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

• 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) $$- \frac{1}{{2x + 1}} + c$$

If f(x) = x^5 + x, then which of the following is true about it.

• A) Its anti – derivative is x^5/5 + 1.
• B) Its anti – derivative is x^6/6 + x^2/2 + 6.
• C) Its anti – derivative is 5x^4 + 1.
• D) None of these.

Which of the following is the sum of 5^ (k+1) where k goes from 1 to 3?

• A) 525
• B) 675
• C) 775
• D) 875

What will be the value of summation of k3 where k goes from 3 to 3?

• A) 9
• B) 1
• C) 27
• D) 3

Newton’s method is not applicable when ‘f’ is a …………

• A) Exponential function.
• B) Trignometric function
• C) Constant function
• D) Polynomial

1+2+3+…+200 equals _____.

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

The derivative of the area under the continuous function f(x)= 2+3Sinx in the interval[-pi,pi] is---------

• A) 2-3Cosx
• B) 2-3Sinx
• C) 2+3Cosx
• D) 2+3Sinx

If f(x)=Tan(x) then mean value theorem can be applied to it on the interval (0,2pi)

• A) False
• B) True
• C)
• D)

By using Newton method, which of the following is the poorest initial approximate solution of equation:x+Cosx=0?

• A) x=-pi/3
• B) x=0
• C) x=pi/2
• D) x=-pi/4

For the given function f(x)= 2(x^2)+1 in the interval [-1,2] , which condition of Rolle’s theorem is not satisfied.

• A) f (-1) = f (2)
• B) The function is continuous in the interval
• C) The function is differentiable in the interval
• D) None of these.

How many subintervals of length ‘2’ will be formed for the interval [4,16] ?

• A) 5
• B) 7
• C) 4
• D) 6

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

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

If a function f has a relative extrema at a point c, then c is a critical point for f .

• A) True
• B) False
• C)
• D)

What is the estimated area under f(x) = x from x = 0 to x = 3 with mid points for n = 3?

• A) 5.0
• B) 3.5
• C) 6.5
• D) 4.5

$\text{A critical point for a function }f\text{ is any value of }x\text{ in the domain of }f\text{ at which}$

• A) $${f}'\left( x \right)=0\,\text{or }f\text{ is not differentiable}$$
• B) $$f\text{ is differentiable }$$
• C) $${f}'\left( x \right)\le 0$$
• D) $${f}'\left( x \right)>0$$

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

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

What will be the sigma notation for 43+63+83+103?

• A) summation of (2k)3 where (k varies from 2 to 5)
• B) summation of (2k)3 where (k varies from 1 to 4)
• C) summation of (2k+1)3 where (k varies from 2 to 5)
• D) summation of (2k3) where (k varies from 2 to 5)

In approximation to an area Rn (where n is subscript) when limit is taken as n goes to infinity, approximation becomes actual area.

• A) True
• B) False
• C)
• D)

If $f''(x) > 0\,\forall \,x \in I$, then the graph of $f$ is___________ on $I$.

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

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

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