JEE Main Mathematics — Algebra previous year questions with solutions.
Let $A$ and $B$ be any two $3 \times 3$ matrices. If $A$ is symmetric and $B$ is skew symmetric, then the matrix $AB-BA$ is
If $A$is a $3\times 3$ non-singular matrix such that $A{A}^{'}={A}^{'}A$ and $B={A}^{-1}{A}^{'},$ then $B{B}^{'}$ equals, where ${X}^{'}$ denotes the transpose of the matrix $X$.
If $B$ is a $3\times 3$ matrix such that ${B}^{2}=0$, then $det.[{(I+B)}^{50}-50B]$ is equal to :
Let A be a $3 \times 3$ matrix such that $$ \mathrm{A}\left[\begin{array}{lll} 1 & 2 & 3 \\ 0 & 2 & 3 \\ 0 & 1 & 1 \end{array}\right]=\left[\begin{array}{lll} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right] $$ Then $\mathrm{A}^{-1}$ is:
Let $\mathrm{f}$ be an odd function defined on the set of real numbers such that for $\mathrm{x} \geq 0$, $f(x)=3 \sin x+4 \cos x$. Then $\mathrm{f}(\mathrm{x})$ at $\mathrm{x}=-\frac{11 \pi}{6}$ is equal to:
If $a\in R$ and the equation $-3(x-[x]{)}^{2}+2(x-[x])+{a}^{2}=0$ (where $[x]$ denotes the greatest integer $\leq$$x$) has no integral solution, then all possible values of $a$ lie in the interval
Let $f(n)=[\frac{1}{3}+\frac{3n}{100}]n$, where $[n]$ denotes the greatest integer less than or equal to $n$. Then $\sum _{n=1}^{56}f(n)$ is equal to
If the coefficients of ${x}^{3}$ and ${x}^{4}$ in the expansion of $(1+ax+b{x}^{2}){(1-2x)}^{18}$ in powers of $x$ are both zero, then $(a,b)$ is equal to
Let $\mathrm{G}$ be the geometric mean of two positive numbers $\mathrm{a}$ and $\mathrm{b}$, and $\mathrm{M}$ be the arithmetic mean of $\frac{1}{\mathrm{a}}$ and $\frac{1}{\mathrm{~b}}$. If $\frac{1}{\mathrm{M}}: \mathrm{G}$ is $4: 5$, then $\mathrm{a}: \mathrm{b}$ can be:
Given an A.P. whose terms are all positive integers. The sum of its first nine terms is greater than $200$ and less than $220.$ If the second term in it is $12$, then its ${4}^{\mathrm{th}}$ term is :
If $1+x^4+x^5=\sum_{i=0}^5 a_i\left(1+x^i\right)$, for all $x$ in $R$, then $a_2$ is:
Let for $\mathrm{i}=1,2,3, \mathrm{p}_{\mathrm{i}}(\mathrm{x})$ be a polynomial of degree 2 in $x, p^{\prime}{ }_i(x)$ and $p^{\prime \prime}{ }_i(x)$ be the first and second order derivatives of $p_i(x)$ respectively. Let, $$ \left.\mathrm{A}(\mathrm{x})=\left[\begin{array}{lll} \mathrm{p}_1(\mathrm{x}) & \mathrm{p}_1^{\prime}(\mathrm{x}) & \mathrm{p}_1^{\prime \prime} \mathrm{x}( \\ \mathrm{p}_2(\mathrm{x}) & \mathrm{p}_2^{\prime}(\mathrm{x}) & \mathrm{p}_2^{\prime \prime}( \\ \mathrm{p}_3(\mathrm{x}) & \mathrm{p}_3^{\prime}(\mathrm{x}) & \mathrm{p}_3^{\prime \prime}(\mathrm{x} \end{array}\right]\right) $$ and $\mathrm{B}(\mathrm{x})=[\mathrm{A}(\mathrm{x})]^{\mathrm{T}} \mathrm{A}(\mathrm{x})$. Then determinant of $\mathrm{B}(\mathrm{x})$ :
Two women and some men participated in a chess tournament in which every participant played two games with each of the other participants. If the number of games that the men played between them-selves exceeds the number of games that the men played with the women by$66$, then the number of men who participated in the tournament lies in the interval
Let $z \neq-i$ be any complex number such that $\frac{\mathrm{z}-\mathrm{i}}{\mathrm{z}+\mathrm{i}}$ is a purely imaginary number. Then $\mathrm{z}+\frac{1}{\mathrm{z}}$ is:
If $a,b,c$ are non - zero real numbers and if the system of equations $(a-1)x=y+z$ $(b-1)y=x+z$ $(c-1)z=x+y$ has a non - trivial solution, then $ab+bc+ca$ equals :
Let $P$ be the relation defined on the set of all real numbers such that $P={(a,b):{\mathrm{sec}}^{2}a-{\mathrm{tan}}^{2}b=1}$. Then, $P$ is
8-digit numbers are formed using the digits 1, 1, 2, $2,2,3,4,4$. The number of such numbers in which the odd digits do no occupy odd places, is:
Let $f:R\rightarrow R$ be defined by $f(x)=\frac{|x|-1}{|x|+1},$ then $f$ is
If $\alpha$ and $\beta$ are roots of the equation, $x^2-4 \sqrt{2} k x+2 e^{4 \ln k}-1=0$ for some $k$, and $\alpha^2+\beta^2=66$, then $\alpha^3+\beta^3$ is equal to:
The sum of the roots of the equation, $\mathrm{x}^2+|2 \mathrm{x}-3|-4=0$, is:
Three positive numbers form an increasing $G.P.$ If the middle term in this $G.P.$ is doubled, the new numbers are in $A.P.$ Then the common ratio of the $G.P.$ is :
If $z$ is a complex number such that $|z|\geq 2,$ then the minimum value of $|z+\frac{1}{2}|$ :
For all complex numbers $z$ of the form $1+i\alpha ,\alpha \in R,$ if ${z}^{2}=x+iy,$ then
The sum of the digits in the unit's place of all the $4$ - digit numbers formed by using the numbers $3,4,5$ and$6$, without repetition is :