JEE Main Mathematics — Algebra previous year questions with solutions.
If $A=[\begin{matrix}1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{matrix}]$ and $M=A+{A}^{2}+{A}^{3}+\ldots +{A}^{20},$ then the sum of all the elements of the matrix $M$ is equal to _______.
Let $A$ and $B$ be two $3\times 3$ real matrices such that $({A}^{2}-{B}^{2})$ is invertible matrix. If ${A}^{5}={B}^{5}$ and ${A}^{3}{B}^{2}={A}^{2}{B}^{3},$ then the value of the determinant of the matrix ${A}^{3}+{B}^{3}$ is equal to :
Let $I$ be an identity matrix of order $2\times 2$ and $P=[\begin{matrix}2 & -1 \\ 5 & -3\end{matrix}]$. Then the value of $n\in N$ for which ${P}^{n}=5I-8P$ is equal to ___ .
The total number of $3\times 3$ matrices $A$ having enteries from the set $(0,1,2,3)$ such that the sum of all the diagonal entries of $A{A}^{T}$ is $9$, is equal to
Let $P=[\begin{matrix}3 & -1 & -2 \\ 2 & 0 & \alpha \\ 3 & -5 & 0\end{matrix}],$ where $\alpha \in R.$ Suppose $Q=[{q}_{ij}]$ is a matrix satisfying $PQ=k{I}_{3}$ for some non-zero $k\in R.$ If ${q}_{23}=-\frac{k}{8}$ and $|Q|=\frac{{k}^{2}}{2}$, then ${\alpha }^{2}+{k}^{2}$ is equal to_________.
If $A=[\begin{matrix}0 & -\mathrm{tan}(\frac{\theta }{2}) \\ \mathrm{tan}(\frac{\theta }{2}) & 0\end{matrix}]$ and $({I}_{2}+A){({I}_{2}-A)}^{-1}=[\begin{matrix}a & -b \\ b & a\end{matrix}],$ then $13({a}^{2}+{b}^{2})$ is equal to _____ .
If $\alpha +\beta +\gamma =2\pi ,$ then the system of equations $x+(\mathrm{cos}\gamma )y+(\mathrm{cos}\beta )z=0$ $(\mathrm{cos}\gamma )x+y+(\mathrm{cos}\alpha )z=0$ $(\mathrm{cos}\beta )x+(\mathrm{cos}\alpha )y+z=0$ has :
If the following system of linear equations $2x+y+z=5$ $x-y+z=3$ $x+y+az=b$ has no solution, then :
Let $f(x)=|\begin{matrix}{\mathrm{sin}}^{2}x & -2+{\mathrm{cos}}^{2}x & \mathrm{cos}2x \\ 2+{\mathrm{sin}}^{2}x & {\mathrm{cos}}^{2}x & \mathrm{cos}2x \\ {\mathrm{sin}}^{2}x & {\mathrm{cos}}^{2}x & 1+\mathrm{cos}2x\end{matrix}|,x\in [0,\pi ].$ Then the maximum value of $f(x)$ is equal to
If the system of linear equations $2x+y-z=3$ $x-y-z=\alpha$ $3x+3y+\beta z=3$ has infinitely many solutions, then $|\alpha +\beta -\alpha \beta |$ is equal to __________.
Two fair dice are thrown. The numbers on them are taken as $\lambda$ and $\mu ,$ and a system of linear equations $x+y+z=5$ $x+2y+3z=\mu$ $x+3y+\lambda z=1$ is constructed. If $p$ is the probability that the system has a unique solution and $q$ is the probability that the system has no solution, then:
The value of $k\in R,$ for which the following system of linear equations $3x-y+4z=3$ $x+2y-3z=-2$ $6x+5y+kz=-3$ has infinitely many solutions, is:
Let $a,b,c,d$ be in arithmetic progression with common difference $\lambda$. If $|\begin{matrix}x+a-c & x+b & x+a \\ x-1 & x+c & x+b \\ x-b+d & x+d & x+c\end{matrix}|=2$, then value of ${\lambda }^{2}$ is equal to________.
In a school, there are three types of games to be played. Some of the students play two types of games, but none play all the three games. Which Venn diagrams can justify the above statement? 
The value of $|\begin{matrix}(a+1)(a+2) & a+2 & 1 \\ (a+2)(a+3) & a+3 & 1 \\ (a+3)(a+4) & a+4 & 1\end{matrix}|$ is
The sum of the roots of the equation, $x+1-2{\mathrm{log}}_{2}(3+{2}^{x})+2{\mathrm{log}}_{4}(10-{2}^{-x})=0,$ is :
Let $A=[\begin{matrix}a & b \\ c & d\end{matrix}]$ and $B=[\begin{matrix}\alpha \\ \beta \end{matrix}]\neq [\begin{matrix}0 \\ 0\end{matrix}]$ such that $AB=B$ and $a+d=2021,$ then the value of $ad-bc$ is equal to ______ .
Let $f(x)$ be a polynomial of degree $3$ such that $f(k)=-\frac{2}{k}$ for $k=2,3,4,5.$ Then the value of $52-10f(10)$ is equal to _____ .
Let $[x]$ denote the greatest integer less than or equal to $x.$ Then, the values of $x\in R$ satisfying the equation ${[{e}^{x}]}^{2}+[{e}^{x}+1]-3=0$ lie in the interval:
Let $f:R-{\frac{\alpha }{6}}\rightarrow R$ be defined by $f(x)=(\frac{5x+3}{6x-\alpha })$. Then the value of $\alpha$ for which $(fof)(x)=x$, for all $x\in R-{\frac{\alpha }{6}},$ is
Let $[x]$ denote the greatest integer $\leq x$, where $x\in R$. If the domain of the real valued function $f(x)=\sqrt{\frac{|[x]|-2}{|[x]|-3}}$ is $(-\infty ,a)\cup [b,c)\cup [4,\infty ),a<b<c$, then the value of $a+b+c$ is:
The inverse of $y={5}^{\mathrm{log}x}$ is:
If $a+\alpha =1,b+\beta =2$ and $af(x)+\alpha f(\frac{1}{x})=bx+\frac{\beta }{x},x\neq 0,$ then the value of the expression $\frac{f(x)+f(\frac{1}{x})}{x+\frac{1}{x}}$ is ___________.
Consider the system of linear equations $-x+y+2z=0$ $3x-ay+5z=1$ $2x-2y-az=7$ Let ${S}_{1}$ be the set of all $a\in R$ for which the system is inconsistent and ${S}_{2}$ be the set of all $a\in R$ for which the system has infinitely many solutions. If $n({S}_{1})$ and $n({S}_{2})$ denote the number of elements in ${S}_{1}$ and ${S}_{2}$ respectively, then