MCQ Questions for Class 12 Maths Chapter 6 Application of Derivatives with Answers

Question 1.
The sides of an equilateral triangle are increasing at the rate of 2cm/sec. The rate at which the are increases, when side is 10 cm is
(a) 10 cm²/s
(b) √3 cm²/s
(c) 10√3 cm²/s
(d) $$\frac{10}{3}$$ cm²/s

Question 2.
A ladder, 5 meter long, standing oh 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) $$\frac{1}{10}$$ radian/sec
(b) $$\frac{1}{20}$$ radian/sec

Answer: (b) $$\frac{1}{20}$$ radian/sec

Question 3.
The curve y – x1/5 at (0, 0) has
(a) a vertical tangent (parallel to y-axis)
(b) a horizontal tangent (parallel to x-axis)
(c) an oblique tangent
(d) no tangent

Answer: (b) a horizontal tangent (parallel to x-axis)

Question 4.
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

Answer: (c) x + 3y ± 8 = 0

Question 5.
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 6.
If y = x4 – 10 and if x changes from 2 to 1.99 what is the change in y
(a) 0.32
(b) 0.032
(c) 5.68
(d) 5.968

Question 7.
The equation of tangent to the curve y (1 + x²) = 2 – x, w here it crosses x-axis is:
(a) x + 5y = 2
(b) x – 5y = 2
(c) 5x – y = 2
(d) 5x + y = 2

Answer: (a) x + 5y = 2

Question 8.
The points at which the tangents to the curve y = x² – 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 9.
The tangent to the curve y = e2x 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 10.
The slope of tangent to the curve x = t² + 3t – 8, y = 2t² – 2t – 5 at the point (2, -1) is
(a) $$\frac{22}{7}$$
(b) $$\frac{6}{7}$$
(c) $$\frac{-6}{7}$$
(d) -6

Answer: (c) $$\frac{-6}{7}$$

Question 11.
The two curves; x³ – 3xy² + 2 = 0 and 3x²y – y³ – 2 = 0 intersect at an angle of
(a) $$\frac{π}{4}$$
(b) $$\frac{π}{3}$$
(c) $$\frac{π}{2}$$
(d) $$\frac{π}{6}$$

Answer: (a) $$\frac{π}{4}$$

Question 12.
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 13.
Let the f: R → R be defined by f (x) = 2x + cos x, then f
(a) has a minimum at x = 3t
(b) has a maximum, at x = 0
(c) is a decreasing function
(d) is an increasing function

Answer: (d) is an increasing function

Question 14.
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 15.
The function f(x) = 4 sin³ x – 6 sin²x + 12 sin x + 100 is strictly
(a) increasing in (π, $$\frac{3π}{2}$$)
(b) decreasing in ($$\frac{π}{2}$$, π)
(c) decreasing in [$$\frac{-π}{2}$$,$$\frac{π}{2}$$]
(d) decreasing in [0, $$\frac{π}{2}$$]

Answer: (c) decreasing in [$$\frac{-π}{2}$$,$$\frac{π}{2}$$]

Question 16.
Which of the following functions is decreasing on(0, $$\frac{π}{2}$$)?
(a) sin 2x
(b) tan x
(c) cos x
(d) cos 3x

Question 17.
The function f(x) = tan x – x
(a) always increases
(b) always decreases
(c) sometimes increases and sometimes decreases
(d) never increases

Question 18.
If x is real, the minimum value of x² – 8x + 17 is
(a) -1
(b) 0
(c) 1
(d) 2

Question 19.
The smallest value of the polynomial x³ – 18x² + 96x in [0, 9] is
(a) 126
(b) 0
(c) 135
(d) 160

Question 20.
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

Answer: (c) one maxima and one minima

Question 21.
The maximum value of sin x . cos x is
(a) $$\frac{1}{4}$$
(b) $$\frac{1}{2}$$
(c) √2
(d) 2√2

Answer: (b) $$\frac{1}{2}$$

Question 22.
At x = $$\frac{5π}{6}$$, f (x) = 2 sin 3x + 3 cos 3x is
(a) maximum
(b) minimum
(c) zero
(d) neither maximum nor minimum

Answer: (d) neither maximum nor minimum

Question 23.
Maximum slope of the curve y = -x³ + 3x² + 9x – 27 is
(a) 0
(b) 12
(c) 16
(d) 32

Question 24.
f(x) = xx has a stationary point at
(a) x = e
(b) x = $$\frac{1}{e}$$
(c) x = 1
(d) x = √e

Answer: (b) x = $$\frac{1}{e}$$

Question 25.
The maximum value of ($$\frac{1}{x}$$)x is
(a) e
(b) e²
(c) e1/x
(d) ($$\frac{1}{e}$$)1/e

Answer: (d) ($$\frac{1}{e}$$)1/e

Question 26.
If the volume of a sphere is increasing at a constant rate, then the rate at which its radius is increasing is
(a) a constant
(c) inversely proportional to the radius
(d) inversely proportional to the surface area

Answer: (d) inversely proportional to the surface area

Question 27.
A particle is moving along the curve x = at² + bt + c. If ac = b², then particle would be moving with uniform
(a) rotation
(b) velocity
(c) acceleration
(d) retardation

Question 28.
The distance Y metres covered by a body in t seconds, is given by s = 3t² – 8t + 5. The body will stop after
(a) 1 s
(b) $$\frac{3}{4}$$ s
(c) $$\frac{4}{3}$$ s
(d) 4 s

Answer: (c) $$\frac{4}{3}$$ s

Question 29.
The position of a point in time Y is given by x = a + bt + ct², y = at + bt². Its acceleration at timet Y is
(a) b – c
(b) b + c
(c) 2b – 2c
(d) 2$$\sqrt{b^2+c^2}$$

Answer: (d) 2$$\sqrt{b^2+c^2}$$

Question 30.
The function f(x) = log (1 + x) – $$\frac{2x}{2+x}$$ is increasing on
(a) (-1, ∞)
(b) (-∞, 0)
(b) (-∞, ∞)
(d) None of these

Question 31.
f(x) = ($$\frac{e^{2x}-1}{e^{2x}+1}$$) is
(a) an increasing function
(b) a decreasing function
(c) an even function
(d) None of these

Question 32.
If f (x) = $$\frac{x}{sin x}$$ and g (x) = $$\frac{x}{tan x}$$, 0 < x ≤ 1, then in the interval
(a) both f (x) and g (x) are increasing functions
(b) both f (x) and g (x) are decreasing functions
(c) f(x) is an increasing function
(d) g (x) is an increasing function

Answer: (c) f(x) is an increasing function

Question 33.
The function f(x) = cot-1 x + x increases in the interval
(a) (1, ∞)
(b) (-1, ∞)
(c) (0, ∞)
(d) (-∞, ∞)

Question 34.
The function f(x) = $$\frac{x}{log x}$$ increases on the interval
(a) (0, ∞)
(b) (0, e)
(c) (e, ∞)
(d) None of these

Question 35.
The value of b for which the function f (x) = sin x – bx + c is decreasing for x ∈ R is given by
(a) b < 1
(b) b ≥ 1
(c) b > 1
(d) b ≤ 1

Question 36.
If f (x) = x³ – 6x² + 9x + 3 be a decreasing function, then x lies in
(a) (-∞, -1) ∩ (3, ∞)
(b) (1, 3)
(c) (3, ∞)
(d) None of these

Question 37.
The function f (x) = 1 – x³ – x5 is decreasing for
(a) 1 < x < 5
(b) x < 1
(c) x > 1
(d) all values of x

Answer: (d) all values of x

Question 38.
Function, f (x) = $$\frac{λ sin x+ 6 cos x}{2 sin x + 3 cos x}$$ is monotonic increasing, if
(a) λ > 1
(b) λ < 1
(c) λ < 4
(d) λ > 4

Question 39.
The length of the longest interval, in which the function 3 sin x – 4 sin³ x is increasing is
(a) $$\frac{π}{3}$$
(b) $$\frac{π}{2}$$
(c) $$\frac{3π}{2}$$
(d) π

Question 40.
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 < –$$\frac{1}{2}$$

Answer: (c) x < -1 or x > 1

Question 41.
The function f(x) = x + cos x is
(a) always increasing
(b) always decreasing
(c) increasing for certain range of x
(d) None of these

Question 42.
The function which is neither decreasing nor increasing in ($$\frac{π}{2}$$, $$\frac{3π}{2}$$) is
(a) cosec x
(b) tan x
(c) x²
(d) |x – 1|

Question 43.
The function /’defined by f(x) = 44 – 2x + 1 is increasing for
(a) x < 1
(b) x > 0
(c) x < $$\frac{1}{2}$$
(d) x > $$\frac{1}{2}$$

Answer: (d) x > $$\frac{1}{2}$$

Question 44.
The interval in which the function y = x³ + 5x² – 1 is decreasing, is
(a) (0, $$\frac{1}{3}$$)
(b) (0, 10)
(c) ($$\frac{-10}{3}$$, 0)
(d) None of these

Answer: (c) ($$\frac{-10}{3}$$, 0)