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

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

To find out critical values of a function, we put it equal to zero and solve for x.

• A) True
• B) False
• C)
• D)

summation of (6) ;where ( j varies from 1 to 8) indicates to add ‘6’ to itself ……….. times.

• A) 8
• B) 9
• C) 10
• D) 11

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

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

\[{\text{For}}\,{\text{Rolle's}}\,{\text{Theorem,}}\,f\,{\text{is}}\,{\text{continuous}}\,{\text{on}}\,{\text{the}}\,{\text{interval __________}}{\text{.}}\]

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

What is the estimated area under f(x) = 10-x2 from x = 0 to x = 3 with left end points for n = 3?

• A) 15
• B) 21
• C) 30
• D) 25

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

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

Integration of “cosecx.cotx” with respect to x is -----

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

If x = 5 + 6 + . . . + 40, then x = ________.

• A) None of these.
• B) 820.
• C) 810.
• D) 850.

\[\text{If }{f}'(x)={{x}^{2}}-x\text{. Then the critical points of the function }f\text{ are}\]

• A) 1, -1
• B) 1, 2
• C) 0, 1
• D) 0, 2

Sum of n-terms of a series whose nth term is ‘n’ = 1/n+1.then what is the sum of the first two terms is -----

• A) 5/6
• B) 6
• C) 6/4
• D) 6/5

A function $f$ has an/a___________ at $c$ if $f(c) \leqslant f(x)\,\forall \,x \in \,D$, where $D$ is the domain of $f$.

• A) absolute minimum/global minimum
• B) local minimum
• C) global minimum
• D) absolute minimum

Why the equation: x^2 + 8 = 0 does not have approximate solution while using Newton's method?

• A) x^2 will always be nonnegative
• B) x^2 will always be negative
• C)
• D)

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

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

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

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

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

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

Which of the following is the sum of (2k-1) where k goes from 0 to 2?

• A) 4
• B) -2
• C) -1
• D) 3

If we subdivide the interval [2,4] into n equal parts, then what will be the length ∆x of each part?

• A) 8/n
• B) 1/n
• C) 2/n
• D) 6/n

\[\begin{align} & \text{A function }f\text{ is said to have a relative minimum at }{{x}_{0}}\text{, if}\,\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\,\text{for all }x\text{ } \\ & \text{in some open interval containing }{{x}_{0}}. \\ \end{align}\]

• A) \[f\left( {{x}_{0}} \right)\le \text{ }f\left( x \right)\]
• B) \[f\left( {{x}_{0}} \right)=f\left( x \right)\]
• C) \[f\left( {{x}_{0}} \right)\ge \text{ }f\left( x \right)\]
• D) \[f\left( {{x}_{0}} \right)>\text{ }f\left( x \right)\]

The estimated area under f(x) = 12 / x from
x = 1 to x = 3 with right end points for
n = 2 is ________.

• A) 20.
• B) 10
• C) None of these.
• D) 18.