Other Maths · JEE Main previous year questions

A square ABCD has all its vertices on the curve $x^2{y}^2=1$. The midpoints of its sides also lie on the same curve. Then, the square of area of ABCD is _______.

Let $f:\mathrm{R}-\{3\}\to \mathrm{R}-\{1\}$ be defined by $f(x)=\frac{x-2}{x-3}$. Let $g:\mathrm{R}\to \mathrm{R}$ be given as $g(x)=2x-3$. Then, the sum of all the values of x for which $f^{-1}(x)+g^{-1}(x)=\frac{13}{2}$ is equal to :

1. A.3
2. B.5
3. C.2
4. D.7

If ${\sum }_{r=1}^{10}r!(r^3+6r^2+2r+5)=\alpha (11!)$, then the value of $\alpha$ is equal to ____.

Let $[x]$ denote greatest integer less than or equal to x. If for $n\in \mathbb{N}$, $(1-x+x^3{)}^n={\sum }_{j=0}^{3n}{a}_j{x}^j$, then ${\sum }_{j=0}^{\left[\frac{3n}{2}\right]}{a}_{2j}+4{\sum }_{j=0}^{\left[\frac{3n-1}{2}\right]}{a}_{2j+1}$ is equal to :

1. A.$2^n-1$
2. B.n
3. C.2
4. D.1

The value of ${\int }_{-\pi /2}^{\pi /2}\frac{{\cos }^{2}x}{1+3^x}dx$ is :

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

The value of ${\sum }_{n=1}^{100}{\int }_{n-1}^n{e}^{x-[x]}dx$, where $[x]$ is the greatest integer $\le x$, is :

1. A.$100e$
2. B.$100(e-1)$
3. C.$100(1+e)$
4. D.$100(1-e)$

The value of $\left|\begin{array}{ccc}(a+1)(a+2)& a+2& 1\\ (a+2)(a+3)& a+3& 1\\ (a+3)(a+4)& a+4& 1\end{array}\right|$ is :

1. A.$-2$
2. B.$0$
3. C.$(a+2)(a+3)(a+4)$
4. D.$(a+1)(a+2)(a+3)$

The number of solutions of the equation ${\log }_4(x-1)={\log }_2(x-3)$ is _________.

Let $m,n\in \mathbb{N}$ and $\gcd (2,n)=1$. If $30\left(\begin{array}{c}30\\ 0\end{array}\right)+29\left(\begin{array}{c}30\\ 1\end{array}\right)+\dots \cdots +2\left(\begin{array}{c}30\\ 28\end{array}\right)+1\left(\begin{array}{c}30\\ 29\end{array}\right)=n\cdot 2^m$, then $n+m$ is equal to _________. (Here $\left(\begin{array}{c}n\\ k\end{array}\right)={}^n{\mathrm{C}}_k$)

Let $F_1(A,B,C)=(A\wedge \sim B)\vee [\sim C\wedge (A\vee B)]\vee \sim A$ and $F_2(A,B)=(A\vee B)\vee (B\to \sim A)$ be two logical expressions. Then :

1. A.Both $F_1$ and $F_2$ are not tautologies
2. B.$F_1$ and $F_2$ both are tautologies
3. C.$F_1$ is not a tautology but $F_2$ is a tautology
4. D.$F_1$ is a tautology but $F_2$ is not a tautology

Let $\alpha$ and $\beta$ be two real numbers such that $\alpha +\beta =1$ and $\alpha \beta =-1$. Let $p_n=(\alpha {)}^n+(\beta {)}^n$, $p_{n-1}=11$ and $p_{n+1}=29$ for some integer $n\ge 1$. Then, the value of $p_n^2$ is _________.

Let the normals at all the points on a given curve pass through a fixed point $(a,b)$. If the curve passes through $(3,-3)$ and $(4,-2\sqrt{2})$, and given that $a-2\sqrt{2}b=3$, then $(a^2+b^2+ab)$ is equal to _________.

Let $\alpha$ and $\beta$ be the roots of $x^2-6x-2=0$. If $a_n={\alpha }^n-{\beta }^n$ for $n\ge 1$, then the value of $\frac{a_{10}-2a_8}{3a_9}$ is :

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

If $\alpha ,\beta \in \mathbb{R}$ are such that $1-2i$ (here $i^2=-1$) is a root of $z^2+\alpha z+\beta =0$, then $(\alpha -\beta )$ is equal to :

1. A.$-7$
2. B.7
3. C.3
4. D.$-3$

If the remainder when x is divided by 4 is 3, then the remainder when $(2020+x{)}^{2022}$ is divided by 8 is ______ .

If the curves $x=y^4$ and $xy=k$ cut at right angles, then $(4k{)}^6$ is equal to ______ .

Let a , b, c , d and p be any non zero distinct real numbers such that $(a^2+b^2+c^2)p^2-2(ab+bc+cd)p+(b^2+c^2+d^2)=0$. Then :

1. A.a, c, p are in G.P.
2. B.a, b, c, d are in G.P.
3. C.a, b, c, d are in A.P.
4. D.a, c, p are in A.P.

If $\{p\}$ denotes the fractional part of the number $p$, then $\left\{\frac{3^{200}}{8}\right\}$, is equal to :

1. A.$\frac{5}{8}$
2. B.$\frac{7}{8}$
3. C.$\frac{1}{8}$
4. D.$\frac{3}{8}$

If $I_1={\int }_0^1(1-x^{50}{)}^{100}dx$ and $I_2={\int }_0^1(1-x^{50}{)}^{101}dx$ such that $I_2=\alpha I_1$ then $\alpha$ equals to :

1. A.$\frac{5051}{5050}$
2. B.$\frac{5050}{5051}$
3. C.$\frac{5050}{5049}$
4. D.$\frac{5049}{5050}$

The value of $(2\cdot {}^1{\mathrm{P}}_0-3\cdot {}^2{\mathrm{P}}_1+4\cdot {}^3{\mathrm{P}}_2\dots \text{up to }51^{\mathrm{th}}\text{term})+(1!-2!+3!-\dots \text{up to }51^{\mathrm{th}}\text{term})$ is equal to :

1. A.1
2. B.$1+(51)!$
3. C.$1-51(51)!$
4. D.$1+(52)!$

The proposition $p\to \sim (p\wedge \sim q)$ is equivalent to :

1. A.$(\sim p)\vee q$
2. B.$q$
3. C.$(\sim p)\wedge q$
4. D.$(\sim p)\vee (\sim q)$

The value of $(0.16{)}^{{\log }_{2.5}\left(\frac{1}{3}+\frac{1}{3^2}+\dots to\infty \right)}$ is equal to _______.

If ${\left(\frac{1+i}{1-i}\right)}^{\frac{m}{2}}={\left(\frac{1+i}{i-1}\right)}^{\frac{n}{3}}=1$, $(m,n\in \mathbb{N})$ then the greatest common divisor of the least values of m and n is _______.

If ${\lim }_{x\to 1}\frac{x+x^2+x^3+\dots +x^n-n}{x-1}=820$, ($n\in \mathbb{N}$) then the value of n is equal to _______.

Let n > 2 be an integer. Suppose that there are n Metro stations in a city located along a circular path. Each pair of stations is connected by a straight track only. Further, each pair of nearest stations is connected by blue line, whereas all remaining pairs of stations are connected by red line. If the number of red lines is 99 times the number of blue lines, then the value of n is :

1. A.201
2. B.199
3. C.101
4. D.200