\[{\text{The}}\,{\text{value}}\,{\text{of}}\,c\,{\text{in}}\,{\text{Rolle's}}\,{\text{Theorem}}\,{\text{for}}\,{\text{the}}\,{\text{function}}\,f(x) = {e^x}\sin x,\,\,x \in [0,\pi ]\,{\text{is}}\_\_\_\_\_\_\_\_\_\_.\]

• A) \[\frac{{3\pi }}{2}\]
• B) \[\frac{\pi }{6}\]
• C) \[\frac{\pi }{4}\]
• D) \[\frac{{3\pi }}{4}\]

$${\text{For}}\,{\text{Rolle's}}\,{\text{Theorem,}}\,f\,{\text{is}}\,{\text{differentiable}}\,{\text{on}}\,{\text{the}}\,{\text{interval __________}}{\text{.}}$$

• A) $$(a,b)$$
• B) $$[a,b)$$
• C) $$(a,b]$$
• D) $$[a,b]$$

\[{\text{The integral }}\int {{{\left( {{x^2} + 1} \right)}^{\frac{5}{2}}}.\,x\,dx\,} {\text{will be equal to ?}}\]

• A) \[{\text{None of these}}\]
• B) \[\frac{{{{\left( {{x^2} + 1} \right)}^{\frac{7}{2}}}}}{7} + c\]
• C) \[\frac{{2{{\left( {{x^2} + 1} \right)}^{\frac{7}{2}}}}}{7} + c\]
• D) \[{\left( {{x^2} + 1} \right)^{\frac{7}{2}}} + c\]

If the function is continuous on the closed interval [a,b], then the function has …………

• A) Only minimum value on the [a,b]
• B) Both maximum and minimum values on the [a,b]
• C) Only minimum value on the [a,b]
• D) Only maximum value on the [a,b]

If $f''(x) < 0\,\forall \,x \in I$, then the graph of $f$ is ___________ on $I$.

• A) None of these
• B) concave downward
• C) constant
• D) concave upward

If f(x) = Cos(x) + x, then which of the following is NOT true about it.

• A) Its anti – derivative is Sin(x) + x^2/2 + 10.
• B) Its anti – derivative is -Sin(x) + x^2/2 + 4.
• C) Its anti – derivative is Sin(x) + x^2/2 + 4.
• D) Its anti – derivative is Sin(x) + x^2/2 + 6.

Integration of (Cosx/Sinx).Cosecx with respect to x…………

• A) Cosecx
• B) Secx
• C) -Cosecx
• D) Cotx

Which geometrical figure is used for approximating the area under the curve?

• A) Pentagon
• B) Right angled triangle
• C) None of these
• D) Circle

Sigma notation is used to write lengthy……….in compact form.

• A) difference
• B) sums
• C) products
• D) quotient

$${\text{The}}\,{\text{value}}\,{\text{of}}\,c\,{\text{in}}\,{\text{Rolle's}}\,{\text{theorem}}\,{\text{for}}\,{\text{the}}\,{\text{function}}\,f(x) = {x^3} - 3x\,{\text{in}}\,{\text{the}}\,{\text{interval}}\,\left[ {0,\sqrt 3 } \right]{\text{.}}$$

• A) $$\frac{1}{3}$$
• B) $$\frac{2}{3}$$
• C) $$1$$
• D) $$- 1$$

In the indefinite integral of x(y^2) w.r.t ‘y’ , the term ….. serve to identify the independent variable in the function.

• A) y^2
• B) x
• C) y
• D) dy

The indefinite integral of ‘sec(x)tan(x)’ is…………….

• A) Sinx+c
• B) Cotx +c
• C) Secx+c
• D) Tanx+c

Summation of 9t where ‘t’ goes from 1 to 30” is same as “summation of 9k where k goes from 1 to 30”.

• A) True
• B) False
• C)
• D)

If $f$ has a local maximum or minimum at c, and if $f'(c)$ exists, then_________.

• A) $f'(c) < 0$
• B) None of these
• C) $f'(c) = 0$
• D) $f'(c) > 0$

1+2+3+…+200 equals _____.

• A) 20012
• B) 21220
• C) 21021
• D) 20100

$${\text{The integral }}\int {\frac{{6x}}{{{{(3{x^2} + 1)}^2}}}\,dx\,} {\text{will be equal to ?}}$$

• A) $$- \frac{1}{{{{(3{x^2} + 1)}^3}}} + c$$
• B) $$- \frac{1}{{6x + 1}} + c$$
• C) $$\frac{1}{{3{x^2} + 1}} + c$$
• D) $$- \frac{1}{{3{x^2} + 1}} + c$$

The vertical asymptotes of a function occur at the points where the denominator of the function becomes

• A) -1
• B) 1
• C) 0
• D) Undefined

$${\text{The integral }}\int {{{\sec }^2}(5{x^2})\,.10x\,dx\,} {\text{will be equal to ?}}$$

• A) $$\tan (5x) + c$$
• B) $$\tan (5{x^2}) + c$$
• C) $${\text{se}}{{\text{c}}^2}(10x) + c$$
• D) $$\sec (5{x^2}).\tan (5{x^2}) + c$$

If f (x) = |x| -2 is defined on the interval
[-2, 2], then which of the following is true about it.

• A) It is discontinuous on the interval [-2, 2].
• B) There is a point in the interval (-2, 2) where f(x) has a horizontal tangent
• C) There is no such point in the interval (-2, 2) where f(x) has a horizontal tangent.
• D) None of these.

Critical point of function f(x) =4x^2-4x+1 on closed interval [0,1] is x = ………

• A) 1,2
• B) 0
• C) 1/2
• D) 2