MCQs For NCERT Class 12 Mathematics Chapter 6 Application of Derivatives

Please refer to the MCQ Questions for Class 12 Mathematics Chapter 6 Application of Derivatives with Answers. The following Application of Derivatives Class 12 Mathematics MCQ Questions have been designed based on the latest syllabus and examination pattern for Class 12. Our experts have designed MCQ Questions for Class 12 Mathematics with Answers for all chapters in your NCERT Class 12 Mathematics book.

Application of Derivatives Class 12 MCQ Questions with Answers

See below Application of Derivatives Class 12 Mathematics MCQ Questions, solve the questions and compare your answers with the solutions provided below.

Question. The equation of normal to the curve 3x² – y² = 8 which is parallel to the line x + 3y = 8 is
(a) 3x – y = 8
(b) 3x + y + 8 = 0
(c) x + 3y ± 8 = 0
(d) x + 3y = 0

Question. If g(x) =min (x,x²), where x is real number, then
(a) g(x) is an increasing function
(b) g(x) is a decreasing function
(c) f(x) is a constant function
(d) g(x) is a continuous function except at x = 0

Question. The function f (x) =x/1+|x| is
(a) strictly increasing
(b) strictly decreasing
(c) neither increasing nor decreasing
(d) not differential at x = 0 

Question.

Question. The function f(x) = x +cos x is
(a) always increasing
(b) always decreasing
(c) increasing for certain range of x
(d) None of the above

Question. 2x³– 6x+5  is an increasing function, if
(a) 0 <x<1
(b) −1<x<1
(c) x < −1 or x > 1
(d) −1 <x < − 1/2

Question. The length of the longest interval, in which the function 3sin x-4 sin³ x is increasing, is
(a) π/3
(b) π/2
(c) 3π/2
(d) π

Question. If f(x)= xe ^x(1-x), then f(x)  is
(a) increasing on −[-1/2,1]
 (b) decreasing on R
(c) increasing on R
(d) decreasing on [-1/2,1]

Question. The function ‘g ’  defined by g(x)= f(x²-2x+8)+ f(14+2x-x²), where f(x) is twice differentiable function, f ′′ (x) ≥ 0 for all real numbers x. The function g(x) is increasing in the interval
(a) [-1,1]∪ [2,∞)
(b) (,∞ ,1] ∪ [1,3]
(c) [-1,1] ∪ [3,∞)
(d) (∞,-2] ∪ [1,∞)

Question. Consider the following statements S and R S : Both sin x and cos x are decreasing function in (π/2,π).
R : If a differentiable function decreases in (a ,b), then its derivative also decreases in ( a, b).
Which of the following is true ?
(a) Both S and R are wrong.
(b) Both S and R are correct but R is not the correct explanation for S.
(c) S is correct and R is the correct explanation for S.
(d) S is correct, R is wrong.

Question.

Question. If the curve ay + x² = 7 and x³ = y, cut orthogonally at (1, 1), then the value of a is
(a) 1
(b) 0
(c) – 6
(d) 6

Question. If the sides of an equilateral triangle are increasing at the rate of 2 cm/s then the rate at which the area increases, when side is 10 cm, is

Question. Two curves x³ 3xy² + 2 = 0  intersect at and 3x² y – y³ – 2 = 0  intersect at an angle of
(a) π/4
(b) π/3
(c) π/2
(d) π/6

Question. The interval on which the function f (x) = 2x³ + 9x² + 12x – 1 is decreasing is
(a) [-1, ∞)
(b) [- 2, -1]
(c) (- ∞, – 2]
(d) [-1,1]

Question. If y = x⁴ – 10 and x changes from 2 to 1.99, then what is the change in y?
(a) 0.32
(b) 0.032
(c) 5.68
(d) 5.968

Question. The equation of tangent to the curve y(1 + x²) = 2 – x, where it crosses X-axis, is
(a) x + 5y = 2
(b) x – 5y = 2
(c) 5x – y = 2
(d) 5x + y = 2

Question. A ladder, 5 m 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/s, then the rate at which the angle between the floor and the ladder is decreasing when lower end of ladder is 2 m from the wall is

Question. The function f (x) = 4sin³ x – 6sin² x + 12sin x + 100 is strictly

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

Question. The points at which the tangents to the curve y = x³ – 12x + 18 are parallel to X-axis are
(a) (2, – 2), (- 2, – 34)
(b) (2, 34), (- 2, 0)
(c) (0, 34), (- 2, 0)
(d) (2, 2), (- 2, 34)

Question. The tangent to the curve y = e^2x at the point (0, 1) meets X-axis at

Question. The slope of tangent to the curve x = t² + 3t – 8 and y = 2t² – 2t – 5 at the point (2, – 1) is
(a) 22/7
(b) 6/7
(c) – 6/7
(d) – 6

Question. If f : R → R be defined by f (x) = 2x + cos x, then f
(a) has a minimum at x = π
(b) has a maximum at x = 0
(c) is a decreasing function
(d) is an increasing function

Question. Which of the following functions is decreasing on

(a) sin²x
(b) tan x
(c) cos x
(d) cos³x

Question. If x is real, then the minimum value of x² – 8x + 17 is
(a) -1
(b) 0
(c) 1
(d) 2

Question. The maximum slope of curve y = – x³ + 3x² + 9x – 27 is
(a) 0
(b) 12
(c) 16
(d) 32

Question. The smallest value of polynomial x³ – 18x² + 96x in [0, 9] is
(a) 126
(b) 0
(c) 135
(d) 160

Question. The function f (x) x^x = has a stationary point at
(a) x = e
(b) x = (1/e)
(c) x =1
(d) x = e

Question. The function f (x) = 2x³ – 3x² – 12x + 4, has
(a) two points of local maximum
(b) two points of local minimum
(c) one maxima and one minima
(d) no maxima or minima

Question. The maximum value of sin x – cos x is
(a) 1/4
(b) 1/2
(c) √2
(d) 2 √2

Question. A function y = f (x) has a second order derivative f ”(x) = 6(x – 1).
If its graph passes through the point (2,1) and at that point the tangent to the graph is y = 3x – 5, then the function is
(a) (x + 1)²
(b) (x – 1)³
(c) (x + 1)³
(d) (x – 1)²

Question. The function f(x) = 2x³ – 3x² – 12x + 4, has
(a) two points of local maximum
(b) two points of local minimum
(c) one maxima and one minima
(d) no maxima or minima

Question. If tangent to the curve x = at², y = 2at is perpendicular to x-axis, then its point of contact is
(a) (a, a)
(b) (0, a)
(c) (0, 0)
(d) (a, 0)

Question. What is the slope of the normal at the point (at², 2at) of the parabola y² = 4ax ?
(a) 1/t
(b) t
(c) – t
(d) -1/t

Question. A circular disc of radius 3 cm is being heated. Due to expansion, its radius increases at the rate of 0.05 cm/s.
The rate, at which its area is increasing when its radius is 3.2 cm, is
(a) 0.320 πcm²/s
(b) 0.160 πcm²/s
(c) 0.260 πcm²/s
(d) 1.2 πcm²/s

Question. The shortest distance between the line y – x = 1 and the curve x = y² is
(a) 3√2/8
(b) 2√3/8
(c) 3√2/5
(d) √3/4

Question. A wire 34 cm long is to be bent in the form of a quadrilateral of which each angle is 90°. What is the maximum area which can be enclosed inside the quadrilateral?
(a) 68 cm²
(b) 70 cm²
(c) 71.25 cm²
(d) 72. 25 cm² 

Question. The curve y – exy + x = 0 has a vertical tangent at the point:
(a) (1, 1)
(b) at no point
(c) (0, 1)
(d) (1, 0)

Question. The length x of a rectangle is decreasing at the rate of 5 cm/min and the width y is increasing at the rate of 4 cm/min. If x = 8 cm and y = 6 cm, then which of the following is correct?
I. The rate of change of the perimeter is – 2 cm/min.
II. The rate of change of the area of the rectangle is 12 cm²/min.
(a) Only I is correct
(b) Only II is correct
(c) Both I and II are correct
(d) Both I and II are incorrect

Question. f(x) = (e2x – 1/e2x + 1) is
(a) an increasing function
(b) a decreasing function
(c) an even function
(d) None of these

Question. The minimum value of e^{(2×2 – 2x +1) sin2} x is
(a) 0
(b) 1
(c) 2
(d) 3

Question. A monotonic function f in an interval I means that f is
(a) increasing in I
(b) dereasing in I
(c) either increasing in I or decreasing in I
(d) neither increasing in I nor decreasing in I

Question. The distance between the point (1, 1) and the tangent to the curve y = e^2x + x² drawn at the point x = 0 is
(a) 1/√5
(b) -1/√5
(c) 2/√5
(d) -2/√5

Question. At what point, the slope of the tangent to the curve x² + y² – 2x – 3 = 0 is zero?
(a) (3, 0), (– 1, 0)
(b) (3, 0), (1, 2)
(c) (– 1, 0), (1, 2)
(d) (1, 2), (1, – 2)

Question. The coordinates of the point on the parabola y² = 8x which is at minimum distance from the circle x2 + (y + 6)2 = 1 are
(a) (2, – 4)
(b) (18, – 12)
(c) (2, 4)
(d) None of these

Question. The approximate change in the volume V of a cube of side x meters caused by increasing the side by 2%, is
(a) 1.06x3m³
(b) 1.26x3m³
(c) 2.50x3m³
(d) 0.06x3m³

Question. Which of the following function is decreasing on (0,π/2)
(a) sin²x
(b) tan x
(c) cos x
(d) cos³x

Question. A point on the parabola y² = 18x at which the ordinate increases at twice the rate of the abscissa is
(a) (9/8 , 9/2)
(b) (2, -4)
(c) (-9/8 , 9/2)
(d) (2, 4)

Question. If f (x) = 3x⁴ + 4x³ – 12x² + 12, then f (x) is
(a) increasing in (– ∞ , – 2) and in (0, 1)
(b) increasing in ( – 2, 0) and in (1, ∞ )
(c) decreasing in ( – 2, 0) and in (0,1)
(d) decreasing in ( – ∞ , – 2) and in (1, ∞ )

Question. For the curve y = 5x – 2x³, if x increases at the rate of 2 units/s, then the rate at which the slope of curve is changing when x = 3, is
(a) – 78 units/s
(b) – 72 units/s
(c) – 36 units/s
(d) – 18 units/s

Question. The function f (x) = tan x – x
(a) always increases
(b) always decreases
(c) never increases
(d) sometimes increases and sometimes decreases

Question. If 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 < 3/2

Question. At x = 5π/6, f (x) = 2sin3x + 3cos3x is
(a) maximum
(b) minimum
(c) zero
(d) neither maximum nor minimum

Question. The maximum value of (1/x)^x is
(a) e
(b) e^e
(c) e^1/e
(d) (1/x)^1/e

Assertion – Reason Type Questions :

(a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
(b) Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion
(c) Assertion is correct, Reason is incorrect
(d) Assertion is incorrect, Reason is correct.

Question. Assertion: The function y² = 4x has no absolute maximum or minimum.
Reason: In the graph of the function the value of increases unboundedly and decreases unboundedly as x increases.

Question. Consider the function

Assertion: f has a local maximum value at x = 0.
Reason: f'(0) = 0 and f ”(0) < 0

Question. Assertion : Let f : R → R be a function such that f(x) = x³ + x² + 3x + sin x. Then f is one-one.
Reason : f(x) neither increasing nor decreasing function.

Question. Assertion : Let f : R → R be a function such that f(x) = x³ + x² + 3x + sin x. Then f is one-one.
Reason : f(x) neither increasing nor decreasing function.

Question. Assertion: If the radius of a sphere is measure as 9 m with an error of 0.03 m, then the approximate error in calculating its surface area is 2.16 πm².
Reason: We have, ΔS = (ds/dr) Δr where, ΔS = Approximate error in calculating the surface area, Δr = Error in measuring radius r.

Question. Assertion : f (x) cos² x cos³ (x+π/3) – cosxcos³ (x+π/3) then f ‘(x) = 0
Reason : Derivative of constant function is zero.