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

(a)

0

(b)

∞

(c)

e

(d)

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

The angle made by any tangent to the curve y = x⁵ + 8x + 1 with the X-axis is a __________

(a)

obtuse

(b)

right angle

(c)

acute angle

(d)

no angle

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

The equation of the tangent to the curve x = t cost, y = t sin t at the origin is __________

(a)

x = 0

(b)

y = 0

(c)

x +y = 0

(d)

x + y = 7

In LMV theorem, we have f'(x₁) = \(\frac { f(b)-f(a) }{ b-a } \) then a < x₁ _________

(a)

<b

(b)

≤b

(c)

=b

(d)

≠b

If the slope of the curve 2y² = ax²+b at (1,-1) is - 1, then the values of a, b is __________

(a)

2, 0

(b)

0, 2

(c)

0, 0

(d)

2, 2

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

The function -3x+12 is ________ function on R.

(a)

decreasing

(b)

strictly decreasing

(c)

increasing

(d)

strictly increasing

The function f(x) = x⁹ + 3x⁷+ 64 is increasing on ________

(a)

R

(b)

(-∞, 0)

(c)

(0, ∞)

(d)

None of these

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

(a)

8

(b)

16

(c)

20

(d)

24

The curve y = e^x is ________

(a)

convex

(b)

concave

(c)

convex upwards

(d)

concave upwards

\(\underset { x\rightarrow 0 }{ lim } \frac { x }{ tanx } \) is _________

(a)

1

(b)

-1

(c)

0

(d)

∞

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

The distance moved by the particle in time t is given by \(x=t^{3}-12 t^{2}+6 t+8\). At the instant when its acceleration is zero, the velocity is ____________

(a)

42

(b)

-42

(c)

48

(d)

-48

lf the rate of change of volume of sphere is equal to the rate of change of its radius, then its radius is equal to ____________

(a)

1 unit

(b)

\(\sqrt{2 \pi} \text { units }\)

(c)

\(\frac{1}{\sqrt{2 \pi}} \text { units }\)

(d)

\(\frac{1}{2 \sqrt{\pi}} \text { units }\)