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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 = bex 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 = e2x 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 = x3 – 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 – 2x3, 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 cm3/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 cm3/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) = x3 – 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 = x3 + 6 at the points (-1, 5) and (1, 7) are parallel. [HOTS]
Answer/Explanation
Answer:
Explaination:
y = x3 + 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 = t3 – 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 = ex 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)=x4 – 62x² + ax + 9 attains its maximum value on the interval [0,2], Find the value of a. [NCERT]
Answer/Explanation
Answer:
Explaination:
f'(x)=4x3 – 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)= x3 + 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) = xx, x > 0, find the stationary point for the function f.
Answer/Explanation
Answer:
Explaination:
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