If the closed interval [-2,2] is divided into ‘50’ equally spaced sub-intervals then the width of each sub-interval is ------------

• A) 2/25
• B) 1/25
• C) 4/25
• D) -4/25

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

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

An Inflection Point is a point where a curve changes from

• A) c. Both a and b
• B) d. None of these
• C) b. Concave down to concave up
• D) a. Concave up to concave down

$$\text{If a function }f\text{ has a relative extrema at }{{x}_{0}}\text{, then}$$

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

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

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

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 + 6.
• B) Its anti – derivative is Sin(x) + x^2/2 + 4.
• C) Its anti – derivative is Sin(x) + x^2/2 + 10.
• D) Its anti – derivative is -Sin(x) + x^2/2 + 4.

$$\int {\tan x} dx = \_\_\_\_\_\_\_\_\_\_\_\_\_\_.$$

• A) $${\sec ^2}x + C$$
• B) $$\ln \left| {\sin x} \right| + C$$
• C) $$\ln |\cos x| + C$$
• D) $$\ln \left| {\sec x} \right| + C$$

Newton’s Method fails to find the approximate solution of an equation if _____________.

• A) the slope of the tangent line(at any approximated point) is non-zero
• B) the tangent line(at any approximated point) is parallel to x-axis.
• C) None of these
• D) the tangent line (at any approximated point) is not parallel to x-axis.

Sum of n-terms of a series whose nth term is ‘n’ = ---

• A) n(n-1)/2
• B) (n+1)/2
• C) n(n+1)/2
• D) n(n+1)

The symbol $\int {}$ was introduced by___________and is called integral sign.

• A) Newton
• B) Leibnitz
• C) Cauchy
• D) Lagrange

If x = (1^2)+(2^2)+(3^2)+(4^2) + . . . + (30^2), then x = ________.

• A) 9455.
• B) None of these.
• C) 900.
• D) 465.

If $f''(x) < 0\,\forall \,x \in I$, then the graph of $f$ is ___________ on $I$.

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

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

The indefinite integral of ‘sec(x)tan(x)’ is…………….

• A) Cotx +c
• B) Sinx+c
• C) Tanx+c
• D) Secx+c

In the indefinite integral of x(y^2) w.r.t ‘y’ , the independent variable is ……..

• A) xy
• B) y
• C) none of these
• D) x

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

• A) 2
• B) 4
• C) 3
• D) 5

What is the estimated area under f(x) = 9 - x2 from x = 0 to x = 4 with mid points for n = 2?

• A) 25
• B) 16
• C) 21
• D) 12

If x=-3 and x=3 are the two critical points of the function: f(x)=81x-3(x^3) then by using the 2nd derivative test, we can conclude that f(x) is relatively maximum at-----

• A) x= -9
• B) x= 3
• C) x=0
• D) x=-3

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

• A) $${\text{se}}{{\text{c}}^2}(2x) + c$$
• B) $$\tan (2{x^2}) + c$$
• C) $$\sec (2{x^2}).\tan (2{x^2}) + c$$
• D) $$\tan (2x) + c$$

Sum of cubes of n-terms of a series whose nth term is ‘n’ = ---

• A) Square of n(n+1)(2n+1)/6
• B) Square of n(n+1)/2
• C) Square of n(n+1)/6
• D) Square of (n+1)/2