Integral Calculus & Area · NDA previous year questions

What is the value of ${\lim }_{x\to \infty }{\left(\frac{x+6}{x+1}\right)}^{x+4}$?

1. A.$e$
2. B.$e^2$
3. C.$e^4$
4. D.$e^5$

What is the area enclosed between the curves $y^2=12x$ and the lines $x=0$ and $y=6$?

1. A.2 square units
2. B.4 square units
3. C.6 square units
4. D.8 square units

What is $\int \frac{1}{1+e^x}dx$ equal to? where c is a constant of integration.

1. A.$x-\ln x+c$
2. B.$x-\ln (\tan x)+c$
3. C.$x-\ln (1+e^x)+c$
4. D.$\ln (1+e^x)+c$

If $\int x^2\ln xdx=\frac{x^3}{m}\ln x+\frac{x^3}{n}+c$, then what are the values of m and n respectively? where c is a constant of integration.

1. A.1/3, -1/9
2. B.3, -9
3. C.3, 9
4. D.3, 3

What is the principal value of ${\mathrm{cosec}}^{-1}(-\sqrt{2})$?

1. A.$\frac{\pi }{4}$
2. B.$\frac{\pi }{2}$
3. C.$-\frac{\pi }{4}$
4. D.0

If $f:R\to R,g:R\to R$ and $g(x)=x+3$ and $(fog)(x)=(x+3{)}^2$, then what is the value of $f(-3)$?

1. A.-9
2. B.0
3. C.9
4. D.3

What is the value of $k$ if the area bounded by the curve $y=\sin kx$, $y=0$, $x=\pi /k$, $x=\pi /(3k)$ is 3 square units ?

1. A.$1/2$
2. B.1
3. C.$3/2$
4. D.2

What is the area bounded by the curve $y=4x-x^2-3$ and the $x$-axis ?

1. A.$2/3\text{ square units}$
2. B.$4/3\text{ square units}$
3. C.$5/3\text{ square units}$
4. D.$4/5\text{ square units}$