If [-8,8] is subdivided into ‘16’ equally spaced subintervals, then the LEFT end point of 13th sub-interval will be--------.

• A) 4
• B) 5
• C) 2
• D) 3

For the area under the curve f(x) = 2x from x = 0 to x = 8 with mid points approximations for n = 2, what will be the values of xk* ?

• A) 3 and 7
• B) 2 and 6
• C) 4 and 6
• D) 1 and 5

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

• A) $$\frac{{\ln \left( {1 - {x^2}} \right)}}{2} + c$$
• B) $$2\ln (1 + {x^2}) + c$$
• C) $${\text{ln(1 + }}{{\text{x}}^2}) + c$$
• D) $$\frac{{\ln \left( {1 + {x^2}} \right)}}{2} + c$$

If $f$ is______________ on a closed interval $[a,b]$, then $f$, attains an absolute maximum value $f(c)$ and an absoulte minimum value $f(d)$ at some numbers $c$ and $d$ in $[a,b]$.

• A) differentiable
• B) constant
• C) None of these
• D) continuous

$\text{A critical point for a function }f,\text{ where }{f}'(x)=0\text{ is called as}\,\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\,\text{ point}\text{.}$

• A) Stationary
• B) Tangent
• C) Mid
• D) Corner

Integral of x^(-10) is

NOTE: x^n means ‘x’ to the power ‘n’

• A) - 1/(9x^9)+C
• B) -10x^(-11)+C
• C) 1/(9x^9)+C
• D) None of these

\[{\text{The integral }}\int {\frac{{4x}}{{{{(2{x^2} + 1)}^2}}}\,dx\,} {\text{will be equal to ?}}\]

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

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

• 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)