Q: If x = a\cos^3\theta and y = a\sin^3\theta, then the value of \sqrt{1 + \left(\frac{dy}{dx}\right)^2} is:

|\sec\theta|. \frac{dy}{dx} = -\tan\theta. \sqrt{1 + \tan^2\theta} = |\sec\theta|.

Q: If f(x) = \frac{\sin 3x}{x} for x \ne 0 and f(0) = k is continuous at x = 0, then the value of k is:

3. \lim_{x \to 0} \frac{\sin 3x}{x} = 3\lim_{x \to 0} \frac{\sin 3x}{3x} = 3.

Question 1

The negation of the logical statement (p \wedge \sim q) 	o (\sim p \vee r) is logically equivalent to:

Accepted Answer

(p \wedge \sim q) \wedge (p \wedge \sim r). By conditional negation: \sim(A 	o B) \equiv A \wedge \sim B. Here A = (p \wedge \sim q) and \sim B = \sim(\sim p \vee r) \equiv p \wedge \sim r. So the negation is (p \wedge \sim q) \wedge (p \wedge \sim r).

Question 2

The dual of the logical statement [\sim (p \vee q)] \wedge [p \vee \sim (q \wedge \sim s)] is:

Accepted Answer

[\sim (p \wedge q)] \vee [p \wedge \sim (q \vee \sim s)]. Dual is obtained by swapping \vee \leftrightarrow \wedge while keeping \sim unchanged. Replacing: [\sim(p \wedge q)] \vee [p \wedge \sim(q \vee \sim s)].

Question 3

If \sin^{-1}x + \sin^{-1}y + \sin^{-1}z = \frac{3\pi}{2}, then the value of x^{100} + y^{100} + z^{100} - \frac{9}{x^{101} + y^{101} + z^{101}} is:

Accepted Answer

0. Each \sin^{-1} has maximum \pi/2, so sum = 3\pi/2 forces x = y = z = 1. Substituting: 3 - 9/3 = 0.

Question 4

In any 	riangle ABC, if \frac{a}{\cos A} = \frac{b}{\cos B} = \frac{c}{\cos C}, then the triangle is:

Accepted Answer

Equilateral. By sine rule a = 2R\sin A, so \frac{\sin A}{\cos A} = \frac{\sin B}{\cos B} = \frac{\sin C}{\cos C}, i.e. 	an A = 	an B = 	an C. Hence A = B = C = 60°, equilateral.

Question 5

If the lines represented by the equation 2x^2 - pxy + 2y^2 = 0 are real and coincident, then the value of the parameter p is:

Accepted Answer

\pm 4. Coincident lines: h^2 - ab = 0. Here a = 2, h = -p/2, b = 2. So p^2/4 = 4 \implies p = \pm 4.

Question 6

If y = \sqrt{\sin x + \sqrt{\sin x + \sqrt{\sin x + \dots \infty}}}, then (2y - 1)\frac{dy}{dx} equals:

Accepted Answer

\cos x. y^2 = \sin x + y. Differentiating: 2y\frac{dy}{dx} = \cos x + \frac{dy}{dx}. So (2y-1)\frac{dy}{dx} = \cos x.

Question 7

If x = a\cos^3	heta and y = a\sin^3	heta, then the value of \sqrt{1 + \left(\frac{dy}{dx}ight)^2} is:

Accepted Answer

|\sec	heta|. \frac{dy}{dx} = -	an	heta. \sqrt{1 + 	an^2	heta} = |\sec	heta|.

Question 8

If f(x) = \frac{\sin 3x}{x} for x 
e 0 and f(0) = k is continuous at x = 0, then the value of k is:

Accepted Answer

3. \lim_{x 	o 0} \frac{\sin 3x}{x} = 3\lim_{x 	o 0} \frac{\sin 3x}{3x} = 3.

Question 9

The point on the curve y = x^2 - 4x + 5 where the tangent is strictly parallel to the line 2x - y + 7 = 0 is:

Accepted Answer

(3, 2). Slope of line = 2. y' = 2x - 4 = 2 \implies x = 3. y = 9 - 12 + 5 = 2. Point (3, 2).

Question 10

The side of an equilateral triangle is expanding at the constant rate of 2 cm/s. The rate at which its area increases when the side length is 10 cm is:

Accepted Answer

10\sqrt{3} cm^2/s. A = \frac{\sqrt{3}}{4}s^2. \frac{dA}{dt} = \frac{\sqrt{3}}{2}s\frac{ds}{dt} = \frac{\sqrt{3}}{2}(10)(2) = 10\sqrt{3} cm^2/s.

Question 11

The value of c guaranteed by Rolle's theorem for the function f(x) = x(x-3)^2 on the closed interval [0, 3] is:

Accepted Answer

1. f'(x) = 3x^2 - 12x + 9 = 3(x-1)(x-3) = 0. Since c \in (0,3), c = 1.

Question 12

The function f(x) = \frac{x}{\ln x} attains its local minimum at x equal to:

Accepted Answer

e. f'(x) = \frac{\ln x - 1}{(\ln x)^2} = 0 \implies \ln x = 1 \implies x = e. Second derivative confirms minimum.

Question 13

The value of the definite integral \int_0^1 x(1-x)^9\,dx is:

Accepted Answer

\frac{1}{110}. Using King's property: \int_0^1 (1-x)x^9 dx = \frac{1}{10} - \frac{1}{11} = \frac{1}{110}.

Question 14

The area bounded by the curve y = \cos x and the x-axis from x = 0 to x = 2\pi is:

Accepted Answer

4 sq. units. Area = \int_0^{2\pi}|\cos x|dx = 4\int_0^{\pi/2}\cos x\,dx = 4.

Question 15

The differential equation corresponding to the family of concentric circles x^2 + y^2 = a^2 (where a is an arbitrary constant) is:

Accepted Answer

x + y\frac{dy}{dx} = 0. Differentiating: 2x + 2y\frac{dy}{dx} = 0 \implies x + y\frac{dy}{dx} = 0.

Question 16

The general solution of the differential equation \frac{dy}{dx} + \frac{y}{x} = x^2 is:

Accepted Answer

yx = \frac{x^4}{4} + C. I.F. = e^{\int (1/x)dx} = x. Multiplying: d(yx) = x^3 dx. Integrating: yx = x^4/4 + C.

Question 17

The general solution of the differential equation \frac{dy}{dx} = e^{x-y} + x^2 e^{-y} is:

Accepted Answer

e^y = e^x + \frac{x^3}{3} + C. \frac{dy}{dx} = e^{-y}(e^x + x^2). Separating: e^y dy = (e^x + x^2)dx. Integrating: e^y = e^x + x^3/3 + C.

Question 18

Bacteria in a culture grow at a rate proportional to the population present. If the population doubles in 3 hours, then the time in which it will become 8 times its initial value is:

Accepted Answer

9 hours. 8 = 2^3, so 3 doublings are needed. Time = 3 	imes 3 = 9 hours.

Question 19

The probability distribution of a random variable X is given by P(X = x) = kx^2 for x = 1, 2, 3 and P(X = x) = 0 otherwise. The expectation E(X) is:

Accepted Answer

\frac{18}{7}. 14k = 1 \implies k = 1/14. E(X) = k(1 + 8 + 27) = 36/14 = 18/7.

Question 20

If the cumulative distribution function (c.d.f.) of a discrete random variable X is F(x) = \frac{x^2 + 1}{26} for x \in \{1, 2, 3, 4, 5\}, then the probability P(X = 3) is:

Accepted Answer

\frac{5}{26}. P(X=3) = F(3) - F(2) = \frac{10}{26} - \frac{5}{26} = \frac{5}{26}.

Mathematics - 2 — MH-CET Full Length Paper MCQs with Solutions

Sample questions with solutions