Integral Calculus & Area · JEE Main previous year questions

If for $x\ge 0$, $y=y(x)$ is the solution of the differential equation $(x+1)dy=((x+1{)}^2+y-3)dx,y(2)=0$, then $y(3)$ is equal to _______.

Given : $$f(x)=\{\begin{array}{llc}x& ,& 0\le x<\frac{1}{2}\\ \frac{1}{2}& ,& x=\frac{1}{2}\\ 1-x& ,& \frac{1}{2}<x\le 1\end{array}$$ and $g(x)={\left(x-\frac{1}{2}\right)}^2,x\in \mathbb{R}$ Then the area (in sq. units) of the region bounded by the curves, $y=f(x)$ and $y=g(x)$ between the lines, $2x=1$ and $2x=\sqrt{3}$, is :

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

If $\int \frac{d\theta }{{\cos }^2\theta (\tan 2\theta +\sec 2\theta )}=\lambda \tan \theta +2{\log }_e|f(\theta )|+C$ where C is a constant of integration, then the ordered pair $(\lambda ,f(\theta ))$ is equal to :

1. A.$(-1,1-\tan \theta )$
2. B.$(1,1+\tan \theta )$
3. C.$(-1,1+\tan \theta )$
4. D.$(1,1-\tan \theta )$

Let $a-2b+c=1$. If $f(x)=\left|\begin{array}{ccc}x+a& x+2& x+1\\ x+b& x+3& x+2\\ x+c& x+4& x+3\end{array}\right|$, then:

1. A.$f(50)=1$
2. B.$f(-50)=-1$
3. C.$f(50)=-501$
4. D.$f(-50)=501$

For $a>0$, let the curves $C_1:y^2=ax$ and $C_2:x^2=ay$ intersect at origin O and a point P. Let the line $x=b$ ($0<b<a$) intersect the chord OP and the x-axis at points Q and R, respectively. If the line $x=b$ bisects the area bounded by the curves, $C_1$ and $C_2$, and the area of $\Delta \mathrm{OQR}=\frac{1}{2}$, then 'a' satisfies the equation :

1. A.$x^6-12x^3+4=0$
2. B.$x^6-12x^3-4=0$
3. C.$x^6+6x^3-4=0$
4. D.$x^6-6x^3+4=0$

If $\int \frac{\cos xdx}{{\sin }^{3}x(1+{\sin }^{6}x{)}^{2/3}}=f(x)(1+{\sin }^{6}x{)}^{1/\lambda }+c$ where c is a constant of integration, then $\lambda f\left(\frac{\pi }{3}\right)$ is equal to

1. A.$\frac{9}{8}$
2. B.2
3. C.-2
4. D.$-\frac{9}{8}$

The area (in sq. units) of the region $\{(x,y)\in {\mathbf{R}}^2:x^2\le y\le 3-2x\}$, is :

1. A.$\frac{34}{3}$
2. B.$\frac{29}{3}$
3. C.$\frac{31}{3}$
4. D.$\frac{32}{3}$

If $I={\int }_1^2\frac{dx}{\sqrt{2x^3-9x^2+12x+4}}$, then :

1. A.$\frac{1}{16}<I^2<\frac{1}{9}$
2. B.$\frac{1}{8}<I^2<\frac{1}{4}$
3. C.$\frac{1}{9}<I^2<\frac{1}{8}$
4. D.$\frac{1}{6}<I^2<\frac{1}{2}$

If $f(a+b+1-x)=f(x)$, for all x, where a and b are fixed positive real numbers, then $\frac{1}{a+b}{\int }_a^{b}x(f(x)+f(x+1))dx$ is equal to:

1. A.${\int }_{a-1}^{b-1}f(x+1)dx$
2. B.${\int }_{a+1}^{b+1}f(x+1)dx$
3. C.${\int }_{a-1}^{b-1}f(x)dx$
4. D.${\int }_{a+1}^{b+1}f(x)dx$

The value of $\alpha$ for which $4\alpha {\int }_{-1}^2{e}^{-\alpha |x|}dx=5$, is:

1. A.${\log }_{e}2$
2. B.${\log }_e\sqrt{2}$
3. C.${\log }_e\left(\frac{4}{3}\right)$
4. D.${\log }_e\left(\frac{3}{2}\right)$

The area (in sq. units) of the region $\{(x,y)\in {\mathbb{R}}^2\mid 4x^2\le y\le 8x+12\}$ is :

1. A.$\frac{125}{3}$
2. B.$\frac{128}{3}$
3. C.$\frac{127}{3}$
4. D.$\frac{124}{3}$

The integral $\int \frac{2x^3-1}{x^4+x}dx$ is equal to : (Here C is a constant of integration)

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

If ${\int }_0^{\frac{\pi }{2}}\frac{\cot x}{\cot x+\mathrm{cosec}x}dx=m(\pi +n)$, then m.n is equal to

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

Consider the differential equation, $y^{2}dx+\left(x-\frac{1}{y}\right)dy=0$. If value of y is 1 when x = 1, then the value of x for which y = 2, is :

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

If the area (in sq. units) of the region $\{(x,y):y^2\le 4x,x+y\le 1,x\ge 0,y\ge 0\}$ is $a\sqrt{2}+b$, then $a-b$ is equal to :

1. A.$\frac{8}{3}$
2. B.$-\frac{2}{3}$
3. C.6
4. D.$\frac{10}{3}$

Let $\alpha \in \left(0,\frac{\pi }{2}\right)$ be fixed. If the integral $\int \frac{\tan x+\tan \alpha }{\tan x-\tan \alpha }dx=\text{A}(x)\cos 2\alpha +\text{B}(x)\sin 2\alpha +\text{C}$, where C is a constant of integration, then the functions A(x) and B(x) are respectively :

1. A.$x-\alpha$ and ${\log }_e|\cos (x-\alpha )|$
2. B.$x+\alpha$ and ${\log }_e|\sin (x-\alpha )|$
3. C.$x+\alpha$ and ${\log }_e|\sin (x+\alpha )|$
4. D.$x-\alpha$ and ${\log }_e|\sin (x-\alpha )|$

The general solution of the differential equation $(y^2-x^3)dx-xydy=0$ ($x\ne 0$) is : (where c is a constant of integration)

1. A.$y^2+2x^3+c{x}^2=0$
2. B.$y^2+2x^2+c{x}^3=0$
3. C.$y^2-2x+c{x}^3=0$
4. D.$y^2-2x^3+c{x}^2=0$

The region represented by $|x-y|\le 2$ and $|x+y|\le 2$ is bounded by a :

1. A.rhombus of area $8\sqrt{2}sq.\mathrm{units}$
2. B.square of side length $2\sqrt{2}\mathrm{units}$
3. C.square of area 16 sq. units
4. D.rhombus of side length 2 units

If $\int \frac{dx}{(x^2-2x+10{)}^2}=A\left({\tan }^{-1}\left(\frac{x-1}{3}\right)+\frac{f(x)}{x^2-2x+10}\right)+C$ where C is a constant of integration then :

1. A.$A=\frac{1}{54}$ and $f(x)=9(x-1{)}^2$
2. B.$A=\frac{1}{54}$ and $f(x)=3(x-1)$
3. C.$A=\frac{1}{81}$ and $f(x)=3(x-1)$
4. D.$A=\frac{1}{27}$ and $f(x)=9(x-1{)}^2$

The value of ${\int }_0^{2\pi }[\sin 2x(1+\cos 3x)]dx$, where [t] denotes the greatest integer function is :

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

The area (in sq.units) of the region bounded by the curves $y=2^x$ and $y=|x+1|$, in the first quadrant is :

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

The integral ${\int }_{\pi /6}^{\pi /3}{\sec }^{2/3}x{\mathrm{cosec}}^{4/3}xdx$ is equal to :

1. A.$3^{\frac{5}{3}}-3^{\frac{1}{3}}$
2. B.$3^{\frac{5}{6}}-3^{\frac{2}{3}}$
3. C.$3^{\frac{4}{3}}-3^{\frac{1}{3}}$
4. D.$3^{\frac{7}{6}}-3^{\frac{5}{6}}$

If $\int x^5{e}^{-x^2}dx=g(x)e^{-x^2}+c$, where c is a constant of integration, then $g(-1)$ is equal to :

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

The value of ${\int }_0^{\pi /2}\frac{{\sin }^{3}x}{\sin x+\cos x}dx$ is

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

The integral $\int {\sec }^{2/3}x{\mathrm{cosec}}^{4/3}xdx$ is equal to (Hence C is a constant of integration)

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