# Applications of Derivatives MCQ for ISC Class-12 Maths

Applications of Derivatives MCQ for ISC Class-12 Maths .These MCQ  / Objective Type Questions is based on latest reduced syllabus according 2021-22 session on bifurcated pattern. Main motto of MCQ Type Question is cracking the next upcoming exam of council. Visit official website CISCE for detail information about ISC Board Class-12 Physics.

## ISC Class-12 Maths , Applications of Derivatives MCQ Type Questions

 Board ISC Class 12th (XII) Subject Maths Ch-Name Applications of Derivatives Syllabus on bifurcated syllabus (after reduction) Bifurcated pattern Semester-1 Session 2021-22 Topic MCQ / Objective Type Question

### Applications of Derivatives MCQ Type Questions  for ISC Class-12 Maths

Question 1: If f and g are differentiable functions on [0, 1] satisfying f(0) = 2 = g(l), g(0) = 0 and f(1) = 6, then for some c ∈ ] 0, 1 :

(a) 2f'(c) = 3g'(c)
(b) f'(c) = g'(c)
(c) f'(c) = 2g'(c)
(d) 2f'(c) = g'(c).

Question 2: Twenty metres of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower bed is:

(a) 25
(b) 30
(c) 12.5
(d) 10.

Question 3: Find all the points of local maxima and local minima of the function f(x) = (x – 1)(x + 1)2

(a) 1, -1, -1/5
(b) 1, -1
(c) 1, -1/5
(d) -1, -1/5

Question 4:  The total revenue in ₹ received from the sale of x units of an article is given by R(x) = 3x² + 36x + 5. The marginal revenue when x = 15 is (in ₹ )

(a) 126
(b) 116
(c) 96
(d) 90

Question 5: The point(s) on the curve y = x², at which y-coordinate is changing six times as fast as x-coordinate is/are

(a) (2, 4)
(b) (3, 9)
(c) (3, 9), (9, 3)
(d) (6, 2)

Question 6: 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) 10/3 cm²/s

Question 7: 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

Question 8:  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 9:  The equation of tangent to curve y = e–|x| at the points where the curve cuts the line x = 1 is

(a)  x+ y = e

(b)  e (x + y) = 1

(c) y + ex = 1

(d) None of these

Question 10:  The number of roots of x3– 3x + 1 = 0 in [1, 2] is

(a) One

(b) Two

(c) Three

(d) None of these

Question 11:  Let h(x) = f(x) – [f (x)]2 + [f (x)]3 for every real number x. Then

(a) h is increasing whenever f is increasing

(b)     h is increasing whenever f is decreasing

(c)h is decreasing whenever f is increasing

(d) Nothing can be said in general

Answer: (a) h is increasing whenever f is increasing

Question 12: Let y = |x| + |x – 2|, then dy/dx at x = 2

(a) 2

(b) 0

(c) Does not exist

(d) None of these

Question 13: The number of solutions of equations 3x2 + xsinx + cosx = 0

(a) 3

(b) 2

(c) 1

(d) 0

Question 14: The maximum value of (1/x)x is​

(a) (1/e)1/e

(b) (e)2/e

(c) (e)-1/e

(d) (e)1/e

Question 15: The global maximum and global minimum of f (x) = 2x3 – 9x2 + 12x + 6 in [0, 2]

(a) (11, 6)

(b) ( 6,11)

(c) ( -6,11)

(d) ( -11, 6)

Question 16: The real number k for which the equation 2x³ + 3x + k = 0 has two distinct real roots in [0,1]:

(a) lies between 2 and 3
(b) lies between -1 and 0
(c) does not exist
(d) lies between 1 and 2.

Question 17: The point on the curve y² = x, where the tangent makes an angle of π/4 with x-axis is:

(a) (1/21/4)
(b) (1/41/2)
(c) (4, 2)
(d) (1, 1).

Question 18: 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 19:  Find the maximum profit that a company can make, if the profit function is given by P(x) = 41 + 24x – 18x2.

(a) 25
(b) 43
(c) 62
(d) 49

Question 20:  If y = x3 + x2 + x + 1, then y

(a) has a local minimum
(b) has a local maximum
(c) neither has a local minimum nor local maximum
(d) None of these

Answer: (c) neither has a local minimum nor local maximum

Question 21:  Find both the maximum and minimum values respectively of 3x4 – 8x3 + 12x2 – 48x + 1 on the interval [1, 4].

(a) -63, 257
(b) 257, -40
(c) 257, -63
(d) 63, -257

Question 22: It is given that at x = 1, the function x4 – 62x2 + ax + 9 attains its maximum value on the interval [0, 2]. Find the value of a.

(a) 100
(b) 120
(c) 140
(d) 160

Question 23: 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 24: The line y = x + 1 is a tangent to the curve y2 = 4x at the point

(a) (-1, 2)
(b) (1, 2)
(c) (1, -2)
(d) (2, 1)

Question 25: The curves y = ae-x and y = bex are orthogonal if

(a) a = b
(b) a = -b
(c) ab = -1
(d) ab = 1

Question 26: If the curves 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 27: 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 28: 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 29:  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 30: The function f(x) = x5 – 5x4 + 5x3 – 1 has

(a) one minima and two maxima
(b) two minima and one maxima
(c) two minima and two maxima
(d) one minima and one maxima

Answer: (d) one minima and one maxima

Question 31: The area of a right-angled triangle of the given hypotenuse is maximum when the triangle is

(a) scalene
(b) equilateral
(c) isosceles
(d) None of these

Question 32: Find the area of the largest isosceles triangle having perimeter 18 metres.

(a) 9√3
(b) 8√3
(c) 4√3
(d) 7√3

Question 33: The absolute maximum value of y = x3 – 3x + 2 in 0 ≤ x ≤ 2 is

(a) 4
(b) 6
(c) 2
(d) 0

Question 34:  The tangent to the curve y = e2x at the point (0, 1) meets x-axis at

(a) (0, 1)
(b) (-1/2, 0)
(c) (2, 0)
(d) (0, 2)

Question 35:  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

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

Question 36:  The slope of the tangent to the curve x = a sin t, y = a{cot t + log(tan t/2)} at the point ‘t’ is

(a) tan t
(b) cot t
(c) tan t/2
(d) None of these

Question 37:  The equation of the normal to the curves y = sin x at (0, 0) is

(a) x = 0
(b) x + y = 0
(c) y = 0
(d) x – y = 0

Answer: (b) x + y = 0

Question 38: 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 39:  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 40:  The maximum and the minimum value of 3x4 – 8x3 + 12x2 – 48x + 1 on the interval [1,4]​

(a) -40,257

(b) -48,258

(c) -49,258

(d) -58,257

Question 41:  Find the maximum and minimum values of f (x) = 2x3 – 24x + 107 in the interval [1, 3].​

(a) 89, 69

(b) 89, 75

(c) 59, 56

(d) 89, -9

Question 42:  The radius of air bubble is increasing at the rate of 0. 25 cm/s. At what rate the volume of the bubble is increasing when the radius is 1 cm.​

(a) 4π cm3/s

(b) 22π cm3/s

(c) 2π cm3/s

(d) π cm3/s

Question 43:  The total revenue in Rupees received from the sale of x units of a product is given by R(x) = 5x2 + 22x + 35. Find the marginal revenue, when x = 7, where by marginal revenue we mean the rate of change of total revenue with respect to the number of items sold at an instant

(a) Rs 7

(b) Rs 127

(c) Rs 92

(d) Rs 48

Question 44:  Find a point on the curve y = (x – 2)2. at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4).

(a) (3, 1)
(b) (4, 1)
(c) (6,1)
(d) (5, 1)

Question 45: Tangents to the curve x2 + y2 = 2 at the points (1, 1) and (-1, 1) are

(a) parallel
(b) perpendicular
(c) intersecting but not at right angles
(d) none of these

Question 46:  If there is an error of 2% in measuring the length of a simple pendulum, then percentage error in its period is

(a) 1%
(b) 2%
(c) 3%
(d) 4%

Question 47: If there is an error of a% in measuring the edge of a cube, then percentage error in its surface area is

(a) 2a%
(b) a/2 %
(c) 3a%
(d) None of these

Question 48: Which of the following functions is decreasing on(0, π/2)?

(a) sin 2x
(b) tan x
(c) cos x
(d) cos 3x

Question 49:  The function f(x) = tan x – x

(a) always increases
(b) always decreases
(c) sometimes increases and sometimes decreases
(d) never increases

Question 50:  The smallest value of the polynomial x³ – 18x² + 96x in [0, 9] is

(a) 126
(b) 0
(c) 135
(d) 160