JEE Main Mathematics — Algebra previous year questions with solutions.
Let $S$ denote the set of all real values of $\lambda$ such that the system of equations $\lambda x+y+z=1$ $x+\lambda y+z=1$ $x+y+\lambda z=1$ is inconsistent, then $\underset{\lambda \in S}{\sum }({|\lambda |}^{2}+|\lambda |)$ is equal to
If domain of the function ${\mathrm{log}}_{e}(\frac{6{x}^{2}+5x+1}{2x-1})+{\mathrm{cos}}^{-1}(\frac{2{x}^{2}-3x+4}{3x-5})$ is $(\alpha ,\beta )\cup (\gamma ,\delta )$, then $18({\alpha }^{2}+{\beta }^{2}+{\gamma }^{2}+{\delta }^{2})$ is equal to
The number of numbers, strictly between $5000$ and $10000$ can be formed using the digits $1,3,5,7,9$ without repetition, is
The number of relations, on the set ${1,2,3}$ containing $(1,2)$ and $(2,3)$ which are reflexive and transitive but not symmetric, is _________.
For the two positive numbers $a,b$, if $a,b$ and $\frac{1}{18}$ are in a geometric progression, while $\frac{1}{a},10$ and $\frac{1}{b}$ are in an arithmetic progression, then, $16a+12b$ is equal to _____ .
Let $f:R\rightarrow R$ be a function such that $f(x)=\frac{{x}^{2}+2x+1}{{x}^{2}+1}$. Then
The number of arrangements of the letters of the word "INDEPENDENCE" in which all the vowels always occur together is
Let $S$ be the set of all values of $\theta \in [-\pi ,\pi ]$ for which the system of linear equations $x+y+\sqrt{3}z=0$ $-x+(\mathrm{tan}\theta )y+\sqrt{7}z=0$ $x+y+(\mathrm{tan}\theta )z=0$ has non-trivial solution.Then $\frac{120}{\pi }\underset{0\in S}{\sum }\theta$ is equal to
Let $f(x)$ be a function such that $f(x+y)=f(x)\cdot f(y)$ for all $x,y\in N$, If $f(1)=3$ and $\sum _{k=1}^{n}f(k)=3279$, then the value of $n$ is
If $A$ is a $3\times 3$ matrix and $|A|=2$, then $|3adj(|3A|{A}^{2})|$ is equal to
Let $f:R-{2,6}\rightarrow R$ be real valued function defined as $f(x)=\frac{x+2x+1}{{x}^{2}-8x+12}$. Then range of $f$ is
Let ${a}_{1},{a}_{2},{a}_{3},\ldots \ldots$. be an A.P. If ${a}_{7}=3$, the product $({a}_{1}{a}_{4})$ is minimum and the sum of its first $n$ terms is zero then $n!-4{a}_{n(n+2)}$ is equal to
The number of ways of selecting two numbers $a$ and $b$, $a\in {2,4,6,\ldots \ldots ,100}$ and $b\in {1,3,5,\ldots \ldots ,99}$ such that $2$ is the remainder when $a+b$ is divided by $23$ is
If a point $P(\alpha ,\beta ,\gamma )$ satisfying $(\begin{matrix}\alpha & \beta & \gamma \end{matrix})(\begin{matrix}2 & 10 & 8 \\ 9 & 3 & 8 \\ 8 & 4 & 8\end{matrix})=(\begin{matrix}0 & 0 & 0\end{matrix})$ lies on the plane $2x+4y+3z=5$, then $6\alpha +9\beta +7\gamma$ is equal to
Let $S={x:x\in \mathbb{R}\text{ and }{(\sqrt{3}+\sqrt{2})}^{{x}^{2}-4}+{(\sqrt{3}-\sqrt{2})}^{{x}^{2}-4}=10}$. Then $n(S)$ is equal to
If $\\alpha$ and $\\beta$ are roots of $x^2 - 5x + 6 = 0$, then $\\alpha + \\beta$ is
Let ${a}_{1}=8,{a}_{2},{a}_{3},\ldots .{a}_{n}$ be an A.P. If the sum of its first four terms is $50$ and the sum of its last four terms is $170$ , then the product of its middle two terms is _____ .
The number of permutations, of the digits $1,2,3,\ldots ,7$ without repetition, which neither contain the string $153$ nor the string $2467$, is _______ .
if the coefficients of three consecutive terms in the expansion of ${(1+x)}^{n}$ are the ratio $1:5:20$then the coefficient of the fourth term is
Let $A$ be a $2\times 2$ matrix with real entries such that ${A}^{'}=\alpha A+1,$ where $\alpha \in \mathbb{R}-{-1,1}$., If det $({A}^{2}-A)=4$, the sum of all possible values of $\alpha$ is equal to
Let $A$ be a symmetric matrix such that $|A|=2$ and $[\begin{matrix}2 & 1 \\ 3 & \frac{3}{2}\end{matrix}]A=[\begin{matrix}1 & 2 \\ \alpha & \beta \end{matrix}]$ If the sum of the diagonal elements of $A$ is $s$, then $\frac{\beta s}{{\alpha }^{2}}$ is equal to _________.
Let $a,b,c>1,{a}^{3},{b}^{3}$ and ${c}^{3}$ be in $A.P.$ and ${\mathrm{log}}_{a}b$, ${\mathrm{log}}_{c}a$ and ${\mathrm{log}}_{b}c$ be in $G.P.$ If the sum of first $20$ terms of an $A.P.$, whose first term is $\frac{a+4b+c}{3}$ and the common difference is $\frac{a-8b+c}{10}$ is $-444$, then $abc$ is equal to
Let $A=[\begin{matrix}m & n \\ p & q\end{matrix}],d=|A|\neq 0$ and $|A-d(AdjA)|=0$. Then
If $A=\frac{1}{5!6!7!}[\begin{matrix}5! & 6! & 7! \\ 6! & 7! & 8! \\ 7! & 8! & 9!\end{matrix}]$, then $|adj(adj(2A))|$ is equal to