JEE Main Mathematics — Algebra previous year questions with solutions.
If the sum of the coefficients of all even powers of $x$ in the product $(1+x+{x}^{2}+\ldots +{x}^{2n})(1-x+{x}^{2}-{x}^{3}+\ldots +{x}^{2n})$ is $61,$ then n is equal to
Let$n>2$ be an integer. Suppose that there are $n$ Metro stations in a city located around a circular path. Each pair of the nearest stations is connected by a straight track only. Further, each pair of the nearest station is connected by blue line, whereas all remaining pairs of stations are connected by red line. If number of red lines is $99$ times the number of blue lines, then the value of $n$ is
If the minimum and the maximum values of the function $f:[\frac{\pi }{4},\frac{\pi }{2}]\rightarrow R,$ defined by $f(\theta )=|\begin{matrix}-{\mathrm{sin}}^{2}\theta & -1-{\mathrm{sin}}^{2}\theta & 1 \\ -{\mathrm{cos}}^{2}\theta & -1-{\mathrm{cos}}^{2}\theta & 1 \\ 12 & 10 & -2\end{matrix}|$ are $m$ and $M$respectively, then the ordered pair $(m,M)$ is equal to :
A survey shows that $63%$ of the people in a city read newspaper $A$ whereas $76%$ read news paper $B$. If $x%$ of the people read both the newspapers, then a possible value of $x$ can be:
If the system of linear equations $2x+2ay+az=0$ $2x+3by+bz=0$ $2x+4cy+cz=0,$ where $a,b,c\in R$ are non-zero and distinct; has a non-zero solution, then
Let $\alpha$ and $\beta$ be the roots of ${x}^{2}-3x+p=0$ and $\gamma$ and $\delta$ be the roots of ${x}^{2}-6x+q=0.$ If $\alpha ,\beta ,\gamma ,\delta$ from a geometric progression. Then ratio $(2q+p):(2q-p)$ is
The value of $\sum _{r=0}^{20}C650-r$ is equal to:
If one real root of the quadratic equation $81 x^{2}+k x+256=0$ is cube of the other root, then a value of $\mathrm{k}$ is :
For $x\in R,$ Let $[x]$ denotes the greatest integer $\leq x,$then the sum of the series $[-\frac{1}{3}]+[-\frac{1}{3}-\frac{1}{100}]+[-\frac{1}{3}-\frac{2}{100}]+.....+[-\frac{1}{3}-\frac{99}{100}]$ is
The greatest value of $c\in R$ for which the system of linear equations $x-cy-cz=0$, $cx-y+cz=0$, $cx+cy-z=0$ has a non-trivial solution, is
The number of$6$ digit number that can be formed using the digits $0, 1, 2, 5, 7$ and $9$ which are divisible by $11$ and no digit is repeated is:
Let $f:[0,1]\rightarrow R$ be such that $f(xy)=f(x).f(y),$ for all $x,y\in [0,1],$ and $f(0)\neq 0.$ If $y=y(x)$ satisfies the differential equation, $\frac{dy}{dx}=f(x)$ with $y(0)=1$ then $y(\frac{1}{4})+y(\frac{3}{4})$ is equal to:
If $\frac{z - \alpha }{z + \alpha }(\alpha \in R)$ is a purely imaginary number and $|z|=2$, then a value of $\alpha$ is :
The sum of all natural numbers $n$ such that $100<n<200$ and $H.C.F.$ $(91,n)>1$ is
Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If $99$ more identical balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square, whose each side contains exactly $2$ balls less than the number of balls each side of the triangle contains. Then the number of balls used to form the equilateral triangle is
Let ${a}_{1}, {a}_{2}, \ldots , {a}_{30}$ be an A.P., $S= \sum _{i=1}^{30}{a}_{i}$ and $T= \sum _{i=1}^{15}{a}_{(2i-1)}.$ If ${a}_{5}=27$ and $S-2T=75,$ then ${a}_{10}$ is equal to:
If ${a}_{1},{a}_{2},{a}_{3},....$ are in A.P. such that ${a}_{1}+{a}_{7}+{a}_{16}=40,$ then the sum of the first $15$ terms of this A.P is:
The number of integral values of $m$ for which the quadratic expression $(1+2m) {x}^{2}-2(1+3m)x+4(1+m), x\in R$ is always positive, is
Let $p, q\in Q .$ If $2-\sqrt{3}$ is a root of the quadratic equation ${x}^{2}+px+q=0,$ then
The number of real roots of the equation $5+|{2}^{x}-1|={2}^{x}({2}^{x}-2)$ is :
If $[\begin{matrix}1 & 1 \\ 0 & 1\end{matrix}][\begin{matrix}1 & 2 \\ 0 & 1\end{matrix}][\begin{matrix}1 & 3 \\ 0 & 1\end{matrix}]\ldots .[\begin{matrix}1 & n-1 \\ 0 & 1\end{matrix}]=[\begin{matrix}1 & 78 \\ 0 & 1\end{matrix}],$ then the inverse of $[\begin{matrix}1 & n \\ 0 & 1\end{matrix}]$ is:
Let $\alpha$ and $\beta$ be the roots of the equation ${x}^{2}+x+1=0.$ Then for $y\neq 0$ in $R,|\begin{matrix}y+1 & \alpha & \beta \\ \alpha & y+\beta & 1 \\ \beta & 1 & y+\alpha \end{matrix}|$ is equal to
The total number of matrices $A=(\begin{matrix}0 & 2y & 1 \\ 2x & y & -1 \\ 2x & -y & 1\end{matrix}),(x,y\in R,x\neq y)$ for which ${A}^{T}A=3{I}_{3}$ is:
The sum of the real roots of the equation $|\begin{matrix}x & -6 & -1 \\ 2 & -3x & x-3 \\ -3 & 2x & x+2\end{matrix}|=0,$ is equal to: