If the radius of a sphere is doubled, how does the volume change?

• A) It becomes eight times of the original volume.
• B) It remains the same of the original volume.
• C) It becomes four times of the original volume.
• D) It becomes half of the original volume.

Which of the following is the indefinite integral of f(x)=3/(x^2)

• A) None of these.
• B) (1/4) x^ (4/3) +c
• C) (-3/x) +c
• D) (3/x)

If f(x)= x and g(x)=2x are integrable functions over the interval [0, a] for all xϵ[0, a], then which of the following expressions is true for f and g?

• A) $$\int\limits_0^a {f(x)dx} > \int\limits_0^a {g(x)dx}$$
• B) $$\int\limits_0^a {f(x)dx} < \int\limits_0^a {g(x)dx}$$
• C) $$\int\limits_0^a {f(x)dx} \geqslant \int\limits_0^a {g(x)dx}$$
• D) $$\int\limits_0^a {f(x)dx} \leqslant \int\limits_0^a {g(x)dx}$$

$\int_a^b {f(x)} dx = 0$ if__________.

• A) $a < b$
• B) $a > b$
• C) $a = b$
• D) None of these

If the definite integral of f(x)=cos x over the interval [-a,0] is equal to ‘-1’ then what will be the value of the definite integral of f(x)= (cos x) -1 over the same interval?

• A) -1-a
• B) 1+a
• C) -2
• D) -1+a

The Volume of a cylindrical shell is _____.

• A) $2\pi .(\frac{{{r_2} + {r_1}}}{2}).h.({r_2} + {r_1})$
• B) $2\pi .(\frac{{{r_2} + {r_1}}}{2}).h.({r_2} - {r_1})$
• C) None of the above
• D) $\pi .(\frac{{{r_2} + {r_1}}}{2}).h.({r_2} - {r_1})$

What is the primary difference between a 1d object and a 2d object?

• A) A 1d object can curve, while a 2d object is always straight.
• B) A 1d object has width, while a 2d object has height.
• C) A 1d object has length, while a 2d object has area.
• D) A 1d object is flat, while a 2d object is three-dimensional

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

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

\[{\text{The integral }}\int {\frac{{10x}}{{{{(5{x^2} + 1)}^2}}}\,dx\,} {\text{will be equal to ?}}\]

• A) \[ - \frac{1}{{{{(5{x^2} + 1)}^3}}} + c\]
• B) \[\frac{1}{{5{x^2} + 1}} + c\]
• C) \[ - \frac{1}{{5{x^2} + 1}} + c\]
• D) \[ - \frac{1}{{5x + 1}} + c\]

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= 3
• B) x= -9
• C) x=0
• D) x=-3

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

• A) 4
• B) 5
• C) 2
• D) 3

Integral of x^101 is

NOTE: x^n means ‘x’ to the power ‘n’

• A) (1/102)x^102+C
• B) None of these
• C) 102x^102+C
• D) 101x^100 +C

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

The estimated area under f(x) = x from
x = 0 to x = 3 with right end points for
n = 3 is ________.

• A) None of these
• B) 7.
• C) 6.
• D) 5.

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

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

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

• A) 4Sinx
• B) - 4Sinx
• C)
• D)

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

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

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

NOTE: x^n means ‘x’ to the power ‘n’

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

If $f'$ changes from negative to positive at c, then $f$ has a _______________ at c.

• A) local maximum
• B) None of these
• C) constant
• D) local minimum

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) 3
• D) 2