Application of Derivatives
(2025)
Q.1 The values of λ so that f(x) = sin x – cos x – λx + C decreases for all real
values of xx are :
(1 Mark) (CBSE 2025 - 65/4/1)
A. 1 < λ < √2
B. λ ≥ √2
C. λ ≥ 1
D. λ < 1
Q.2 If f(x) = 2x + cos x, then f(x) :
(1 Mark) (CBSE 2025 - 65/4/1)
A. is an increasing function
B. has a minima at x = π
C. is a decreasing function
D. has a maxima at x = π
Q.3 Let f(x) = |x|, x ∈ R. Then, which of the following statements is incorrect?
(1 Mark) (CBSE 2025 - 65/6/1)
A. f has a minimum value at x = 0.
B. f is differentiable at x = 0.
C. f has no maximum value in R.
D. f is continuous at x = 0.
Q.4 The function f(x) = x2 − 4x + 6 is increasing in the interval
(1 Mark) (CBSE 2025 - 65/2/1)
,A. (−∞, 2]
B. [1,2]
C. (0, 2)
D. [2, ∞)
Q.5 A cylindrical tank of radius 10 cm is being filled with sugar at the rate
of 100π cm3/s. The rate, at which the height of the sugar inside the tank is
increasing, is :
(1 Mark) (CBSE 2025 - 65/2/1)
A. 1 cm/s
B. 1.1 cm/s
C. 0.5 cm/s
D. 0.1 cm/s
Q.6 A spherical ball has a variable diameter 5/2(3x + 1). The rate of change of
its volume w.r.t. x, when x = 1, is :
(1 Mark) (CBSE 2025 - 65/7/1)
A. 300π
B. 375π
C. 125π
D. 225π
Q.7 If f : R → R is defined as f(x) = 2x – sin x, then f is :
(1 Mark) (CBSE 2025 - 65/7/1)
A. a decreasing function
B. maximum at x = 0
C. an increasing function
D. maximum at x = π/2
,Q.8 The slope of the curve y = −x3 + 3x2 + 8x − 20 is maximum at :
(1 Mark) (CBSE 2025 - 65/5/1)
A. (−10, 1)
B. (1, −10)
C. (1, 10)
D. (10, 1)
Q.9 The absolute maximum value of function f(x) = x3 − 3x + 2 in [0,2] is :
(1 Mark) (CBSE 2025 - 65/1/1)
A. 2
B. 4
C. 5
D. 0
Q.10 Find the least value of 'a' so that f(x) = 2x2 – ax + 3 is an increasing
function on [2,4].
(2 Mark) (CBSE 2025 - 65/4/1)
Q.11
(2 Mark) (CBSE 2025 - 65/4/1)
Q.12 For the curve y = 5x − 2x3, if x increases at the rate of 2 units /s, then how
fast is the slope of the curve changing when x=2 ?
(2 Mark) (CBSE 2025 - 65/4/1)
Q.13 Determine the values of x for which f(x) = is an increasing or
a decreasing function.
(2 Mark) (CBSE 2025 - 65/6/1)
, Q.14 Find the values of 'a' for which f(x) = sin x – ax + b is increasing on R.
(2 Mark) (CBSE 2025 - 65/6/1)
Q.15 Surface area of a balloon (spherical), when air is blown into it, increases at
a rate of 5 mm2/s. When the radius of the balloon is 8 m, find the rate at which
the volume of the balloon is increasing.
(2 Mark) (CBSE 2025 - 65/5/1)
Q.16 Find the intervals in which function is (i) increasing (ii)
decreasing.
(2 Mark) (CBSE 2025 - 65/1/1)
Q.17 Amongst all pairs of positive integers with product as 289, find which of
the two numbers add up to the least.
(3 Mark) (CBSE 2025 - 65/7/1)
Q.18 Find the value of 'a' for which f(x) = √3sin x – cos x − 2ax + 6 is
decreasing in R.
(3 Mark) (CBSE 2025 - 65/5/1)
Q.19 The side of an equilateral triangle is increasing at the rate of 3 cm/s. At
what rate its area increasing when the side of the triangle is 15 cm ?
(3 Mark) (CBSE 2025 - 65/1/1)
Q.20
(2025)
Q.1 The values of λ so that f(x) = sin x – cos x – λx + C decreases for all real
values of xx are :
(1 Mark) (CBSE 2025 - 65/4/1)
A. 1 < λ < √2
B. λ ≥ √2
C. λ ≥ 1
D. λ < 1
Q.2 If f(x) = 2x + cos x, then f(x) :
(1 Mark) (CBSE 2025 - 65/4/1)
A. is an increasing function
B. has a minima at x = π
C. is a decreasing function
D. has a maxima at x = π
Q.3 Let f(x) = |x|, x ∈ R. Then, which of the following statements is incorrect?
(1 Mark) (CBSE 2025 - 65/6/1)
A. f has a minimum value at x = 0.
B. f is differentiable at x = 0.
C. f has no maximum value in R.
D. f is continuous at x = 0.
Q.4 The function f(x) = x2 − 4x + 6 is increasing in the interval
(1 Mark) (CBSE 2025 - 65/2/1)
,A. (−∞, 2]
B. [1,2]
C. (0, 2)
D. [2, ∞)
Q.5 A cylindrical tank of radius 10 cm is being filled with sugar at the rate
of 100π cm3/s. The rate, at which the height of the sugar inside the tank is
increasing, is :
(1 Mark) (CBSE 2025 - 65/2/1)
A. 1 cm/s
B. 1.1 cm/s
C. 0.5 cm/s
D. 0.1 cm/s
Q.6 A spherical ball has a variable diameter 5/2(3x + 1). The rate of change of
its volume w.r.t. x, when x = 1, is :
(1 Mark) (CBSE 2025 - 65/7/1)
A. 300π
B. 375π
C. 125π
D. 225π
Q.7 If f : R → R is defined as f(x) = 2x – sin x, then f is :
(1 Mark) (CBSE 2025 - 65/7/1)
A. a decreasing function
B. maximum at x = 0
C. an increasing function
D. maximum at x = π/2
,Q.8 The slope of the curve y = −x3 + 3x2 + 8x − 20 is maximum at :
(1 Mark) (CBSE 2025 - 65/5/1)
A. (−10, 1)
B. (1, −10)
C. (1, 10)
D. (10, 1)
Q.9 The absolute maximum value of function f(x) = x3 − 3x + 2 in [0,2] is :
(1 Mark) (CBSE 2025 - 65/1/1)
A. 2
B. 4
C. 5
D. 0
Q.10 Find the least value of 'a' so that f(x) = 2x2 – ax + 3 is an increasing
function on [2,4].
(2 Mark) (CBSE 2025 - 65/4/1)
Q.11
(2 Mark) (CBSE 2025 - 65/4/1)
Q.12 For the curve y = 5x − 2x3, if x increases at the rate of 2 units /s, then how
fast is the slope of the curve changing when x=2 ?
(2 Mark) (CBSE 2025 - 65/4/1)
Q.13 Determine the values of x for which f(x) = is an increasing or
a decreasing function.
(2 Mark) (CBSE 2025 - 65/6/1)
, Q.14 Find the values of 'a' for which f(x) = sin x – ax + b is increasing on R.
(2 Mark) (CBSE 2025 - 65/6/1)
Q.15 Surface area of a balloon (spherical), when air is blown into it, increases at
a rate of 5 mm2/s. When the radius of the balloon is 8 m, find the rate at which
the volume of the balloon is increasing.
(2 Mark) (CBSE 2025 - 65/5/1)
Q.16 Find the intervals in which function is (i) increasing (ii)
decreasing.
(2 Mark) (CBSE 2025 - 65/1/1)
Q.17 Amongst all pairs of positive integers with product as 289, find which of
the two numbers add up to the least.
(3 Mark) (CBSE 2025 - 65/7/1)
Q.18 Find the value of 'a' for which f(x) = √3sin x – cos x − 2ax + 6 is
decreasing in R.
(3 Mark) (CBSE 2025 - 65/5/1)
Q.19 The side of an equilateral triangle is increasing at the rate of 3 cm/s. At
what rate its area increasing when the side of the triangle is 15 cm ?
(3 Mark) (CBSE 2025 - 65/1/1)
Q.20
