Integral Calculus & Area · NDA previous year questions

What is the area bounded by $y=\sqrt{16-x^2}$, $y\ge 0$ and the x-axis?

1. A.$16\pi$ square units.
2. B.$8\pi$ square units.
3. C.$4\pi$ square units.
4. D.$2\pi$ square units.

What is the area of the region enclosed between the curve $y^2=2x$ and the straight line $y=x$ ?

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

If $f(x)=2x-x^2$, then what is the value of $f(x+2)+f(x-2)$ when $x=0$ ?

1. A.$-8$
2. B.$-4$
3. C.8
4. D.4

If $x^m{y}^n=a^{m+n}$, then what is $\frac{dy}{dx}$ equal to?

1. A.$\frac{my}{nx}$
2. B.$-\frac{my}{nx}$
3. C.$\frac{mx}{ny}$
4. D.$-\frac{ny}{mx}$

What is $\int \frac{dx}{x(x^n+1)}$ equal to?

1. A.$\frac{1}{n}\ln \left(\frac{x^n}{x^n+1}\right)+c$
2. B.$\ln \left(\frac{x^n+1}{x^n}\right)+c$
3. C.$\ln \left(\frac{x^n}{x^n+1}\right)+c$
4. D.$\frac{1}{n}\ln \left(\frac{x^n+1}{x^n}\right)+c$

What is the value of $k$ such that integration of $\frac{3x^2+8-4k}{x}$ with respect to $x$, may be a rational function?

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

Directions for the following two (02) items : Read the following information and answer the two items that follow : Consider the integrals $I_1={\int }_0^{\pi }\frac{xdx}{1+\sin x}$ and $I_2={\int }_0^{\pi }\frac{(\pi -x)dx}{1-\sin (\pi +x)}$ What is the value of $I_1$?

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

Directions for the following two (02) items : Read the following information and answer the two items that follow : Consider the integrals $I_1={\int }_0^{\pi }\frac{xdx}{1+\sin x}$ and $I_2={\int }_0^{\pi }\frac{(\pi -x)dx}{1-\sin (\pi +x)}$ What is the value of $I_1$? What is the value of $I_1+I_2$?

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

What is the area of the region enclosed between the curve $y^2=2x$ and the straight line $y=x$?

1. A.$\frac{2}{3}$ square units
2. B.$\frac{4}{3}$ square units
3. C.$\frac{1}{3}$ square units
4. D.1 square unit

What is the area of the region bounded by $|x|<5$, $y=0$ and $y=8$?

1. A.40 square units
2. B.80 square units
3. C.120 square units
4. D.160 square units

What is $\int {\sin }^{3}x\cos xdx$ equal to? where c is the constant of integration.

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

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

1. A.$c+\frac{1}{ab}{\tan }^{-1}\left(\frac{a\tan x}{b}\right)$
2. B.$c-\frac{1}{ab}{\tan }^{-1}\left(\frac{b\tan x}{a}\right)$
3. C.$c+\frac{1}{ab}{\tan }^{-1}\left(\frac{b\tan x}{a}\right)$
4. D.None of the above

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

1. A.$\ln |\tan x|+c$
2. B.$\ln |\sec x|+c$
3. C.$\tan x+c$
4. D.$e^{\tan x}+c$

The number of non-zero integral solutions of the equation $|1-2i{|}^x=5^x$ is

1. A.Zero (No solution)
2. B.One
3. C.Two
4. D.Three

What is the area of the region bounded by the parabolas $y^2=6(x-1)$ and $y^2=3x$?

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

Let $f(n)=\left[\frac{1}{4}+\frac{n}{1000}\right]$, where $[x]$ denotes the integral part of $x$. Then the value of ${\sum }_{n=1}^{1000}f(n)$ is

1. A.251
2. B.250
3. C.1
4. D.0

If $f(x)=\frac{4x+x^4}{1+4x^3}$ and $g(x)=\ln \left(\frac{1+x}{1-x}\right)$, then what is the value of $f\circ g\left(\frac{e-1}{e+1}\right)$ equal to?

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

The area bounded by the curve $|x|+|y|=1$ is

1. A.1 square unit
2. B.$2\sqrt{2}$ square units
3. C.2 square units
4. D.$2\sqrt{3}$ square units

Let $f(x)$ be an indefinite integral of ${\sin }^{2}x$. Consider the following statements: Statement 1: The function $f(x)$ satisfies $f(x+\pi )=f(x)$ for all real $x$. Statement 2: ${\sin }^2(x+\pi )={\sin }^{2}x$ for all real $x$. Which one of the following is correct in respect of the above statements?

1. A.Both the statements are true and Statement 2 is the correct explanation of Statement 1
2. B.Both the statements are true but Statement 2 is not the correct explanation of Statement 1
3. C.Statement 1 is true but Statement 2 is false
4. D.Statement 1 is false but Statement 2 is true

1. A.$\frac{1}{2}$
2. B.1
3. C.$\sqrt{3}$
4. D.None of the above

What is the area bounded by the curves $|y|=1-x^2$?

1. A.4/3 square units
2. B.8/3 square units
3. C.4 square units
4. D.16/3 square units

For the next five (5) items that follow : Consider the function $$f(x)=|x-1|+x^2$$ where $x\in \mathbb{R}$. What is the area of the region bounded by $x$-axis, the curve $y=f(x)$ and the two ordinates $x=\frac{1}{2}$ and $x=1$?

1. A.$\frac{5}{12}$ square unit
2. B.$\frac{5}{6}$ square unit
3. C.$\frac{7}{6}$ square units
4. D.2 square units

For the next five (5) items that follow : Consider the function $$f(x)=|x-1|+x^2$$ where $x\in \mathbb{R}$. What is the area of the region bounded by $x$-axis, the curve $y=f(x)$ and the two ordinates $x=1$ and $x=\frac{3}{2}$?

1. A.$\frac{5}{12}$ square unit
2. B.$\frac{7}{12}$ square unit
3. C.$\frac{2}{3}$ square unit
4. D.$\frac{11}{12}$ square unit

For the next two (2) items that follow : Consider the curves $f(x)=x|x|-1$ and $g(x)=\{\begin{array}{ll}\frac{3x}{2},& x>0\\ 2x,& x\le 0\end{array}$ What is the area bounded by the curves?

1. A.$\frac{17}{6}$ square units
2. B.$\frac{8}{3}$ square units
3. C.2 square units
4. D.$\frac{1}{3}$ square unit

For the next two (2) items that follow : Consider the lines $y=3x$, $y=6x$ and $y=9$ What is the area of the triangle formed by these lines?

1. A.$\frac{27}{4}$ square units
2. B.$\frac{27}{2}$ square units
3. C.$\frac{19}{4}$ square units
4. D.$\frac{19}{2}$ square units