$${\text{The integral }}\int {\cot (2x)\,dx\,} {\text{will be equal to ?}}$$

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

If ‘n’ goes from 1 to 3 and the summation of ‘na’ = 6a, then the value of ‘a’ is ----------

• A) -6
• B) 6
• C) Undetermined
• D) 1

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

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

Given f(x)= 1/(x-1) on the interval (0,3) , then ‘f ’ has no relative maxima or relative minima due to………

• A) None of these
• B) Discontinuity in the interval
• C) Rational function
• D) Open interval

What is the estimated area under f(x) = x2+2 from x = 1 to x = 5 with mid points for n = 2

• A) 48
• B) 76
• C) 26
• D) 38

\[{\text{What}}\,{\text{does}}\,{\text{the}}\,{\text{indefinite}}\,{\text{integral }}\int_{}^{} {f(x)} dx\,{\text{represent?}}\]

• A) \[{\text{Area}}\,{\text{under}}\,{\text{the}}\,{\text{curve}}\]
• B) \[{\text{Families}}\,{\text{of}}\,{\text{antiderivative}}\,{\text{of}}\,{\text{the}}\,{\text{function }}f(x)\]
• C) \[{\text{None}}\,{\text{of}}\,{\text{these}}\]
• D) \[{\text{Curvature}}\,{\text{of}}\,{\text{the}}\,{\text{curve}}\]

Summation of ‘kx’ where k goes from 1 to 5 equals

• A) 15k
• B) 55
• C) None of these
• D) 15x

What are critical points of the function f(x) =x-1?

• A) x=0
• B) x=1
• C) None of these
• D) No critical point

According to the Newton’s method, the equation Cosx=x (where x is in radian) has a solution between ……………

• A) 0 and -1
• B) Solution not exist
• C) 0 and 1
• D) 1 and -1

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

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

The approximate solution is possible to generate using Newton’s Method if ___________.

• A) the tangent line(at approximated points) must crosses the x - axis.
• B) the slope of tangent line(at any approximated point) is zero.
• C) the tangent line(at any approximated point) is parallel to x- axis.
• D) None of these.

Let y = f(x) be a discontinuous function on a finite closed interval, then which of the following is true about it.

• A) It must have absolute extreme values.
• B) It may or may not have absolute extreme values.
• C) None of these.
• D) It has only absolute minimum value.

13+23+33+…+153 equals _____.

• A) 15(16)/22
• B) [15(16)]2/2
• C) [15(16)/2]2
• D) 15(16)2/2

\[\text{The vertical asymptote of the function }f(x)=\frac{3{{x}^{2}}+1}{2-x}\text{ is}\]

• A) $\infty $
• B) -2
• C) 2
• D) 0

What will be the value of summation of k2 where k goes from 8 to 8?

• A) 16
• B) 8
• C) 1
• D) 64

If f(x) = x^4, then which of the following is Not true about it.

• A) Its anti – derivative is x^5/5.
• B) Its anti – derivative is x^5/5 + 10.
• C) Its anti – derivative is x^5/5 + 2.
• D) Its anti – derivative is x^5 + 5.

If f (x) = x^3 is defined on the interval [1, 3], then which of the following is true about it.

• A) Its absolute minimum value exists at the critical point.
• B) Its relative minimum value does not exist at the critical point
• C) None of these.
• D) Its relative minimum value exists at the critical point.

$${\text{The integral }}\int {\sqrt {4x - 3} \,dx\,} {\text{will be equal to ?}}$$

• A) $$\frac{{{{(4x + 3)}^{\frac{3}{2}}}}}{6} + c$$
• B) $${\text{None of these}}$$
• C) $$\frac{{{{(4x - 3)}^{\frac{3}{2}}}}}{6} + c$$
• D) $$\frac{{{{(4x - 3)}^{\frac{3}{2}}}}}{3} + c$$

What is the estimated area under f(x) = 9 - x2 from x = 0 to x = 4 with mid points for n = 1?

• A) 20
• B) 24
• C) 28
• D) 21

Subdivide the interval [0, 1] into 2 equal parts, and then the width of each sub-interval will have length ………

• A) -1/2
• B) 1/2
• C) 0
• D) 1