JEE Main Mathematics — Algebra previous year questions with solutions.
Let $p,q$ and $r$ be real numbers $(p\neq q,r\neq 0)$, such that the roots of the equation $\frac{1}{x+p}+\frac{1}{x+q}=\frac{1}{r}$ are equal in magnitude but opposite in sign, then the sum of squares of these roots is equal to
Let $N$ denote the set of all natural numbers. Define two binary relations on $N$ as ${R}_{1}={(x,y)\in N\times N:2x+y=10}$ and ${R}_{2}={(x,y)\in N\times N:x+2y=10}$. Then
If $b$ is the first term of an infinite G. P whose sum is five, then $b$ lies in the interval.
Let $A$ be a matrix such that $A\cdot [\begin{matrix}1 & 2 \\ 0 & 3\end{matrix}]$ is a scalar matrix and $|3A|=108$. Then, ${A}^{2}$ equals :
The number of values of $k$ for which the system of linear equations $(k+2)x+10y=k&$ $kx+(k+3)y=k-1$ has no solution is
Let $f: \mathrm{A} \rightarrow$ B be a function defined as $f(x)=$ $\frac{x-1}{x-2}$, where $A=R-\{2\}$ and $B=R-\{1\}$. Then $f$ is
The number of numbers between $2,000$ and $5,000$ that can be formed with the digits $0,1,2,3,4$ (repetition of digits is not allowed) and are multiple of $3$ is
If $n$ is the degree of the polynomial, ${[\frac{2}{\sqrt{5{x}^{3}+1}-\sqrt{5{x}^{3}-1}}]}^{8}+ {[\frac{2}{\sqrt{5{x}^{3}+1}+\sqrt{5{x}^{3}-1}}]}^{8}$ and $m$ is the coefficient of ${x}^{n}$ in it, then the ordered pair $(n,m)$ is equal to
If $n$ is the degree of the polynomial, $$ \left[\frac{1}{\sqrt{5 x^3+1}-\sqrt{5 x^3-1}}\right]^8+\left[\frac{1}{\sqrt{5 x^3+1}+\sqrt{5 x^3-1}}\right]^8 $$ and $\mathrm{m}$ is the coefficient of $\mathrm{x}^{\mathrm{n}}$ in it, then the ordered pair $(\mathrm{n}, \mathrm{m})$ is equal to
$n$-digit numbers are formed using only three digits $2,5$ and $7$. The smallest value of $n$ for which $900$ such distinct numbers can be formed is :
If $\lambda \in \mathrm{R}$ is such that the sum of the cubes of the roots of the equation, $x^2+(2-\lambda) x+(10-\lambda)=0$ is minimum, then the magnitude of the difference of the roots of this equation is
If $a, b, c$ are in A.P. and $a^2, b^2, c^2$ are in G.P. such that $a < b < c$ and $a+b+c=\frac{3}{4}$, then the value of $a$ is
Suppose $A$ is any $3 \times 3$ non-singular matrix and $(A-3 I)(A-5 I)=O$, where $I=I_3$ and $O=O_3$. If $\alpha A+$ $\beta A^{-1}=4 I$, then $\alpha+\beta$ is equal to
If $b$ is the first term of an infinite geometric progression whose sum is five, then $b$ lies in the interval
Let $S={x\in R :x\geq 0 & 2|\sqrt{x}-3|+\sqrt{x} (\sqrt{x}-6)+6=0}$ . Then $S$:
If $\lambda \in R$ is such that the sum of the cubes of the roots of the equation ${x}^{2}+(2-\lambda )x+(10-\lambda )=0$ is minimum, then the magnitude of the difference of the roots of this equation is :
If $|z-3+2 i| \leq 4$ then the difference between the greatest value and the least value of $|z|$ is
If $\alpha , \beta \in C$ are the distinct roots of the equation ${x}^{2}-x+1=0,$ then ${\alpha }^{101}+{\beta }^{107}$ is equal to
The set of all $\alpha \in R$, for which $w=\frac{1+(1-8\alpha )z}{1-z}$ is a purely imaginary number, for all $z\in C$ satisfying $|z|=1$ and $Re(z)\neq 1$, is :
The number of four letter words that can be formed using the letters of the word BARRACK is
Consider the following two binary relations on the set $A={a,b,c}:{R}_{1}={(c,a),(b,b),(a,c),(c,c),(b,c),(a,a)}$ and ${R}_{2}={(a,b),(b,a),(c,c),(c,a),(a,a),(b,b),(a,c)}$, then :
If $x=a, y=b, z=c$ is a solution of the system of linear equations $x+8y+7z=0$ $9x+2y+3z=0$ $x+y+z=0$ Such that the point $(a,b,c)$ lies on the plane $x+2y+z=6$ , then $2a+b+c$ equals:
For two $3\times 3$ matrices $A$ and $B$, let $A+B=2{B}^{'}$ and $3A+2B={I}_{3},$ where ${B}^{'}$ is the transpose of $B$ and ${I}_{3}$ is $3\times 3$ identity matrix. Then :
Let ${S}_{n}=\frac{1}{{1}^{3}}+\frac{1+2}{{1}^{3}+{2}^{3}}+\frac{1+2+3}{{1}^{3}+{2}^{3}+{3}^{3}}+\ldots +\frac{1+2+\ldots ,+n}{{1}^{3}+{2}^{3}+\ldots {n}^{3}}$ . If $100 {S}_{n}=n,$ then $n$ is equal to: