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

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

The polynomial function f(x)=6x^2-30x+36 has the critical point over the real line is ………

• A) 2/5
• B) 5/2
• C) 2
• D) 5

Let f(x)=x^3 and a=-3 ,b=3 and n = no. of partition of [-3,3]= 4
Let the widths of first, second, third and fourth intervals are 2, 1, 1, 2 respectively. Then “Mesh size” of this partition of [-3,3] is-------

• A) -1
• B) 2
• C) 1
• D) 0

If x =(1+2+3+...+10)+{(1^3)+(2^3)+(3^3)+...+ (10^3)}, then x = _______.

• A) 55.
• B) 110.
• C) 3080.
• D) 3025.

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

If x = (4^2) + (5^2) + (6^ 2) + . . . + (30^2), then x = ________.

• A) 465.
• B) 9455.
• C) 9441.
• D) 400.

What is the estimated area under f(x) = 12 / x from x = 1 to x = 3 with mid points for n = 2?

• A) 12.8
• B) 18.0
• C) 13.1
• D) 10.5

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

• A) -1
• B) 3
• C) 4
• D) -2

The function f(x)= 2x^3-15x^2+36x have the critical points ……………on the interval [1,5]

• A) 2,3
• B) 3,4
• C) 1,5
• D) 2,4

The estimated area under f(x) = x^2 from
x = 1 to x = 3 with left end points for
n = 2 is ________

• A) 13.
• B) 6.
• C) 5.
• D) None of these.

If the interval [3,7] is divided into ‘4’ equal subintervals ,then right endpoint of each subinterval will be………

• A) 3,4,5,6
• B) 5,6,7,8
• C) 4,5,6,7
• D) 3,5,6,8

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

• A) 3, 7 and 11
• B) 2, 5 and 8
• C) 2, 6 and10
• D) 1, 5 and 9

The estimated area under f(x) = x^2 + 2 from
x = 1 to x = 5 with left end points for
n = 2 is ________.

• A) 76.
• B) None of these.
• C) 25.
• D) 28.

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

• A) 1
• B) 16
• C) 64
• D) 8

What are critical points of the function f(x) =x^2-1?
NOTE: x^n means ‘x’ to the power ‘n’

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

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

• A) 18
• B) 16
• C) 8
• D) 10

The dimensions of a rectangle are given to be 8ft by 12ft. The perimeter of rectangle will be…………..

• A) 40 feet
• B) 60 feet
• C) 30 feet
• D) 50feet

Subdivide the interval [3, 5] into n equal parts, and then the width of each subinterval is ----

• A) n
• B) -2/n
• C) 1/n
• D) 2/n

summation of (ai) (i varies from 1 to n) , summation of (aj) (j varies from 1 to n),summation of (ak) (k varies from 1 to n)
All these three represents same summation.

• A) False
• B) True
• C)
• D)

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

• A) $$\cos ec(x) + c$$
• B) $${\text{None of these}}$$
• C) $$\sec (x) + c$$
• D) $$- \ln \left| {\cos (x)} \right| + c$$