Maths MCQs for Class 12 with Answers Chapter 6 Application of Derivatives

## Application of Derivatives Class 12 Maths MCQs Pdf

1. 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

**Answer/Explanation**

Answer: a

Explaination:

(a), as R'(x) = 6x + 36

⇒ R'(15) = 90 + 36 = 126

2. The side of an equilateral triangle is increasing at the rate of 2 cm/s. The rate at which area increases when the side is 10 is

(a) 10 cm²/s

(b) √3 cm²/s

(c) 10√3 cm²/s

(d) \(\frac{10}{3}\)cm²/s

**Answer/Explanation**

Answer: c

Explaination:

3. 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)

**Answer/Explanation**

Answer: b

Explaination:

4. 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/Explanation**

Answer: c

Explaination:

5. The point on the curve where tangent to the curve y2 = x, makes an angle of 45° clockwise with the x-axis is

**Answer/Explanation**

Answer: b

Explaination:

6. 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)

**Answer/Explanation**

Answer: b

Explaination:

(b), as slope of line y = x + 1 is 1

and \(\frac{dy}{dx}\) = \(\frac{2}{y}\) = 1

⇒ y = 2 and x = 1.

Therefore point is (1, 2).

7. The curves y = ae^{-x} and y = be^{x} are orthogonal if

(a) a = b

(b) a = -b

(c) ab = -1

(d) ab = 1

**Answer/Explanation**

Answer: d

Explaination:

8. 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

**Answer/Explanation**

Answer: d

Explaination:

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

(a) (0, 1)

(b) (2, 0)

(c) (-\(\frac{1}{2}\), 0)

(d) (-2, 0)

**Answer/Explanation**

Answer: c

Explaination:

10. The angle between the curve y² = x and x² =y at (1, 1) is

(a) 60°

(b) tan^{-1}\(\frac{4}{3}\)

(c) cot^{-1}\(\frac{4}{3}\)

(d) 90°

**Answer/Explanation**

Answer: c

Explaination:

11. The absolute maximum value of y = x^{3} – 3x + 2 in 0 ≤ x ≤ 2 is

(a) 4

(b) 6

(c) 2

(d) 0

**Answer/Explanation**

Answer: a

Explaination:

(a), as y’ = 3x² – 3, for a point of absolute maximum or minimum y’=0 ⇒ x = ± 1.

y]_{x=0} = 2,

y]_{x=1 }= 1 – 3 + 2 = 0,

y]_{x=-1 }= -1 +3+ 2 = 4,

y]_{x=2} = 8 – 6 + 2 = 4

12. Diameter of a sphere is \(\frac{3}{2}\)(2x + 5), the rate of change of its surface area with respect to x is ____ .

**Answer/Explanation**

Answer:

Explaination:

13. The edge of a cube is increasing at the rate of 0.3 cm/s, the rate of change of its surface area when edge is 3 cm is ____ .

**Answer/Explanation**

Answer:

Explaination:

14. The rate of change of area of a circle with respect to its radius is _______ .

**Answer/Explanation**

Answer:

Explaination:

2πr, as A = πr²

⇒ \(\frac{dA}{dr}\) = 2πr

15. Radius of a variable circle is changing at the rate of 5 cm/s. What is the radius of the circle at a time when its area is changing at the rate of 100 cm²/s? [HOTS]

**Answer/Explanation**

Answer:

Explaination:

16. Find the point on the curve y=x², where the rate of change of x-coordinate is equal to the rate of change of y-coordinate.

**Answer/Explanation**

Answer:

Explaination:

17. The side of an equilateral triangle is increasing at the rate of 0.5 cm/s. Find the rate of increase of its perimeter.

**Answer/Explanation**

Answer:

Explaination:

18. If the rate of change of volume of a sphere is equal to the rate of change of its radius, then find the radius.

**Answer/Explanation**

Answer:

Explaination:

19. The volume of a sphere is increasing at the rate of 3 cubic centimeter per second. Find the rate of increase of its surface area, when the radius is 2 cm. [Delhi 2017]

**Answer/Explanation**

Answer:

Explaination:

20. For the curve y = 5x – 2x^{3}, if x increases at the rate of 2 units/s, then find the rate of change of the slope of the curve when x = 3. [Delhi 2017]

**Answer/Explanation**

Answer:

Explaination:

21. The volume of a cube is increasing at the rate of 9 cm^{3}/s. How fast is its surface area increasing when the length of an edge is 10 cm? [AI 2017]

**Answer/Explanation**

Answer:

Explaination:

22. The volume of a sphere is increasing at the rate of 8 cm^{3}/s. Find the rate at which its surface area is increasing when the radius of the sphere is 12 cm. [AI 2017]

**Answer/Explanation**

Answer:

Explaination:

23. The radius r of a right circular cylinder is decreasing at the rate of 3 cm/min. and its height h is increasing at the rate of 2 cm/ min. When r = 7 cm and h = 2 cm, find the rate of change of the volume of cylinder.

[Use π = \(\frac{22}{7}\)] [Foreign 2017]

**Answer/Explanation**

Answer:

Explaination:

24. The radius r of the base of a right circular cone is decreasing at the rate of 2 cm/min and its height h is increasing at the rate of 3 cm/min. When r = 3.5 cm and h = 6 cm, find the rate of change of the volume of the cone. [Use π = \(\frac{22}{7}\)]

**Answer/Explanation**

Answer:

Explaination:

25. The function f(x) = 4x + 3, x ∈ R is an increasing function. State true or false.

**Answer/Explanation**

Answer:

Explaination: True, as f'(x) = 4 > 0. Hence, increasing.

26. The function f(x) = log(cos x) is increasing function for [0, \(\frac{\pi}{2}\)] State true or false.

**Answer/Explanation**

Answer:

Explaination:

27. Show that function y = 4x – 9 is increasing for all x ∈ R.

**Answer/Explanation**

Answer:

Explaination:

Given y = 4x – 9

\(\frac{dy}{dx}\)= 4 > 0 for all x ∈ R.

Hence, function is increasing for all x ∈ R.

28. Show that the function given by f(x) = sin x is strictly decreasing in (\(\frac{\pi}{2}\), π).

**Answer/Explanation**

Answer:

Explaination:

Consider f(x) = sin x

f(x) = cos x …(i)

cos x < 0 for each x ∈ (\(\frac{\pi}{2}\), π)

∴ f(x) < 0 [from (i)] Hence, function is strictly decreasing in (\(\frac{\pi}{2}\), π).

29. Show that the function y = \(\frac{3}{x}\) + 7 is strictly decreasing for x ∈ R (x ≠ 0).

**Answer/Explanation**

Answer:

Explaination:

y’= \(\frac{3}{x²}\) < 0, for x ∈ R, x ≠ 0.

Hence, function is strictly decreasing.

30. Show that the function f(x) = log |cos x| is strictly decreasing in (0, \(\frac{\pi}{2}\)) [HOTS]

**Answer/Explanation**

Answer:

Explaination:

f'(x)= \(\frac{1}{\cos x}\).(-sinx) = -tan x., tan

x > 0 for (0, \(\frac{\pi}{2}\)) f'(x) < 0

Hence, function is strictly decreasing.

31. Prove that the function given by f(x) = x^{3} – 3x² + 3x – 100 is increasing in R. [NCERT]

**Answer/Explanation**

Answer:

Explaination:

f'(x) = 3x² – 6x + 3 = 3(x² – 2x + 1)

= 3(x – 1)² > 0. Hence, f is increasing in R.

32. Find the interval for which the function f(x) = cot^{-1} x + x increases.

**Answer/Explanation**

Answer:

Explaination:

33. ShoW that the function fix) = 4×3 – 18X2 + 27x – 7 is always increasing on R. [Delhi 2017]

**Answer/Explanation**

Answer:

Explaination:

34. Show that the function f given by f(x)=tan^{-1}(sinx+cos x) is decreasing for all (\(\frac{\pi}{4}\), \(\frac{\pi}{2}\)). [Foreign 2017]

**Answer/Explanation**

Answer:

Explaination:

35. Tangent to the curve given by x = sec θ and y = cosec θ, at θ = \(\frac{\pi}{2}\) makes an angle _______ with the x-axis.

**Answer/Explanation**

Answer:

Explaination:

36. Prove that the tangents to the curve y = x^{3} + 6 at the points (-1, 5) and (1, 7) are parallel. [HOTS]

**Answer/Explanation**

Answer:

Explaination:

y = x^{3} + 6

⇒ y’ = 3x²

y’]_{(-1, 5)} = 3(-1)² = 3 and y’]_{(1, 7)}

= 3(1)² = 3

As slope at these points are equal. Hence, tangents are parallel.

37. At what point on the curve y = x² does the tangent make an angle of 45° with the x-axis? [HOTS]

**Answer/Explanation**

Answer:

Explaination:

Slope of the tangent = tan 45° = 1.

y’ = 2x

⇒ 2x = 1

⇒ x = \(\frac{1}{2}\). Substituting in curve, we get

y = \(\frac{1}{4}\). Point is (\(\frac{1}{2}\). \(\frac{1}{4}\)).

38. Find the slope of the tangent to the curve x = 3t² + 1, y = t^{3} – 1 at x = 1.

**Answer/Explanation**

Answer:

Explaination:

39. If the radius of a sphere is measured as 9 cm with an error of 0.03 cm, then find the approximate error in calculating its

**Answer/Explanation**

Answer:

Explaination:

Let r be the radius of the sphere and Δr be the error in measuring the radius.

Then r = 9, Δr = 0.03, Surface area of a sphere is given by S = 4πr²

ΔS = \(\frac{dS}{dr}\).Δr = 8πr.Δr = 8π × 9 × 0.03

= 2.16 π cm²

40. If radius of a circle is increased from 5 cm to 5.1 cm. Find the approximate increase in area.

**Answer/Explanation**

Answer:

Explaination:

Area of a circle of radius r is given by, A = πr²

Then, r = 5, r + Δr = 5.1, Δr = 0.1

ΔA =\(\frac{dA}{dr}\) Δr 2πr × 0.1

= 2π × 5 × 0.1 = π cm²

41. If f(x) = \(\frac{1}{4 x^{2}+2 x+1}\), then its maximum value is _____ .

**Answer/Explanation**

Answer:

Explaination:

42. Show that y = e^{x} has no local maxima or local minima. [HOTS]

**Answer/Explanation**

Answer:

Explaination:

y’ ≠ 0, for any x, so no solution of y’ = 0.

Hence, no local maximum or local minimum.

43. It is given that at x = 1, the function f(x)=x^{4} – 62x² + ax + 9 attains its maximum value on the interval [0,2], Find the value of a. [NCERT]

**Answer/Explanation**

Answer:

Explaination:

f'(x)=4x^{3} – 124x + a, for a point of maximum

f'(1) = 0 ⇒ 4 – 124 + a = 0

⇒ a = 120

44. Find the maximum and minimum values if any of the function given by f(x) = -(x – 1)² + 10. [NCERT]

**Answer/Explanation**

Answer:

Explaination:

-(x – 1)² < 0 for x ∈ R

⇒ -(x – 1)²+ 10 ≤ 10

⇒ f(x) ≤ 10,

Maximum value = 10.

Minimum value = nil.

45. Find the maximum and minimum values if any of the function given by f(x) = sin 2x + 5. [NCERT]

**Answer/Explanation**

Answer:

Explaination:

-1 ≤ sin 2x ≤ 1

⇒ -1 + 5 ≤ sin 2x+5 ≤ 1 + 5

⇒ 4 ≤ sin 2x + 5 ≤ 6.

Maximum value = 6, Minimum value = 4.

46. Prove that the function f(x)= x^{3} + x² + x + 1 does not have a maxima or minima.

**Answer/Explanation**

Answer:

Explaination:

f'(x) = 3x² + 2x + 1,

f'(x) = 0

⇒ 3x² + 2x + 1

As f'(x) ≠ 0 for any real x.

Hence, no maxima or minima.

47. Find the maximum and minimum values, if any, of the function given by f(x) = |sin 4x + 3| [NCERT]

**Answer/Explanation**

Answer:

Explaination:

f(x) = |sin 4x + 3|

-1 ≤ sin 4x ≤ 1

⇒ 2 ≤ sin 4x + 3 ≤ 4

⇒ 2 ≤ |sin 4x + 3| ≤ 4.

Minimum value = 2, Maximum value = 4.

48. Find the maximum and minimum value of the function y = |x – 3| + 7, x ∈ R.

**Answer/Explanation**

Answer:

Explaination:

∀ x ∈ R

|x – 3| ≥ 0

⇒ |x – 3| + 7 ≥ 7

⇒ y ≥ 7

⇒ minimum value = 7, no maximum value.

49. Find the number which exceeds its square by the greatest possible number.

**Answer/Explanation**

Answer:

Explaination:

Let number x exceeds its square by the greatest possible number

y = x – x²

y’ = 1 – 2x,

For maxima y, y’ = 0

⇒ 1 – 2x = 0

50. Given the function f(x) = x^{x}, x > 0, find the stationary point for the function f.

**Answer/Explanation**

Answer:

Explaination: