{and}\ P(A\cup B)=\frac{1}{2},\ \text{then}\) \(P(A|B’) + P(B|A’|)\) is equal to
(A) \(\frac{3}{4}\)
(B) \(\frac{5}{8}\)
(C) \(\frac{5}{4}\)
(D) \(\frac{7}{8}\)

Answer: (B) \(\frac{5}{8}\)

Question 28

Let [t] denote the greatest integer less than or equal to t. Then the value of the integral \(\int_{-3}^{101}\left([\sin (\pi x)]+e^{[\cos(2\pi x)]}\right)dx\) is equal to
(A) \(\frac{52(1-e)}{e}\)
(B) \(\frac{52}{e}\)
(C) \(\frac{52(2+e))}{e}\)
(D) \(\frac{104}{e}\)

Answer: (B) \(\frac{52}{e}\)

Question 29

Let the point P(α, β) be at a unit distance from each of the two lines L1 : 3x – 4y + 12 = 0 and L2 : 8x + 6y + 11 = 0. If P lies below L1 and above L2, then 100(α + β) is equal to
(A) –14
(B) 42
(C) –22
(D) 14

Answer: (D) 14

Question 30

Let a smooth curve y = f(x) be such that the slope of the tangent at any point (x, y) on it is directly proportional to (-y/x). If the curve passes through the points (1, 2) and (8, 1), then \(\left|y\left(\frac{1}{8}\right)\right|\ \text{ is equal to}\)
(A) 2 loge2
(B) 4
(C) 1
(D) 4 loge2

Answer: (B) 4

Question 31

If the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\ \text{meets the line}\ \frac{x}{7}+\frac{y}{2\sqrt{6}} =1\)  on the x-axis and the line \(\frac{x}{7}-\frac{y}{2\sqrt{6}} =1\) on the y-axis, then the eccentricity of the ellipse is
(A) \(\frac{5}{7}\)
(B) \(\frac{2\sqrt{6}}{7}\)
(C) \(\frac{3}{7}\)
(D) \(\frac{2\sqrt{5}}{7}\)

Answer: (A) \(\frac{5}{7}\)

Question 32

The tangents at the points A(1, 3) and B(1, –1) on the parabola y^2 – 2x – 2y = 1 meet at the point P. Then the area (in unit^2) of the triangle PAB is :
(A) 4
(B) 6
(C) 7
(D) 8

Answer: (D) 8

Question 33

Let the foci of the ellipse \(\frac{x^2}{16}+\frac{y^2}{7}=1\ \text{and the hyperbola}\ \frac{x^2}{144}-\frac{y^2}{\alpha}=\frac{1}{25}\) coincide. Then the length of the latus rectum of the hyperbola is :
(A) \(\frac{32}{9}\)
(B) \(\frac{18}{5}\)
(C) \(\frac{27}{4}\)
(D) \(\frac{27}{10}\)

Answer: (D) \(\frac{27}{10}\)

Question 34

A plane E is perpendicular to the two planes 2x – 2y + z = 0 and x – y + 2z = 4, and passes through the point P(1, –1, 1). If the distance of the plane E from the point Q(a, a, 2) is 3√2, then (PQ)^2 is equal to
(A) 9
(B) 12
(C) 21
(D) 33

Answer: (C) 21

Question 35

The shortest distance between the lines \(\frac{x+7}{-6}=\frac{y-6}{7}=z\ \text{and}\ \frac{7-x}{2}=y-2=z-6\) is
(A) \(2\sqrt{29}\)
(B) 1
(C) \(\sqrt{\frac{37}{29}}\)
(D) \(\sqrt{\frac{29}{2}}\)

Answer: (A) \(2\sqrt{29}\)

Question 36

Let \(\vec{a}=\hat{i}-\hat{j}+2\hat{k}\) \(\text{and let}\ \vec{b}\ \text{be a vector such that}\)  \(\vec{a}\times \vec{b} =2\hat{i}-\hat{k}\ \text{and}\ \vec{a}\cdot\vec{b}=3\)  Then the projection of \(\vec{b}\ \text{on the vector}\ \vec{a}- \vec{b}\ \text{is}:\)  
(A) \(\frac{2}{\sqrt{21}}\)
(B) \(2\sqrt{\frac{3}{7}}\)
(C) \(\frac{2}{3}\sqrt{\frac{7}{3}}\)
(D) \(\frac{2}{3}\)

Answer: (A) \(\frac{2}{\sqrt{21}}\)

Question 37

If the mean deviation about median for the number 3, 5, 7, 2k, 12, 16, 21, 24 arranged in the ascending order, is 6 then the median is
(A) 11.5
(B) 10.5
(C) 12
(D) 11

Answer: (D) 11

Question 38

\(2\sin\left(\frac{\pi}{22}\right)\sin\left(\frac{3\pi}{22}\right)\sin\left(\frac{5\pi}{22}\right)\sin\left(\frac{7\pi}{22}\right)\sin\left(\frac{9\pi}{22}\right)\) is equal to :
(A) \(\frac{3}{16}\\\)
(B) \(\frac{1}{16}\\\)
(C) \(\frac{1}{32}\\\)
(D) \(\frac{9}{32}\)

Answer: (B) \(\frac{1}{16}\\\)

Question 39

Consider the following statements :
P : Ramu is intelligent.
Q : Ramu is rich.
R : Ramu is not honest.
The negation of the statement “Ramu is intelligent and honest if and only if Ramu is not rich” can be expressed as :
(A) ((P ∧ (~ R)) ∧ Q) ∧ ((~ Q) ∧ ((~ P) ∨ R))
(B) ((P ∧ R) ∧ Q) ∨ ((~ Q) ∧ ((~ P) ∨ (~ R)))
(C) ((P ∧ R) ∧ Q) ∧ ((~ Q) ∧ (( ~ P) ∨ (~ R)))
(D) ((P ∧ (~ R)) ∧ Q) ∨ ((~ Q) ∧ ((~ P) ∧ R))

Answer: (D) ((P ∧ (~ R)) ∧ Q) ∨ ((~ Q) ∧ ((~ P) ∧ R))

Question 40

Let f :R→R be a continuous function such that f(3x) – f(x) = x. If f(8) = 7, then f(14) is equal to
(A) 4
(B) 10
(C) 11
(D) 16

Answer: (B) 10

Question 41

Let O be the origin and A be the point z1 = 1 + 2i. If B is the point z2, Re(z2) < 0, such that OAB is a right angled isosceles triangle with OB as hypotenuse, then which of the following is NOT true?
(A) \( argz_2 = \pi – tan^{-1}3\)
(B) \((\text{arg})\left(z_1-2z_2\right)=-\tan^{-1}\frac{4}{3}\)
(C) \(\left|z_2\right|=\sqrt{10}\)
(D) \(\left|2z_1-z_2\right|=5\)

Answer: (D) \(\left|2z_1-z_2\right|=5\)

Question 42

If the system of linear equations.
8x + y + 4z = –2
x + y + z = 0
\(\lambda x – 3y = \mu\)
has infinitely many solutions, then the distance of the point (λ, μ, -1/2) from the plane 8x + y + 4z + 2 = 0 is
(A) \(3\sqrt{5}\)
(B) \( 4\)
(C) \(\frac{26}{9}\)
(D) \(\frac{10}{3}\)

Answer: (D) \(\frac{10}{3}\)

Question 43

Let A be a 2 × 2 matrix with det (A) = –1 and det((A + I) (Adj (A) + I)) = 4. Then the sum of the diagonal elements of A can be
(A) -1
(B) 2
(C) 1
(D) -√2

Answer: (B) 2

Question 44

The odd natural number a, such that the area of the region bounded by y = 1, y = 3, x = 0, x = y^a is 364/3, is equal to
(A) 3
(B) 5
(C) 7
(D) 9

Answer: (B) 5

Question 45

Consider two G.Ps. 2, 2^2, 2^3, …. and 4, 4^2, 4^3, … of 60 and n terms respectively. If the geometric mean of all the 60 + n terms is \(\left(2\right)^\frac{225}{8}\ \text{then}\ \sum_{k=1}^{n}k\left(n-k\right)\) is equal to
(A) 560
(B) 1540
(C) 1330
(D) 2600

Answer: (C) 1330

Question 46

If the function \(f\left(x\right)=\left\{\frac{\text{log}_e\left(1-x+x^2\right)+\text{log}_e\left(1+x+x^2\right)}{\underset{k,}{\sec x}-\cos x} \right.,\underset{x=0}{x\in\left(\frac{-\pi}{2},\frac{\pi}{2}\right)}-\left\{0 \right\}\)
is continuous at x = 0, then k is equal to
(A) 1
(B) –1
(C) e
(D) 0

Answer: (A) 1

Question 47

If
\(f\left(x\right)=\left\{\begin{matrix}x+a, & x\leq 0 \\\left|x-4\right|, & x>0 \\\end{matrix}\right.\text{ and }g\left(x\right)=\left\{\begin{matrix}x+1, & x<0 \\\left(x-4\right)^2+b, & x\geq 0 \\\end{matrix}\right.\) are continuous on R, then (gof) (2) + (fog) (–2) is equal to
(A) –10
(B) 10
(C) 8
(D) –8

Answer: (D) –8

Question 48

Let \(f\left(x\right)=\left\{\begin{matrix}x^3-x^2+10x-7, & x\leq1 \\-2x+\text{log}_2\left(b^2-4\right), & x>1 \\\end{matrix}\right.\)
Then the set of all values of b, for which f(x) has maximum value at x = 1, is
(A) \( \left(-6, -2\right)\\ \)
(B) \( \left(2, 6\right)\\ \)
(C) \( \left[-6, -2) \cup (2, 6\right]\)
(D) \(\left[-\sqrt{6},-2\right]\cup \left(2,\sqrt{6}\right] \)

Answer: (C) \( \left[-6, -2) \cup (2, 6\right]\)

Question 49

\(If~ a=\displaystyle \lim_{n \to \infty}\sum_{k=1}^{n}\frac{2n}{n^2+k^2}\text{and}\)
\(f\left(x\right)=\sqrt{\frac{1-\cos x}{1+\cos x}},x\in\left(0,1\right),then\)
(A) \(2\sqrt{2}f\left(\frac{a}{2}\right)=f’\left(\frac{a}{2}\right)\)
(B) \(f\left(\frac{a}{2}\right)f’\left(\frac{a}{2}\right)=\sqrt{2}\)
(C) \(\sqrt{2}f\left(\frac{a}{2}\right)=f’\left(\frac{a}{2}\right)\)
(D) \(f\left(\frac{a}{2}\right)=\sqrt{2}f’\left(\frac{a}{2}\right)\)

Answer: (C) \(\sqrt{2}f\left(\frac{a}{2}\right)=f’\left(\frac{a}{2}\right)\)

Question 50

If \(\frac{dy}{dx}+2y\tan x=\sin x,0(A) \(\frac{1}{8}\)
(B) \(\frac{3}{4}\)
(C) \(\frac{1}{4}\)
(D) \(\frac{3}{8}\)

Answer: (A) \(\frac{1}{8}\)

Question 51

A point P moves so that the sum of squares of its distances from the points (1, 2) and (–2, 1) is 14. Let f(x, y) = 0 be the locus of P, which intersects the x-axis at the points A, B and the y-axis at the points C, D. Then the area of the quadrilateral ACBD is equal to
(A) \(\frac{9}{2} \)
(B) \(\frac{3\sqrt{17}}{2} \)
(C) \(\frac{3\sqrt{17}}{4} \)
(D) \(9\)

Answer: (B) \(\frac{3\sqrt{17}}{2} \)

Question 52

Let the tangent drawn to the parabola y^2 = 24x at the point (α, β) is perpendicular to the line 2x + 2y = 5. Then the normal to the hyperbola \(\frac{x^2}{\alpha^2}-\frac{y^2}{\beta^2}=1 \) at the point (α + 4, β + 4) does NOT pass through the point
(A) (25, 10)
(B) (20, 12)
(C) (30, 8)
(D) (15, 13)

Answer: (D) (15, 13)

Question 53

The length of the perpendicular from the point (1, –2, 5) on the line passing through (1, 2, 4) and parallel to the line x + y – z = 0 = x – 2y + 3z – 5 is
(A) \(\sqrt{\frac{21}{2}}\)
(B) \(\sqrt{\frac{9}{2}}\)
(C) \(\sqrt{\frac{73}{2}}\)
(D) \(1 \)

Answer: (A) \(\sqrt{\frac{21}{2}}\)

Question 54

Let \(\vec{a}=\alpha\hat{i}+\hat{j}-\hat{k}\ \text{and}\ \vec{b}=2\hat{i}+\hat{j}-\alpha\hat{k},\alpha >0.\)  \(\text{If the projection of}\ \vec{a}\times \vec{b}\ \text{on the vector}\ -\hat{i}+2\hat{j}-2\hat{k}\) is 30, then α is equal to
(A) \(\frac{15}{2} \)
(B) \(8\)
(C) \(\frac{13}{2} \)
(D) \(7\)

Answer: (D) \(7\)

Question 55

The mean and variance of a binomial distribution are α and α/3, respectively. If P(X = 1) = 4/243 then P(X = 4 or 5) is equal to :
(A) \(\frac{5}{9} \)
(B) \(\frac{64}{81} \)
(C) \(\frac{16}{27} \)
(D) \(\frac{145}{243} \)

Answer: (C) \(\frac{16}{27} \)

Question 56

Let E1, E2, E3 be three mutually exclusive events such that \(P\left(E_1\right)=\frac{2+3p}{6},P\left(E_2\right)=\frac{2-p}{8}\ \text{and}\ P\left(E_3\right)=\frac{1-p}{2}.\) If the maximum and minimum values of p are p1 and p2, then (p1 + p2) is equal to :
(A) \(\frac{2}{3}\)
(B) \(\frac{5}{3}\)
(C) \(\frac{5}{4}\)
(D) \(1\)

Answer: (B) \(\frac{5}{3}\)

Question 57

\(\text{Let } S=\left\{\theta\in\left[0,2\pi\right]:8^{2\sin^2\theta}+8^{2\cos^2\theta} =16\right\} \). \(\text{Then}~ n\left(S\right)+\underset{\theta\in S}{\sum}\left(\sec\left(\frac{\pi}{4}+2\theta\right)\text{cosec}\left(\frac{\pi}{4}+2\theta\right)\right)\text{is equal to }:\)
(A) 0
(B) – 2
(C) – 4
(D) 12

Answer: (C) – 4

Question 58

\(\tan\left(2\tan^{-1}\frac{1}{5}+\sec^{-1}\frac{\sqrt{5}}{2}+2\tan^{-1}\frac{1}{8}\right)\) is equal to :
(A) \(1\)
(B) \(2\)
(C) \(\frac{1}{4} \)
(D) \(\frac{5}{4} \)

Answer: (B) \(2\)

Question 59

The statement \(\left(\sim\left(p\ \Leftrightarrow \sim q\right)\right)\wedge q\) is :
(A) A tautology
(B) A contradiction
(C) \(\text{Equivalent to} \left ( p \Rightarrow q \right) \wedge q\)
(D) \(\text{Equivalent to} \left ( p \Rightarrow q \right) \wedge p\)

Answer: (D) \(\text{Equivalent to} \left ( p \Rightarrow q \right) \wedge p\)

Question 60

The minimum value of the sum of the squares of the roots of x^2 + (3 – a)x + 1 = 2a is
(A) 4
(B) 5
(C) 6
(D) 8

Answer: (C) 6

Question 61

If z = x + iy satisfies | z | – 2 = 0 and |z – i| – | z + 5i| = 0, then
(A) x + 2y – 4 = 0
(B) x^2 + y – 4 = 0
(C) x + 2y + 4 = 0
(D) x^2 – y + 3 = 0

Answer: (C) x + 2y + 4 = 0

Question 62

Let \(A=\begin{bmatrix}1 \\ 1\\1\end{bmatrix}\text{ and }B=\begin{bmatrix}9^2 & -10^2 & 11^2 \\12^2 & 13^2 & -14^2 \\-15^2 & 16^2 & 17^2 \\\end{bmatrix},\) then the value of A′BA is
(A) 1224
(B) 1042
(C) 540
(D) 539

Answer: (D) 539

Question 63

\(\sum_{\underset{i\neq j}{i, j=0}}^{n}{^n}C_i\ ^nC_j\) is equal to
(A) \( 2^{2n} – ^{2n}C_n \)
(B) \( 2^{2n-1} – ^{2n-1}C_{n-1} \)
(C) \(2^{2n}-\frac{1}{2}\ ^{2n}C_n\)
(D) \(2^{n-1}+2^{2n-1}C_n\)

Answer: (A) \( 2^{2n} – ^{2n}C_n \)

Question 64

Let P and Q be any points on the curves (x – 1)^2 + (y + 1)^2 = 1 and y = x^2, respectively. The distance between P and Q is minimum for some value of the abscissa of P in the interval
(A) \(\left(0,\frac{1}{4}\right) \)
(B) \(\left(\frac{1}{2},\frac{3}{4}\right) \)
(C) \(\left(\frac{1}{4},\frac{1}{2}\right)\)
(D) \(\left(\frac{3}{4},1\right)\)

Answer: (C) \(\left(\frac{1}{4},\frac{1}{2}\right)\)

Question 65

Let \(\beta=\displaystyle \lim_{x \to 0}\frac{\alpha x-\left(e^{3x}-1\right)}{\alpha x\left(e^{3x}-1\right)}\ \text{for some}\ \alpha\ \in\ \mathbb{R}.\) Then the value of α + β is
(A) \(\frac{14}{5} \)
(B) \(\frac{3}{2} \)
(C) \(\frac{5}{2} \)
(D) \(\frac{7}{2} \)

Answer: (C) \(\frac{5}{2} \)

Question 66

The value of \(\text{log}_e2\frac{d}{dx}\left(\text{log}_{\cos x}\text{cosec x}\right) \textup{ at }x=\frac{\pi}{4}\) is
(A) \(-2\sqrt{2}\)
(B) \(2\sqrt{2}\)
(C) \(-4\)
(D) \(4\)

Answer: (D) \(4\)

Question 67

\(\displaystyle\int\limits_0^{20\pi}\left(\left|\sin x\right|+\left|\cos x\right|\right)^2 dx\) is equal to
(A) \(10\left(\pi+4\right)\)
(B) \(10\left(\pi+2\right)\)
(C) \(20\left(\pi-2\right)\)
(D) \(20\left(\pi+2\right)\)

Answer: (D) \(20\left(\pi+2\right)\)

Question 68

Let the solution curve y = f(x) of the differential equation \(\frac{dy}{dx}+\frac{xy}{x^2-1}=\frac{x^4+2x}{\sqrt{1-x^2}},x \in \left(-1, 1\right)\) pass through the origin. Then \(\displaystyle\int\limits_{-\frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}}f\left(x\right)dx\) is
(A) \(\frac{\pi}{3}-\frac{1}{4}\)
(B) \(\frac{\pi}{3}-\frac{\sqrt{3}}{4}\)
(C) \(\frac{\pi}{6}-\frac{\sqrt{3}}{4}\)
(D) \(\frac{\pi}{6}-\frac{\sqrt{3}}{2}\)

Answer: (B) \(\frac{\pi}{3}-\frac{\sqrt{3}}{4}\)

Question 69

The acute angle between the pair of tangents drawn to the ellipse 2x^2 + 3y^2 = 5 from the point (1, 3) is
(A) \(\tan^{-1}\left(\frac{16}{7\sqrt{5}}\right)\)
(B) \(\tan^{-1}\left(\frac{24}{7\sqrt{5}}\right)\)
(C) \(\tan^{-1}\left(\frac{32}{7\sqrt{5}}\right)\)
(D) \(\tan^{-1}\left(\frac{3+8\sqrt{5}}{35}\right)\)

Answer: (B) \(\tan^{-1}\left(\frac{24}{7\sqrt{5}}\right)\)

Question 70

The equation of a common tangent to the parabolas y = x^2 and y = –(x – 2)^2 is
(A) y = 4(x – 2)
(B) y = 4(x – 1)
(C) y = 4(x + 1)
(D) y = 4(x + 2)

Answer: (B) y = 4(x – 1)

Question 71

Let the abscissae of the two points P and Q on a circle be the roots of x^2 – 4x – 6 = 0 and the ordinates of P and Q be the roots of y^2 + 2y – 7 = 0. If PQ is a diameter of the circle x^2 + y^2 + 2ax + 2by + c = 0, then the value of (a + b – c) is
(A) 12
(B) 13
(C) 14
(D) 16

Answer: (A) 12

Question 72

If the line x – 1 = 0 is a directrix of the hyperbola kx^2 – y^2 = 6, then the hyperbola passes through the point
(A) \(\left(-2\sqrt{5},6\right)\)
(B) \(\left(-\sqrt{5},3\right)\)
(C) \(\left(\sqrt{5},-2\right)\)
(D) \(\left(2\sqrt{5},3\sqrt{6}\right)\)

Answer: (C) \(\left(\sqrt{5},-2\right)\)

Question 73

A vector \(\vec{a}\)is parallel to the line of intersection of the plane determined by the vectors \(\hat{i},\hat{i}+\hat{j}\)and the plane determined by the vectors \(\hat{i}-\hat{j},\hat{i}+\hat{k}. \) The obtuse angle between\(\vec{a}\ \text{and the vector}\ \vec{b}=\hat{i}-2\hat{j}+2\hat{k}\)  is
(A) \(\frac{3\pi}{4} \)
(B) \(\frac{2\pi}{3} \)
(C) \(\frac{4\pi}{5} \)
(D) \(\frac{5\pi}{6}\)

Answer: (A) \(\frac{3\pi}{4} \)

Question 74

If \(0(A) \(4\sqrt{\left(1-x^2\right)}\left(1-2x^2\right)\)
(B) \(4x\sqrt{\left(1-x^2\right)}\left(1-2x^2\right)\)
(C) \(2x\sqrt{\left(1-x^2\right)}\left(1-4x^2\right)\)
(D) \(4\sqrt{\left(1-x^2\right)}\left(1-4x^2\right)\)

Answer: (B) \(4x\sqrt{\left(1-x^2\right)}\left(1-2x^2\right)\)

Question 75

Negation of the Boolean expression p⇔ (q⇒p) is
(A) (~ p) ∧q
(B) p∧ (~ q)
(C) (~ p) ∨ (~ q)
(D) (~ p) ∧ (~ q)

Answer: (D) (~ p) ∧ (~ q)

Question 76

Let X be a binomially distributed random variable with mean 4 and variance 4/3. Then, 54 P(X≤ 2) is equal to
(A) \(\frac{73}{27}\)
(B) \(\frac{146}{27}\)
(C) \(\frac{146}{81}\)
(D) \(\frac{126}{81}\)

Answer: (B) \(\frac{146}{27}\)

Question 77

The integral \(\int\frac{\left(1-\frac{1}{\sqrt{3}}\right)\left(\cos x-\sin x\right)}{\left(1+\frac{2}{\sqrt{3}}\sin 2x\right)}dx\) is equal to
(A) \(\frac{1}{2}\text{log}_e\left|\frac{\tan\left(\frac{x}{2}+\frac{\pi}{12}\right)}{\tan\left(\frac{x}{2}+\frac{\pi}{6}\right)}\right|+C\)
(B) \(\frac{1}{2}\text{log}_e\left|\frac{\tan\left(\frac{x}{2}+\frac{x}{6}\right)}{\tan\left(\frac{x}{2}+\frac{\pi}{3}\right)}\right|+C\)
(C) \(\text{log}_e\left|\frac{\tan\left(\frac{x}{2}+\frac{\pi}{6}\right)}{\tan\left(\frac{x}{2}+\frac{\pi}{12}\right)}\right|+C\)
(D) \(\frac{1}{2}\text{log}_e\left|\frac{\tan\left(\frac{x}{2}-\frac{\pi}{12}\right)}{\tan\left(\frac{x}{2}-\frac{\pi}{6}\right)}\right|+C\)

Answer: (A) \(\frac{1}{2}\text{log}_e\left|\frac{\tan\left(\frac{x}{2}+\frac{\pi}{12}\right)}{\tan\left(\frac{x}{2}+\frac{\pi}{6}\right)}\right|+C\)

Question 78

The area bounded by the curves y = |x^2 – 1| and y = 1 is
(A) \(\frac{2}{3}\left(\sqrt{2}+1\right)\)
(B) \(\frac{4}{3}\left(\sqrt{2}-1\right)\)
(C) \(2\left(\sqrt{2}-1\right)\)
(D) \(\frac{8}{3}\left(\sqrt{2}-1\right)\)

Answer: (D) \(\frac{8}{3}\left(\sqrt{2}-1\right)\)

Question 79

Let R1 and R2 be two relations defined on ℝ by a R1b ⇔ ab ≥ 0 and aR2b ⇔ a ≥ b. Then,
(A) R1 is an equivalence relation but not R2
(B) R2 is an equivalence relation but not R1
(C) Both R1 and R2 are equivalence relations
(D) Neither R1 nor R2 is an equivalence relation

Answer: (D) Neither R1 nor R2 is an equivalence relation

Question 80

Let \(f,g : \mathbb{N} = \{1\} \rightarrow \mathbb{N}\ \text{be functions defined by}\)
f(a) = α, where α is the maximum of the powers of those primes p such that p^α divides a, and g(a) = a + 1, for all a ∈ N – {1}. Then, the function  f + g is
(A) One-one but not onto
(B) Onto but not one-one
(C) Both one-one and onto
(D) Neither one-one nor onto

Answer: (D) Neither one-one nor onto

Question 81

Let the minimum value v0 of \(v= \left|z\right|^2 + \left|z – 3\right|^2 + \left|z – 6i\right|^2, z \in \mathbb {C}\) is attained at z = z0. Then \(\left|2z_0^2 – \bar{z}_0^3 + 3 \right|^2 + v_0^2\) is equal to
(A) 1000
(B) 1024
(C) 1105
(D) 1196

Answer: (A) 1000

Question 82

Let \(A = \begin{pmatrix}1 & 2 \\-2 & -5 \\\end{pmatrix}\). Let α, β ∈ ℝ be such that αA^2 + βA = 2I. Then α + β is equal to
(A) –10
(B) –6
(C) 6
(D) 10

Answer: (D) 10

Question 83

The remainder when (2021)^2022 + (2022)^2021 is divided by 7 is
(A) 0
(B) 1
(C) 2
(D) 6

Answer: (A) 0

Question 84

Suppose a1, a2, … an, … be an arithmetic progression of natural numbers. If the ration of the sum of first five terms to the sum of first nine terms of the progression is 5 : 17 and 110 < a15 < 120, then the sum of the first ten terms of the progression is equal to
(A) 290
(B) 380
(C) 460
(D) 510

Answer: (B) 380

Question 85

Let ℝ → ℝ be function defined as \(f(x)=a\sin \left(\frac{\pi [x]}{2}\right) + [2-x], a\in \mathbb{R}\) where [t] is the greatest integer less than or equal to t. \(\text{If}\ \displaystyle \lim_{ x \to 1}f(x)\ \text{exists, then the value of}\ \int_{0}^{4}f(x) dx\) is equal to
(A) –1
(B) –2
(C) 1
(D) 2

Answer: (B) –2

Question 86

The area of the smaller region enclosed by the curves y^2 = 8x + 4 and \(x^2 +y^2 + 4\sqrt{3}x-4 =0\) is equal to
(A) \(\frac{1}{3}(2-12\sqrt{3} + 8\pi)\)
(B) \(\frac{1}{3}(2-12\sqrt{3} + 6\pi)\)
(C) \(\frac{1}{3}(4-12\sqrt{3} + 8\pi)\)
(D) \(\frac{1}{3}(4-12\sqrt{3} + 6\pi)\)

Answer: (C) \(\frac{1}{3}(4-12\sqrt{3} + 8\pi)\)

Question 87

Let y = y1(x) and y = y2(x) be two distinct solution of the differential equation \(\frac{dy}{dx}=x+y,\) with y1(0) = 0 and y2(0) = 1 respectively. Then, the number of points of intersection of y = y1 (x) and y = y2(x) is
(A) 0
(B) 1
(C) 2
(D) 3

Answer: (A) 0

Question 88

Let P(a, b) be a point on the parabola y^2 = 8x such that the tangent at P passes through the centre of the circle x^2 + y^2 – 10x – 14y + 65 = 0. Let A be the product of all possible values of a and B be the product of all possible values of b. Then the value of A + B is equal to
(A) 0
(B) 25
(C) 40
(D) 65

Answer: (D) 65

Question 89

\(\text{Let}~\vec{a} = \alpha \hat{i} + \hat{j} + \beta \hat{k}~\text{and}~ \vec{b} = 3 \hat{i} + 5\hat{j} + 4 \hat{k}~ \text{be two vectors, such that }\vec{a} \times \vec{b} = -\hat{i} + 9\hat{j} + 12 \hat{k}.~\text{Then the projection of } \vec{b}-2\vec{a} ~\text{on}~ \vec{b}+ \vec{a}~\text {is equal to}\)
(A) \(2\)
(B) \(\frac{39}{5}\)
(C) \(9\)
(D) \(\frac{46}{5}\)

Answer: (D) \(\frac{46}{5}\)

Question 90

\(\text{Let}~\vec{a} = 2\hat{i}-\hat{j}+5\hat{k}~\text{and}~ \vec{b} = \alpha \hat{i} + \beta \hat{j}+2\hat{k}. \text{If}~((\vec{a}\times \vec{b}) \times \hat{i})\cdot \hat{k}=\frac{23}{2},~ \text{then} \left|\vec{b}\times 2\hat{j}\right| \text{is equal to} \)
(A) \(4\)
(B) \(5\)
(C) \(\sqrt{21}\)
(D) \(\sqrt{17}\)

Answer: (B) \(5\)

Question 91

Let S be the sample space of all five digit numbers. It p is the probability that a randomly selected number from S, is multiple of 7 but not divisible by 5, then 9p is equal to
(A) 1.0146
(B) 1.2085
(C) 1.0285
(D) 1.1521

Answer: (C) 1.0285

Question 92

Let a vertical tower AB of height 2h stands on a horizontal ground. Let from a point P on the ground a man can see upto height h of the tower with an angle of elevation 2α.  \(\text{When from P, he moves a distance d in the direction of}\ \overrightarrow{AP},\) he can see the top B of the tower with an angle of elevation α. if d = √7 h, then tan α is equal to
(A) \(\sqrt{5}-2\)
(B) \(\sqrt{3}-1\)
(C) \(\sqrt{7}-2\)
(D) \(\sqrt{7}-\sqrt{3}\)

Answer: (C) \(\sqrt{7}-2\)

Question 93

\((p \land r )\Leftrightarrow ( p \land (\sim q))\) is equivalent to (~ p) when r is
(A) p
(B) ~p
(C) q
(D) ~q

Answer: (C) q

Question 94

If the plane P passes through the intersection of two mutually perpendicular planes 2x + ky – 5z = 1 and 3kx – ky + z = 5, k < 3 and intercepts a unit length on positive x-axis, then the intercept made by the plane P on the y-axis is
(A) \(\ \frac{1}{11} \)
(B) \(\ \frac{5}{11} \)
(C) \(\ 6\)
(D) \(\ 7 \)

Answer: (D) \(\ 7 \)

Question 95

Let A(1, 1), B(-4, 3) and C(-2, -5) be vertices of a triangle ABC, P be a point on side BC, and Δ1 and Δ2 be the areas of triangles APB and ABC, respectively. If Δ1 : Δ2 = 4 : 7, then the area enclosed by the lines AP, AC and the x-axis is
(A) \(\frac{1}{4}\)
(B) \(\frac{3}{4}\)
(C) \(\frac{1}{2}\)
(D) \(1\)

Answer: (C) \(\frac{1}{2}\)

Question 96

If the circle \(x^2 + y^2 – 2gx + 6y – 19c = 0, g, c \in \mathbb {R} \) passes through the point (6, 1) and its centre lies on the line x – 2cy = 8, then the length of intercept made by the circle on x-axis is
(A) \(\sqrt{11}\)
(B) \(4\)
(C) \(3 \)
(D) \(2\sqrt{23}\)

Answer: (D) \(2\sqrt{23}\)

Question 97

Let a function f: ℝ → ℝ be defined as :
\(f(x) = \left\{\begin{matrix}\int_{0}^{x}(5-|t-3|)dt, & x>4 \\x^2 + bx, & x \le4 \\\end{matrix}\right.\) where b ∈ ℝ. If f is continuous at x = 4 then which of the following statements is NOT true?
(A) \(\text{f is not differentiable at x} = 4 \)
(B) \(f'(3)+f'(5)=\frac{35}{4}\)
(C) \(f \text{ is increasing in } \left(-\infty, \frac{1}{8}\right) \cup (8, \infty)\)
(D) \(f \text{ has a local minima at } x = \frac{1}{8}\)

Answer: (C) \(f \text{ is increasing in } \left(-\infty, \frac{1}{8}\right) \cup (8, \infty)\)

Question 98

The domain of the function \(f\left(x\right)=\sin^{-1}\left[2x^2-3\right]+\text{log}_2\left(\text{log}_\frac{1}{2}\left(x^2-5x+5\right)\right) \)
where [t] is the greatest integer function, is
(A) \(\left(-\sqrt{\frac{5}{2}},\frac{5-\sqrt{5}}{2}\right)\)
(B) \(\left(\frac{5-\sqrt{5}}{2},\frac{5+\sqrt{5}}{2}\right)\)
(C) \(\left(1, \frac{5-\sqrt{5}}{2}\right)\)
(D) \(\left(1, \frac{5+\sqrt{5}}{2}\right)\)

Answer: (C) \(\left(1, \frac{5-\sqrt{5}}{2}\right)\)

Question 99

Let S be the set of (α, β), π < α, β < 2π, for which the complex number \(\frac{1-i\sin\alpha}{1+2i\sin\alpha}\ \text{is purely imaginary and}\ \frac{1+i\cos\beta}{1-2i\cos\beta}\ \text{is purely real},\) \(\text{Let}\ Z_{\alpha \beta} = sin 2\alpha + i cos 2\beta, \left(\alpha, \beta\right) \in S.\ \text{Then}\) \(\sum_{\left(\alpha,\beta\right)\in S}\left(iZ_{\alpha\beta}+\frac{1}{i\overline{Z}_{\alpha\beta}}\right)\) is equal to
(A) 3
(B) 3i
(C) 1
(D) 2 – i

Answer: (C) 1

Question 100

If α, β are the roots of the equation \(x^2-\left(5+3^{\sqrt{\text{log}_35}}-5^{\sqrt{\text{log}_53}}\right)+3\left(3^{\left(\text{log}_35\right)^{\frac{1}{3}}}-5^{\left(\text{log}_53\right)^{\frac{2}{3}}}-1\right)=0\)
then the equation, whose roots are α + 1/β and β + 1/α , is
(A) 3x^2 – 20x – 12 = 0
(B) 3x^2 – 10x – 4 = 0
(C) 3x^2 – 10x + 2 = 0
(D) 3x^2 – 20x + 16 = 0

Answer: (B) 3x^2 – 10x – 4 = 0

Question 101

Let \(A=\begin{pmatrix}4 & -2 \\\alpha & \beta \\\end{pmatrix}\)
If A^2 + γA + 18I = 0, then det (A) is equal to ______.
(A) –18
(B) 18
(C) –50
(D) 50

Answer: (B) 18

Question 102

If for p ≠ q ≠ 0, the function \(f\left(x\right)=\frac{\sqrt[7]{p\left(729+x\right)}-3}{\sqrt[3]{729+qx}-9}\) is continuous at x = 0, then
(A) \( 7pq~f\left(0\right) – 1 = 0\)
(B) \( 63q f\left(0\right) – p^2 = 0\)
(C) \( 21qf\left(0\right) – p^2 = 0\)
(D) \( 7pq f\left(0\right) – 9 = 0\)

Answer: (B) \( 63q f\left(0\right) – p^2 = 0\)

Question 103

Let \(f\left(x\right)=2+\left|x\right|-\left|x-1\right|+\left|x+1\right| ,x\in R.\)
\(\text{Consider}\left(S1\right) :f’\left(-\frac{3}{2}\right)+f’\left(-\frac{1}{2}\right)+f’\left(\frac{1}{2}\right)+f’\left(\frac{3}{2}\right)=2\)
\(\left(S2\right):\displaystyle\int\limits_{-2}^2f\left(x\right)dx=12\)
Then,
(A) Both (S1) and (S2) are correct
(B) Both (S1) and (S2) are wrong
(C) Only (S1) is correct
(D) Only (S2) is correct

Answer: (D) Only (S2) is correct

Question 104

Let the sum of an infinite G.P., whose first term is a and the common ratio is r, be 5. Let the sum of its first five terms be 98/25. Then the sum of the first 21 terms of an AP, whose first term is 10ar, n^th term is an and the common difference is 10ar^2, is equal to
(A) 21 a11
(B) 22 a11
(C) 15 a16
(D) 14 a16

Answer: (A) 21 a11

Question 105

The area of the region enclosed by \(y\leq 4x^2, x^2\leq 9y\text{ and }y\leq 4,\) is equal to
(A) \(\frac{40}{3}\)
(B) \(\frac{56}{3}\)
(C) \(\frac{112}{3}\)
(D) \(\frac{80}{3}\)

Answer: (D) \(\frac{80}{3}\)

Question 106

\(\displaystyle\int\limits_0^2\left(\left|2x^2-3x\right|+\left[x-\frac{1}{2}\right]\right)dx, \) where [t] is the greatest integer function, is equal to
(A) \(\frac{7}{6}\)
(B) \(\frac{19}{12}\)
(C) \(\frac{31}{12}\)
(D) \(\frac{3}{2}\)

Answer: (B) \(\frac{19}{12}\)

Question 107

Consider a curve y = y(x) in the first quadrant as shown in the figure. Let the area A1 is twice the area A2. Then the normal to the curve perpendicular to the line 2x – 12y = 15 does NOT pass through the point.
(A) \( \left(6, 21\right)\\\)
(B) \( \left(8, 9\right)\\ \)
(C) \( \left(10, -4\right)\\ \)
(D) \( \left(12, -15\right)\)

Answer: (C) \( \left(10, -4\right)\\ \)

Question 108

The equations of the sides AB, BC and CA of a triangle ABC are 2x + y = 0, x + py = 39 and x – y = 3, respectively and P(2, 3) is its circumcentre. Then which of the following is NOT true?
(A) \( \left(AC\right)^2 = 9p\\ \)
(B) \( \left(AC\right)^2 + p^2 = 136\\\)
(C) \( 32 < area\left(\triangle ABC\right) < 36\\ \)
(D) \(34 < area\left(\triangle ABC\right) < 38\)

Answer: (D) \(34 < area\left(\triangle ABC\right) < 38\)

Question 109

A circle C1 passes through the origin O and has diameter 4 on the positive x-axis. The line y = 2x gives a chord OA of circle C1. Let C2 be the circle with OA as a diameter. If the tangent to C2 at the point A meets the x-axis at P and y-axis at Q, then QA :AP is equal to
(A) 1 : 4
(B) 1 : 5
(C) 2 : 5
(D) 1 : 3

Answer: (A) 1 : 4

Question 110

If the length of the latus rectum of a parabola, whose focus is (a, a) and the tangent at its vertex is x + y = a, is 16, then |a| is equal to :
(A) \(2\sqrt{2}\)
(B) \(2\sqrt{3}\)
(C) \(4\sqrt{2}\)
(D) \(4\)

Answer: (C) \(4\sqrt{2}\)

Question 111

If the length of the perpendicular drawn from the point P(a, 4, 2), a> 0 on the line \(\frac{x+1}{2}=\frac{y-3}{3} =\frac{z-1}{-1}\text{is }2\sqrt{6}\ \text{units}\ \text{and}\ Q\left(\alpha_1,\alpha_2,\alpha_3\right)\) is the image of the point P in this line, then \(a+\sum_{i=1}^{3}\alpha_i \) is equal to :
(A) 7
(B) 8
(C) 12
(D) 14

Answer: (B) 8

Question 112

If the line of intersection of the planes ax + by = 3 and ax + by + cz = 0, a> 0 makes an angle 30° with the plane y – z + 2 = 0, then the direction cosines of the line are :
(A) \(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0\)
(B) \(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}},0\)
(C) \(\frac{1}{\sqrt{5}},-\frac{2}{\sqrt{5}},0\)
(D) \(\frac{1}{2},-\frac{\sqrt{3}}{2},0\)

Answer: (B) \(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}},0\)

Question 113

Let X have a binomial distribution B(n, p) such that the sum and the product of the mean and variance of X are 24 and 128 respectively. If \(P\left(X>n – 3\right) = \frac{k}{2^n},\) then k is equal to :
(A) 528
(B) 529
(C) 629
(D) 630

Answer: (B) 529

Question 114

A six faced die is biased such that
3 × P (a prime number) = 6 × P (a composite number) = 2 × P (1).
Let X be a random variable that counts the number of times one gets a perfect square on some throws of this die. If the die is thrown twice, then the mean of X is :
(A) \(\frac{3}{11}\)
(B) \(\frac{5}{11}\)
(C) \(\frac{7}{11}\)
(D) \(\frac{8}{11}\)

Answer: (D) \(\frac{8}{11}\)

Question 115

The angle of elevation of the top P of a vertical tower PQ of height 10 from a point A on the horizontal ground is 45°, Let R be a point on AQ and from a point B, vertically above R, the angle of elevation of P is 60°. If \(\angle BAQ = 30^\circ\), AB = d and the area of the trapezium PQRB is α, then the ordered pair (d, α) is :
(A) \(\left(10\left(\sqrt{3}-1\right),25\right) \)
(B) \(\left(10\left(\sqrt{3}-1\right),\frac{25}{2}\right) \)
(C) \(\left(10\left(\sqrt{3}+1\right),25\right) \)
(D) \(\left(10\left(\sqrt{3}+1\right),\frac{25}{2}\right) \)

Answer: (A) \(\left(10\left(\sqrt{3}-1\right),25\right) \)

Question 116

\(\text{ Let } S=\left\{0\in\left(0,\frac{\pi}{2}\right):\sum_{m=1}^{9}\sec\left(\theta+\left(m-1\right)\frac{\pi}{6}\right)\sec\left(\theta+\frac{m\pi}{6}\right)=-\frac{8}{\sqrt{3}} \right\} \) Then
(A) \(S=\left\{\frac{\pi}{12}\right\}\)
(B) \(S=\left\{\frac{2\pi}{3}\right\}\)
(C) \(\sum_{\theta\in S}\theta= \frac{\pi}{2}\)
(D) \(\sum_{\theta\in S}\theta= \frac{3\pi}{4}\)

Answer: (C) \(\sum_{\theta\in S}\theta= \frac{\pi}{2}\)

Question 117

If the truth value of the statement \(\left(P\wedge\left(\sim R\right)\right)\rightarrow\left(\left(\sim R\right)\wedge Q\right)\) is F, then the truth value of which of the following is F?
(A) \( P\vee Q ~\rightarrow ~\sim R\)
(B) \( R\vee Q ~\rightarrow ~\sim P\)
(C) \( \sim \left(P\vee Q \right)~\rightarrow ~\sim R\)
(D) \( \sim \left(R\vee Q \right)~\rightarrow ~\sim P\)

Answer: (D) \( \sim \left(R\vee Q \right)~\rightarrow ~\sim P\)

Question 118

Let the solution curve of the differential equation \(xdy=\left(\sqrt{x^2 + y^2 }+y\right)dx, x>0\) intersect the line x = 1 at y = 0 and the line x = 2 at y = α. Then the value of α is
(A) \(\frac{1}{2}\)
(B) \(\frac{3}{2}\)
(C) \(-\frac{3}{2}\)
(D) \(\frac{5}{2}\)

Answer: (B) \(\frac{3}{2}\)

Question 119

Considering only the principal values of the inverse trigonometric functions, the domain of the function \(f(x)=\cos^{-1}\left(\frac{x^2-4x+2}{x^2+3}\right)\) is
(A) \(\left(-\infty, \frac{1}{4}\right]\)
(B) \(\left[-\frac{1}{4}, \infty \right)\)
(C) \(\left(\frac{-1}{3}, \infty \right)\)
(D) \(\left(-\infty, \frac{1}{3} \right]\)

Answer: (B) \(\left[-\frac{1}{4}, \infty \right)\)

Question 120

Let the vectors \(\vec{a}=(1+t)\hat{i}+(1-t)\hat{j}+\hat{k},\ \vec{b}=(1-t)\hat{i}+(1+t)\hat{j}+2\hat{k}\) and \(\vec{c}=t\hat{i}-t\hat{j}+\hat{k}, t\in \mathbf{R}\) be such that for \(\alpha, \beta, \gamma \in \mathbf R,\ \alpha \vec{a}+\beta \vec{b}+\gamma \vec{c} =\vec{0}\ \Rightarrow \alpha =\beta = \gamma =0.\) Then, the set of all values of t is
(A) A non-empty finite set
(B) Equal to N
(C) \( \text{Equal to}~ \mathbf{R} – \{0\}\)
(D) \( \text{Equal to}~ \mathbf{R}\)

Answer: (C) \( \text{Equal to}~ \mathbf{R} – \{0\}\)

Question 121

Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation \(cos^{-1}\left(x\right) – 2sin^{-1}\left(x\right) = cos^{-1}\left(2x\right)\) is equal to
(A) \(0\)
(B) \(1\)
(C) \(\frac{1}{2}\)
(D) \(-\frac{1}{2}\)

Answer: (A) \(0\)

Question 122

Let the operations *, ◉ ∈ {∧, ∨}. If (p * q) ◉ (p ◉ ~q) is a tautology, then the ordered pair (*, ◉) is
(A) (∨, ∧)
(B) (∨, ∨)
(C) (∧, ∧)
(D) (∧, ∨)

Answer: (B) (∨, ∨)

Question 123

Let a vector \(\vec{a}\ \text{has magnitude}\ 9.\ \text{Let a vector}\ \vec{b}\) be such that for every \(\left(x, y\right) \in \mathbf R \times \mathbf R – \{\left(0, 0\right)\},\ \text{the vector}\ (x\vec{a} + y\vec{b})\)  \(\text{is perpendicular to the vector}\ (6y\vec{a} – 18x\vec{b}).\) \(\text{Then the value of}\  |\vec{a}\times \vec{b}|\) is equal to
(A) \(9\sqrt{3}\)
(B) \(27\sqrt{3}\)
(C) \(9\)
(D) \(81\)

Answer: (B) \(27\sqrt{3}\)

Question 124

For t ∈ (0, 2π), if ABC is an equilateral triangle with vertices A(sint, – cost), B(cost, sint) and C(a, b) such that its orthocentre lies on a circle with centre (1, 1/3), then (a^2 – b^2) is equal to
(A) \(\frac{8}{3}\)
(B) \(8\)
(C) \(\frac{77}{9}\)
(D) \(\frac{80}{9}\)

Answer: (B) \(8\)

Question 125

For α ∈ N, consider a relation R on N given by R = {(x, y) : 3x + αy is a multiple of 7}. The relation R is an equivalence relation if and only if
(A) α = 14
(B) α is a multiple of 4
(C) 4 is the remainder when α is divided by 10
(D) 4 is the remainder when α is divided by 7

Answer: (D) 4 is the remainder when α is divided by 7

Question 126

If y = y(x), x ∈ (0, π/2) be the solution curve of the differential equation \((\sin^22x)\frac{dy}{dx}+ (8\sin^22x + 2\sin 4x)y = 2e^{-4x}(2\sin 2x + \cos 2x),\) \(\text{with}\ y\left(\frac{\pi}{4}\right)=e^{-\pi},\ \text{then}\ y\left(\frac{\pi}{6}\right)\) is equal to
(A) \(\frac{2}{\sqrt{3}}e^{-2\pi/3}\)
(B) \(\frac{2}{\sqrt{3}}e^{2\pi/3}\)
(C) \(\frac{1}{\sqrt{3}}e^{-2\pi/3}\)
(D) \(\frac{1}{\sqrt{3}}e^{2\pi/3}\)

Answer: (A) \(\frac{2}{\sqrt{3}}e^{-2\pi/3}\)

Question 127

If the tangents drawn at the points P and Q on the parabola y^2 = 2x – 3 intersect at the point R(0, 1), then the orthocentre of the triangle PQR is :
(A) (0, 1)
(B) (2, –1)
(C) (6, 3)
(D) (2, 1)

Answer: (B) (2, –1)

Question 128

Let C be the centre of the circle \(x^2+y^2-x+2y=\frac{11}{4}\) and P be a point on the circle. A line passes through the point C, makes an angle of π/4with the line CP and intersects the circle at the Q and R. Then the area of the triangle PQR (in unit^2) is :
(A) \(2\)
(B) \(2\sqrt{2}\)
(C) \(8\sin\left(\frac{\pi}{8}\right)\)
(D) \(8\cos\left(\frac{\pi}{8}\right)\)

Answer: (B) \(2\sqrt{2}\)

Question 129

The remainder 7^2022 + 3^2022 is divided by 5 is:
(A) 0
(B) 2
(C) 3
(D) 4

Answer: (C) 3

Question 130

Let the matrix \(A= \begin{vmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\\end{vmatrix}\) and the matrix \(B_0 = A^{49} + 2A^{98}. ~\text{If} ~B_n = Adj(B_{n – 1})~\text{for all}~n \geq 1,\) then det(B4) is equal to :
(A) 3^28
(B) 3^30
(C) 3^32
(D) 3^36

Answer: (C) 3^32

Question 131

Let \(S_1= \left\{z_1 \in C : |z_1 – 3| = \frac{1}{2}\right\}\ \text{and}\ S_2= \left\{z_2 \in C : |z_2 – |z_2 + 1|| = |z_2 + |z_2 – 1||\right\}.\) Then, for z1 ∈ S1 and z2 ∈ S2, the least value of |z2 – z1| is :
(A) \(0\)
(B) \(\frac{1}{2}\)
(C) \(\frac{3}{2}\)
(D) \(\frac{5}{2}\)

Answer: (C) \(\frac{3}{2}\)

Question 132

The foot of the perpendicular from a point on the circle x^2 + y^2 = 1, z = 0 to the plane 2x + 3y + z = 6 lies on which one of the following curves?
(A) \( \left(6x + 5y – 12\right)^2 + 4\left(3x + 7y – 8\right)^2 = 1, z = 6 – 2x – 3y\)
(B) \(\left(5x + 6y – 12\right)^2 + 4\left(3x + 5y – 9\right)^2 = 1, z = 6- 2x – 3y\)
(C) \( \left(6x + 5y – 14\right)^2 + 9\left(3x + 5y – 7\right)^2 = 1, z = 6 – 2x – 3y\)
(D) \( \left(5x + 6y – 14\right)^2 + 9\left(3x + 7y – 8\right)^2 = 1, z = 6 – 2x – 3y\)

Answer: (B) \(\left(5x + 6y – 12\right)^2 + 4\left(3x + 5y – 9\right)^2 = 1, z = 6- 2x – 3y\)

Question 133

If the minimum value of \(f(x)=\frac{5x^2}{2}+\frac{\alpha}{x^5}, x>0\) is 14, then the value of α is equal to
(A) 32
(B) 64
(C) 128
(D) 256

Answer: (C) 128

Question 134

Let α, β and γ be three positive real numbers. \(Let f\left(x\right) = \alpha x^5 + \beta x^3 + \gamma x,x \in \mathbb {R} ~\text{and } g:\mathbb{R} \rightarrow \mathbb{R}~\text{be such that}~ g\left(f(x)\right) = x~\text{for all}~ x \in \mathbb{R}. \) If a1, a2, a3, …, an be in arithmetic progression with mean zero, then the value of \(f\left( g\left( \frac{1}{n} \sum_{i=1}^{n}f(a_i)\right)\right)\) is equal to
(A) 0
(B) 3
(C) 9
(D) 27

Answer: (A) 0

Question 135

Consider the sequence a1, a2, a3, … such that a1 = 1, a2 = 2 and \(a_{n+2}=\frac{2}{a_{n+1}}+a_n \text{ for } n = 1,2,3,…\). If \(\left(\frac{a_1+\frac{1}{a_2}}{a_3}\right)\left(\frac{a_2+\frac{1}{a_3}}{a_4}\right)\left(\frac{a_3+\frac{1}{a_4}}{a_5}\right)\cdots\left(\frac{a_{30}+\frac{1}{a_{31}}}{a_{32}}\right)=2^\alpha (^{61}C_{31}), \) then α is equal to
(A) –30
(B) –31
(C) –60
(D) –61

Answer: (C) –60

Question 136

The minimum value of the twice differentiable function \(f(x)=\int_{0}^{x}e^{x-t}f'(t)dt-(x^2-x+1)e^x, x\in \mathbb{R}\) is
(A) \(\ -\frac{2}{\sqrt{e}}\)
(B) \(\ -2\sqrt{e}\)
(C) \(\ -\sqrt{e}\)
(D) \(\ \frac{2}{\sqrt{e}}\)

Answer: (A) \(\ -\frac{2}{\sqrt{e}}\)

Question 137

Let \(S=\left\{x\in\left[-6, 3\right]-\left\{-2,2 \right\}:\frac{\left|x+3\right|-1}{\left|x\right|-2}\geq 0 \right\}\ \text{and} T=\left\{x\in \mathbb{Z}:x^2-7\left|x\right|+9\leq 0\right\}.\) Then the number of elements in S ⋂ T is
(A) 7
(B) 5
(C) 4
(D) 3

Answer: (D) 3

Question 138

Let α, β be the roots of the equation \(x^2-\sqrt{2}x+\sqrt{6}=0\ \text{and}\ \frac{1}{\alpha^2}+1,\frac{1}{\beta^2}+1\) be the roots of the equation x^2 + ax + b = 0 . Then the roots of the equation x^2 – (a + b – 2)x + (a + b + 2) = 0 are
(A) Non-real complex number
(B) Real and both negative
(C) Real and both positive
(D) Real and exactly one of them is positive

Answer: (B) Real and both negative

Question 139

Let A and B be any two 3 × 3 symmetric and skew symmetric matrices, respectively. Then Which of the following is NOT true?
(A) A^4 – B^4 is a symmetric matrix
(B) AB – BA is a symmetric matrix
(C) B^5 – A^5 is a skew-symmetric matrix
(D) AB + BA is a skew-symmetric matrix

Answer: (C) B^5 – A^5 is a skew-symmetric matrix

Question 140

Let f(x) = ax^2 + bx + c be such that f(1) = 3, f(-2) = λ and f(3) = 4. If f(0) + f(1) + f(-2) + f(3) = 14, then λ is equal to
(A) \( -4\\ \)
(B) \( \frac{13}{2} \\ \)
(C) \(\frac{23}{2}\\ \)
(D) \(4\)

Answer: (D) \(4\)

Question 141

The function f: ℝ → ℝ defined by \(f\left(x\right)=\displaystyle \lim_{n \to \infty}\frac{\cos\left(2\pi x\right)-x^{2n}\sin\left(x-1\right)}{1+x^{2n+1}-x^{2n}}\) is continuous for all x in
(A) \(\mathbb{R} – \left\{-1 \right\}\)
(B) \(\mathbb{R} – \left\{-1,1 \right\}\)
(C) \(\mathbb{R} – \left\{1 \right\}\)
(D) \(\mathbb{R} – \left\{0 \right\}\)

Answer: (B) \(\mathbb{R} – \left\{-1,1 \right\}\)

Question 142

The function \(f\left(x\right)=xe^{x\left(1-x\right)}, x\in \mathbb{R} \) is
(A) \(\text{Increasing in}\left(-\frac{1}{2},1\right)\)
(B) \(\text{Decreasing in}\left(\frac{1}{2},2\right)\)
(C) \(\text{Increasing in}\left(-1,-\frac{1}{2}\right)\)
(D) \(\text{Decreasing in}\left(-\frac{1}{2},\frac{1}{2}\right)\)

Answer: (A) \(\text{Increasing in}\left(-\frac{1}{2},1\right)\)

Question 143

The sum of the absolute maximum and absolute minimum values of the function \(f\left(x\right)=\tan^{-1}\left(\sin x-\cos x\right) \) in the interval [0, π] is
(A) \(0\)
(B) \(\tan^{-1}\left(\frac{1}{\sqrt{2}}\right)-\frac{\pi}{4}\)
(C) \(\cos^{-1}\left(\frac{1}{\sqrt{3}}\right)-\frac{\pi}{4}\)
(D) \(\frac{-\pi}{12}\)

Answer: (C) \(\cos^{-1}\left(\frac{1}{\sqrt{3}}\right)-\frac{\pi}{4}\)

Question 144

Let \(x\left(t\right)=2\sqrt{2}\cos t\sqrt{\sin 2t}\ \text{and}\ y\left(t\right)=2\sqrt{2}\sin t\sqrt{\sin 2t},t\in \left(0,\frac{\pi}{2}\right).\) \(\text{Then}\ \frac{1+\left(\frac{dy}{dx}\right)^2}{\frac{d^2y}{dx^2}}\ \text{at}\ t=\frac{\pi}{4}\) is equal to
(A) \(\frac{-2\sqrt{2}}{3}\)
(B) \(\frac{2}{3}\)
(C) \(\frac{1}{3}\)
(D) \(\frac{-2}{3}\)

Answer: (D) \(\frac{-2}{3}\)

Question 145

Let \(I_n\left(x\right)=\int_0^x\frac{1}{\left(t^2+5\right)^n}dt, n=1, 2, 3,\cdots\) Then
(A) \(50I_6-9I_5=x\overset{‘}{I}_5 \)
(B) \(50I_6-11I_5=x\overset{‘}{I}_5 \)
(C) \(50I_6-9I_5=\overset{‘}{I}_5 \)
(D) \(50I_6-11I_5=\overset{‘}{I}_5 \)

Answer: (A) \(50I_6-9I_5=x\overset{‘}{I}_5 \)

Question 146

The area enclosed by the curves \(y=\text{log}_e\left(x+e^2\right),\ x=\text{log}_e\left(\frac{2}{y}\right)\text{ and }x=\text{ log }_e\ 2,\) above the line y = 1 is
(A) \( 2 + e – log_e2 \\\)
(B) \( 1 + e – log_e2\\ \)
(C) \( e – log_e2 \\\)
(D) \( 1 + log_e2\)

Answer: (B) \( 1 + e – log_e2\\ \)

Question 147

Let y = y(x) be the solution curve of the differential equation \(\frac{dy}{dx}+\frac{1}{x^2-1}y=\left(\frac{x-1}{x+1}\right)^{1/2},x>1 \) passing through the point (2, √ (1/3)). Then √ 7 y(8) is
(A) \( 11 + 6 log_e3\\ \)
(B) \( 19\\ \)
(C) \( 12 – 2 log_e3\\ \)
(D) \( 19 – 6 log_e3\)

Answer: (D) \( 19 – 6 log_e3\)

Question 148

The differential equation of the family of circles passing through the points (0, 2) and (0, –2) is
(A) \(2xy\frac{dy}{dx}+\left(x^2-y^2+4\right)=0 \)
(B) \(2xy\frac{dy}{dx}+\left(x^2+y^2-4\right)=0\)
(C) \(2xy\frac{dy}{dx}+\left(y^2-x^2+4\right)=0\)
(D) \(2xy\frac{dy}{dx}-\left(x^2-y^2+4\right)=0\)

Answer: (A) \(2xy\frac{dy}{dx}+\left(x^2-y^2+4\right)=0 \)

Question 149

Let the tangents at two points A and B on the circle x^2 + y^2 – 4x + 3 = 0 meet at origin O(0, 0). Then the area of the triangle OAB is
(A) \(\frac{3\sqrt{3}}{2} \)
(B) \(\frac{3\sqrt{3}}{4} \)
(C) \(\frac{3}{2\sqrt{3}} \)
(D) \(\frac{3}{4\sqrt{3}}\)

Answer: (B) \(\frac{3\sqrt{3}}{4} \)

Question 150

Let the hyperbola \(H:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\ \text{pass through the point}\ \left(2\sqrt{2},-2\sqrt{2}\right).\) A parabola is drawn whose focus is same as the focus of H with positive abscissa and the directrix of the parabola passes through the other focus of H. If the length of the latus rectum of the parabola is e times the length of the latus rectum of H, where e is the eccentricity of H, then which of the following points lies on the parabola?
(A) \(\left(2\sqrt{3},3\sqrt{2}\right)\)
(B) \(\left(3\sqrt{3},-6\sqrt{2}\right)\)
(C) \(\left(\sqrt{3},-\sqrt{6}\right)\)
(D) \(\left(3\sqrt{6},6\sqrt{2}\right)\)

Answer: (B) \(\left(3\sqrt{3},-6\sqrt{2}\right)\)

Question 151

Let the lines \(\frac{x-1}{\lambda}=\frac{y-2}{1}=\frac{z-3}{2}\ \text{and}\ \frac{x+26}{-2}=\frac{y+18}{3}=\frac{z+28}{\lambda}\) be coplanar and P be the plane containing these two lines. Then which of the following points does NOT lie on P?
(A) (0, -2, -2)
(B) (-5, 0, -1)
(C) (3, -1, 0)
(D) (0, 4, 5)

Answer: (D) (0, 4, 5)

Question 152

A plane P is parallel to two lines whose direction rations are –2, 1, –3 and –1, 2, –2 and it contains the point (2, 2, –2). Let P intersect the co-ordinate axes at the points A, B, C making the intercepts α, β, γ. If V is the volume of the tetrahedron OABC, where O is the origin and p = α + β + γ, then the ordered pair (V, p) is equal to :
(A) (48, –13)
(B) (24, –13)
(C) (48, 11)
(D) (24, –5)

Answer: (B) (24, –13)

Question 153

Let S be the set of all a∈ R for which the angle between the vectors \(\overrightarrow{u}=a\left(\text{log}_e b\right)\hat{i}-6\hat{j}+3\hat{k}\) and \(\overrightarrow{v}=\left(\text{log}_e b\right)\hat{i}+2\hat{j}+2a\left(\text{log}_e b\right)\hat{k},\left(b>1\right)\) is acute. Then S is equal to
(A) \(\left(-\infty, -\frac{4}{3}\right)\)
(B) \(Phi \)
(C) \(\left(-\frac{4}{3},0\right)\)
(D) \(\left(\frac{12}{7},\infty\right)\)

Answer: (B) \(Phi \)

Question 154

A horizontal park is in the shape of a triangle OAB with AB = 16. A vertical lamp post OP is erected at the point O such that \(\angle PAO = \angle PBO = 15^\circ ~\text{and}~ \angle PCO = 45^\circ,\) where C is the midpoint of AB. Then (OP)^2 is equal to
(A) \(\frac{32}{\sqrt{3}}\left(\sqrt{3}-1\right) \)
(B) \(\frac{32}{\sqrt{3}}\left(2-\sqrt{3}\right) \)
(C) \(\frac{16}{\sqrt{3}}\left(\sqrt{3}-1\right) \)
(D) \(\frac{16}{\sqrt{3}}\left(2-\sqrt{3}\right) \)

Answer: (B) \(\frac{32}{\sqrt{3}}\left(2-\sqrt{3}\right) \)

Question 155

Let A and B be two events such that \(P\left(B/A\right)\frac{2}{5},\ P\left(A/B\right)=\frac{1}{7}\ \text{and}\ P\left(A\cap B\right)=\frac{1}{9}.\) Consider
\(\left(S1\right)P\left(A’\cup B\right)=\frac{5}{6},\)
\(\left(S2\right)P\left(A’\cap B’\right)=\frac{1}{18}.\) Then
(A) Both (S1) and (S2) are true
(B) Both (S1) and (S2) are false
(C) Only (S1) is true
(D) Only (S2) is true

Answer: (A) Both (S1) and (S2) are true

Question 156

Let
p : Ramesh listens to music.
q :Ramesh is out of his village.
r : It is Sunday.
s : It is Saturday.
Then the statement “Ramesh listens to music only if he is in his village and it is Sunday or Saturday” can be expressed as
(A) \( \left(\left(\sim q\right) \wedge \left(r\vee s\right)\right) \Rightarrow p\\ \)
(B) \( \left(q\wedge \left(r\vee s \right)\right) \Rightarrow p\\ \)
(C) \( p\Rightarrow \left(q\wedge \left(r\vee s\right)\right)\\ \)
(D) \(p\Rightarrow \left(\sim q \right) \wedge \left(r\vee s\right)\)

Answer: (D) \(p\Rightarrow \left(\sim q \right) \wedge \left(r\vee s\right)\)

Question 157

Let R be a relation from the set {1, 2, 3, ….., 60} to itself such that R = {(a, b) : b = pq, where p, q≥ 3 are prime numbers}. Then, the number of elements in R is :
(A) 600
(B) 660
(C) 540
(D) 720

Answer: (B) 660

Question 158

If z = 2 + 3i, then \(z^5+(\bar{z})^5\) is equal to :
(A) 244
(B) 224
(C) 245
(D) 265

Answer: (A) 244

Question 159

Let A and B be two 3 × 3 non-zero real matrices such that AB is a zero matrix. Then
(A) the system of linear equations AX = 0 has a unique solution
(B) the system of linear equations AX = 0 has infinitely many solutions
(C) B is an invertible matrix
(D) adj(A) is an invertible matrix

Answer: (B) the system of linear equations AX = 0 has infinitely many solutions

Question 160

If \(\frac{1}{\left(20-a\right)\left(40-a\right)}+\frac{1}{\left(40-a\right)\left(60-a\right)}+\cdots + \frac{1}{\left(180-a\right)\left(200-a\right)}=\frac{1}{256},\) then the maximum value of a is :
(A) 198
(B) 202
(C) 212
(D) 218

Answer: (C) 212

Question 161

If \(\displaystyle \lim_{x \to 0}\frac{\alpha e^x+\beta e^{-x}+\gamma\sin x}{x\sin^2 x}=\frac{2}{3},\) where α, β, γ∈R, then which of the following is NOT correct?
(A) α^2 + β^2 + γ^2 = 6
(B) αβ + βγ + γα + 1 = 0
(C) αβ^2 + βγ^2 + γα^2 + 3 = 0
(D) α^2 – β^2 + γ^2 = 4

Answer: (C) αβ^2 + βγ^2 + γα^2 + 3 = 0

Question 162

The integral \(\displaystyle\int\limits_0^\frac{\pi}{2}\frac{1}{3+2\sin x+\cos x}dx \) is equal to
(A) tan^–1(2)
(B) \(\tan^{-1}\left(2\right)-\frac{\pi}{4}\)
(C) \(\frac{1}{2}\tan^{-1}\left(2\right)-\frac{\pi}{8}\)
(D) \(\frac{1}{2}\)

Answer: (B) \(\tan^{-1}\left(2\right)-\frac{\pi}{4}\)

Question 163

Let the solution curve y = y(x) of the differential equation \(\left(1+e^{2x}\right)\left(\frac{dy}{dx}+y\right)=1\) pass through the point (0, π/2). Then, \(\displaystyle \lim_{x \to \infty}e^xy\left(x\right)\) is equal to
(A) \(\frac{\pi}{4} \)
(B) \(\frac{3\pi}{4} \)
(C) \(\frac{\pi}{2} \)
(D) \(\frac{3\pi}{2} \)

Answer: (B) \(\frac{3\pi}{4} \)

Question 164

Let a line L pass through the point intersection of the lines bx + 10y – 8 = 0 and \(2x – 3y = 0, b \in R – \left\{ \frac{4}{3}\right\}.\) If the line L also passes through the point (1, 1) and touches the circle 17(x^2 + y^2) = 16, then the eccentricity of the ellipse (x^2/5) + (y^2/5) = 1 is
(A) \(\frac{2}{\sqrt{5}} \)
(B) \(\sqrt{\frac{3}{5}} \)
(C) \(\frac{1}{\sqrt{5}}\)
(D) \(\sqrt{\frac{2}{5}}\)

Answer: (B) \(\sqrt{\frac{3}{5}} \)

Question 165

If the foot of the perpendicular from the point A(–1, 4, 3) on the plane P : 2x + my + nz = 4, is (-2, 7/2, 3/2), then the distance of the point A from the plane P, measured parallel to a line with direction ratios 3, –1, –4, is equal to
(A) 1
(B) \(\sqrt{26}\)
(C) \(2\sqrt{2}\)
(D) \(\sqrt{14}\)

Answer: (B) \(\sqrt{26}\)

Question 166

Let \(\vec{a}=3\hat{i}+\hat{j}\ \text{and}\ \vec{b}=\hat{i}+2\hat{j}+\hat{k}.\) \(\text{Let}\ \vec{c}\ \text{be a vector satisfying}\ \vec{a}\times\left(\vec{b}\times\vec{c}\right)=\vec{b}+\lambda\vec{c}.\) \(\text{If}\ \vec{b}\ \text{and}\ \vec{c}\ \text{are non-parallel},\) then the value of λ is
(A) –5
(B) 5
(C) 1
(D) –1

Answer: (A) –5

Question 167

The angle of elevation of the top of a tower from a point A due north of it is α and from a point B at a distance of 9 units due west of A is \(\cos^{-1}\left(\frac{3}{\sqrt{13}}\right).\) If the distance of the point B from the tower is 15 units, then cot α is equal to :
(A) \(\frac{6}{5}\)
(B) \(\frac{9}{5}\)
(C) \(\frac{4}{3}\)
(D) \(\frac{7}{3}\)

Answer: (A) \(\frac{6}{5}\)

Question 168

The statement (p∧q) ⇒ (p∧r) is equivalent to :
(A) q⇒ (p∧r)
(B) p⇒ (p∧r)
(C) (p∧r) ⇒ (p∧q)
(D) (p∧q) ⇒ r

Answer: (D) (p∧q) ⇒ r

Question 169

Let the circumcentre of a triangle with vertices A(a, 3), B(b, 5) and C(a, b), ab > 0 be P(1, 1). If the line AP intersects the line BC at the point Q(k1, k2), then k1 + k2 is equal to :
(A) 2
(B) \(\frac{4}{7} \)
(C) \(\frac{2}{7} \)
(D) 4

Answer: (B) \(\frac{4}{7} \)

Question 170

Let \(\hat{a}\ \text{and}\ \hat{b}\) be two unit vectors such that the angle between them is π/4. If θ is the angle between the vectors \(\left(\hat{a}+\hat{b}\right)\ \text{and}\ \left(\hat{a}+2\hat{b}+2\left(\hat{a}\times\hat{b}\right)\right),\) then the value of 164 cos^2θ is equal to :
(A) \(90+27\sqrt{2}\)
(B) \(45+18\sqrt{2}\)
(C) \(90+3\sqrt{2}\)
(D) \(54+90\sqrt{2}\)

Answer: (A) \(90+27\sqrt{2}\)

Question 171

If \(f\left(\alpha\right)=\displaystyle\int\limits_1^\alpha\frac{\text{log}_{10}t}{1+t}dt, \alpha > 0,\) then f(e^3) + f(e^–3) is equal to :
(A) 9
(B) \(\frac{9}{2}\)
(C) \(\frac{9}{\text{log}_e\left(10\right)}\)
(D) \(\frac{9}{2\text{log}_e\left(10\right)}\)

Answer: (D) \(\frac{9}{2\text{log}_e\left(10\right)}\)

Question 172

The area of the region \(\left\{\left(x,y\right);\left|x-1\right|\leq y \leq \sqrt{5-x^2} \right\}\) is equal to
(A) \(\frac{5}{2}\sin^{-1}\left(\frac{3}{5}\right)-\frac{1}{2}\)
(B) \(\frac{5\pi}{4}-\frac{3}{2}\)
(C) \(\frac{3\pi}{4}+\frac{3}{2}\)
(D) \(\frac{5\pi}{4}-\frac{1}{2}\)

Answer: (D) \(\frac{5\pi}{4}-\frac{1}{2}\)

Question 173

Let the focal chord of the parabola P :y^2 = 4x along the line L : y = mx + c, m> 0 meet the parabola at the points M and N. Let the line L be a tangent to the hyperbola H :x^2 – y^2 = 4. If O is the vertex of P and F is the focus of H on the positive x-axis, then the area of the quadrilateral OMFN is
(A) \(2\sqrt{6} \)
(B) \(2\sqrt{14} \)
(C) \(4\sqrt{6} \)
(D) \(4\sqrt{14} \)

Answer: (B) \(2\sqrt{14} \)

Question 174

The number of points, where the function f: ℝ → ℝ, f(x) = |x – 1|cos|x – 2|sin|x – 1| + (x – 3)|x^2 – 5x + 4|, is NOT differentiable, is
(A) 1
(B) 2
(C) 3
(D) 4

Answer: (B) 2

Question 175

Let S = {1, 2, 3, …, 2022}. Then the probability that a randomly chosen number n from the set S such that HCF (n, 2022) = 1, is
(A) \(\frac{128}{1011} \)
(B) \(\frac{166}{1011} \)
(C) \(\frac{127}{337} \)
(D) \(\frac{112}{337} \)

Answer: (D) \(\frac{112}{337} \)

Question 176

Let \(f\left(x\right)=3^{\left(x^2-2\right)^3+4},x\in \mathbb{R}.\) Then which of the following statements are true?
P :x = 0 is a point of local minima of f
Q: x = √2 is a point of inflection of f
R :f ′ is increasing for x > √2
(A) Only P and Q
(B) Only P and R
(C) Only Q and R
(D) All P, Q and R

Answer: (D) All P, Q and R

Question 177

If z ≠ 0 be a complex number such that \(\left|z-\frac{1}{z}\right|=2,\) then the maximum value of |z| is
(A) √2
(B) 1
(C) √2 – 1
(D) √2 + 1

Answer: (D) √2 + 1

Question 178

Which of the following matrices can NOT be obtained from the matrix \(\begin{bmatrix} -1& 2 \\1 & -1 \\\end{bmatrix}\) by a single elementary row operation?
(A) \(\begin{bmatrix} 0& 1 \\1 & -1 \\\end{bmatrix}\)
(B) \(\begin{bmatrix} 1& -1 \\-1 & 2\\\end{bmatrix}\)
(C) \(\begin{bmatrix} -1& 2 \\-2 & 7\\\end{bmatrix}\)
(D) \(\begin{bmatrix} -1& 2 \\-1 & 3\\\end{bmatrix}\)

Answer: (C) \(\begin{bmatrix} -1& 2 \\-2 & 7\\\end{bmatrix}\)

Question 179

If the system of equations \(x + y + z = 6\\ 2x + 5y + \alpha z = \beta\\ x + 2y + 3z = 14\) has infinitely many solutions, then α + β is equal to
(A) 8
(B) 36
(C) 44
(D) 48

Answer: (C) 44

Question 180

Let the function \(f(x)= \left\{\begin{matrix}\frac{\log_e(1+5x)-\log_e(1+\alpha x)}{x} &; \text{if } x\in0 \\10 & ; \text{if } x=0 \\\end{matrix}\right.\) be continuous at x = 0. Then α is equal to
(A) 10
(B) –10
(C) 5
(D) –5

Answer: (D) –5

Question 181

If [t] denotes the greatest integer ≤ t, then the value of \(\int_{0}^{1}[2x-|3x^2 -5x + 2| + 1]dx \) is
(A) \(\frac{\sqrt{37} + \sqrt{13}-4}{6}\)
(B) \(\frac{\sqrt{37} – \sqrt{13}-4}{6}\)
(C) \(\frac{-\sqrt{37} – \sqrt{13}+4}{6}\)
(D) \(\frac{-\sqrt{37} + \sqrt{13}+4}{6}\)

Answer: (A) \(\frac{\sqrt{37} + \sqrt{13}-4}{6}\)

Question 182

Let \(\{a_n\}_{n=0}^\infty\ \text{be a sequence such that}\ a_0 = a_1 = 0\ \text{and}\)\(a_{n+2}=3a_{n+1}-2a_{n} + 1, \forall \ n \ge 0.\ \text{Then}\ a_{25}a_{23}-2a_{25}a_{22}-2a_{23}a_{24}+4a_{22}a_{24}\) is equal to
(A) 483
(B) 528
(C) 575
(D) 624

Answer: (B) 528

Question 183

\(\sum_{r=1}^{20}(r^2+1)(r!)\) is equal to
(A) \( 22! – 21!\\ \)
(B) \( 22! – 2(21!)\\ \)
(C) \( 21! – 2(20!)\\ \)
(D) \( 21! – 20!\)

Answer: (B) \( 22! – 2(21!)\\ \)

Question 184

For \(I(x)=\int\frac{\sec^2 x – 2022}{\sin^{2022}x} dx,\ \text{if}\ I\left(\frac{\pi}{4}\right)=2^{1011},\) then
(A) \(\ 3^{1010}I\left(\frac{\pi}{3}\right)-I\left(\frac{\pi}{6}\right)=0\)
(B) \(\ 3^{1010}I\left(\frac{\pi}{6}\right)-I\left(\frac{\pi}{3}\right)=0\)
(C) \(\ 3^{1011}I\left(\frac{\pi}{3}\right)-I\left(\frac{\pi}{6}\right)=0\)
(D) \(\ 3^{1011}I\left(\frac{\pi}{6}\right)-I\left(\frac{\pi}{3}\right)=0\)

Answer: (A) \(\ 3^{1010}I\left(\frac{\pi}{3}\right)-I\left(\frac{\pi}{6}\right)=0\)

Question 185

if the solution curve of the differential equation \(\frac{dy}{dx}=\frac{x+y-2}{x-y} \) passes through the points (2, 1) and (k + 1, 2), k > 0, then
(A) \(2\tan^{-1}\left(\frac{1}{k}\right)=\log_e(k^2+1)\)
(B) \(\tan^{-1}\left(\frac{1}{k}\right)=\log_e(k^2+1)\)
(C) \(2\tan^{-1}\left(\frac{1}{k+1}\right)=\log_e(k^2+2k +2)\)
(D) \(2\tan^{-1}\left(\frac{1}{k}\right)=\log_e\left(\frac{k^2+1}{k^2}\right)\)

Answer: (A) \(2\tan^{-1}\left(\frac{1}{k}\right)=\log_e(k^2+1)\)

Question 186

Let y = y(x) be the solution curve of the differential equation \(\frac{dy}{dx}+\left(\frac{2x^2+11x+13}{x^3+6x^2+11x+6}\right)y = \frac{(x+3)}{x+1}, x > -1\) which passes through the point (0, 1). Then y(1) is equal to
(A) 1/2
(B) 3/2
(C) 5/2
(D) 7/2

Answer: (B) 3/2

Question 187

Let m1, m2 be the slopes of two adjacent sides of a square of side a such that \(a^2 + 11a + 3(m_1^2 + m_2^2) = 220. \) If one vertex of the square is \(\left(10\left(cos~\alpha – sin~\alpha\right),10\left(sin~\alpha + cos~ \alpha\right)\right),\ \text{where}\ \alpha \in \left(0, \frac{\pi}{2}\right)\) and the equation of one diagonal is \((\cos \alpha – \sin \alpha)x + (\sin \alpha + \cos \alpha)y = 10,\ \text{then}\ 72(\sin^4 \alpha + \cos^4 \alpha) + a^2 -3a + 13 \) is equal to :
(A) 119
(B) 128
(C) 145
(D) 155

Answer: (B) 128

Question 188

The number of elements in the set
\(S=\left\{x \in \mathbb{R} : 2\cos \left(\frac{x^2 + x}{6}\right)=4^x + 4^{-x}\right\}\)
(A) 1
(B) 3
(C) 0
(D) infinite

Answer: (A) 1

Question 189

Let A(α, -2), B(α, 6) and C(α/4, -2) be vertices of a ΔABC. If (5, α/4) is the circumcentre of ΔABC, then which of the following is NOT correct about ΔABC?
(A) Area is 24
(B) Perimeter is 25
(C) Circumradius is 5
(D) Inradius is 2

Answer: (B) Perimeter is 25

Question 190

Let Q be the foot of perpendicular drawn from the point P(1, 2, 3) to the plane x + 2y + z = 14. If R is a point on the plane such that ∠PRQ = 60°, then the area of ΔPQR is equal to :
(A) \(\frac{\sqrt{3}}{2}\)
(B) \(\sqrt{3}\)
(C) \(2\sqrt{3}\)
(D) \(3\)

Answer: (B) \(\sqrt{3}\)

Question 191

If (2, 3, 9), (5, 2, 1), (1, λ, 8) and (λ, 2, 3) are coplanar, then the product of all possible values of λ is :
(A) \(\frac{21}{2}\)
(B) \(\frac{59}{8}\)
(C) \(\frac{57}{8}\)
(D) \(\frac{95}{8}\)

Answer: (D) \(\frac{95}{8}\)

Question 192

Bag I contains 3 red, 4 black and 3 white balls and Bag II contains 2 red, 5 black and 2 white balls. One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball so drawn is found to be black in colour. Then the probability, that the transferred ball is red, is :
(A) \(\frac{4}{9}\)
(B) \(\frac{5}{18}\)
(C) \(\frac{1}{6}\)
(D) \(\frac{3}{10}\)

Answer: (B) \(\frac{5}{18}\)

Question 193

Let \(S = \{z = x + iy : \left|z – 1 + i\right| \geq \left|z\right|, \left|z\right| < 2, \left|z + i\right| = \left|z – 1\right|\}.\) Then the set of all values of x, for which w = 2x + iy ∈ S for some y ∈ R is
(A) \(\left(-\sqrt{2}, \frac{1}{2\sqrt{2}}\right]\)
(B) \(\left(-\frac{1}{\sqrt{2}}, \frac{1}{4}\right]\)
(C) \(\left(-\sqrt{2}, \frac{1}{2}\right]\)
(D) \(\left(-\frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}\right]\)

Answer: (B) \(\left(-\frac{1}{\sqrt{2}}, \frac{1}{4}\right]\)

Question 194

Let \(\vec{a},\vec{b},\vec{c} \) be three coplanar concurrent vectors such that angles between any two of them is same. If the product of their magnitudes is 14 and \((\vec{a}\times \vec{b}) \cdot (\vec{b} \times \vec{c}) + (\vec{b} \times \vec{c})\cdot (\vec{c} \times \vec{a}) + (\vec{c} \times \vec{a})\cdot (\vec{a} \times \vec{b}) = 168,\ \text{then}\ |\vec{a}| + |\vec{b}| + |\vec{c}|\) is equal to :
(A) 10
(B) 14
(C) 16
(D) 18

Answer: (C) 16

Question 195

The domain of the function \(f(x)=\sin^{-1}\left(\frac{x^2-3x+2}{x^2+2x+7}\right)\) is :
(A) \( \left[1, \infty\right)\\ \)
(B) \( \left[-1, 2\right]\\ \)
(C) \( \left[-1, \infty\right)\\ \)
(D) \( \left(-\infty , 2\right]\)

Answer: (C) \( \left[-1, \infty\right)\\ \)

Question 196

The statement \(\left(p \Rightarrow q\right) \vee \left(p \Rightarrow r\right)\) is NOT equivalent to
(A) \((p \wedge (\sim r))\Rightarrow q\)
(B) \((\sim q)\Rightarrow ((\sim r)\vee p)\)
(C) \(p\Rightarrow (q\vee r)\)
(D) \((p\wedge (\sim q))\Rightarrow r\)

Answer: (B) \((\sim q)\Rightarrow ((\sim r)\vee p)\)

Question 197

Let A = {z ∈ C : 1 ≤ |z – (1 + i)| ≤ 2} and B = {z ∈ A : |z – (1 – i)| = 1}. Then, B :
(A) Is an empty set
(B) Contains exactly two elements
(C) Contains exactly three elements
(D) Is an infinite set

Answer: (D) Is an infinite set

Question 198

The remainder when 3^2022 is divided by 5 is :
(A) 1
(B) 2
(C) 3
(D) 4

Answer: (D) 4

Question 199

The surface area of a balloon of spherical shape being inflated, increases at a constant rate. If initially, the radius of balloon is 3 units and after 5 seconds, it becomes 7 units, then its radius after 9 seconds is :
(A) 9
(B) 10
(C) 11
(D) 12

Answer: (A) 9

Question 200

Bag A contains 2 white, 1 black and 3 red balls and bas B contains 3 black, 2 red and n white balls. One bag is chosen at random and 2 balls drawn from it at random, are found to be 1 red and 1 black. If the probability that both balls come from Bag A is \(\frac{6}{11}\) then n is equal to _______.
(A) 13
(B) 6
(C) 4
(D) 3

Answer: (C) 4

Question 201

Let \(x^2+y^2+Ax+By+C=0\) be a circle passing through (0, 6) and touching the parabola y = x^2 at (2, 4). Then A + C is equal to ________.
(A) 16
(B) \(\frac{88}{5}\)
(C) 72
(D) –8

Answer: (A) 16

Question 202

The number of values of α for which the system of equations :
x + y + z = α
αx + 2αy + 3z = –1
x + 3αy + 5z = 4
is inconsistent, is
(A) 0
(B) 1
(C) 2
(D) 3

Answer: (B) 1

Question 203

If the sum of the squares of the reciprocals of the roots α and β of the equation 3x^2 + λx – 1 = 0 is 15, then 6(α^3 + β^3)^2 is equal to :
(A) 18
(B) 24
(C) 36
(D) 96

Answer: (B) 24

Question 204

The set of all values of k for which \(\left ( \tan^{-1}x \right )^3+\left ( \cot^{-1}x \right )^3=k\pi ^3, x \in R,\) is the interval:
(A) \(\left [ \frac{1}{32},\frac{7}{8} \right )\)
(B) \(\left ( \frac{1}{24},\frac{13}{16} \right )\)
(C) \(\left [ \frac{1}{48},\frac{13}{16} \right ]\)
(D) \(\left [ \frac{1}{32},\frac{9}{8} \right )\)

Answer: (A) \(\left [ \frac{1}{32},\frac{7}{8} \right )\)

Question 205

Let \(S=\left\{ \sqrt{n}:1\le n \le 50 \text{ and } n \text{ is odd} \right\}\).
Let \(a\in S \text{ and } A=\begin{bmatrix}1 & 0 & a \\-1 & 1 & 0 \\-a & 0 & 1 \\\end{bmatrix}\)
If \(\sum_{a~\in~S} \det \left ( adj A \right ) = 100 \lambda\), then λ is equal to :
(A) 218
(B) 221
(C) 663
(D) 1717

Answer: (B) 221

Question 206

For the function
f(x) = 4 loge(x – 1) – 2x^2 + 4x + 5, x > 1, which one of the following is NOT correct?
(A) f is increasing in (1, 2) and decreasing in (2, )
(B) f(x) = –1 has exactly two solutions
(C) f′(e) – f′′(2) < 0
(D) f(x) = 0 has a root in the interval (e, e + 1)

Answer: (C) f′(e) – f′′(2) < 0

Question 207

If the tangent at the point (x1, y1) on the curve y = x^3 + 3x^2 + 5 passes through the origin, then (x1, y1) does NOT lie on the curve :
(A) \(x^2+\frac{y^2}{81}=2\)
(B) \(\frac{y^2}{9}-x^2=8 \)
(C) y = 4x^2 + 5
(D) \(\frac{x}{3}-y^2=2\)

Answer: (D) \(\frac{x}{3}-y^2=2\)

Question 208

The sum of absolute maximum and absolute minimum values of the function f(x) = |2x^2 + 3x – 2| + sinx cosx in the interval [0, 1] is :
(A) \(3+\frac{\sin(1)\cos^2\left ( \frac{1}{2} \right )}{2} \)
(B) \(3+\frac{1}{2}\left( 1+2\cos(1)\right )\sin(1)\)
(C) \(5+\frac{1}{2}\left ( \sin(1)+sin(2) \right )\)
(D) \(2+\sin\left ( \frac{1}{2} \right )\cos\left ( \frac{1}{2} \right )\)

Answer: (B) \(3+\frac{1}{2}\left( 1+2\cos(1)\right )\sin(1)\)

Question 209

If \(\left\{ a_i\right\}_{i=1}^n\), where n is an even integer, is an arithmetic progression with common difference 1, and \(\sum_{i=1}^{n}a_i = 192,\sum_{i=1}^{n/2}a_{2i}= 120\), then n is equal to :
(A) 48
(B) 96
(C) 92
(D) 104

Answer: (B) 96

Question 210

If x = x(y) is the solution of the differential equation \(y\frac{dx}{dy}=2x+y^3(y+1)e^y, x(1)=0;\) then x(e) is equal to :
(A) e^3(e^e – 1)
(B) e^e(e^3 – 1)
(C) e^2(e^e + 1)
(D) e^e(e^2 – 1)

Answer: (A) e^3(e^e – 1)

Question 211

Let λx – 2y = μ be a tangent to the hyperbola a^2x^2 – y^2 = b^2. Then \(\left ( \frac{\lambda}{a} \right )^2-\left ( \frac{\mu}{b} \right )^2\) is equal to :
(A) –2
(B) –4
(C) 2
(D) 4

Answer: (D) 4

Question 212

Let \(\hat{a}, \hat{b}\)be unit vectors. If \(\vec{c}\) be a vector such that the angle between \(\hat{a}\) and \(\vec{c}\) is \(\frac{\pi}{12}\) and \(\hat{b}=\vec{c}+2\left ( \vec {c}\times\hat{a} \right )\) then \(\left|6 \vec{c}\right|^2\) is equal to:
(A) \(6\left ( 3-\sqrt{3} \right )\)
(B) \(3+\sqrt{3}\)
(C) \(6\left ( 3+\sqrt{3} \right )\)
(D) \(6\left ( \sqrt{3} + 1 \right )\)

Answer: (C) \(6\left ( 3+\sqrt{3} \right )\)

Question 213

If a random variable X follows the Binomial distribution B(33, p) such that 3P(X = 0) = P(X = 1), then the value of \(\frac{P\left ( X=15 \right )}{P\left ( X=18 \right )}-\frac{P\left ( X=16 \right )}{P\left ( X=17 \right )}\) is equal to:
(A) 1320
(B) 1088
(C) \(\frac{120}{1331}\)
(D) \(\frac{1088}{1089}\)

Answer: (A) 1320

Question 214

The domain of the function \(f(x)=\frac{\cos^{-1}\left ( \frac{x^2-5x+6}{x^2-9} \right )}{\log_e\left ( x^2-3x+2 \right )}\) is:
(A) \(\left ( -\infty,1 \right )\cup\left ( 2, \infty \right )\)
(B) (2, ∞)
(C) \(\left [ -\frac{1}{2},1 \right )\cup\left ( 2,\infty \right )\)
(D) \(\left [ -\frac{1}{2},1 \right )\cup\left ( 2,\infty \right )-\left\{ \frac{3+\sqrt{5}}{2},\frac{3-\sqrt{5}}{2}\right\}\)

Answer: (D) \(\left [ -\frac{1}{2},1 \right )\cup\left ( 2,\infty \right )-\left\{ \frac{3+\sqrt{5}}{2},\frac{3-\sqrt{5}}{2}\right\}\)

Question 215

Let \(S=\left\{ \theta \in \left [ -\pi, \pi \right ]-\left\{ \pm\frac{\pi}{2}\right\}: \sin\theta\tan\theta + \tan\theta=\sin2\theta\right\}\). If \(T=\sum_{\theta = S}\cos2\theta\) then T + n(S) is equal to:
(A) \(7+\sqrt{3}\)
(B) 9
(C) \(8+\sqrt{3}\)
(D) 10

Answer: (B) 9

Question 216

The number of choices for \(\Delta \in \left\{ \land, \lor, \Rightarrow, \Leftrightarrow \right\}\) such that (p Δ q) ⇒ ((p Δ ~ q) ∨ ((~p) Δ q)) is a tautology, is
(A) 1
(B) 2
(C) 3
(D) 4

Answer: (B) 2

Question 217

Let x * y = x^2 + y^3 and (x * 1) * 1 = x * (1 * 1). Then a value of \(2\ sin^{-1}\left(\frac{x^4+x^2-2}{x^4+x^2+2}\right)\) is
(A) \(\frac{\pi}{4}\)
(B) \(\frac{\pi}{3}\)
(C) \(\frac{\pi}{2}\)
(D) \(\frac{\pi}{6}\)

Answer: (B) \(\frac{\pi}{3}\)

Question 218

The sum of all the real roots of the equation (e^2^x – 4)(6e^2^x – 5e^x + 1) = 0 is
(A) loge3
(B) –loge3
(C) loge6
(D) –loge6

Answer: (B) –loge3

Question 219

Let the system of linear equations
x + y + az = 2
3x + y + z = 4
x + 2z = 1
have a unique solution (x*, y*, z*). If (α, x*), (y*, α) and (x*, –y*) are collinear points, then the sum of absolute values of all possible values of α is
(A) 4
(B) 3
(C) 2
(D) 1

Answer: (C) 2

Question 220

Let x, y > 0. If x^3y^2 = 2^15, then the least value of 3x + 2y is
(A) 30
(B) 32
(C) 36
(D) 40

Answer: (D) 40

Question 221

Let \(f(x)\left\{\begin{matrix} \frac{\sin(x-[x])}{x-[x]},& x \in (-2,-1)\\ \max{\left\{2x,3[\left|x \right|]\right\}},&\left|x \right|<1 \\ 1& ,\text{otherwise}& \\\end{matrix}\right. \)
Where [t] denotes greatest integer t. If m is the number of points where f is not continuous and n is the number of points where f is not differentiable, then the ordered pair (m, n) is
(A) (3, 3)
(B) (2, 4)
(C) (2, 3)
(D) (3, 4)

Answer: (C) (2, 3)

Question 222

The value of the integral
\(\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}}\frac{dx}{(1+e^x)(sin^6x+cos^6x)}\) is equal to
(A) 2π
(B) 0
(C) π
(D) π/2

Answer: (C) π

Question 223

\(\displaystyle \lim_{n\to \infty}\left(\frac{n^2}{(n^2+1)(n+1)}+\frac{n^2}{(n^2+4)(n+2)}+\frac{n^2}{(n^2+9)(n+3)}……+\frac{n^2}{(n^2+n^2)(n+n)}\right)\)
is equal to
(A) \(\frac{\pi}{8}+\frac{1}{4}log_e2\)
(B) \(\frac{\pi}{4}+\frac{1}{8}log_e2\)
(C) \(\frac{\pi}{4}-\frac{1}{8}log_e2\)
(D) \(\frac{\pi}{8}+\frac{1}{8}log_e\sqrt{2}\)

Answer: (A) \(\frac{\pi}{8}+\frac{1}{4}log_e2\)

Question 224

A particle is moving in the xy-plane along a curve C passing through the point (3, 3). The tangent to the curve C at the point P meets the x-axis at Q. If the y-axis bisects the segment PQ, then C is a parabola with
(A) Length of latus rectum 3
(B) Length of latus rectum 6
(C) \(Focus\left(\frac{4}{3},0\right )\)
(D) \(Focus\left(0,\frac{3}{4}\right)\)

Answer: (A) Length of latus rectum 3

Question 225

Let the maximum area of the triangle that can be inscribed in the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{4}=1, a>2,\) having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to the y-axis, be 6√3. Then the eccentricity of the ellipse is
(A) \(\frac{\sqrt{3}}{2}\)
(B) \(\frac{1}{2}\)
(C) \(\frac{1}{\sqrt{2}}\)
(D) \(\frac{\sqrt{3}}{4}\)

Answer: (A) \(\frac{\sqrt{3}}{2}\)

Question 226

Let the area of the triangle with vertices A(1, α), B(α, 0) and C(0, α) be 4 sq. units. If the points (α, –α), (–α, α) and (α^2, β) are collinear, then β is equal to
(A) 64
(B) –8
(C) –64
(D) 512

Answer: (C) –64

Question 227

The number of distinct real roots of the equation x^7 – 7x – 2 = 0 is
(A) 5
(B) 7
(C) 1
(D) 3

Answer: (D) 3

Question 228

A random variable X has the following probability distribution :
X
0
1
2
3
4
P(X)
k
2k
4k
6k
8k
The value of P(1 < X < 4 | x ≤ 2) is equal to
(A) \(\frac{4}{7}\)
(B) \(\frac{2}{3}\)
(C) \(\frac{3}{7}\)
(D) \(\frac{4}{5}\)

Answer: (A) \(\frac{4}{7}\)

Question 229

The number of solutions of the equation \(cos\left ( x+\frac{\pi}{3} \right)cos\left (\frac{\pi}{3}-x\right)=\frac{1}{4}cos^22x,x\in[-3\pi,3\pi]\) is:
(A) 8
(B) 5
(C) 6
(D) 7

Answer: (D) 7

Question 230

If the shortest distance between the lines \(\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{\lambda}~\text{and}~ \frac{x-2}{1}=\frac{y-4}{4}=\frac{z-5}{5}\) is 1/√3, then the sum of all possible values of λ is :
(A) 16
(B) 6
(C) 12
(D) 15

Answer: (A) 16

Question 231

Let the points on the plane P be equidistant from the points (–4, 2, 1) and (2, –2, 3). Then the acute angle between the plane P and the plane 2x + y + 3z = 1 is
(A) \(\frac{\pi}{6}\)
(B) \(\frac{\pi}{4}\)
(C) \(\frac{\pi}{3}\)
(D) \(\frac{5\pi}{12}\)

Answer: (C) \(\frac{\pi}{3}\)

Question 232

\(\text{Let}\ \hat{a}\ \text{and}\ \hat{b}\ \text{be two unit vectors such that}\)  \(\left|(\hat{a}+\hat{b})+2(\hat{a}\times\hat{b}) \right|=2. \) If θ ∈ (0, π) is the angle between \(\hat{a}\ \text{and}\ \hat{b},\ \text{then among the statements}:\)
(S1): \(2\left|\hat{a}\times\hat{b}\right|=\left|\hat{a}-\hat{b} \right| \)
(S2): The projection of \(\hat{a}\ \text{on}\ (\hat{a}+\hat{b})is \frac{1}{2}\)
(A) Only (S1) is true
(B) Only (S2) is true
(C) Both (S1) and (S2) are true
(D) Both (S1) and (S2) are false

Answer: (C) Both (S1) and (S2) are true

Question 233

If \(y=tan^{-1}(sec~ x^3-tan~x^3),\frac{\pi}{2}(A) xy′′ + 2y′ = 0
(B) \(x^2y”-6y+\frac{3\pi}{2}=0\)
(C) x^2y″ – 6y + 3π = 0
(D) xy″ – 4y′ = 0

Answer: (B) \(x^2y”-6y+\frac{3\pi}{2}=0\)

Question 234

Consider the following statements:
A : Rishi is a judge.
B : Rishi is honest.
C : Rishi is not arrogant.
The negation of the statement “if Rishi is a judge and he is not arrogant, then he is honest” is
(A) B → (A ∨ C)
(B) (~ B) ∧ (A ∧ C)
(C) B → ((~ A) ∨ (~ C))
(D) B → (A ∧ C)

Answer: (B) (~ B) ∧ (A ∧ C)

Question 235

The slope of normal at any point (x, y), x > 0, y > 0 on the curve y = y(x) is given by \(\frac{x^2}{xy-x^2y^2-1}.\) If the curve passes through the point (1, 1), then e · y(e) is equal to
(A) \(\frac{1-tan(1)}{1+tan(1)}\)
(B) tan(1)
(C) 1
(D) \(\frac{1+tan(1)}{1-tan(1)}\)

Answer: (D) \(\frac{1+tan(1)}{1-tan(1)}\)

Question 236

Let λ* be the largest value of λ for which the function fλ(x) = 4λx^3 – 36λx^2 + 36x + 48 is increasing for all x ∈ ℝ. Then fλ* (1) + fλ* (– 1) is equal to :
(A) 36
(B) 48
(C) 64
(D) 72

Answer: (D) 72

Question 237

Let a circle C touch the lines L1 : 4x – 3y +K1 = 0 and L2 : 4x – 3y + K2 = 0, K1, K2∈R. If a line passing through the centre of the circle C intersects L1 at (–1, 2) and L2 at (3, –6), then the equation of the circle Cis :
(A) (x – 1)^2 + (y – 2)^2 = 4
(B) (x + 1)^2 + (y – 2)^2 = 4
(C) (x – 1)^2 + (y + 2)^2 = 16
(D) (x – 1)^2 + (y – 2)^2 = 16

Answer: (C) (x – 1)^2 + (y + 2)^2 = 16

Question 238

The value of \(\displaystyle\int\limits_0^\pi\frac{e^{\cos x}\sin x}{\left ( 1+\cos^2x \right )\left ( e^{\cos x}+e^{-\cos x} \right )}dx\) is equal to :
(A) \(\frac{\pi^2}{4}\)
(B) \(\frac{\pi^2}{2}\)
(C) \(\frac{\pi}{4}\)
(D) \(\frac{\pi}{2}\)

Answer: (C) \(\frac{\pi}{4}\)

Question 239

Let a, b and c be the length of sides of a triangle ABC such that \(\frac{a+b}{7}=\frac{b+c}{8}=\frac{c+a}{9}\). If r and R are the radius of incircle and radius of circumcircle of the triangle ABC, respectively, then the value of \(\frac{R}{r}\) is equal to :
(A) \(\frac{5}{2}\)
(B) 2
(C) \(\frac{3}{2}\)
(D) 1

Answer: (A) \(\frac{5}{2}\)

Question 240

Let ƒ : N→R be a function such that ƒ(x + y) = 2ƒ(x) ƒ(y) for natural numbers x and y. If ƒ(1) = 2, then the value of α for which
\(\sum_{k=1}^{10}f\left ( \alpha+k \right )=\frac{512}{3}\left ( 2^{20}-1 \right )|\)
holds, is :
(A) 2
(B) 3
(C) 4
(D) 6

Answer: (C) 4

Question 241

Let A be a 3 × 3 real matrix such that
\(A\begin{pmatrix}1 \\1 \\0\end{pmatrix}= \begin{pmatrix}1 \\1 \\0\end{pmatrix};~~A \begin{pmatrix}1 \\0 \\1\end{pmatrix}= \begin{pmatrix}-1 \\0 \\1\end{pmatrix}\) and \(A\begin{pmatrix}0 \\0 \\1\end{pmatrix}= \begin{pmatrix}1 \\1 \\2\end{pmatrix}\)
If X = (x1, x2, x3)^T and I is an identity matrix of order 3, then the system \(\left ( A-2I \right )X=\begin{pmatrix} 4\\ 1\\1\end{pmatrix}\) has :
(A) No solution
(B) Infinitely many solutions
(C) Unique solution
(D) Exactly two solutions

Answer: (B) Infinitely many solutions

Question 242

Let ƒ : R→R be defined as
ƒ(x) = x^3 + x – 5
If g(x) is a function such that ƒ(g(x)) = \(x,\forall^{‘}x^{‘}\epsilon \textbf{R}\) then g′(63) is equal to_____.
(A) \(\frac{1}{49}\)
(B) \(\frac{3}{49}\)
(C) \(\frac{43}{49}\)
(D) \(\frac{91}{49}\)

Answer: (A) \(\frac{1}{49}\)

Question 243

Consider the following two propositions :
P1 : ~ (p → ~ q)
P2: (p ∧ ~q) ∧ ((-~p) ∨ q)
If the proposition p → ((~p) ∨ q) is evaluated as FALSE, then :
(A) P1 is TRUE and P2 is FALSE
(B) P1 is FALSE and P2 is TRUE
(C) Both P1 and P2 are FALSE
(D) Both P1 and P2 are TRUE

Answer: (C) Both P1 and P2 are FALSE

Question 244

If \(\frac{1}{2\cdot3^{10}}+\frac{1}{2^2\cdot3^9}+\dots+\frac{1}{2^{10}\cdot3}=\frac{K}{2^{10}\cdot3^{10}}\) then the remainder when K is divided by 6 is :
(A) 1
(B) 2
(C) 3
(D) 5

Answer: (D) 5

Question 245

Let ƒ(x) be a polynomial function such that ƒ(x) + ƒ′(x) + ƒ′′(x) = x^5 + 64. Then, the value of \(\displaystyle \lim_{ x\to 1}\frac{f\left ( x \right )}{x-1}\) is equal to :
(A) –15
(B) –60
(C) 60
(D) 15

Answer: (A) –15

Question 246

Let E1 and E2 be two events such that the conditional probabilities \(P\left ( E_1|E_2 \right )=\frac{1}{2},\) \(P\left ( E_2|E_1 \right )=\frac{3}{4}\) and \(P\left ( E_1\cap E_2 \right )=\frac{1}{8}\cdot\) Then:
(A) \(P\left ( E_1\cap E_2 \right )=P\left ( E_1 \right )\cdot P\left ( E_2 \right )\)
(B) \(P\left ( E_1^{‘}\cap E_2^{‘} \right )=P\left ( E_1^{‘} \right )\cdot P\left ( E_2 \right )\)
(C) \(P\left ( E_1\cap E_2^{‘} \right )=P\left ( E_1 \right )\cdot P\left ( E_2 \right )\)
(D) \(P\left ( E_1^{‘}\cap E_2 \right )=P\left ( E_1 \right )\cdot P\left ( E_2 \right )\)

Answer: (C) \(P\left ( E_1\cap E_2^{‘} \right )=P\left ( E_1 \right )\cdot P\left ( E_2 \right )\)

Question 247

Let \(A=\begin{bmatrix}0 & -2 \\2 & 0 \\\end{bmatrix}\) If M and N are two matrices given by \(M=\displaystyle\sum\limits_{k=1}^{10} A^{2k}\) and \(N=\displaystyle\sum\limits_{k=1}^{10}A^{2k-1}\) then MN^2is :
(A) a non-identity symmetric matrix
(B) a skew-symmetric matrix
(C) neither symmetric nor skew-symmetric matrix
(D) an identity matrix

Answer: (A) a non-identity symmetric matrix

Question 248

Let g : (0, ∞) →R be a differentiable function such that \(\int\left ( \frac{x\left ( \cos x-\sin x \right )}{e^x+1}+\frac{g\left ( x \right )\left ( e^x+1-xe^x \right )}{\left ( e^x+1 \right )^2} \right ) dx=\frac{xg\left ( x \right )}{e^x+1}+c,\) for all x> 0, where c is an arbitrary constant. Then :
(A) g is decreasingin \(\left ( 0, \frac{\pi}{4} \right )\)
(B) g′ is increasing in \(\left ( 0, \frac{\pi}{4} \right )\)
(C) g + g′ is increasing in \(\left ( 0, \frac{\pi}{2} \right )\)
(D) g – g′ is increasing in \(\left ( 0, \frac{\pi}{2} \right )\)

Answer: (D) g – g′ is increasing in \(\left ( 0, \frac{\pi}{2} \right )\)

Question 249

Let f :R→R and g : R→R be two functions defined by f(x) = loge(x^2 + 1) – e^–^x + 1 and \(g\left ( x \right )=\frac{1-2e^{2x}}{e^x}\). Then, for which of the following range of α, the inequality \(f\left ( g\left ( \frac{\left ( \alpha-1 \right )^2}{3} \right ) \right )>f\left ( g\left ( \alpha-\frac{5}{3} \right ) \right )\) holds?
(A) (2, 3)
(B) (–2, –1)
(C) (1, 2)
(D) (–1, 1)

Answer: (A) (2, 3)

Question 250

Let \(\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}a_i>0, i=1,~2,~3\) be a vector which makes equal angles with the coordinate axes OX, OY and OZ. Also, let the projection of \(\vec{a}\) on the vector \(3\hat{i}+4\hat{j}\) be 7. Let \(\vec{b}\) be a vector obtained by rotating \(\vec{a}\) with 90°. If \(\vec{a},\vec{b}\) and x-axis are coplanar, then projection of a vector \(\vec{b}\) on \(3\hat{i}+4\hat{j}\) is equal to :
(A) \(\sqrt{7}\)
(B) \(\sqrt{2}\)
(C) 2
(D) 7

Answer: (B) \(\sqrt{2}\)

Question 251

Let y = y(x) be the solution of the differential equation (x + 1)y′ – y = e^3^x(x + 1)^2, with \(y\left ( 0 \right )=\frac{1}{3}\) Then, the point \(x=-\frac{4}{3}\) for the curve y = y(x)is :
(A) not a critical point
(B) a point of local minima
(C) a point of local maxima
(D) a point of inflection

Answer: (B) a point of local minima

Question 252

If y = m1x + c1 and y = m2x + c2, m1≠m2 are two common tangents of circle x^2 + y^2 = 2 and parabola y^2 = x, then the value of 8|m1m2| is equal to :
(A) \(3+4\sqrt{2}\)
(B) \(-5+6\sqrt{2}\)
(C) \(-4+3\sqrt{2}\)
(D) \(7+6\sqrt{2}\)

Answer: (C) \(-4+3\sqrt{2}\)

Question 253

Let Q be the mirror image of the point P(1, 0, 1) with respect to the plane S: x + y + z = 5. If a line L passing through (1, –1, –1), parallel to the line PQ meets the plane S at R, then QR^2 is equal to :
(A) 2
(B) 5
(C) 7
(D) 11

Answer: (B) 5

Question 254

If the solution curve y = y(x) of the differential equation y^2dx + (x^2 – xy + y^2)dy = 0, which passes through the point (1,1) and intersects the line \(y=\sqrt{3}x\) at the point \(\left ( \alpha,\sqrt{3}\alpha \right )\), then value of \(log_e\left ( \sqrt{3}\alpha \right )\) is equal to :
(A) \(\frac{\pi}{3}\)
(B) \(\frac{\pi}{2}\)
(C) \(\frac{\pi}{12}\)
(D) \(\frac{\pi}{6}\)

Answer: (C) \(\frac{\pi}{12}\)

Question 255

Let \(x=2t,y=\frac{t^2}{3}\) be a conic. Let S be the focus and B be the point on the axis of the conic such that SA⊥BA, where A is any point on the conic. If k is the ordinate of the centroid of the ΔSAB, then \(\displaystyle \lim_{ t\to 1}k\) is equal to
(A) \(\frac{17}{18}\)
(B) \(\frac{19}{18}\)
(C) \(\frac{11}{18}\)
(D) \(\frac{13}{18}\)

Answer: (D) \(\frac{13}{18}\)

Question 256

Let a circle C in complex plane pass through the points z1 = 3 + 4i, z2 = 4 + 3i and z3 = 5i. If z(≠z1) is a point on C such that the line through z and z1 is perpendicular to the line through z2 and z3, then arg(z) is equal to:
(A) \(\tan^{-1}\left (\frac{2}{\sqrt{5}} \right )-\pi\)
(B) \(\tan^{-1}\left ( \frac{24}{7} \right )-\pi\)
(C) tan^–1 (3) – π
(D) \(\tan^{-1}\left (\frac{3}{4} \right )-\pi\)

Answer: (B) \(\tan^{-1}\left ( \frac{24}{7} \right )-\pi\)

Question 257

Let A = {x ∈ R : | x + 1 | < 2} and B = {x ∈ R : | x – 1| ≥ 2}. Then which one of the following statements is NOT true?
(A) A – B = (–1, 1)
(B) B – A = R – (–3, 1)
(C) A ⋂ B = (–3, –1]
(D) A U B = R – [1, 3)

Answer: (B) B – A = R – (–3, 1)

Question 258

Let a, b ∈ R be such that the equation ax^2 – 2bx + 15 = 0 has a repeated root α. If α and β are the roots of the equation x^2 – 2bx + 21 = 0, then α^2 + β^2 is equal to
(A) 37
(B) 58
(C) 68
(D) 92

Answer: (B) 58

Question 259

Let z1 and z2 be two complex numbers such that \(\overline{z_1}=i\overline{z_2}\ \text{and}\ arg\left( \frac{z_1}{\overline{z_2}} \right )=\pi.\) Then
(A) \(arg~ z_2=\left (\frac{\pi}{4}\right )\)
(B) \(arg z_2=-\frac{3\pi}{4}\)
(C) \(arg~ z_1=\frac{\pi}{4}\)
(D) \(arg z_1=-\frac{3\pi}{4}\)

Answer: (C) \(arg~ z_1=\frac{\pi}{4}\)

Question 260

The system of equations
–kx + 3y – 14z = 25
–15x + 4y – kz = 3
–4x + y + 3z = 4
is consistent for all k in the set
(A) R
(B) R – {–11, 13}
(C) R – {13}
(D) R – {–11, 11}

Answer: (D) R – {–11, 11}

Question 261

\(\displaystyle \lim_{x\to \frac{\pi}{2}}tan^2x\left((2sin^2x+3sinx+4)^{\frac{1}{2}}-(sin^2x+6sinx+2)^{\frac{1}{2}}\right)\) is equal to
(A) 1/12
(B) -1/18
(C) -1/12
(D) 1/6

Answer: (A) 1/12

Question 262

The area of the region enclosed between the parabolas y^2 = 2x – 1 and y^2 = 4x – 3 is
(A) ⅓
(B) ⅙
(C) ⅔
(D) ¾

Answer: (A) ⅓

Question 263

The coefficient of x^101 in the expression (5 + x)^500 + x(5 + x)^499 + x^2(5 + x)^498 + ……+ x^500, x > 0, is
(A) ^501C101 (5)^399
(B) ^501C101 (5)^400
(C) ^501C100 (5)^400
(D) ^500C101 (5)^399

Answer: (A) ^501C101 (5)^399

Question 264

The sum 1 + 2 ⋅ 3 + 3 ⋅ 3^2 + …. + 10 ⋅ 3^9 is equal to
(A) \(\frac{2.3^{12}+10}{4}\)
(B) \(\frac{19.3^{10}+1}{4}\)
(C) \(5.3^{10}-2\)
(D) \(\frac{9.3^{10}+1}{2}\)

Answer: (B) \(\frac{19.3^{10}+1}{4}\)

Question 265

Let P be the plane passing through the intersection of the planes \(\overrightarrow{r}.(\hat{i}+3\hat{j}-\hat{k})=5 ~and~ \overrightarrow{r}~.(2\hat{i}-\hat{j}+\hat{k})=3,\) and the point (2, 1, –2). Let the position vectors of the points X and Y be \(\hat{i}-2\hat{j}+4\hat{k}\ \text{and}\ 5\hat{i}-\hat{j}+2\hat{k}\) respectively. Then the points
(A) X and X + Y are on the same side of P
(B) Y and Y – X are on the opposite sides of P
(C) X and Y are on the opposite sides of P
(D) X + Y and X – Y are on the same side of P

Answer: (C) X and Y are on the opposite sides of P

Question 266

A circle touches both the y-axis and the line x + y = 0. Then the locus of its center is
(A) \(y=\sqrt{2}x\)
(B) \(x=\sqrt{2}y\)
(C) \(y^2-x^2=2xy\)
(D) \(x^2-y^2=2xy\)

Answer: (D) \(x^2-y^2=2xy\)

Question 267

Water is being filled at the rate of 1 cm^3/sec in a right circular conical vessel (vertex downwards) of height 35 cm and diameter 14 cm. When the height of the water level is 10 cm, the rate (in cm^2/sec) at which the wet conical surface area of the vessel increase, is
(A) 5
(B) \(\frac{\sqrt{21}}{5}\)
(C) \(\frac{\sqrt{26}}{5}\)
(D) \(\frac{\sqrt{26}}{10}\)

Answer: (C) \(\frac{\sqrt{26}}{5}\)

Question 268

If \(b_n=\int_{0}^{\frac{\pi}{2}}\frac{cos^2nx}{sinx}dx,n\epsilon N,\) then
(A) b3 – b2, b4 – b3, b5 – b4 are in an A.P. with a common difference –2
(B) \(\frac{1}{b_3-b_2},\frac{1}{b_4-b_3},\frac{1}{b_5-b_4}\ \text{are in an A. P. with common difference 2}\)
(C) b3 – b2, b4 – b3, b5 – b4 are in a G.P.
(D) \(\frac{1}{b_3-b_2},\frac{1}{b_4-b_3},\frac{1}{b_5-b_4}\ \text{are in an A.P. with common difference –2}\)

Answer: (D) \(\frac{1}{b_3-b_2},\frac{1}{b_4-b_3},\frac{1}{b_5-b_4}\ \text{are in an A.P. with common difference –2}\)

Question 269

If y = y(x) is the solution of the differential equation \(2x^2\frac{dy}{dx}-2xy+3y^2=0\ \text{such that}\ y(e)=\frac{e}{3},\) then y(1) is equal to
(A) ⅓
(B) ⅔
(C) 3/2
(D) 3

Answer: (B) ⅔

Question 270

If the angle made by the tangent at the point (x0, y0) on the curve x = 12(t + sin t cos t), \(y=12(1+\sin t)^2,0(A) \(6(3+2\sqrt{2})\)
(B) \(3(7+4\sqrt{3})\)
(C) 27
(D) 48

Answer: (C) 27

Question 271

The value of 2 sin(12°) – sin(72°) is :
(A) \(\frac{\sqrt{5}(1-\sqrt{3})}{4}\)
(B) \(\frac{1-\sqrt{5})}{8}\)
(C) \(\frac{\sqrt{3}(1-\sqrt{5})}{2}\)
(D) \(\frac{\sqrt{3}(1-\sqrt{5})}{4}\)

Answer: (D) \(\frac{\sqrt{3}(1-\sqrt{5})}{4}\)

Question 272

A biased die is marked with numbers 2, 4, 8, 16, 32, 32 on its faces and the probability of getting a face with mark n is 1/n. If the die is thrown thrice, then the probability, that the sum of the numbers obtained is 48, is :
(A) \(\frac{7}{2^{11}}\)
(B) \(\frac{7}{2^{12}}\)
(C) \(\frac{3}{2^{10}}\)
(D) \(\frac{13}{2^{12}}\)

Answer: (D) \(\frac{13}{2^{12}}\)

Question 273

The negation of the Boolean expression ((~ q) ∧ p) ⇒ ((~ p) ∨ q) is logically equivalent to :
(A) p ⇒ q
(B) q ⇒ p
(C) ~ (p ⇒ q)
(D) ~ (q ⇒ p)

Answer: (C) ~ (p ⇒ q)

Question 274

If the line y = 4 + kx, k > 0, is the tangent to the parabola y = x – x^2 at the point P and V is the vertex of the parabola, then the slope of the line through P and V is :
(A) \(\frac{3}{2}\)
(B) \(\frac{26}{9}\)
(C) \(\frac{5}{2}\)
(D) \(\frac{23}{6}\)

Answer: (C) \(\frac{5}{2}\)

Question 275

The value of \(tan^{-1}\left(\frac{cos\frac{15\pi}{4}-1}{sin\left(\frac{\pi}{4}\right)}\right )\) is equal to :
(A) \(-\frac{\pi}{4}\)
(B) \(-\frac{\pi}{8}\)
(C) \(-\frac{5\pi}{12}\)
(D) \(-\frac{4\pi}{9}\)

Answer: (B) \(-\frac{\pi}{8}\)

Question 276

The line y = x + 1 meets the ellipse \(\frac{x^2}{4}+\frac{y^2}{2}=1\) at two points P and Q. If r is the radius of the circle with PQ as diameter then (3r)^2 is equal to :
(A) 20
(B) 12
(C) 11
(D) 8

Answer: (A) 20

Question 277

Let \(f\left ( x \right )=\frac{x-1}{x+1},x\epsilon R-\left\{0,-1,1 \right\}\) If ƒ^n^+1(x) = ƒ(ƒ^n(x)) for all n∈N, then ƒ^6(6) + ƒ^7(7) is equal to :
(A) \(\frac{7}{6}\)
(B) \(-\frac{3}{2}\)
(C) \(\frac{7}{12}\)
(D) \(-\frac{11}{12}\)

Answer: (B) \(-\frac{3}{2}\)

Question 278

Let \(A=\left\{\textbf{z}~\epsilon~\textbf{C}:\left|\frac{z+1}{z-1} \right| <1\right\}\)
and \(B=\left\{\textbf{z}~\epsilon~\textbf{C}:arg\left ( \frac{z-1}{z+1} \right ) = \frac{2\pi}{3}\right\}\)
Then A∩Bis :
(A) A portion of a circle centred at \(\left ( 0,~-\frac{1}{\sqrt{3}} \right )\) that lies in the second and third quadrants only
(B) A portion of a circle centred at \(\left ( 0,~-\frac{1}{\sqrt{3}} \right )\) that lies in the second quadrant only
(C) An empty set
(D) A portion of a circle of radius \(\frac{2}{\sqrt{3}}\) that lies in the third quadrant only

Answer: (B) A portion of a circle centred at \(\left ( 0,~-\frac{1}{\sqrt{3}} \right )\) that lies in the second quadrant only

Question 279

Let A be a 3 × 3 invertible matrix. If |adj (24A)| = |adj (3 adj (2A))|, then |A|^2 is equal to :
(A) 6^6
(B) 2^12
(C) 2^6
(D) 1

Answer: (C) 2^6

Question 280

The ordered pair (a, b), for which the system of linear equations
3x – 2y + z = b
5x – 8y + 9z = 3
2x + y + az = –1
has no solution, is :
(A) \(\left ( 3,\frac{1}{3} \right )\)
(B) \(\left (- 3,\frac{1}{3} \right )\)
(C) \(\left (- 3,-\frac{1}{3} \right )\)
(D) \(\left ( 3,-\frac{1}{3} \right )\)

Answer: (C) \(\left (- 3,-\frac{1}{3} \right )\)

Question 281

The remainder when (2021)^2023 is divided by 7 is :
(A) 1
(B) 2
(C) 5
(D) 6

Answer: (C) 5

Question 282

\(\displaystyle \lim_{x\rightarrow\frac{1}{\sqrt{2}}}\frac{\sin\left ( \cos^{-1}x \right )-x}{1-\tan\left ( \cos^{-1}x \right )}\) is equal to :
(A) \(\sqrt{2}\)
(B) \(-\sqrt{2}\)
(C) \(\frac{1}{\sqrt{2}}\)
(D) \(-\frac{1}{\sqrt{2}}\)

Answer: (D) \(-\frac{1}{\sqrt{2}}\)

Question 283

g :R→R be two real valued functions defined as \(f\left ( x \right )=\left\{\begin{matrix}-\left|x+3 \right|, & x<0 \\e^x, & x\geq 0 \\\end{matrix}\right.\) and
\(g\left ( x \right )=\left\{\begin{matrix}x^2+k_1x, & x<0 \\4x+k_2, & x\geq 0 \\\end{matrix}\right.\) where k1 and k2 are real constants. If (goƒ) is differentiable at x = 0, then (goƒ) (–4) + (goƒ) (4) is equal to :
(A) 4(e^4 + 1)
(B) 2(2e^4 + 1)
(C) 4e^4
(D) 2(2e^4 – 1)

Answer: (D) 2(2e^4 – 1)

Question 284

The sum of the absolute minimum and the absolute maximum values of the function ƒ(x) = |3x – x^2 + 2| – x in the interval [–1, 2] is :
(A) \(\frac{\sqrt{17}+3}{2}\)
(B) \(\frac{\sqrt{17}+5}{2}\)
(C) 5
(D) \(\frac{9-\sqrt{17}}{2}\)

Answer: (A) \(\frac{\sqrt{17}+3}{2}\)

Question 285

Let S be the set of all the natural numbers, for which the line \(\frac{x}{a}+\frac{y}{b}=2\) is a tangent to the curve \(\left ( \frac{x}{a} \right )^n+\left ( \frac{y}{b} \right )^n=2\) at the point (a, b), ab ≠ 0. Then :
(A) S = ɸ
(B) n(S) = 1
(C) S = {2k : k ∈ N }
(D) S = N

Answer: (D) S = N

Question 286

Let R be the point (3, 7) and let P and Q be two points on the line x + y = 5 such that PQR is an equilateral triangle, Then the area of ΔPQRis :
(A) \(\frac{25}{4\sqrt{3}}\)
(B) \(\frac{25\sqrt{3}}{2}\)
(C) \(\frac{25}{\sqrt{3}}\)
(D) \(\frac{25}{2\sqrt{3}}\)

Answer: (D) \(\frac{25}{2\sqrt{3}}\)

Question 287

Let C be a circle passing through the points A(2, –1) and B (3, 4). The line segment AB is not a diameter of C. If r is the radius of C and its centre lies on the circle \(\left ( x-5 \right )^2+\left ( y-1 \right )^2=\frac{13}{2}\) then r^2 is equal to :
(A) 32
(B) \(\frac{65}{2}\)
(C) \(\frac{61}{2}\)
(D) 30

Answer: (B) \(\frac{65}{2}\)

Question 288

Let the normal at the point P on the parabola y^2 = 6x pass through the point (5, –8). If the tangent at P to the parabola intersects its directrix at the point Q, then the ordinate of the point Q is :
(A) –3
(B) \(-\frac{9}{4}\)
(C) \(-\frac{5}{2}\)
(D) –2

Answer: (B) \(-\frac{9}{4}\)

Question 289

If the two lines \(l_1:\frac{x-2}{3}=\frac{y+1}{-2},z=2\) and \(l_2:\frac{x-1}{1}=\frac{2y+3}{\alpha}=\frac{z+5}{2}\) are perpendicular, then an angle between the lines l2 and \(l_3:\frac{1-x}{3}=\frac{2y-1}{-4}=\frac{z}{4}\)is :
(A) \(\cos^{-1}\left ( \frac{29}{4}\right )\)
(B) \(\sec^{-1}\left ( \frac{29}{4}\right )\)
(C) \(\cos^{-1}\left ( \frac{2}{29} \right )\)
(D) \(\cos^{-1}\left ( \frac{2}{\sqrt{29}} \right )\)

Answer: (B) \(\sec^{-1}\left ( \frac{29}{4}\right )\)

Question 290

Let the plane 2x + 3y + z + 20 = 0 be rotated through a right angle about its line of intersection with the plane x – 3y + 5z = 8. If the mirror image of the point \(\left ( 2,-\frac{1}{2},2 \right )\) in the rotated plane is B( a, b, c),then :
(A) \(\frac{a}{8}=\frac{b}{5}=\frac{c}{-4}\)
(B) \(\frac{a}{4}=\frac{b}{5}=\frac{c}{-2}\)
(C) \(\frac{a}{8}=\frac{b}{-5}=\frac{c}{4}\)
(D) \(\frac{a}{4}=\frac{b}{5}=\frac{c}{2}\)

Answer: (A) \(\frac{a}{8}=\frac{b}{5}=\frac{c}{-4}\)

Question 291

If \(\vec{a}\cdot\vec{b}=1,\vec{b}\cdot\vec{c}=2~\textup{and}~\vec{c}\cdot\vec{a}=3\), then the value of \(\left [ \vec{a}\times\left ( \vec{b}\times\vec{c} \right ),\vec{b}\times\left ( \vec{c}\times\vec{a} \right ),\vec{c}\times\left ( \vec{b}\times\vec{a} \right ) \right ]\)is :
(A) 0
(B) \(-6\vec{a}\cdot\left ( \vec{b}\times\vec{c} \right )\)
(C) \(12\vec{c}\cdot\left ( \vec{a}\times\vec{b} \right )\)
(D) \(-12\vec{b}\cdot\left ( \vec{c}\times\vec{a} \right )\)

Answer: (A) 0

Question 292

Let a biased coin be tossed 5 times. If the probability of getting 4 heads is equal to the probability of getting 5 heads, then the probability of getting atmost two heads is:
(A) \(\frac{275}{6^5}\)
(B) \(\frac{36}{5^4}\)
(C) \(\frac{181}{5^5}\)
(D) \(\frac{46}{6^4}\)

Answer: (D) \(\frac{46}{6^4}\)

Question 293

The mean of the numbers a, b, 8, 5, 10 is 6 and their variance is 6.8. If M is the mean deviation of the numbers about the mean, then 25 M is equal to:
(A) 60
(B) 55
(C) 50
(D) 45

Answer: (A) 60

Question 294

Let \(f\left ( x \right )=2\cos^{-1}x+4\cot^{-1}x-3x^2-2x+10,\chi\epsilon\left [ -1,1 \right ]\) If [a, b] is the range of the function,f then 4a – b is equal to :
(A) 11
(B) 11 – π
(C) 11 + π
(D) 15 – π

Answer: (B) 11 – π

Question 295

Let \(\Delta,\triangledown \epsilon\left\{ \wedge ,\vee \right\}\) be such that \(p\triangledown q\Rightarrow\left ( \left ( p\Delta q \right )\triangledown r \right ) \) is a tautology. Then \(\left ( p\triangledown q \right )\Delta r\) is logically equivalent to :
(A) \(\left ( p~\Delta~r \right )\vee q\)
(B) \(\left ( p~\Delta~r \right )\wedge q\)
(C) \(\left ( p \wedge r \right )\Delta q\)
(D) \(\left ( p \triangledown r \right )\wedge q\)

Answer: (A) \(\left ( p~\Delta~r \right )\vee q\)

Question 296

Let \(f:\mathbb{R}\rightarrow \mathbb{R}\) be defined as f(x) = x – 1 and \(g:\mathbb{R}-\left\{1,-1 \right\}\to \mathbb{R}\) be defined as\(g(x)=\frac{x^2}{x^2-1}.\)
Then the function fog is:
(A) One-one but not onto
(B) Onto but not one-one
(C) Both one-one and onto
(D) Neither one-one nor onto

Answer: (D) Neither one-one nor onto

Question 297

If the system of equations αx + y + z = 5, x + 2y + 3z = 4, x + 3y +5z = β has infinitely many solutions, then the ordered pair (α, β) is equal to:
(A) (1, –3)
(B) (–1, 3)
(C) (1, 3)
(D) (–1, –3)

Answer: (C) (1, 3)

Question 298

\(\text{If}\ A=\sum_{n=1 }^{\infty}\frac{1}{\left(3+(-1)^n\right)^n}\ \text{and}\ B=\sum_{n=1 }^{\infty}\frac{(-1)^n}{\left(3+(-1)^n\right)^n},\) then A/B is equal to:
(A) 11/9
(B) 1
(C) -11/9
(D) -11/3

Answer: (C) -11/9

Question 299

\(\displaystyle \lim_{x\to 0}\frac{cos(sin~x)-cos~x}{x^4}\ \text{is equal to}:\)
(A) 1/3
(B) 1/4
(C) 1/6
(D) 1/12

Answer: (C) 1/6

Question 300

Let f(x) = min {1, 1 + x sin x}, 0 ≤ x ≤ 2π. If m is the number of points, where f is not differentiable, and n is the number of points, where f is not continuous, then the ordered pair (m, n) is equal to
(A) (2, 0)
(B) (1, 0)
(C) (1, 1)
(D) (2, 1)

Answer: (B) (1, 0)

Question 301

Consider a cuboid of sides 2x, 4x and 5x and a closed hemisphere of radius r. If the sum of their surface areas is a constant k, then the ratio x : r, for which the sum of their volumes is maximum, is
(A) 2 : 5
(B) 19 : 45
(C) 3 : 8
(D) 19 : 15

Answer: (B) 19 : 45

Question 302

The area of the region bounded by y^2 = 8x and y^2 = 16(3 – x) is equal to
(A) \(\frac{32}{3}\)
(B) \(\frac{40}{3}\)
(C) 16
(D) 19

Answer: (C) 16

Question 303

\(\text{If}\ \int\frac{1}{x}\sqrt{\frac{1-x}{1+x}}dx=g(x)+c,g(1)=0,\ \text{then}\ g\left(\frac{1}{2}\right )\) is equal to
(A) \(log_e\left(\frac{\sqrt{3}-1}{\sqrt{3}+1}\right)+\frac{\pi}{3}\)
(B) \(log_e\left(\frac{\sqrt{3}+1}{\sqrt{3}-1}\right)+\frac{\pi}{3}\)
(C) \(log_e\left(\frac{\sqrt{3}+1}{\sqrt{3}-1}\right)-\frac{\pi}{3}\)
(D) \(\frac{1}{2}log_e\left(\frac{\sqrt{3}-1}{\sqrt{3}+1}\right)-\frac{\pi}{6}\)

Answer: (A) \(log_e\left(\frac{\sqrt{3}-1}{\sqrt{3}+1}\right)+\frac{\pi}{3}\)

Question 304

If y = y(x) is the solution of the differential equation \(x\frac{dy}{dx}+2y=xe^x,y(1)=0\) then the local maximum value of the function \(z(x)=x^2y(x)-e^x,x\in R\) is
(A) 1 – e
(B) 0
(C) 1/2
(D) \(\frac{4}{e}-e\)

Answer: (D) \(\frac{4}{e}-e\)

Question 305

If the solution of the differential equation \(\frac{dy}{dx}+e^x(x^2-2)y=(x^2-2x)(x^2-2)e^{2x}\) satisfies y(0) = 0, then the value of y(2) is ______.
(A) –1
(B) 1
(C) 0
(D) e

Answer: (C) 0

Question 306

If m is the slope of a common tangent to the curves \(\frac{x^2}{16}+\frac{y^2}{9}=1\) and x^2 + y^2 = 12, then 12m^2 is equal to:
(A) 6
(B) 9
(C) 10
(D) 12

Answer: (B) 9

Question 307

The locus of the mid-point of the line segment joining the point (4, 3) and the points on the ellipse x^2 + 2y^2 = 4 is an ellipse with eccentricity:
(A) \(\frac{\sqrt{3}}{2}\)
(B) \(\frac{1}{2\sqrt{2}}\)
(C) \(\frac{1}{\sqrt{2}}\)
(D) \(\frac{1}{2}\)

Answer: (C) \(\frac{1}{\sqrt{2}}\)

Question 308

The normal to the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{9}=1\) at the point (8, 3√3) on it passes through the point:
(A) \((15,-2\sqrt{3})\)
(B) \((9,2\sqrt{3})\)
(C) \((-1,9\sqrt{3})\)
(D) \((-1,6\sqrt{3})\)

Answer: (C) \((-1,9\sqrt{3})\)

Question 309

If the plane 2x + y – 5z = 0 is rotated about its line of intersection with the plane 3x – y + 4z – 7 = 0 by an angle of π/2, then the plane after the rotation passes through the point:
(A) (2, –2, 0)
(B) (–2, 2, 0)
(C) (1, 0, 2)
(D) (–1, 0, –2)

Answer: (C) (1, 0, 2)

Question 310

If the lines \(\vec{r}=(\hat{i}-\hat{j}+\hat{k})+\lambda (3\hat{j}-\hat{k})\) and \(\vec{r}=(\alpha \hat{i}-\hat{j})+\mu(2\hat{j}-3\hat{k})\) are co-planar, then the distance of the plane containing these two lines from the point (α, 0, 0) is :
(A) \(\frac{2}{9}\)
(B) \(\frac{2}{11}\)
(C) \(\frac{4}{11}\)
(D) 2

Answer: (B) \(\frac{2}{11}\)

Question 311

Let \(\vec{a}=\hat{i}+\hat{j}+2\hat{k},\vec{b}=2\dot{i}-3\hat{j}+\hat{k}\) and \(\vec{c}=\hat{i}-\hat{j}+\hat{k}\) be three given vectors. \(\text{Let}\ \vec{v}\ \text{be a vector in the plane of}\ \vec{a}\ \text{and}\ \vec{b}\ \text{whose projection on}\ \vec{c}\ \text{is}\ \frac{2}{\sqrt{3}}.\)
\(\text{If}\ \vec{v}.\hat{j}=7,\ \text{then}\ \vec{v}.(\hat{i}+\hat{k})\ \text{is equal to}:\)
(A) 6
(B) 7
(C) 8
(D) 9

Answer: (D) 9

Question 312

The mean and standard deviation of 50 observations are 15 and 2 respectively. It was found that one incorrect observation was taken such that the sum of correct and incorrect observations is 70. If the correct mean is 16, then the correct variance is equal to :
(A) 10
(B) 36
(C) 43
(D) 60

Answer: (C) 43

Question 313

16 sin(20°) sin(40°) sin(80°) is equal to :
(A) √3
(B) 2√3
(C) 3
(D) 4√3

Answer: (B) 2√3

Question 314

If the inverse trigonometric functions take principal values, then \(cos^{-1}\left(\frac{3}{10}cos\left(tan^{-1}\left(\frac{4}{3}\right )\right )+\frac{2}{5}sin\left ( \tan^{-1}\left (\frac{4}{3} \right ) \right ) \right )\) is equal to :
(A) 0
(B) \(\frac{\pi}{4}\)
(C) \(\frac{\pi}{3}\)
(D) \(\frac{\pi}{6}\)

Answer: (C) \(\frac{\pi}{3}\)

Question 315

Let r ∈ {p, q, ~p, ~q} be such that the logical statement r ∨ (~p) ⇒ (p ∧ q) ∨ r is a tautology. Then r is equal to :
(A) p
(B) q
(C) ~p
(D) ~q

Answer: (C) ~p

Question 316

The area of the polygon, whose vertices are the non-real roots of the equation \(\overline{z}=iz^2\) is :
(A) \(\frac{3\sqrt{3}}{4}\)
(B) \(\frac{3\sqrt{3}}{2}\)
(C) \(\frac{3}{2}\)
(D) \(\frac{3}{4}\)

Answer: (A) \(\frac{3\sqrt{3}}{4}\)

Question 317

Let the system of linear equations x + 2y + z = 2, αx + 3y – z = α, –αx + y + 2z = –α be inconsistent. Then α is equal to :
(A) \(\frac{5}{2}\)
(B) \(-\frac{5}{2}\)
(C) \(\frac{7}{2}\)
(D) \(-\frac{7}{2}\)

Answer: (D) \(-\frac{7}{2}\)

Question 318

If \(x=\displaystyle\sum\limits_{n=0}^\infty a^n,y=\displaystyle\sum\limits_{n=0}^\infty b^n, z=\displaystyle\sum\limits_{n=0}^\infty c^n,\) where a, b, c are in A.P. and |a| < 1, |b| < 1, |c| < 1, abc≠ 0,
then :
(A) x, y, zare in A.P.
(B) x, y, zare in G.P.
(C) \(\frac{1}{x},\frac{1}{y},\frac{1}{z}\) are in A.P.
(D) \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1-\left ( a+b+c \right )\)

Answer: (C) \(\frac{1}{x},\frac{1}{y},\frac{1}{z}\) are in A.P.

Question 319

Let \(\frac{dy}{dx}=\frac{ax-by+a}{bx+cy+a}\) where a, b, c are constants, represent a circle passing through the point (2, 5). Then the shortest distance of the point (11, 6) from this circle is
(A) 10
(B) 8
(C) 7
(D) 5

Answer: (B) 8

Question 320

Let a be an integer such that \(\lim\limits_{x\rightarrow7}\frac{18-\left [ 1-x \right ]}{\left [ x-3a \right ]}\) exists, where [t] is greatest integer ≤ t. Then a is equal
to :
(A) –6
(B) –2
(C) 2
(D) 6

Answer: (A) –6

Question 321

The number of distinct real roots of x^4 – 4x + 1 = 0 is :
(A) 4
(B) 2
(C) 1
(D) 0

Answer: (B) 2

Question 322

The lengths of the sides of a triangle are 10 + x^2, 10 + x^2 and 20 – 2x^2. If for x = k, the area of the triangle is maximum, then 3k^2 is equal to :
(A) 5
(B) 8
(C) 10
(D) 12

Answer: (C) 10

Question 323

If \(\cos^{-1}\left ( \frac{y}{2} \right )=\textup{log}_e\left ( \frac{x}{5} \right )^5,\left|y \right|<2\) then :
(A) x^2y′′ + xy′ – 25y = 0
(B) x^2y′′ – xy′ – 25y = 0
(C) x^2y′′ – xy′+ 25y = 0
(D) x^2y′′ + xy′+ 25y = 0

Answer: (D) x^2y′′ + xy′+ 25y = 0

Question 324

If \(\int\frac{\left ( x^2+1 \right )e^x}{\left ( x+1 \right )^2}dx=f\left ( x \right )e^x+C\) where C is a constant, then \(\frac{d^3f}{dx^3}\) at x = 1 is equal to :
(A) \(-\frac{3}{4} \)
(B) \(\frac{3}{4} \)
(C) \(-\frac{3}{2} \)
(D) \(\frac{3}{2} \)

Answer: (B) \(\frac{3}{4} \)

Question 325

The value of the integral \(\displaystyle\int\limits_{-2}^2\frac{\left|x^3+x \right|}{\left (e^{x\left|x\right|}+1 \right ) }dx\) is equal to:
(A) 5e^2
(B) 3e^–2
(C) 4
(D) 6

Answer: (D) 6

Question 326

If \(\frac{dy}{dx}+\frac{2^{x-y}\left ( 2^y-1 \right )}{2^x-1}=0,x,y>0,y\left ( 1 \right ) =1\), then y(2) is equal to :
(A) 2 + log2 3
(B) 2 + log3 2
(C) 2 – log3 2
(D) 2 – log2 3

Answer: (D) 2 – log2 3

Question 327

In an isosceles triangle ABC, the vertex A is (6, 1) and the equation of the base BC is 2x + y = 4. Let the point B lie on the line x + 3y = 7. If (α, β) is the centroid of ΔABC, then 15(α + β) is equal to :
(A) 39
(B) 41
(C) 51
(D) 63

Answer: (C) 51

Question 328

Let the eccentricity of an ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b, \) be \(\frac{1}{4} \). If this ellipse passes through the point \(\left ( -4\sqrt{\frac{2}{5}},3 \right )\), then a^2 + b^2 is equal to :
(A) 29
(B) 31
(C) 32
(D) 34

Answer: (B) 31

Question 329

If two straight lines whose direction cosines are given by the relations l + m – n = 0, 3l^2 + m^2 + cnl = 0 are parallel, then the positive value of cis :
(A) 6
(B) 4
(C) 3
(D) 2

Answer: (A) 6

Question 330

Let \(\vec{a}=\hat{i}+\hat{j}-\hat{k} \) and \(\vec{c}=2\hat{i}-3\hat{j}+2\hat{k} \). Then the number of vectors \(\vec{b}\) such that \(\vec{b}\times\vec{c}=\vec{a}\) and \(\left|\vec{b} \right|\in\left\{1,2,\dots,10 \right\}\) is :
(A) 0
(B) 1
(C) 2
(D) 3

Answer: (A) 0

Question 331

Five numbers, x1, x2, x3, x4, x5 are randomly selected from the numbers 1, 2, 3,….., 18 and are arranged in the increasing order (x1(A) \(\frac{1}{136} \)
(B) \(\frac{1}{72} \)
(C) \(\frac{1}{68} \)
(D) \(\frac{1}{34} \)

Answer: (C) \(\frac{1}{68} \)

Question 332

Let X be a random variable having binomial distribution B(7, p). If P(X = 3) = 5P(X = 4), then the sum of the mean and the variance of X is:
(A) \(\frac{105}{16}\)
(B) \(\frac{7}{16}\)
(C) \(\frac{77}{36}\)
(D) \(\frac{49}{16}\)

Answer: (C) \(\frac{77}{36}\)

Question 333

The value of \(\cos\left ( \frac{2\pi}{7} \right )+\cos\left ( \frac{4\pi}{7} \right )+\cos\left ( \frac{6\pi}{7} \right )\) is equal to:
(A) –1
(B) \(-\frac{1}{2} \)
(C) \(-\frac{1}{3} \)
(D) \(-\frac{1}{4} \)

Answer: (B) \(-\frac{1}{2} \)

Question 334

\(\sin^{-1}\left ( \sin\frac{2\pi}{3} \right )+\cos^{-1}\left ( \cos\frac{7\pi}{6} \right )+\tan^{-1}\left ( \tan\frac{3\pi}{4} \right )\) is equal to:
(A) \(\frac{11\pi}{12}\)
(B) \(\frac{17\pi}{12}\)
(C) \(\frac{31\pi}{12}\)
(D) \(-\frac{3\pi}{4}\)

Answer: (A) \(\frac{11\pi}{12}\)

Question 335

The boolean expression (~(p ∧q)) ∨q is equivalent to:
(A) q→ (p ∧q)
(B) p→q
(C) p→ (p→q)
(D) p→ (p∨q)

Answer: (D) p→ (p∨q)

Question 336

The number of points of intersection of |z – (4 + 3i)| = 2 and |z| + |z – 4| = 6, z ∈ C, is
(A) 0
(B) 1
(C) 2
(D) 3

Answer: (C) 2

Question 337

Let \(f(x) = \begin{vmatrix}a & -1 & 0 \\ax & a & -1 \\ax^2 & ax & a \\\end{vmatrix},\) a ∈ R. Then the sum of the square of all the values of a, for which 2f′(10) –f′(5) + 100 = 0, is
(A) 117
(B) 106
(C) 125
(D) 136

Answer: (C) 125

Question 338

Let for some real numbers α and β, a = α – iβ. If the system of equations 4ix + (1 + i) y = 0 and \(8\left ( cos \frac{2\pi}{3} + i~sin\frac{2\pi}{3} \right )x + \overline{a} y = 0\) has more than one solution, then α/β is equal to
(A) -2 + √3
(B) 2 – √3
(C) 2 + √3
(D) -2 – √3

Answer: (B) 2 – √3

Question 339

Let A and B be two 3 × 3 matrices such that AB = I and |A| = ⅛. Then |adj (B adj(2A))| is equal to
(A) 16
(B) 32
(C) 64
(D) 128

Answer: (B) 64

Question 340

Let \(S=2+\frac{6}{7}+\frac{12}{7^2}+\frac{20}{7^3}+\frac{30}{7^4}+…\) Then 4S is equal to
(A) \(\left ( \frac{7}{3} \right )^2\)
(B) \(\left ( \frac{7^3}{3^2} \right )\)
(C) \(\left ( \frac{7}{3} \right )^3\)
(D) \(\left ( \frac{7^2}{3^3} \right )\)

Answer: (C) \(\left ( \frac{7}{3} \right )^3\)

Question 341

If a1, a2, a3 ….. and b1, b2, b3 ….. are A.P., and a1 = 2, a10 = 3, a1b1 = 1 = a10b10, then a4b4 is equal to
(A) \(\frac{35}{27}\)
(B) 1
(C) \(\frac{27}{28}\)
(D) \(\frac{28}{27}\)

Answer: (D) \(\frac{28}{27}\)

Question 342

If m and n respectively are the number of local maximum and local minimum points of the function \(f(x) = \int_{0}^{x^2} \frac{t^2 – 5t + 4}{2 + e^t}dt,\) then the ordered pair (m, n) is equal to
(A) (3, 2)
(B) (2, 3)
(C) (2, 2)
(D) (3, 4)

Answer: (B) (2, 3)

Question 343

Let f be a differentiable function in (0, π/2). \(\text{If}\ \int_{cos x}^{1}t^2 f(t)dt = sin^3 x + cos x,\ \text{then}\ \frac{1}{\sqrt{3}} f’\left ( \frac{1}{\sqrt{3}} \right )\) is equal to
(A) \(6-9\sqrt{2}\)
(B) \(6-\frac{9}{\sqrt{2}}\)
(C) \(\frac{9}{2}-6\sqrt{2}\)
(D) \(\frac{9}{\sqrt{2}}-6\)

Answer: (B) \(6-\frac{9}{\sqrt{2}}\)

Question 344

The integral \(\int_{0}^{1} \frac{1}{7^{\left [ \frac{1}{x} \right ]}}dx\) where [⋅] denotes the greatest integer function, is equal to
(A) \(1+6log_e\left (\frac{6}{7} \right )\)
(B) \(1- 6log_e\left (\frac{6}{7} \right )\)
(C) \(log_e\left (\frac{7}{6} \right )\)
(D) \(1 -7log_e\left (\frac{6}{7} \right )\)

Answer: (A) \(1+6log_e\left (\frac{6}{7} \right )\)

Question 345

If the solution curve of the differential equation \(\left ( \left ( tan^{-1}y \right )-x \right )dy = \left ( 1+ y^2 \right )dx\) passes through the point (1, 0), then the abscissa of the point on the curve whose ordinate is tan(1), is
(A) 2e
(B) 2/e
(C) 2
(D) 1/e

Answer: (B) 2/e

Question 346

If the equation of the parabola, whose vertex is at (5, 4) and the directrix is 3x + y – 29 = 0, is x^2 + ay^2 + bxy + cx + dy + k = 0, then a + b + c + d + k is equal to
(A) 575
(B) –575
(C) 576
(D) –576

Answer: (D) –576

Question 347

The set of values of k, for which the circle C : 4x^2 + 4y^2 – 12x + 8y + k = 0 lies inside the fourth quadrant and the point (1, -1/3) lies on or inside the circle C, is
(A) An empty set
(B) \(\left(6, \frac{65}{9}\right]\)
(C) \(\left( \frac{80}{9},10\right]\)
(D) \(\left(9, \frac{92}{9}\right]\)

Answer: (D) \(\left(9, \frac{92}{9}\right]\)

Question 348

Let the foot of the perpendicular from the point (1, 2, 4) on the line \(\frac{x+2}{4}=\frac{y-1}{2}=\frac{z+1}{3}\) be P, Then the distance of P from the plane 3x + 4y + 12z + 23 = 0 is
(A) 5
(B) \(\frac{50}{13}\)
(C) 4
(D) \(\frac{63}{13}\)

Answer: (A) 5

Question 349

The shortest distance between the lines \(\frac{x-3}{2}=\frac{y-2}{3}=\frac{z-1}{-1}\) and \(\frac{x+3}{2}=\frac{y-6}{1}=\frac{z-5}{3},\ \text{is}\)
(A) \(\frac{18}{\sqrt{5}}\)
(B) \(\frac{22}{3 \sqrt{5}}\)
(C) \(\frac{46}{3 \sqrt{5}}\)
(D) \(6 \sqrt{3}\)

Answer: (A) \(\frac{18}{\sqrt{5}}\)

Question 350

Let \(\vec{a}\ \text{and}\ \vec{b}\) be the vectors along the diagonals of a parallelogram having area 2√2. Let the angle between \(\vec{a}\ \text{and}\ \vec{b}\ \text{be acute,}\ |\vec{a}|=1,\ \text{and}\ |\vec{a} \cdot \vec{b}|=|\vec{a} \times \vec{b}|\) 
\(\text{If}\ \vec{c}=2 \sqrt{2}(\vec{a} \times \vec{b})-2 \vec{b}\ \text{then an angle between}\ \vec{b}\ \text{and}\ \vec{c}\ \text{is}\) 
(A) \(\frac{\pi}{4}\)
(B) \(-\frac{\pi}{4}\)
(C) \(\frac{5\pi}{6}\)
(D) \(\frac{3\pi}{4}\)

Answer: (D) \(\frac{3\pi}{4}\)

Question 351

The mean and variance of the data 4, 5, 6, 6, 7, 8, x, y, where x < y, are 6 and 9/4, respectively. Then x^4 + y^2 is equal to
(A) 162
(B) 320
(C) 674
(D) 420

Answer: (B) 320

Question 352

If a point A(x, y) lies in the region bounded by the y-axis, straight lines 2y + x = 6 and 5x – 6y = 30, then the probability that y < 1 is
(A) \(\frac{1}{6}\)
(B) \(\frac{5}{6}\)
(C) \(\frac{2}{3}\)
(D) \(\frac{6}{7}\)

Answer: (B) \(\frac{5}{6}\)

Question 353

The value of \(\cot \left(\sum_{n=1}^{50} \tan ^{-1}\left(\frac{1}{1+n+n^{2}}\right)\right)\) is
(A) \(\frac{26}{25}\)
(B) \(\frac{25}{26}\)
(C) \(\frac{50}{51}\)
(D) \(\frac{52}{51}\)

Answer: (A) \(\frac{26}{25}\)

Question 354

α = sin 36º is a root of which of the following equation?
(A) 16x^4 – 10x^2 – 5 = 0
(B) 16x^4 + 20x^2 – 5 = 0
(C) 16x^4 – 20x^2 + 5 = 0
(D) 16x^4 – 10x^2 + 5 = 0

Answer: (C) 16x^4 – 20x^2 + 5 = 0

Question 355

Which of the following statement is a tautology?
(A) ((~ q) ∧ p) ∧ q
(B) ((~ q) ∧ p) ∧ (p ∧ (~ p))
(C) ((~ q) ∧ p) ∨ (p ∨ (~p))
(D) (p ∧ q) ∧ (~ (p ∧ q))

Answer: (C) ((~ q) ∧ p) ∨ (p ∨ (~p))

Question 356

If \(\displaystyle\sum\limits_{k=1}^{31}\left ( ^{31}C_k \right )\left ( ^{31}C_{k-1} \right )-\displaystyle\sum\limits_{k=1}^{30}\left ( ^{30}C_k \right )\left ( ^{30}C_{k-1} \right )=\frac{\alpha\left ( 60! \right )}{\left ( 30! \right )\left ( 31! \right )}\) where α ∈ R, then the value of 16α is equal to
(A) 1411
(B) 1320
(C) 1615
(D) 1855

Answer: (A) 1411

Question 357

Let a function ƒ : N →N be defined by \(f\left ( n \right )=\left[\begin{matrix}2n & n=2,~4,~6,~8,\dots\\n-1, & n=3,~7,~11,~15,\dots \\\frac{n+1}{2}, & n=1,~5,~9,~13,\dots \\\end{matrix}\right. \) then, ƒ is
(A) One-one but not onto
(B) Onto but not one-one
(C) Neither one-one nor onto
(D) One-one and onto

Answer: (D) One-one and onto

Question 358

If the system of linear equations
2x + 3y – z = –2
x + y + z = 4
x – y + |λ|z = 4λ – 4
where λ∈ R, has no solution, then
(A) λ = 7
(B) λ = –7
(C) λ = 8
(D) λ^2 = 1

Answer: (B) λ = –7

Question 359

Let A be a matrix of order 3 × 3 and det (A) = 2. Then det (det (A) adj (5 adj (A^3))) is equal to ______.
(A) 512 × 10^6
(B) 256 × 10^6
(C) 1024 × 10^6
(D) 256 × 10^11

Answer: (A) 512 × 10^6

Question 360

The total number of 5-digit numbers, formed by using the digits 1, 2, 3, 5, 6, 7 without repetition, which are multiple of 6, is
(A) 36
(B) 48
(C) 60
(D) 72

Answer: (D) 72

Question 361

Let A1, A2, A3, … be an increasing geometric progression of positive real numbers. If A1A3A5A7 = 1/1296 and A2 + A4 = 7/36 then, the value of A6 + A8 + A10 is equal to
(A) 33
(B) 37
(C) 43
(D) 47

Answer: (C) 43

Question 362

Let [t] denote the greatest integer less than or equal to t. Then, the value of the integral \(\displaystyle\int\limits_0^1\left [-8x^2+6x-1 \right ]dx \) is equal to
(A) –1
(B) \(\frac{-5}{4}\)
(C) \(\frac{\sqrt{17}-13}{8} \)
(D) \(\frac{\sqrt{17}-16}{8} \)

Answer: (C) \(\frac{\sqrt{17}-13}{8} \)

Question 363

Let f: ℝ → ℝ be defined as \(f\left( x \right )=\left[\begin{matrix}\left [ e^x \right ], & x<0 \\ae^x+\left [ x-1 \right ], & 0\leq x<1 \\b+\left [ \sin\left ( \pi x \right ) \right ],&1\leq x<2 \\\left [ e^{-x} \right ]-c, & x\geq 2 \\\end{matrix}\right.\)
Where a, b, c ∈  ℝ and [t] denotes greatest integer less than or equal to t. Then, which of the following statements is true?
(A) There exists a, b, c ∈  ℝ  such that ƒiscontinuous on ∈  ℝ .
(B) If ƒ is discontinuous at exactly one point, then a + b + c = 1
(C) If ƒ is discontinuous at exactly one point, then a + b + c ≠ 1
(D) ƒ is discontinuous at atleast two points, for any values of a, b and c

Answer: (C) If ƒ is discontinuous at exactly one point, then a + b + c ≠ 1

Question 364

The area of the region\(\left\{\left ( x,y \right ):y^2\leq 8x,y\geq \sqrt{2}x,x\geq 1 \right\}\) is
(A) \(\frac{13\sqrt{2}}{6}\)
(B) \(\frac{11\sqrt{2}}{6}\)
(C) \(\frac{5\sqrt{2}}{6}\)
(D) \(\frac{19\sqrt{2}}{6}\)

Answer: (B) \(\frac{11\sqrt{2}}{6}\)

Question 365

Let the solution curve y = y(x) of the differential equation
\(\left [\frac{x}{\sqrt{x^2-y^2}}+e^\frac{y}{x} \right ]x\frac{dy}{dx}=x+\left [\frac{x}{\sqrt{x^2-y^2}}+e^\frac{y}{x} \right ]y\) pass through the points (1, 0) and (2α, α), α> 0. Then α is equal to
(A) \(\frac{1}{2}\textup{exp}\left ( \frac{\pi}{6}+\sqrt{e}-1 \right )\)
(B) \(\frac{1}{2}\textup{exp}\left ( \frac{\pi}{3}+e-1 \right )\)
(C) \(\textup{exp}\left ( \frac{\pi}{6}+\sqrt{e}+1 \right )\)
(D) \(2~\textup{exp}\left ( \frac{\pi}{3}+\sqrt{e}-1 \right )\)

Answer: (A) \(\frac{1}{2}\textup{exp}\left ( \frac{\pi}{6}+\sqrt{e}-1 \right )\)

Question 366

Let y = y(x) be the solution of the differential equation \(x\left ( 1-x^2 \right )\frac{dy}{dx}+\left ( 3x^2y-y-4x^3 \right )=0,~x>1\) with y(2) = –2. Then y(3) is equal to
(A) –18
(B) –12
(C) –6
(D) –3

Answer: (A) –18

Question 367

The number of real solutions of x^7 + 5x^3 + 3x + 1 = 0 is equal to ______.
(A) 0
(B) 1
(C) 3
(D) 5

Answer: (B) 1

Question 368

Let the eccentricity of the hyperbola \(H:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) be √(5/2) and length of its latus rectum be 6√2, If y = 2x + c is a tangent to the hyperbola H. then the value of c^2 is equal to
(A) 18
(B) 20
(C) 24
(D) 32

Answer: (B) 20

Question 369

If the tangents drawn at the points O(0, 0) and P(1 + √5, 2) on the circle x^2 + y^2 – 2x – 4y = 0 intersect at the point Q, then the area of the triangle OPQ is equal to
(A) \(\frac{3+\sqrt{5}}{2}\)
(B) \(\frac{4+2\sqrt{5}}{2}\)
(C) \(\frac{5+3\sqrt{5}}{2}\)
(D) \(\frac{7+3\sqrt{5}}{2}\)

Answer: (C) \(\frac{5+3\sqrt{5}}{2}\)

Question 370

If two distinct points Q, R lie on the line of intersection of the planes –x + 2y – z = 0 and 3x – 5y + 2z = 0 and \(PQ=PR=\sqrt{18}\) where the point P is (1, –2, 3), then the area of the triangle PQR is equal to
(A) \(\frac{2}{3}\sqrt{38}\)
(B) \(\frac{4}{3}\sqrt{38}\)
(C) \(\frac{8}{3}\sqrt{38} \)
(D) \(\sqrt{\frac{152}{3}} \)

Answer: (B) \(\frac{4}{3}\sqrt{38}\)

Question 371

The acute angle between the planes P1 and P2, when P1 and P2 are the planes passing through the intersection of the planes 5x + 8y + 13z – 29 = 0 and 8x – 7y + z – 20 = 0 and the points (2, 1, 3) and (0, 1, 2), respectively, is
(A) \(\frac{\pi}{3} \)
(B) \(\frac{\pi}{4} \)
(C) \(\frac{\pi}{6} \)
(D) \(\frac{\pi}{12} \)

Answer: (A) \(\frac{\pi}{3} \)

Question 372

Let the plane \(P:\vec{r}\cdot\vec{a}=d \) contain the line of intersection of two planes \(\vec{r}.\left ( \hat{i}+3\hat{j}-\hat{k} \right )=6\) and \(\vec{r}\cdot\left ( -6\hat{i}+5\hat{j}-\hat{k} \right )=7\). If the plane P passes through the point (2, 3, 1/2),  \(\text{then the value of}\ \frac{\left|13\vec{a} \right|^2}{d^2}\ \text{is equal to}\)
(A) 90
(B) 93
(C) 95
(D) 97

Answer: (B) 93

Question 373

The probability, that in a randomly selected 3-digit number at least two digits are odd, is
(A) \(\frac{19}{36} \)
(B) \(\frac{15}{36} \)
(C) \(\frac{13}{36} \)
(D) \(\frac{23}{36} \)

Answer: (A) \(\frac{19}{36} \)

Question 374

Let AB and PQ be two vertical poles, 160 m apart from each other. Let C be the middle point of B and Q, which are feet of these two poles. Let π/8 and θ be the angles of elevation from C to P and A, respectively. If the height of pole PQ is twice the height of pole AB, then tan^2θ is equal to
(A) \(\frac{3-2\sqrt{2}}{2}\)
(B) \(\frac{3+\sqrt{2}}{2}\)
(C) \(\frac{3-2\sqrt{2}}{4}\)
(D) \(\frac{3-\sqrt{2}}{4}\)

Answer: (C) \(\frac{3-2\sqrt{2}}{4}\)

Question 375

Let p, q, r be three logical statements. Consider the compound statements
S1 : ((~p) ∨q) ∨ ((~p) ∨r) and
S2 :p→ (q∨r)
Then, which of the following is NOT true?
(A) If S2 is True, then S1 is True
(B) If S2is False, then S1 is False
(C) If S2 is False, then S1 is True
(D) If S1 is False, then S2 is False

Answer: (C) If S2 is False, then S1 is True

Question 376

Let R1 = {(a, b) ∈ N × N : |a – b| ≤ 13} and R2 = {(a, b) ∈ N × N : |a – b| ≠ 13}. Then on N:
(A) Both R1 and R2 are equivalence relations
(B) Neither R1 nor R2 is an equivalence relation
(C) R1 is an equivalence relation but R2 is not
(D) R2 is an equivalence relation but R1 is not

Answer: (B) Neither R1 nor R2 is an equivalence relation

Question 377

Let f(x) be a quadratic polynomial such that f(–2) + f(3) = 0. If one of the roots of f(x) = 0 is –1, then the sum of the roots of f(x) = 0 is equal to:
(A) \(\frac{11}{3}\)
(B) \(\frac{7}{3}\)
(C) \(\frac{13}{3}\)
(D) \(\frac{14}{3}\)

Answer: (A) \(\frac{11}{3}\)

Question 378

The number of ways to distribute 30 identical candies among four children C1, C2, C3 and C4 so that C2 receives atleast 4 and atmost 7 candies, C3 receives atleast 2 and atmost 6 candies, is equal to:
(A) 205
(B) 615
(C) 510
(D) 430

Answer: (D) 430

Question 379

The term independent of x in the expansion of \(\left(1-x^{2}+3 x^{3}\right)\left(\frac{5}{2} x^{3}-\frac{1}{5 x^{2}}\right)^{11}, x \neq 0\)is:
(A) \(\frac{7}{40}\)
(B) \(\frac{33}{200}\)
(C) \(\frac{39}{200}\)
(D) \(\frac{11}{50}\)

Answer: (B) \(\frac{33}{200}\)

Question 380

If n arithmetic means are inserted between a and 100 such that the ratio of the first mean to the last mean is 1 : 7 and a + n = 33, then the value of n is:
(A) 21
(B) 22
(C) 23
(D) 24

Answer: (C) 23

Question 381

Let f,g : R → R be functions defined by
\(f(x)=\left\{\begin{array}{ll}{[x],} & x<0 \\ |1-x|, & x \geq 0\end{array}\right.$ and $g(x)= \begin{cases}e^{x}-x, & x<0 \\ (x-1)^{2}-1, & x \geq 0\end{cases}\)
Where [x] denotes the greatest integer less than or equal to x. Then, the function fog is discontinuous at exactly :
(A) one point
(B) two points
(C) three points
(D) four points

Answer: (B) two points

Question 382

Let f : R → R be a differentiable function such that \(f\left(\frac{\pi}{4}\right)=\sqrt{2}, f\left(\frac{\pi}{2}\right)=0 \textup{and} f^{\prime}\left(\frac{\pi}{2}\right)=1\) and let \(g(x)=\int_{x}^{\frac{\pi}{4}}\left(f^{\prime}(t) \sec t+\tan t \operatorname{sec~t} f(t)\right) d t\) \(\text{for}\ x \in\left[\frac{\pi}{4}, \frac{\pi}{2}\right)\ \text{Then}\ \lim _{x \rightarrow\left(\frac{\pi}{2}\right)^{-}} g(x)\ \text{is equal to}\) 
(A) 2
(B) 3
(C) 4
(D) –3

Answer: (B) 3

Question 383

Let f : R → R be a continuous function satisfying f(x) + f(x + k) = n, for all x ∈ R where k > 0 and n is a positive integer. If \(l_{1}=\int_{0}^{4 n k} f(x) d x \quad and \quad I_{2}=\int_{-k}^{3 k} f(x) d x\), then
(A) \(I_{1}+2 I_{2}=4 n k\)
(B) \(I_{1}+2 I_{2}=2 n k\)
(C) \(I_{1}+n I_{2}=4 n^{2} k\)
(D) \(l_{1}+n l_{2}=6 n^{2} k\)

Answer: (C) \(I_{1}+n I_{2}=4 n^{2} k\)

Question 384

The area of the bounded region enclosed by the curve \(y=3-\left|x-\frac{1}{2}\right|-|x+1|\) and the x-axis is
(A) \(\frac{9}{4}\)
(B) \(\frac{45}{16}\)
(C) \(\frac{27}{8}\)
(D) \(\frac{63}{16}\)

Answer: (C) \(\frac{27}{8}\)

Question 385

Let x = x(y) be the solution of the differential equation \(2 y e^{\frac{x}{y^{2}}} d x+\left(y^{2}-4 x e^{\frac{x}{y^{2}}}\right) d y=0\) such that x(1) = 0. Then, x(e) is equal to
(A) e loge(2)
(B) -e loge(2)
(C) e^2 loge(2)
(D) -e^2 loge(2)

Answer: (D) -e^2 loge(2)

Question 386

Let the slope of the tangent to a curve y = f(x) at (x, y) be given by 2 tanx(cosx – y). If the curve passes through the point (π/4, 0) then the value of \(\int_{0}^{\pi / 2} y d x\) is equal to :
(A) \((2-\sqrt{2})+\frac{\pi}{\sqrt{2}}\)
(B) \(2-\frac{\pi}{\sqrt{2}}\)
(C) \((2+\sqrt{2})+\frac{\pi}{\sqrt{2}}\)
(D) \(2+\frac{\pi}{\sqrt{2}}\)

Answer: (B) \(2-\frac{\pi}{\sqrt{2}}\)

Question 387

Let a triangle be bounded by the lines L1 : 2x + 5y = 10; L2 : –4x + 3y = 12 and the line L3, which passes through the point P(2, 3), intersects L2 at A and L1 at B. If the point P divides the line-segment AB, internally in the ratio 1 : 3, then the area of the triangle is equal to
(A) \(\frac{110}{13}\)
(B) \(\frac{132}{13}\)
(C) \(\frac{142}{13}\)
(D) \(\frac{151}{13}\)

Answer: (B) \(\frac{132}{13}\)

Question 388

Let a > 0, b > 0. Let e and l respectively be the eccentricity and length of the latus rectum of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) Let e′ and l′ respectively be the eccentricity and length of the latus rectum of its conjugate hyperbola. If \(\mathrm{e}^{2}=\frac{11}{14} l\ \text{and}\ \left(\mathrm{e}^{\prime}\right)^{2}=\frac{11}{8} l^{\prime}\) then the value of 77a + 44b is equal to :
(A) 100
(B) 110
(C) 120
(D) 130

Answer: (D) 130

Question 389

Let, \(\vec{a}=\alpha \hat{i}+2 \hat{j}-\hat{k}$ and $\vec{b}=-2 \hat{i}+\alpha \hat{j}+\hat{k}\), where α ∈ R. If the area of the parallelogram whose adjacent sides are represented by the vectors \(\vec{a}\ \text{and}\ \vec{b}\ is\ \sqrt{15\left(\alpha^{2}+4\right)}\ \text{, then the value of}\ 2|\vec{a}|^{2}+(\vec{a} \cdot \vec{b})|\vec{b}|^{2}\) is equal to :
(A) 10
(B) 7
(C) 9
(D) 14

Answer: (D) 14

Question 390

If vertex of a parabola is (2, –1) and the equation of its directrix is 4x – 3y = 21, then the length of its latus rectum is :
(A) 2
(B) 8
(C) 12
(D) 16

Answer: (B) 8

Question 391

Let the plane ax + by + cz = d pass through (2, 3, –5) and is perpendicular to the planes 2x + y – 5z = 10 and 3x + 5y – 7z = 12. If a, b, c, d are integers d > 0 and gcd (|a|, |b|, |c|, d) = 1, then the value of a + 7b + c + 20d is equal to :
(A) 18
(B) 20
(C) 24
(D) 22

Answer: (D) 22

Question 392

The probability that a randomly chosen one-one function from the set {a, b, c, d} to the set {1, 2, 3, 4, 5} satisfies f(a) + 2f(b) – f(c) = f(d) is :
(A) \(\frac{1}{24}\)
(B) \(\frac{1}{40}\)
(C) \(\frac{1}{30}\)
(D) \(\frac{1}{20}\)

Answer: (D) \(\frac{1}{20}\)

Question 393

The value of \(\lim _{n \rightarrow \infty} 6 \tan \left\{\sum_{r=1}^{n} \tan ^{-1}\left(\frac{1}{r^{2}+3 r+3}\right)\right\}\) is equal to :
(A) 1
(B) 2
(C) 3
(D) 6

Answer: (C) 3

Question 394

Let \(\vec{a}\ \text{be a vector which is perpendicular to the vector}\)  \(3 \hat{i}+\frac{1}{2} \hat{j}+2 \hat{k}.\ \text{If }\ \vec{a} \times(2 \hat{i}+\hat{k})=2 \hat{i}-13 \hat{j}-4 \hat{k}\), then the projection of the vector on the vector \(2 \hat{i}+2 \hat{j}+\hat{k}\ \text{is}:\) 
(A) 1/3
(B) 1
(C) 5/3
(D) 7/3

Answer: (C) 5/3

Question 395

If \(cot~\alpha = 1\ \text{and}\ sec ~\beta = -\frac{5}{3}\ \text{where}\ \pi<\alpha<\frac{3\pi}{2}\ \text{and}\ \frac{\pi}{2}<\beta<\pi,\) then the value of tan(α + β) and the quadrant in which α + β lies, respectively are :
(A) -1/7 and IV^th quadrant
(B) 7 and I^st quadrant
(C) – 7 and IV^th quadrant
(D) 1/7 and I^st quadrant

Answer: (A) -1/7 and IV^th quadrant

Question 396

The probability that a randomly chosen 2 × 2 matrix with all the entries from the set of first 10 primes, is singular, is equal to :
(A) \(\frac{133}{10^4} \)
(B) \(\frac{18}{10^3} \)
(C) \(\frac{19}{10^3} \)
(D) \(\frac{271}{10^4} \)

Answer: (C) \(\frac{19}{10^3} \)

Question 397

Let the solution curve of the differential equation \(x\frac{dy}{dx}-y=\sqrt{y^2+16x^2},\) y(1) = 3 be y = y(x). Then y(2) is equal to :
(A) 15
(B) 11
(C) 13
(D) 17

Answer: (A) 15

Question 398

If the mirror image of the point (2, 4, 7) in the plane 3x – y + 4z = 2 is (a, b, c), then 2a + b + 2c is equal to:
(A) 54
(B) 50
(C) –6
(D) –42

Answer: (C) –6

Question 399

Let ƒ : R ⇒ R be a function defined by :
\(f\left ( x \right )=\left\{\begin{matrix}\underset{t\leq x}{\max}\left\{t^3-3t \right\} & :&x\leq 2 \\x^2+2x-6 &:&25\end{matrix}\right. \)
where [t] is the greatest integer less than or equal to t. Let m be the number of points where ƒ is not differentiable and \(l=\displaystyle\int\limits_{-2}^2f\left ( x \right )dx\) Then the ordered pair (m, I) is equal to :
(A) \(\left ( 3,\frac{27}{4} \right ) \)
(B) \(\left ( 3,\frac{23}{4} \right ) \)
(C) \(\left ( 4,\frac{27}{4} \right ) \)
(D) \(\left ( 4,\frac{23}{4} \right ) \)

Answer: (C) \(\left ( 4,\frac{27}{4} \right ) \)

Question 400

Let \(\vec{a}=\alpha\hat{i}+3\hat{j}-\hat{k},\vec{b}=3\hat{i}-\beta\hat{j}+4\hat{k}\ \text{and}\ \vec{c}=\hat{i}+2\hat{j}-2\hat{k}\) where α, β ∈ R, be three vectors. If the projection of \(\vec{a}\ \text{on}\ \vec{c}\ is\ \frac{10}{3}\ \text{and}\ \vec{b}\times\vec{c}=-6\hat{i}+10\hat{j}+7\hat{k}\) then the value of α + β is equal to :
(A) 3
(B) 4
(C) 5
(D) 6

Answer: (A) 3

Question 401

The area enclosed by y^2 = 8x and y = √2x that lies outside the triangle formed by \(y=\sqrt{2}x,~x=1,~y=2\sqrt{2}\) is equal to :
(A) \(\frac{16\sqrt{2}}{6}\)
(B) \(\frac{11\sqrt{2}}{6}\)
(C) \(\frac{13\sqrt{2}}{6}\)
(D) \(\frac{5\sqrt{2}}{6}\)

Answer: (C) \(\frac{13\sqrt{2}}{6}\)

Question 402

If the system of linear equations
2x + y – z = 7
x – 3y + 2z = 1
x + 4y + δz = k, where δ, k ∈ R
has infinitely many solutions, then δ + k is equal to:
(A) –3
(B) 3
(C) 6
(D) 9

Answer: (B) 3

Question 403

Let α and β be the roots of the equation x^2 + (2i – 1) = 0. Then, the value of |α^2 + β^2| is equal to:
(A) 50
(B) 250
(C) 1250
(D) 1500

Answer: (A) 50

Question 404

Let \(\Delta\in \left\{\wedge,\vee,\Rightarrow,\Leftrightarrow \right\}\ \text{be such that}\left( p\wedge q \right)\Delta \left( \left( p\vee q \right)\Rightarrow q \right)\) is a tautology. Then Δ is equal to :
(A) \(\wedge \)
(B) \(\vee\)
(C) \(\Rightarrow\)
(D) \(\Leftrightarrow\)

Answer: (C) \(\Rightarrow\)

Question 405

Let A = [aij] be a square matrix of order 3 such that aij = 2^j^–^i, for all i, j = 1, 2, 3. Then, the matrix A^2 + A^3 + … + A^10 is equal to :
(A) \(\left ( \frac{3^{10}-3}{2} \right )A\)
(B) \(\left ( \frac{3^{10}-1}{2} \right )A\)
(C) \(\left ( \frac{3^{10}+1}{2} \right )A\)
(D) \(\left ( \frac{3^{10}+3}{2} \right )A\)

Answer: (A) \(\left ( \frac{3^{10}-3}{2} \right )A\)

Question 406

Let a set A = A1 ⋃ A2 ⋃ …⋃ Ak, where Ai ⋂ Aj = Φ for i ≠ j, 1 ≤ i, j ≤ k. Define the relation R from A to A by R = {(x, y) : y ∈ Ai if and only if x ∈ Ai, 1 ≤ i ≤ k}. Then, R is :
(A) reflexive, symmetric but not transitive
(B) reflexive, transitive but not symmetric
(C) reflexive but not symmetric and transitive
(D) an equivalence relation

Answer: (D) an equivalence relation

Question 407

Let \(\left\{a_n \right\}_{n=0}^\infty\) be a sequence such that a0 = a1 = 0 and an + 2 = 2an + 1 – an + 1 for all n ≥ 0. \(\text{Then}\ \displaystyle\sum\limits_{n=2}^\infty\frac{a_n}{7^n}\ \text{is equal to}:\) 
(A) \(\frac{6}{343}\)
(B) \(\frac{7}{216}\)
(C) \(\frac{8}{343}\)
(D) \(\frac{49}{216}\)

Answer: (B) \(\frac{7}{216}\)

Question 408

The distance between the two points A and A′ which lie on y = 2 such that both the line segments AB and A′B (where B is the point (2, 3)) subtend angle π/4 at the origin, is equal to
(A) 10
(B) 48/5
(C) 52/5
(D) 3

Answer: (C) 52/5

Question 409

A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is
(A) \(\frac{22}{9+4\sqrt{3}}\)
(B) \(\frac{66}{9+4\sqrt{3}}\)
(C) \(\frac{22}{4+9\sqrt{3}}\)
(D) \(\frac{66}{4+9\sqrt{3}}\)

Answer: (B) \(\frac{66}{9+4\sqrt{3}}\)

Question 410

The domain of the function \(\cos^{-1}\left ( \frac{2\sin^{-1}\left ( \frac{1}{4x^2-1} \right )}{\pi} \right )\)is :
(A) \(\textbf{R}-\left\{-\frac{1}{2},\frac{1}{2} \right\}\)
(B) \(\left (-\infty,-1 \left . \right ]\cup\left [1,\infty \right . \right )\cup\left\{ 0\right\}\)
(C) \(\left ( -\infty,\frac{-1}{2} \right )\cup\left ( \frac{1}{2},\infty \right ) \cup\left\{0 \right\}\)
(D) \(\left (-\infty,\frac{-1}{\sqrt{2}} \right ]\cup\left [ \frac{1}{\sqrt{2}},\infty \right )\cup\left\{0 \right\}\)

Answer: (D) \(\left (-\infty,\frac{-1}{\sqrt{2}} \right ]\cup\left [ \frac{1}{\sqrt{2}},\infty \right )\cup\left\{0 \right\}\)

Question 411

If the constant term in the expansion of \(\left ( 3x^3-2x^2+\frac{5}{x^5} \right ) ^{10}\) is 2^k·l, where l is an odd integer, then the value of k is equal to
(A) 6
(B) 7
(C) 8
(D) 9

Answer: (D) 9

Question 412

\(\displaystyle\int\limits_0^5\cos\left ( \pi\left ( x-\left [ \frac{x}{2} \right ] \right ) \right )dx\), where [t] denotes greatest integer less than or equal to t, is equal to
(A) –3
(B) –2
(C) 2
(D) 0

Answer: (D) 0

Question 413

Let PQ be a focal chord of the parabola y^2 = 4x such that it subtends an angle of π/2 at the point (3, 0). Let the line segment PQ be also a focal chord of the ellipse \(E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a^2>b^2\). If e is the eccentricity of the ellipse E, then the value of 1/e^2 is equal to
(A) 1 + √2
(B) 3 + 2√2
(C) 1 + 2√3
(D) 4 + 5√3

Answer: (B) 3 + 2√2

Question 414

Let the tangent to the circle C1: x^2 + y^2 = 2 at the point M(–1, 1) intersect the circle C2: (x – 3)^2 + (y – 2)^2 = 5, at two distinct points A and B. If the tangents to C2 at the points A and B intersect at N, then the area of the triangle ANB is equal to
(A) \(\frac{1}{2} \)
(B) \(\frac{2}{3} \)
(C) \(\frac{1}{6} \)
(D) \(\frac{5}{3} \)

Answer: (C) \(\frac{1}{6} \)

Question 415

Let the mean and the variance of 5 observations x1, x2, x3, x4, x5 be 24/5 and 194/25, respectively. If the mean and variance of the first 4 observations are 7/2 and a, respectively, then (4a + x5) is equal to
(A) 13
(B) 15
(C) 17
(D) 18

Answer: (B) 15

Question 416

Let α be a root of the equation 1 + x^2 + x^4 = 0. Then the value of α^1011 + α^2022 – α^3033 is equal to
(A) 1
(B) α
(C) 1 + α
(D) 1 + 2α

Answer: (A) 1

Question 417

Let arg(z) represent the principal argument of the complex number z. Then, |z| = 3 and arg(z – 1) – arg(z + 1) = π/4 intersect
(A) exactly at one point
(B) exactly at two points
(C) nowhere
(D) at infinitely many points

Answer: (C) nowhere

Question 418

Let \(A = \begin{bmatrix}2 & -1 \\0 & 2 \\\end{bmatrix}\) If B = I – ^5C1(adjA) + ^5C2(adjA)^2 – …. – ^5C5(adjA)^5, then the sum of all elements of the matrix B is
(A) –5
(B) –6
(C) –7
(D) –8

Answer: (C) –7

Question 419

The sum of the infinite series
\(1+\frac{5}{6}+\frac{12}{6^{2}}+\frac{22}{6^{3}}+\frac{35}{6^{4}}+\frac{51}{6^{5}}+\frac{70}{6^{6}}+\ldots . .\) is equal to
(A) \(\frac{425}{216}\)
(B) \(\frac{429}{216}\)
(C) \(\frac{288}{125}\)
(D) \(\frac{280}{125}\)

Answer: (C) \(\frac{288}{125}\)

Question 420

The value of \(\lim _{x \rightarrow 1} \frac{\left(x^{2}-1\right) \sin ^{2}(\pi x)}{x^{4}-2 x^{3}+2 x-1}\) is equal to
(A) \(\frac{\pi^{2}}{6}\)
(B) \(\frac{\pi^{2}}{3}\)
(C) \(\frac{\pi^{2}}{2}\)
(D) \(\pi^{2}\)

Answer: (D) \(\pi^{2}\)

Question 421

Let f : R → R be a function defined by;
\(f(x)=(x-3)^{n_{1}}(x-5)^{n_{2}}, n_{1}, n_{2} \in N\) Then, which of the following is NOT true?
(A) For n1 = 3, n2 = 4, there exists α ∈ (3, 5) where f attains local maxima.
(B) For n1 = 4, n2 = 3, there exists α ∈ (3, 5) where f attains local minima.
(C) For n1 = 3, n2 = 5, there exists α ∈ (3, 5) where f attains local maxima.
(D) For n1 = 4, n2 = 6, there exists α ∈ (3, 5) where f attains local maxima.

Answer: (C) For n1 = 3, n2 = 5, there exists α ∈ (3, 5) where f attains local maxima.

Question 422

Let f be a real valued continuous function on [0, 1] and \(f(x)=x+\int_{0}^{1}(x-t) f(t) d t\) Then, which of the following points (x, y) lies on the curve y = f(x)?
(A) (2, 4)
(B) (1, 2)
(C) (4, 17)
(D) (6, 8)

Answer: (D) (6, 8)

Question 423

If \(\int_{0}^{2}\left(\sqrt{2 x}-\sqrt{2 x-x^{2}}\right) d x=\int_{0}^{1}\left(1-\sqrt{1-y^{2}}-\frac{y^{2}}{2}\right) d y+\int_{1}^{2}\left(2-\frac{y^{2}}{2}\right) d y+I\) then I equal is
(A) \(\int_{0}^{1}\left(1+\sqrt{1-y^{2}}\right) d y\)
(B) \(\int_{0}^{1}\left(\frac{y^{2}}{2}-\sqrt{1-y^{2}}+1\right) d y\)
(C) \(\int_{0}^{1}\left(1-\sqrt{1-y^{2}}\right) d y\)
(D) \(\int_{0}^{1}\left(\frac{y^{2}}{2}+\sqrt{1-y^{2}}+1\right) d y\)

Answer: (C) \(\int_{0}^{1}\left(1-\sqrt{1-y^{2}}\right) d y\)

Question 424

If y = y (x) is the solution of the differential equation \(\left(1+e^{2 x}\right) \frac{d y}{d x}+2\left(1+y^{2}\right) e^{x}=0\) and y(0) = 0, then \(6\left(y^{\prime}(0)+\left(y\left(\log _{e} \sqrt{3}\right)\right)^{2}\right)\) is equal to
(A) 2
(B) –2
(C) –4
(D) –1

Answer: (C) –4

Question 425

Let P : y^2 = 4ax, a > 0 be a parabola with focus S. Let the tangents to the parabola P make an angle of π/4 with the line y = 3x + 5 touch the parabola P at A and B. Then the value of a for which A, B and S are collinear is
(A) 8 only
(B) 2 only
(C) \(\frac{1}{4}~\text{only}\)
(D) any a > 0

Answer: (D) any a > 0

Question 426

Let \(\frac{x-2}{3}=\frac{y+1}{-2}=\frac{z+3}{-1}\) lie on the plane px – qy + z = 5, for some p, q ∈ ℝ. The shortest distance of the plane from the origin is :
(A) \(\sqrt{\frac{3}{109}}\)
(B) \(\sqrt{\frac{5}{142}}\)
(C) \(\frac{5}{\sqrt{71}}\)
(D) \(\frac{1}{\sqrt{142}}\)

Answer: (B) \(\sqrt{\frac{5}{142}}\)

Question 427

The distance of the origin from the centroid of the triangle whose two sides have the equations
x – 2y + 1 = 0 and 2x – y – 1 = 0 and whose orthocenter is (7/3, 7/3) is :
(A) √2
(B) 2
(C) 2√2
(D) 4

Answer: (C) 2√2

Question 428

Let Q be the mirror image of the point P(1, 2, 1) with respect to the plane x + 2y + 2z = 16. Let T be a plane passing through the point Q and contains the line \(\vec{r}=-\hat{k}+\lambda(\hat{i}+\hat{j}+2 \hat{k}), \lambda \in \mathbb{R}\). Then, which of the following points lies on T?
(A) (2, 1, 0)
(B) (1, 2, 1)
(C) (1, 2, 2)
(D) (1, 3, 2)

Answer: (B) (1, 2, 1)

Question 429

Let A, B, C be three points whose position vectors respectively are
\(\vec{a}=\hat{i}+4 \hat{j}+3 \hat{k}\)
\(\vec{b}=2 \hat{i}+\alpha \hat{j}+4 \hat{k}, \alpha \in \mathbb{R}\)
\(\vec{c}=3 \hat{i}-2 \hat{j}+5 \hat{k}\)
\(\text{If}\ \alpha\ \text{ is the smallest positive integer for which}\ \vec{a}, \vec{b}, \vec{c}\) are non collinear, then the length of the median, in ΔABC, through A is:
(A) \(\frac{\sqrt{82}}{2}\)
(B) \(\frac{\sqrt{62}}{2}\)
(C) \(\frac{\sqrt{69}}{2}\)
(D) \(\frac{\sqrt{66}}{2}\)

Answer: (A) \(\frac{\sqrt{82}}{2}\)

Question 430

The probability that a relation R from {x, y} to {x, y} is both symmetric and transitive, is equal to
(A) 5/16
(B) 9/16
(C) 11/16
(D) 13/16

Answer: (A) 5/16

Question 431

The number of values of a ∈ N such that the variance of 3, 7, 12, a, 43 – a is a natural number is :
(A) 0
(B) 2
(C) 5
(D) Infinite

Answer: (A) 0

Question 432

From the base of a pole of height 20 meter, the angle of elevation of the top of a tower is 60°. The pole subtends an angle 30° at the top of the tower. Then the height of the tower is :
(A) 15√3
(B) 20√3
(C) 20 + 10√3
(D) 30

Answer: (D) 30

Question 433

Negation of the Boolean statement (p ∨ q) ⇒ ((~ r) ∨ p) is equivalent to
(A) p ∧ (~ q) ∧ r
(B) (~ p) ∧ (~ q) ∧ r
(C) (~p) ∧ q ∧ r
(D) p ∧ q ∧ (~ r)

Answer: (C) (~p) ∧ q ∧ r

Question 434

Let n ≥ 5 be an integer. If 9^n – 8n – 1 = 64α and 6^n – 5n – 1 = 25β, then α – β is equal to
(A) \(1+{ }^{n} C_{2}(8-5)+{ }^{n} C_{3}\left(8^{2}-5^{2}\right)+\ldots+{ }^{n} C_{n}\left(8^{n-1}-5^{n-1}\right)\)
(B) \(1+{ }^{n} C_{3}(8-5)+{ }^{n} C_{4}\left(8^{2}-5^{2}\right)+\ldots+{ }^{n} C_{n}\left(8^{n-2}-5^{n-2}\right)\)
(C) \({ }^{n} C_{3}(8-5)+{ }^{n} C_{4}\left(8^{2}-5^{2}\right)+\ldots+{ }^{n} C_{n}\left(8^{n-2}-5^{n-2}\right)\)
(D) \({ }^{n} C_{4}(8-5)+{ }^{n} C_{5}\left(8^{2}-5^{2}\right)+\ldots+{ }^{n} C_{n}\left(8^{n-3}-5^{n-3}\right)\)

Answer: (C) \({ }^{n} C_{3}(8-5)+{ }^{n} C_{4}\left(8^{2}-5^{2}\right)+\ldots+{ }^{n} C_{n}\left(8^{n-2}-5^{n-2}\right)\)