Integral Calculus & Area · JEE Main previous year questions

If $\int \left(\frac{1-5{\cos }^{2}x}{{\sin }^{5}x{\cos }^{2}x}\right)dx=f(x)+\mathrm{C}$, where C is the constant of integration, then $f\left(\frac{\pi }{6}\right)-f\left(\frac{\pi }{4}\right)$ is equal to

1. A.$\frac{1}{\sqrt{3}}(26-\sqrt{3})$
2. B.$\frac{4}{\sqrt{3}}(8-\sqrt{6})$
3. C.$\frac{1}{\sqrt{3}}(26+\sqrt{3})$
4. D.$\frac{2}{\sqrt{3}}(4+\sqrt{6})$

Given below are two statements : Statement I : $25^{13}+20^{13}+8^{13}+3^{13}$ is divisible by 7. Statement II : The integral part of $(7+4\sqrt{3}{)}^{25}$ is an odd number. In the light of the above statements, choose the correct answer from the options given below :

1. A.Statement I is false but Statement II is true
2. B.Both Statement I and Statement II are false
3. C.Both Statement I and Statement II are true
4. D.Statement I is true but Statement II is false

Let [.] denote the greatest integer function. Then ${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\left(\frac{12(3+[x])}{3+[\sin x]+[\cos x]}\right)dx$ is equal to :

1. A.$12\pi +5$
2. B.$11\pi +2$
3. C.$15\pi +4$
4. D.$13\pi +1$

Let $f(t)=\int \left(\frac{1-\sin ({\log }_{e}t)}{1-\cos ({\log }_{e}t)}\right)dt,t>1$. If $f\left(e^{\pi /2}\right)=-e^{\pi /2}$ and $f\left(e^{\pi /4}\right)=\alpha e^{\pi /4}$, then $\alpha$ equals

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

Let a differentiable function $f$ satisfy the equation ${\int }_0^{36}f\left(\frac{tx}{36}\right)dt=4\alpha f(x)$. If $y=f(x)$ is a standard parabola passing through the points $(2,1)$ and $(-4,\beta )$, then ${\beta }^{\alpha }$ is equal to ____ .

Let $f(x)=\int \frac{(2-x^2)\cdot {\mathrm{e}}^x}{(\sqrt{1+x})(1-x{)}^{3/2}}\mathrm{d}x$. If $f(0)=0$, then $f\left(\frac{1}{2}\right)$ is equal to:

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

The value of the integral ${\int }_{\frac{\pi }{24}}^{\frac{5\pi }{24}}\frac{\mathrm{d}x}{1+\sqrt[3]{\tan 2x}}$ is :

1. A.$\frac{\pi }{3}$
2. B.$\frac{\pi }{18}$
3. C.$\frac{\pi }{6}$
4. D.$\frac{\pi }{12}$

Let $y=y(x)$ be the solution of the differential equation $x^{4}dy+(4x^{3}y+2\sin x)dx=0$, $x>0$, $y\left(\frac{\pi }{2}\right)=0$. Then ${\pi }^{4}y\left(\frac{\pi }{3}\right)$ is equal to :

1. A.92
2. B.72
3. C.64
4. D.81

Let $f$ be a twice differentiable non-negative function such that $(f(x){)}^2=25+{\int }_0^x\left((f(t){)}^2+(f^{\prime }(t){)}^2\right)dt$. Then the mean of $f({\log }_e(1)),f({\log }_e(2)),\dots ,f({\log }_e(625))$ is equal to ____ .

Let $\mathrm{I}(x)=\int \frac{3dx}{(4x+6)\left(\sqrt{4x^2+8x+3}\right)}$ and $\mathrm{I}(0)=\frac{\sqrt{3}}{4}+20$. If $\mathrm{I}\left(\frac{1}{2}\right)=\frac{a\sqrt{2}}{b}+c$, where $a,b,c\in \mathbb{N}$, $\gcd (a,b)=1$, then $a+b+c$ is equal to :

1. A.30
2. B.29
3. C.28
4. D.31

The number of elements in the set $S=\left\{x:x\in [0,100]\text{ and }{\int }_0^x{t}^2\sin (x-t)dt=x^2\right\}$ is _____.

Let the solution curve of the differential equation $xdy-ydx=\sqrt{x^2+y^2}dx,x>0,y(1)=0$, be $y=y(x)$. Then $y(3)$ is equal to

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

The value of ${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\left(\frac{1}{[x]+4}\right)dx$, where $[\cdot ]$ denotes the greatest integer function, is

1. A.$\frac{1}{60}(21\pi -1)$
2. B.$\frac{1}{60}(\pi -7)$
3. C.$\frac{7}{60}(\pi -3)$
4. D.$\frac{7}{60}(3\pi -1)$

Let the line $x=-1$ divide the area of the region $\left\{(x,y):1+x^2\le y\le 3-x\right\}$ in the ratio $m:n,\gcd (m,n)=1$. Then $m+n$ is equal to

1. A.27
2. B.28
3. C.25
4. D.26

Let $f:[1,\infty )\to \mathbb{R}$ be a differentiable function. If $6{\int }_1^{x}f(t)dt=3xf(x)+x^3-4$ for all $x\ge 1$, then the value of $f(2)-f(3)$ is :

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

If the image of the point $\mathrm{P}(1,2,a)$ in the line $\frac{x-6}{3}=\frac{y-7}{2}=\frac{7-z}{2}$ is $\mathrm{Q}(5,b,\mathrm{c})$, then $a^2+b^2+c^2$ is equal to

1. A.298
2. B.264
3. C.293
4. D.283

If $\int (\sin x{)}^{\frac{-11}{2}}(\cos x{)}^{\frac{-5}{2}}dx=-\frac{p_1}{q_1}(\cot x{)}^{\frac{9}{2}}-\frac{p_2}{q_2}(\cot x{)}^{\frac{5}{2}}-\frac{p_3}{q_3}(\cot x{)}^{\frac{1}{2}}+\frac{p_4}{q_4}(\cot x{)}^{\frac{-3}{2}}+\mathrm{C}$, where $p_i$ and $q_i$ are positive integers with $\gcd (p_i,q_i)=1$ for $i=1,2,3,4$ and C is the constant of integration, then $\frac{15p_1{p}_2{p}_3{p}_4}{q_1{q}_2{q}_3{q}_4}$ is equal to ____

The value of $\underset{-\pi /6}{\overset{\pi /6}{\int }}\left(\frac{\pi +4x^{11}}{1-\sin (|x|+\pi /6)}\right)dx$ is equal to:

1. A.$8\pi$
2. B.$4\pi$
3. C.$2\pi$
4. D.$6\pi$

$6{\int }_0^{\pi }|(\sin 3x+\sin 2x+\sin x)|dx$ is equal to ____ .

If the area of the region $\{(x,y):1-2x\le y\le 4-x^2,x\ge 0,y\ge 0\}$ is $\frac{\alpha }{\beta },\alpha ,\beta \in \mathbb{N},\gcd (\alpha ,\beta )=1$, then the value of $(\alpha +\beta )$ is:

1. A.73
2. B.85
3. C.67
4. D.91

Let $y=y(x)$ be the solution of the differential equation $\sec x\frac{dy}{dx}-2y=2+3\sin x$, $x\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$, $y(0)=-\frac{7}{4}$. Then $y\left(\frac{\pi }{6}\right)$ is equal to :

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

The integral $\underset{-1}{\overset{\frac{3}{2}}{\int }}(|{\pi }^{2}x\sin (\pi x)|)dx$ is equal to:

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

The number of integral terms in the expansion of ${\left(5^{\frac{1}{2}}+7^{\frac{1}{8}}\right)}^{1016}$ is:

1. A.$127$
2. B.$128$
3. C.$130$
4. D.$129$

Let the area of the bounded region $\left\{(x,y):0\le 9x\le y^2,y\ge 3x-6\right\}$ be $A$. Then $6A$ is equal to $\underline{}$.

Consider the equation $x^2+4x-n=0$, where $n\in [20,100]$ is a natural number. Then the number of all distinct values of $n$, for which the given equation has integral roots, is equal to

1. A.$6$
2. B.$5$
3. C.$8$
4. D.$7$