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 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)

\(y=\pm 3\)

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

(a)

0

(b)

3

(c)

6

(d)

9

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

(a)

(0, 0)

(b)

(0, 1)

(c)

(1, 0)

(d)

(1, 1)

The point on the curve y = x² is the tangent parallel to X-axis is __________

(a)

(1, 1)

(b)

(2, 2)

(c)

(4, 4)

(d)

(0, 0)

The value of \(\underset { x\rightarrow \infty }{ lim } { e }^{ -x }\) is __________

(a)

0

(b)

∞

(c)

e

(d)

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

The critical points of the function f(x) = \((x-2)^{ \frac { 2 }{ 3 } }(2x+1)\) are __________

(a)

-1, 2

(b)

1, \(\frac { 1 }{ 2 } \)

(c)

1, 2

(d)

none

If the curves y = 2e^x and y = ae^-x intersect orthogonally, then a = _________

(a)

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

(b)

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

(c)

2

(d)

2e²

If x + y = 8, then the maximum value of xy is _________

(a)

8

(b)

16

(c)

20

(d)

24

The statement "If f has a local extremum at c and if f'(c) exists then f'(c) = 0" is ________

(a)

the extreme value theorem

(b)

Fermat's theorem

(c)

Law of mean

(d)

Rolle's theorem