Integral Calculus & Area · JEE Main previous year questions

The integral $\int \sqrt{1+2\cot x(\mathrm{cosec}x+\cot x)}dx$ $(0<x<\frac{\pi }{2})$ is equal to : (where C is a constant of integration)

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

The integral ${\int }_{\frac{\pi }{12}}^{\frac{\pi }{4}}\frac{8\cos 2x}{(\tan x+\cot x{)}^3}dx$ equals :

1. A.$\frac{15}{128}$
2. B.$\frac{15}{64}$
3. C.$\frac{13}{32}$
4. D.$\frac{13}{256}$

The integral $\int \frac{2x^{12}+5x^9}{(x^5+x^3+1{)}^3}dx$ is equal to :

1. A.$-\frac{x^5}{2(x^5+x^3+1{)}^2}+C$
2. B.$\frac{-x^{10}}{2(x^5+x^3+1{)}^2}+C$
3. C.$\frac{-x^5}{(x^5+x^3+1{)}^2}+C$
4. D.$\frac{x^{10}}{2(x^5+x^3+1{)}^2}+C$

The area (in sq. units) of the region $\left\{(x,y):y^2\ge 2x\text{ and }{x}^2+y^2\le 4x,x\ge 0,y\ge 0\right\}$ is :

1. A.$\pi -\frac{4\sqrt{2}}{3}$
2. B.$\frac{\pi }{2}-\frac{2\sqrt{2}}{3}$
3. C.$\pi -\frac{4}{3}$
4. D.$\pi -\frac{8}{3}$

The integral $\int \frac{dx}{x^2(x^4+1{)}^{3/4}}$ equals :

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

The integral ${\int }_2^4\frac{\log x^2}{\log x^2+\log (36-12x+x^2)}dx$ is equal to :

1. A.$1$
2. B.$6$
3. C.$2$
4. D.$4$

The area (in sq. units) of the region described by $\left\{(x,y):y^2\le 2x\text{ and }y\ge 4x-1\right\}$ is :

1. A.$\frac{15}{64}$
2. B.$\frac{9}{32}$
3. C.$\frac{7}{32}$
4. D.$\frac{5}{64}$

If $a\in R$ and the equation $-3(x-[x]{)}^2+2(x-[x])+a^2=0$ (where $[x]$ denotes the greater integer $\le x$) has no integral solution, then all possible values of a lie in the interval :

1. A.$(-2,-1)$
2. B.$(-\infty ,-2)\cup (2,\infty )$
3. C.$(-1,0)\cup (0,1)$
4. D.$(1,2)$

Let $a,b,c$ and $d$ be non-zero numbers. If the point of intersection of the lines $4ax+2ay+c=0$ and $5bx+2by+d=0$ lies in the fourth quadrant and is equidistant from the two axes then :

1. A.$3bc-2ad=0$
2. B.$3bc+2ad=0$
3. C.$2bc-3ad=0$
4. D.$2bc+3ad=0$

The integral $\int (1+x-\frac{1}{x})e^{x+\frac{1}{x}}dx$ is equal to

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

The area of the region described by $A=\{(x,y):x^2+y^2\le 1\text{ and }{y}^2\le 1-x\}$ is :

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

The integral ${\int }_0^{\pi }\sqrt{1+4{\sin }^2\frac{x}{2}-4\sin \frac{x}{2}}dx$ equals:

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

If $a,b,c\in \mathrm{R}$ and 1 is a root of equation $a{x}^2+bx+c=0$, then the curve $y=4a{x}^2+3bx+2c$, $a\ne 0$ intersect $x$-axis at

1. A.two distinct points whose coordinates are always rational numbers
2. B.no point
3. C.exactly two distinct points
4. D.exactly one point

If $[x]$ is the greatest integer $\le x$, then the value of the integral ${\int }_{-0.9}^{0.9}\left([x^2]+\log \left(\frac{2-x}{2+x}\right)\right)dx$ is

1. A.$0.486$
2. B.$0.243$
3. C.$1.8$
4. D.0

If $a,b,c,d$ and $p$ are distinct real numbers such that $(a^2+b^2+c^2)p^2-2p(ab+bc+cd)+(b^2+c^2+d^2)\le 0$, then

1. A.$a,b,c,d$ are in A.P.
2. B.$ab=cd$
3. C.$ac=bd$
4. D.$a,b,c,d$ are in G.P.

The integral of $\frac{x^2-x}{x^3-x^2+x-1}$ w.r.t. $x$ is

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

If $\frac{d}{dx}G(x)=\frac{e^{\tan x}}{x},x\in (0,\pi /2)$, then ${\int }_{1/4}^{1/2}\frac{2}{x}\cdot e^{\tan (\pi x^2)}dx$ is equal to

1. A.$G(\pi /4)-G(\pi /16)$
2. B.$2[G(\pi /4)-G(\pi /16)]$
3. C.$\pi [G(1/2)-G(1/4)]$
4. D.$G(1/\sqrt{2})-G(1/2)$

The area enclosed by the curves $y=x^2,y=x^3,x=0$ and $x=p$, where $p>1$, is $1/6$. Then $p$ equals

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

If a straight line $y-x=2$ divides the region $x^2+y^2\le 4$ into two parts, then the ratio of the area of the smaller part to the area of the greater part is

1. A.$3\pi -8:\pi +8$
2. B.$\pi -3:3\pi +3$
3. C.$3\pi -4:\pi +4$
4. D.$\pi -2:3\pi +2$

Statement 1: The degrees of the differential equations $\frac{dy}{dx}+y^2=x$ and $\frac{d^{2}y}{d{x}^2}+y=\sin x$ are equal. Statement 2: The degree of a differential equation, when it is a polynomial equation in derivatives, is the highest positive integral power of the highest order derivative involved in the differential equation, otherwise degree is not defined.

1. A.Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation of Statement 1.
2. B.Statement 1 is false, Statement 2 is true.
3. C.Statement 1 is true, Statement 2 is false.
4. D.Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation of Statement 1.

If $\left|\begin{array}{ccc}-2a& a+b& a+c\\ b+a& -2b& b+c\\ c+a& b+c& -2c\end{array}\right|=\alpha (a+b)(b+c)(c+a)\ne 0$ then $\alpha$ is equal to

1. A.$a+b+c$
2. B.$abc$
3. C.4
4. D.1

Let $f(x)$ be an indefinite integral of ${\cos }^{3}x$. Statement 1: $f(x)$ is a periodic function of period $\pi$. Statement 2: ${\cos }^{3}x$ is a periodic function.

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

If ${\int }_e^{x}tf(t)dt=\sin x-x\cos x-\frac{x^2}{2}$, for all $x\in R-\{0\}$, then the value of $f\left(\frac{\pi }{6}\right)$ is

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

Solution of the differential equation $$\cos xdy=y(\sin x-y)dx,0<x<\frac{\pi }{2}\text{ is :}$$

1. A.$y\sec x=\tan x+c$
2. B.$y\tan x=\sec x+c$
3. C.$\tan x=(\sec x+c)y$
4. D.$\sec x=(\tan x+c)y$

Let $a,b,c$ be such that $b(a+c)\ne 0$ if $$\left|\begin{array}{ccc}a& a+1& a-1\\ -b& b+1& b-1\\ c& c-1& c+1\end{array}\right|+\left|\begin{array}{ccc}a+1& b+1& c-1\\ a-1& b-1& c+1\\ (-1{)}^{n+2}a& (-1{)}^{n+1}b& (-1{)}^{n}c\end{array}\right|=0$$ then the value of $n$ :

1. A.any even integer
2. B.any odd integer
3. C.any integer
4. D.zero