Integral Calculus & Area · NDA previous year questions

The area bounded by the curve $x=f(y)$, the y-axis and the two lines $y=a$ and $y=b$ is equal to :

1. A.${\int }_a^{b}ydx$
2. B.${\int }_a^b{y}^{2}dx$
3. C.${\int }_a^{b}xdy$
4. D.None of the above

If $y=\frac{x+1}{x-1}$, then what is $\frac{dy}{dx}$ equal to ?

1. A.$\frac{-2}{x-1}$
2. B.$\frac{-2}{(x-1{)}^2}$
3. C.$\frac{2}{(x-1{)}^2}$
4. D.$\frac{2}{x-1}$

Consider the following : 1. $\int \ln 10dx=x+c$ 2. $\int 10^{x}dx=10^x+c$ where c is the constant of integration. Which of the above is/are correct ?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

What is the area enclosed by the equation $x^2+y^2=2$?

1. A.$4\pi$ square units
2. B.$2\pi$ square units
3. C.$4{\pi }^2$ square units
4. D.4 square units

What is $\int \frac{dx}{{\sin }^{2}x{\cos }^{2}x}$ equal to? where $c$ is the constant of integration.

1. A.$\tan x+\cot x+c$
2. B.$\tan x-\cot x+c$
3. C.$(\tan x+\cot x{)}^2+c$
4. D.$(\tan x-\cot x{)}^2+c$

The smallest positive integral value of $n$ for which ${\left(\frac{1-i}{1+i}\right)}^n$ is purely imaginary with positive imaginary part is

1. A.1
2. B.3
3. C.4
4. D.5

What is the area bounded by the curve $\sqrt{x}+\sqrt{y}=\sqrt{a}(x,y\ge 0)$ and the coordinate axes?

1. A.$\frac{5a^2}{6}$
2. B.$\frac{a^2}{3}$
3. C.$\frac{a^2}{2}$
4. D.$\frac{a^2}{6}$

What is $\int {\sin }^{-1}(\cos x)dx$ equal to? where $k$ is a constant of integration.

1. A.$\frac{x\pi }{2}-\frac{x^2}{2}+k$
2. B.$\frac{\pi }{2}+\frac{x^2}{2}+k$
3. C.$-\frac{x\pi }{2}-\frac{x^2}{2}+kx$
4. D.$\frac{\pi }{2}-\frac{x^2}{2}+k$

What is $\int \frac{\sin \sqrt{x}}{\sqrt{x}}dx$ equal to? where $c$ is a constant of integration.

1. A.$\frac{\cos \sqrt{x}}{2}+c$
2. B.$2\cos \sqrt{x}+c$
3. C.$\frac{-\cos \sqrt{x}}{2}+c$
4. D.$-2\cos \sqrt{x}+c$

What is the area bounded by the curves $y=e^x$, $y=e^{-x}$ and the straight line $x=1$?

1. A.$\left(e+\frac{1}{e}\right)$ square units
2. B.$\left(e-\frac{1}{e}\right)$ square units
3. C.$\left(e+\frac{1}{e}-2\right)$ square units
4. D.$\left(e-\frac{1}{e}-2\right)$ square units

What is $\int e^x\left(\sqrt{x}+\frac{1}{2\sqrt{x}}\right)dx$ equal to ? where $c$ is a constant of integration.

1. A.$x{e}^x+c$
2. B.$e^x(\sqrt{x})+c$
3. C.$2e^x(\sqrt{x})+c$
4. D.$2x{e}^x+c$

What is the area of the region bounded by the curve $f(x)=1-\frac{x^2}{4},x\in [-2,2]$ and the $x$-axis?

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

What is ${\lim }_{x\to 0}\frac{a^x-b^x}{x}$ equal to?

1. A.$\ln (ab)$
2. B.$\frac{\ln a}{\ln b}$
3. C.$\ln \left(\frac{a}{b}\right)$
4. D.$\ln \left(\frac{b}{a}\right)$

If $\sqrt{x}+\sqrt{y}=2$, then what is $\frac{dy}{dx}$ at $y=1$ equal to?

1. A.5
2. B.2
3. C.4
4. D.$-1$

For a point of inflection of $y=f(x)$, which one of the following is correct?

1. A.$\frac{dy}{dx}$ must be zero
2. B.$\frac{d^{2}y}{d{x}^2}$ must be zero
3. C.$\frac{dy}{dx}$ must be non-zero
4. D.$\frac{d^{2}y}{d{x}^2}$ must be non-zero

What is the value of the integral ${\int }_{-1}^1|x|dx$ ?

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

What is $\int \frac{e^x(1+x)}{{\cos }^2(x{e}^x)}dx$ equal to? (where $c$ is a constant of integration.)

1. A.$x{e}^x+c$
2. B.$\cos (x{e}^x)+c$
3. C.$\tan (x{e}^x)+c$
4. D.$x\mathrm{cosec}(x{e}^x)+c$

What is $\int \sqrt{x}{e}^{\sqrt{x}}dx$ equal to? (where $c$ is a constant of integration.)

1. A.$2e^{\sqrt{x}}\left(x-2\sqrt{x}+2\right)+c$
2. B.$2e^{\sqrt{x}}\left(x+2\sqrt{x}+2\right)+c$
3. C.$2e^{\sqrt{x}}\left(x+2\sqrt{x}-2\right)+c$
4. D.$2e^{\sqrt{x}}\left(x-2\sqrt{x}-2\right)+c$

What is the general solution of $(1+e^x)ydy=e^{x}dx$ ? (where $c$ is a constant of integration.)

1. A.$y^2=\ln \left[c^2(e^x+1{)}^2\right]$
2. B.$y=\ln \left[c(e^x+1)\right]$
3. C.$y^2=\ln \left[c(e^x+1)\right]$
4. D.None of the above

What is the value of ${\int }_{\pi /6}^{\pi /4}\frac{dx}{\sin x\cos x}$ ?

1. A.$2\ln \sqrt{3}$
2. B.$\ln \sqrt{3}$
3. C.$2\ln 3$
4. D.$4\ln 3$

What is $\int {\sec }^{n}x\tan xdx$ equal to? (where $c$ is a constant of integration.)

1. A.$\frac{{\sec }^{n}x}{n}+c$
2. B.$\frac{{\sec }^{n-1}x}{n-1}+c$
3. C.$\frac{{\tan }^{n}x}{n}+c$
4. D.$\frac{{\tan }^{n-1}x}{n-1}+c$

What is the area bounded by the curve $y=x^2$ and the line $y=16$ ?

1. A.$32/3$
2. B.$64/3$
3. C.$256/3$
4. D.$128/3$

Consider the equation $(x-p)(x-6)+1=0$ having integral coefficients. If the equation has integral roots, then what values can p have?

1. A.4 or 8
2. B.5 or 10
3. C.6 or 12
4. D.3 or 6

What is $\int e^{\ln x}\sin xdx$ equal to? where $c$ is a constant of integration.

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

What is $\int \frac{x^4+1}{x^2+1}dx$ equal to? where $c$ is a constant of integration.

1. A.$\frac{x^3}{3}-x+4{\tan }^{-1}x+c$
2. B.$\frac{x^3}{3}+x+4{\tan }^{-1}x+c$
3. C.$\frac{x^3}{3}-x+2{\tan }^{-1}x+c$
4. D.$\frac{x^3}{3}-x-4{\tan }^{-1}x+c$