CBSE Class 12 Mathematics Application Of Derivatives MCQs Set 06

Question. The abscissa of the point on the curve \( 3y = 6x - 5x^3 \), the normal at which passes through origin is
(a) 1
(b) \( \frac{1}{3} \)
(c) 2
(d) \( \frac{1}{2} \)
Answer: (a) 1

Question. The curve \( y = x^{1/5} \) has at (0, 0)
(a) a vertical tangent (parallel to y-axis)
(b) a horizontal tangent (parallel to x-axis)
(c) an oblique tangent
(d) no tangent
Answer: (a) a vertical tangent (parallel to y-axis)

Question. The equation of normal to the curve \( 3x^2 - y^2 = 8 \) which is parallel to the line \( x + 3y = 8 \) is
(a) \( 3x - y = 8 \)
(b) \( 3x + y + 8 = 0 \)
(c) \( x + 3y \pm 8 = 0 \)
(d) \( x + 3y = 0 \)
Answer: (c) \( x + 3y \pm 8 = 0 \)

Question. The tangent to the curve \( y = e^{2x} \) at the point (0, 1) meets x-axis at
(a) (0, 1)
(b) \( (-\frac{1}{2}, 0) \)
(c) (2, 0)
(d) (0, 2)
Answer: (b) \( (-\frac{1}{2}, 0) \)

Question. \( f(x) = x^x \) has a stationary point at
(a) \( x = e \)
(b) \( x = \frac{1}{e} \)
(c) \( x = 1 \)
(d) \( x = \sqrt{e} \)
Answer: (b) \( x = \frac{1}{e} \)

Question. The maximum value of \( \left( \frac{1}{x} \right)^x \) is
(a) \( e \)
(b) \( e^e \)
(c) \( e^{1/e} \)
(d) \( \left( \frac{1}{e} \right)^{1/e} \)
Answer: (c) \( e^{1/e} \)

Question. The slope of normal to the curve \( y = 2x^2 + 3 \sin x \) at \( x = 0 \) is
(a) 3
(b) \( \frac{1}{3} \)
(c) \( -3 \)
(d) \( -\frac{1}{3} \)
Answer: (d) \( -\frac{1}{3} \)

Question. The line \( y = x + 1 \) is a tangent to the curve \( y^2 = 4x \) at the point
(a) (1, 2)
(b) (2, 1)
(c) (1, -2)
(d) (-1, 2)
Answer: (a) (1, 2)

Question. The point on the curve \( x^2 = 2y \) which is nearest to the point (0, 5) is
(a) \( (2\sqrt{2}, 4) \)
(b) \( (2\sqrt{2}, 0) \)
(c) (0, 0)
(d) (2, 2)
Answer: (a) \( (2\sqrt{2}, 4) \)

Question. The maximum value of \( [x(x - 1) + 1]^{1/3}, 0 \leq x \leq 1 \) is
(a) \( \left(\frac{1}{3}\right)^{1/3} \)
(b) \( \frac{1}{2} \)
(c) 1
(d) 0
Answer: (c) 1

Question. The rate of change of the area of a circle with respect to its radius \( r \) at \( r = 6 \) cm is
(a) \( 10\pi \)
(b) \( 12\pi \)
(c) \( 8\pi \)
(d) \( 11\pi \)
Answer: (b) \( 12\pi \)

Question. The total revenue in rupees received from the sale of \( x \) units of a product is given by \( R(x) = 3x^2 + 36x + 5 \). The marginal revenue, when \( x = 15 \) is
(a) 116
(b) 96
(c) 90
(d) 126
Answer: (d) 126

Question. The two curves \( x^3 - 3xy^2 + 2 = 0 \) and \( 3x^2y - y^3 = 2 \)
(a) touch each other
(b) cut at right angle
(c) cut at an angle \( \frac{\pi}{3} \)
(d) cut at an angle \( \frac{\pi}{4} \)
Answer: (b) cut at right angle

Question. The tangent to the curve given by \( x = e^t . \cos t, y = e^t . \sin t \) at \( t = \frac{\pi}{4} \) makes with x-axis an angle
(a) 0
(b) \( \frac{\pi}{4} \)
(c) \( \frac{\pi}{3} \)
(d) \( \frac{\pi}{2} \)
Answer: (d) \( \frac{\pi}{2} \)

Question. The equation of the normal to the curve \( y = \sin x \) at (0, 0) is
(a) \( x = 0 \)
(b) \( y = 0 \)
(c) \( x + y = 0 \)
(d) \( x - y = 0 \)
Answer: (c) \( x + y = 0 \)

Question. The point on the curve \( y^2 = x \), where the tangent makes an angle of \( \frac{\pi}{4} \) with x-axis is
(a) \( \left(\frac{1}{2}, \frac{1}{4}\right) \)
(b) \( \left(\frac{1}{4}, \frac{1}{2}\right) \)
(c) (4, 2)
(d) (1, 1)
Answer: (b) \( \left(\frac{1}{4}, \frac{1}{2}\right) \)

Question. The sides of an equilateral triangle are increasing at the rate of 2 cm/sec. The rate at which the area increases, when side is 10 cm is
(a) \( 10 \text{ cm}^2/\text{s} \)
(b) \( \sqrt{3} \text{ cm}^2/\text{s} \)
(c) \( 10\sqrt{3} \text{ cm}^2/\text{s} \)
(d) \( \frac{10}{3} \text{ cm}^2/\text{s} \)
Answer: (c) \( 10\sqrt{3} \text{ cm}^2/\text{s} \)

Question. A ladder, 5 meter long, standing on a horizontal floor, leans against a vertical wall. If the top of the ladder slides downwards at the rate of 10 cm/sec, then the rate at which the angle between the floor and the ladder is decreasing when lower end of ladder is 2 metres from the wall is
(a) \( \frac{1}{10} \text{ radian/sec} \)
(b) \( \frac{1}{20} \text{ radian/sec} \)
(c) 20 radian/sec
(d) 10 radian/sec
Answer: (b) \( \frac{1}{20} \text{ radian/sec} \)

Question. \( y = x(x - 3)^2 \) decreases for the values of \( x \) given by
(a) \( 1 < x < 3 \)
(b) \( x < 0 \)
(c) \( x > 0 \)
(d) \( 0 < x < \frac{3}{2} \)
Answer: (a) \( 1 < x < 3 \)

Question. The maximum value of \( \sin x . \cos x \) is
(a) \( \frac{1}{4} \)
(b) \( \frac{1}{2} \)
(c) \( \sqrt{2} \)
(d) \( 2\sqrt{2} \)
Answer: (b) \( \frac{1}{2} \)

Assertion-Reason Questions

Question. Assertion (A): The rate of change of area of a circle with respect to its radius \( r \) when \( r = 6 \) cm is \( 12\pi \text{ cm}^2/\text{cm} \).
Reason (R): Rate of change of area of a circle with respect to its radius \( r \) is \( \frac{dA}{dr} \), where A is the area of the circle.
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (a) Both A and R are true and R is the correct explanation for A.

Question. Assertion (A): \( f(x) = \tan x - x \) always increases.
Reason (R): Any function \( y = f(x) \) is increasing if \( \frac{dy}{dx} > 0 \).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (a) Both A and R are true and R is the correct explanation for A.

Question. Assertion (A): \( f(x) = x^4 \) is decreasing in the interval \( (0, \infty) \).
Reason (R): Any function \( y = f(x) \) is decreasing if \( \frac{dy}{dx} < 0 \).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (d) A is false but R is true.

Question. Assertion (A): The slope of the tangent to the curve \( y = x^3 \) where it cuts x-axis, is 0.
Reason (R): Slope of tangent to the curve \( y = f(x) \) at point \( (x_0, y_0) \) is \( \frac{dy}{dx} \) at \( (x_0, y_0) \).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (a) Both A and R are true and R is the correct explanation for A.