Class 12 application of derivatives multiple choice

Multiple choice(Application of derivatives)

application of derivatives multiple choice

Choose and write the correct option in the following question:(application of derivatives multiple choice)

1.) The interval in which the function f given by f(x)  = x²e^{-x} is strictly increasing, is

(a) (-∞, ∞)                             (b) (-∞, 0)

(c) (2, ∞)                               (d) (0, 2)

Answer (d)

2.) y =x(x-3)^2 decreasing for the values of x given by

(a) 1 < x < 3                           (b)  x < 0

(c) x > 0                                 (d) 0 < x < 3/2

Answer (a)

3.) f(x) = x^x has a stationary point at

(a) x = e                                 (b) x = 1/e

(c) x = 1                                 (d) x = √e

Answer (b)

4.) the maximum value of (\frac{1}{x})^x is

(a) e                                   (b) e^e

(c) e^{1/e}                  (d) (\frac{1}{e})^{1/e}

Answer (c)

5.) the point on the curve x² = 2y which is nearest to the point (0, 5) is

(a) (2√2, 4)                      (b) (2√2, 0)

(c) (0, 0)                            (d) (2, 2)

Answer (a)

6.) The maximum value of [x(x-1)+1]^{1/3}, 0 \leq x\leq 1 is

(a) (\frac{1}{3})^{1/3}            (b) 1/2

(c) 1                                                    (d) 0

Answer (c)

7.) A ladder, 5 meter long, dtanding on 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) 1/10 radyan/sec                       (b) 1/20 radian/sec

(c) 20 radian/sec                             (d) 10 radian/sec

Answer (b)

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

(a) 10 π                                 (b) 12 π

(c) 8 π                                   (d) 11 π

Answer (b)

9.) The total revenue in rupees received from the sale of x units of a product is given by R(x) =3x² + 36x + 5. The marginal revenue, when x = 15 is

(a) 116                                (b) 96

(c) 90                                  (d) 126

Answer (d)

10.) If x is real, the minimum value of x² – 8x + 17 is

(a) -1                                    (b) 0

(c) 1                                       (d) 2

Answer (c)

11.) The function f(x) = 2x³ – 15x² + 36x + 6 is increasing in the interval

(a) (-∞, 2)∪(3, ∞)               (b) (-∞, 2)

(c)  (-∞, 2]∪[3, ∞)               (d) [3, ∞)

Answer (c)      

 12.) In a sphere of radius r, a right circular cone of height h having maximum curved surface area is inscribed. The expression for the square of curved surface of cone is   

(a) 2πr²rh(2rh + h²)           (b) π²rh(2rh + h²)

(c)2π²r(2rh² – h²)                (d) 2π²r²(2rh – h²)

Answer (c)

13.) If f(x) = \frac{x}{\sin x} and g(x)= \frac{x}{\tanx}, where 0< x ≤ 1, then in the interval

(a) Both f(x) and g(x) are increasing functions

(b) Both f(x) and g(x) are decreasing functions

(c) f(x) is an increasing function.

(d) g(x) is an increasing function

Answer (c)

14.) The number of values of x where th function f(x)  = \cos x + \cos(\sqrt{2}x ) attains its maximum is

(a) 0                                (b) 1

(c) 2                                  (d) infinite

Answer (b)

15.) If xy = a² and S = b² x + c²y where a, b, and c are positive constants then the minimum value of S is

(a) abc                            (b) bc√a

(c) 2abc                           (d) None of these

Answer (c)

16.) The least value of the function f(x) = ax + \frac{b}{x}(a>0,b>0,x>0) is

(a) a/b                             (b) 2√ab

(c) 0                                   (d) None of these

Answer (b)

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

(a) 0                                (b) 4/3

(c) ±5                               (d) Maximum value does not exist.

Answer (b)

18.) Let f(x) =2\sin^3x -3\sin^2x +12\sin x+8,0\leq x \leq \frac{\pi}{2}. then f(x) is

(a) Decreasing in [0, π/2]

(b) Increasing in [0, π/2]

(c) increasing in [0, π/4] and decreasing in [π/4, π/2]

(d) None of these

Answer (b)

19.) Let f(x) be a function such that f'(a) ≠ 0. Then at x = a,f(x)

(a) Cannot have a maximum

(b) Cannot have a minimum

(c) Must have neither a maximum nor a minimum

(d) None of these


20.) The global minimum value of f(x) = x^4-x^2-2x+6

(a) 6                       (b) 8

(c) 4                         (d) Does not exist

Answer (c)

21.) The rate change in volume of sphere wrt r when radius is (1/2) is

(a) π                   (b) 2π

(c) 4π                  (d) None of these

22.) The maximum value of slope of the curve y = -x^3 + 3x^2+12x -5 is

(a) 15                            (b) 12

(c) 9                               (d) 0


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