NCERT Class 12 Chapter 6 Maths Application of Derivatives MCQ

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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. 103 cm²/s

2. 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

3. 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

4. The total revenue in Rupees received from the sale of x units of a product is given by R(x) = 5x^{2 + 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. 7

B. Rs 127

C. Rs 92

D. Rs 48

5. 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. 110 radian/sec

B. 120 radian/sec

C. 20 radiah/sec

D. 10 radiah/sec

6. 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

7. The curve y – x^1/5 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

8. Find the area of the largest isosceles triangle having perimeter 18 metres.

A. 9√3

B. 8√3

C. 4√3

D. 7√3

9. Find the approximate change in total surface area of a cube of side x metre caused by increase in side by 1%.

A. 12 m²

B. 0.12x² m²

C. 1.2x m²

D. 12x m²

10. 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

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11. 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

12. The volume of cube is increasing at the constant rate of cm³/S Find the rate of change of edge of the cube when its edge is 5 cm.

A. 25 cm³/sec

B. 25 cm/s

C. 1/25 cm/s

D. 1/25 cm³/s

13. 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

14. 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

15. The total cost associated with the production of x units of a product is given by C(x)=5x²+14x+6 Find marginal cost when 5 units are produced

A. Rs.64

B. Rs.70

C. Rs.50

D. Rs.(10x + 14)

16. 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

17. The radius of a cylinder is increasing at the rate of 3 m/s and its height is decreasing at the rate of 4 m/s. The rate of change of volume when the radius is 4 m and height is 6 m, is

A. 80π cu m/s

B. 144π cu m/s

C. 80 cu m/s

D. 64 cu m/s

18. Use differentials to approximate the value of cube root of 66

A. 4.042

B. 4.3

C. 2.021

D. 8.084

19. 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

20. The rate of change of the area of a circle with respect to its radius r at r = 6 cm is:

A. 10π

B. 12π

C. 8π

D. 11π

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21. A stone is dropped into a quiet lake and waves move in circles at a speed of 2cm per second. At the instant, when the radius of the circular wave is 12 cm, how fast is the enclosed area changing ?

A. Decreasing at the rate of 48π cm² / sec

B. Increasing at the rate of 24π cm² / sec

C. Increasing at the rate of 48π cm,sup>2 / sec

D. Decreasing at the rate of 24π cm2 / sec

22. 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).

23. A cylindrical tank of radius 10 mis being filled with wheat at the rate of 314 cubic m per minute. Then the depth of the wheat is increasing at the rate of:

A. 1 m/minute

B. 0 × 1 m/minute

C. 1 × 1 m/minute

D. 0 × 5 m/minute

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

A. (0, 1)

B. (-12, 0)

C. (2, 0)

D. (0, 2)

25. The function f(x) = tan x – x

A. always increases

B. always decreases

C. sometimes increases and sometimes decreases

D. never increases