‘x.ln(x/e)’ is the anti-derivative of ‘lnx’ if and only if the derivative of ‘x.ln(x/e)’ is euqal to --------

• A) x
• B) xln(x/e)
• C) ln(x/e)
• D) lnx

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

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

If f'(r) =0 at some approximation “r” then we cannot proceed on Newton’s method.

• A) False
• B) True
• C)
• D)

Integration of 4Cosx with respect to x is………..

• A) 4Sinx
• B) - 4Sinx
• C)
• D)

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

• A) None of these.
• B) Its anti – derivative is Cos(x) + x^2/2 + 10.
• C) Its anti – derivative is –Cos(x) + x^2/2 + 10.
• D) Its anti – derivative is Cos(x) + 1.

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

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

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

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

If Newton’s Method succeeded to get the approximate solution of an equation, then which of the following is NOT true about it.

• A) The slope of the tangent line (at any approximated point) must be non zero.
• B) None of these.
• C) The sequence of approximated points not convergent to the exact solution
• D) The tangent line (at any approximated point) is not parallel to x-axis.

What is the summation of ‘kx’ where k goes from 1 to 30?

• A) 419x
• B) 465x
• C) 523x
• D) 414x

To get better approximation to actual area under a continuous curve over a closed interval, we have to increase ………

• A) Total area
• B) Size of the interval
• C) Number of subintervals
• D) Width of the subintervals

The critical value of the function y=100x – x^2 is

NOTE: x^n means ‘x’ to the power ‘n’

• A) x=0
• B) x=50
• C) None of these
• D) x=25

Which of the following is the absolute minima of the function: f(x)=-x in the interval [-1,1]?

• A) -1
• B) 0.5
• C) 0
• D) 1

Which of the following will be left end points if the interval [-2,2] is divided into 4 equal subintervals.

• A) -2,-1,1,2
• B) -1,0,1,2
• C) -2,-1,0,1
• D) None of these

Integration of 3x^2 with respect to x is…………

• A) x^4
• B) x
• C) x^3
• D) x^2

Critical point of f(x)=-Sinx in the interval [0,pi] is-----

• A) pi/4
• B) pi
• C) 0
• D) pi/2

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

• A) False
• B) True
• C)
• D)

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

• A) 5
• B) 4
• C) 2
• D) 3

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

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

In [0,pi],if the area under the curve f(x)=Sinx is divided into ‘3’ equally spaced rectangles of width ‘pi/3’ then by choosing middle point for height, the area of middle rectangle is ---------.

• A) pi/4
• B) pi/2
• C) pi/3
• D) pi/6

For the area under the curve f(x) = 2x from x = 0 to x = 12 with left points approximations for n = 3, what will be the values of xk* ?

• A) 0, 5 and 10
• B) 0, 6 and 12
• C) 0, 3 and 6
• D) 0, 4 and 8