Exercise 7.10: Choose the correct or the most suitable answer

Choose the correct or the most suitable answer from the given four alternatives :

1. The volume of a sphere is increasing in volume at the rate of 3Ï€ cm3 / sec .

The rate of change of its radius when radius is 1/2 cm

(1) 3 cm/s

(2) 2 cm/s

(3) 1 cm/s

(4) 1/2 cm/s

2. A balloon rises straight up at 10 m/s. An observer is 40 m away from the spot where the balloon left the ground. The rate of change of the balloon’s angle of elevation in radian per second when the balloon is 30 metres above the ground.

(1) 3/25 radians/sec

(2) 4/25 radians/sec

(3) 1/5 radians/sec

(4) 1/3 radians/sec

3. The position of a particle moving along a horizontal line of any time t is given by s (t) = 3t2 âˆ’ 2t âˆ’ 8 . The time at which the particle is at rest is

(1) t = 0

(2) t = 1/3

(3) t = 1

(4) t = 3

4. A stone is thrown up vertically. The height it reaches at time t seconds is given by x = 80t âˆ’16t2 . The stone reaches the maximum height in time t seconds is given by

(1) 2

(2) 2.5

(3) 3

(4) 3.5

5. The point on the curve 6y = x3 + 2 at which y-coordinate changes 8 times as fast as x-coordinate is

(1) (4,11)

(2) (4, −11)

(3) (−4,11)

(4) (−4, −11)

6. The abscissa of the point on the curve f (x) = √(8−2x) at which the slope of the tangent is −0.25 ?

(1) −8

(2) −4

(3) −2

(4) 0

7. The slope of the line normal to the curve f (x) = 2 cos 4x at x = Ï€/12 is

(1) −4√3

(2) −4

(3) √3 /  12

(4) 4√3

8. The tangent to the curve y2 âˆ’ xy + 9 = 0 is vertical when

(1) y = 0

(2) y = ± √3

(3) y = 1/2

(4) y = ±3

9. Angle between y2 = x and x2 = y at the origin is

(1) tan−1 (3/4)

(2) tan-1 (4/3)

(3) π/2

(4) π/4

10. The value of the limit  or limx →0 ( cot x â€“ 1/x )is

(1) 0

(2) 1

(3) 2

(4) ∞

11. The function sin4 x + cos4 x is increasing in the interval

Answer: (3)

12. The number given by the Rolle’s theorem for the function x3 âˆ’ 3x2 , x âˆˆ[0, 3] is

(1) 1

(2) √2

(3) 3/2

(4) 2

13. The number given by the Mean value theorem for the function 1/x , x âˆˆ[1, 9] is

(1) 2

(2) 2.5

(3) 3

(4) 3.5

14. The minimum value of the function | 3 − x | + 9 is

(1) 0

(2) 3

(3) 6

(4) 9

15. The maximum slope of the tangent to the curve y = ex sin x , x âˆˆ[0, 2Ï€ ] is at

(1) x = π/4

(2) x = Ï€/2

(3) x = Ï€

(4) x = 3Ï€/2

16. The maximum value of the function x2 e−2x , x > 0 is

(1) 1/e

(2) 1/2e

(3) 1/e2

(4) 4/e4

17. One of the closest points on the curve x2 âˆ’ y2 = 4 to the point (6, 0) is

(1) (2, 0)

(2) (√5,1)

(3) (3, √5 )

(4) (√13, −√3)

18. The maximum value of the product of two positive numbers, when their sum of the squares is 200, is

(1) 100

(2) 25√7

(3) 28

(4) 24√14

19. The curve y = ax4 + bx2 with ab > 0

(1) has no horizontal tangent

(2) is concave up

(3) is concave down

(4) has no points of inflection

20. The point of inflection of the curve y = (x âˆ’1)3 is

(1) (0, 0)

(2) (0,1)

(3) (1, 0)

(4) (1,1)