The volume of a sphere is increasing in volume at the rate of 3 πcm³ / sec. The rate of change of its radius when radius is \(\frac { 1 }{ 2 } \) cm

(a)

3 cm/s

(b)

2 cm/s

(c)

1 cm/s

(d)

\(\cfrac { 1 }{ 2 } cm/s\)

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.

(a)

\(\frac{3}{25} \text { radians } / \mathrm{sec}\)

(b)

\(\frac{4}{25} \text { radians } / \mathrm{sec}\)

(c)

\(\frac{1}{5} \text { radians } / \mathrm{sec}\)

(d)

\(\frac{1}{3} \text { radians } / \mathrm{sec}\)

The position of a particle moving along a horizontal line of any time t is given by s(t) = 3t² -2t- 8. The time at which the particle is at rest is

(a)

t = 0

(b)

\(\\ \\ \\ t=\cfrac { 1 }{ 3 } \)

(c)

t =1

(d)

t = 3

A stone is thrown up vertically. The height it reaches at time t seconds is given by x = 80t -16t². The stone reaches the maximum height in time t seconds is given by

(a)

2

(b)

2.5

(c)

3

(d)

3.5

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

(a)

(4, 11)

(b)

(4, -11)

(c)

(-4, 11)

(d)

(-4,-11)

The abscissa of the point on the curve \(f\left( x \right) =\sqrt { 8-2x } \) at which the slope of the tangent is -0.25 ?

(a)

-8

(b)

-4

(c)

-2

(d)

0

The slope of the line normal to the curve f(x) = 2cos 4x at \(x=\cfrac { \pi }{ 12 } \) is

(a)

\(-4\sqrt { 3 } \)

(b)

-4

(c)

\(\cfrac { \sqrt { 3 } }{ 12 } \)

(d)

\(4\sqrt { 3 } \)

The tangent to the curve y² - xy + 9 = 0 is vertical when 

(a)

y = 0

(b)

\(\\ \\ y=\pm \sqrt { 3 } \)

(c)

\(y=\frac { 1 }{ 2 } \)

(d)

\(y=\pm 3\)

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

(a)

\({ tan }^{ -1 }\cfrac { 3 }{ 4 } \)

(b)

\({ tan }^{ -1 }\left( \cfrac { 4 }{ 3 } \right) \)

(c)

\(\cfrac { \pi }{ 2 } \)

(d)

\(\cfrac { \pi }{ 4 } \)

What is the value of the limit \(\lim _{x \rightarrow 0}\left(\cot x-\frac{1}{x}\right) \text { is }\) 

(a)

0

(b)

1

(c)

2

(d)

∞

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

(a)

\(\left[ \frac { 5\pi }{ 8 } ,\frac { 3\pi }{ 4 } \right] \)

(b)

\(\left[ \frac { \pi }{ 2 } ,\frac { 5\pi }{ 8 } \right] \)

(c)

\(\left[ \frac { \pi }{ 4 } ,\frac { \pi }{ 2 } \right] \)

(d)

\(\left[ 0,\frac { \pi }{ 4 } \right] \)

The number given by the Rolle's theorem for the functlon x³ - 3x², x ∈ [0, 3] is

(a)

1

(b)

\(\sqrt { 2 } \)

(c)

\(\frac { 3 }{ 2 } \)

(d)

2

The number given by the Mean value theorem for the function \(\frac { 1 }{ x } \), x ∈ [1, 9] is

(a)

2

(b)

2.5

(c)

3

(d)

3.5

The minimum value of the function |3 - x| + 9 is

(a)

0

(b)

3

(c)

6

(d)

9

The maximum slope of the tangent to the curve y = e^x sin x, x ∈ [0, 2π] is at

(a)

\(x=\frac { \pi }{ 4 } \)

(b)

\(x=\frac { \pi }{ 2 } \)

(c)

\(x=\pi \)

(d)

\(x=\frac { 3\pi }{ 2 } \)

The maximum value of the function \(x^{2} e^{-2 x}, x>0\) is

(a)

\(\frac { 1 }{ e } \)

(b)

\(\frac { 1 }{ 2e } \)

(c)

\(\frac { 1 }{ { e }^{ 2 } } \)

(d)

\(\frac { 4 }{ { e }^{ 4 } } \)

One of the closest points on the curve x² - y² = 4 to the point (6, 0) is

(a)

(2,0)

(b)

\(\left( \sqrt { 5 } ,1 \right) \)

(c)

\(\left( 3,\sqrt { 5 } \right) \)

(d)

\(\left( \sqrt { 13 } ,-\sqrt { 3 } \right) \)

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

(a)

100

(b)

\(25\sqrt { 7 } \)

(c)

28

(d)

\(24\sqrt { 14 } \)

The curve y= ax⁴ + bx² with ab > 0

(a)

has, no horizontal tangent

(b)

is concave up

(c)

is concave down

(d)

has no points of inflection

The point of inflection of the curve y = (x - 1)³ is

(a)

(0, 0)

(b)

(0, 1)

(c)

(1, 0)

(d)

(1, 1)