JEE Main Mathematics — Algebra previous year questions with solutions.
Let $\frac{1}{{x}_{1}},\frac{1}{{x}_{2}},\ldots ,\frac{1}{{x}_{n}}({x}_{i}\neq 0$ for $i=1,2,\ldots .,n)$ be in A.P. such that ${x}_{1}=4$ and ${x}_{21}=20$$.$ If $n$ is the least positive integer for which ${x}_{n}>50$, then $\sum _{i=1}^{n}(\frac{1}{{x}_{i}})$ is equal to
If $x_1, x_2, \ldots ., x_n$ and $\frac{1}{h_1}, \frac{1}{h^2}, \ldots . . \frac{1}{h_n}$ are two A.P's such that $x_3=h_2=8$ and $x_8=h_7=20$, then $x_5 . h_{10}$ equals.
The least positive integer $n$ for which ${(\frac{1+i\sqrt{3}}{1-i\sqrt{3}})}^{n}=1$ is
From $6$ different novels and $3$ different dictionaries, $4$ novels and $1$ dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. The number of such arrangements is:
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 $\operatorname{Re} z \neq 1$, is
If the system of linear equations $x+ky+3z=0$ $3x+ky-2z=0$ $2x+4y-3z=0$ has a non-zero solution $(x, y, z),$ then $\frac{xz}{{y}^{2}}$ is equal to:
If an angle $A$ of a $\Delta ABC$ satisfies $5\mathrm{cos}A+3=0$, then the roots of the quadratic equation $9{x}^{2}+27x+20=0$ are
If ${x}_{1},{x}_{2},\ldots ..,{x}_{n}$ and $\frac{1}{{h}_{1}},\frac{1}{{h}_{2}},\ldots ..,\frac{1}{{h}_{n}}$ are two A.P.s such that ${x}_{3}={h}_{2}=8&{x}_{8}={h}_{7}=20$, then ${x}_{5}\cdot {h}_{10}$ is equal to
If $\mathrm{tan}A$ and $\mathrm{tan}B$ are the roots of the quadratic equation $3{x}^{2}-10x-25=0$, then the value of $3{\mathrm{sin}}^{2}(A+B)-10\mathrm{sin}(A+B)\mathrm{cos}(A+B)-25{\mathrm{cos}}^{2}(A+B)$ is :
$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
The sum of the first $20$ terms of the series $1+\frac{3}{2}+\frac{7}{4}+\frac{15}{8}+\frac{31}{16}+\ldots$ is
Let ${a}_{1}, {a}_{2}, {a}_{3},\ldots \ldots ,{a}_{49}$ be in $A.P.$ such that $\Sigma _{ k = 0 }^{ 1 2 } {a}_{4k+1}=416$ and ${a}_{9}+{a}_{43}=66.$ If ${a}_{1}^{2}+{a}_{2}^{2}+\ldots +{a}_{17}^{2}=140m,$ then $m$ is equal to:
If $f(x)=\left|\begin{array}{ccc}\cos x & x & 1 \\ 2 \sin x & x^2 & 2 x \\ \tan x & x & 1\end{array}\right|$, then $\lim _{x \rightarrow 0} \frac{f^{\prime}(x)}{x}$
Let $A=[\begin{matrix}1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1\end{matrix}]$ and $B={A}^{20}$. Then the sum of the elements of the first column of $B$ is
Let $A$ be the sum of the first $20$ terms and $B$ be the sum of the first $40$ terms of the series ${1}^{2}+{2\cdot 2}^{2}+{3}^{2}+{2\cdot 4}^{2}+{5}^{2}+{2\cdot 6}^{2}+\ldots$ If $B-2A=100\lambda ,$ then $\lambda$ is equal to :
Consider the following two binary relations on the set $A=\{a, b, c\}: R_1=\{(\mathrm{c}, a)(b, b),(\mathrm{a}, c),(c$, $c),(b, c),(a, a)\}$ and $\mathrm{R}_2=\{(\mathrm{a}, \mathrm{b}),(\mathrm{b}, \mathrm{a}),(\mathrm{c}, \mathrm{c})$, (c, a), (a, a), (b, b), (a, c). Then
Let $A_n=\left(\frac{3}{4}\right)-\left(\frac{3}{4}\right)^2+\left(\frac{3}{4}\right)^3-\ldots+(-1)^{n-1}\left(\frac{3}{4}\right)^n$ and $B_n=1-A_n$. Then, the least odd natural number $p$, so that $B_n>A_n$, for all $n \geq p$ is
The coefficient of ${x}^{2}$ in the expansion of the product $(2-{x}^{2}){{(1+2x+3{x}^{2})}^{6}+{(1-4{x}^{2})}^{6}}$ is
Let $S$ be the set of all real values of $k$ for which the system of linear equations $x+y+z=2$ $2x+y-z=3$ $3x+2y+kz=4$ has a unique solution. Then, $S$is :
Let $S$ be the set of all real values of $k$ for which the system of linear equations $$ \begin{aligned} &x+y+z=2 \\ &2 x+y-z=3 \\ &3 x+2 y+k z=4 \end{aligned} $$ has a unique solution. Then $S$ is
If $f(x)=|\begin{matrix}\mathrm{cos}x & x & 1 \\ 2\mathrm{sin}x & {x}^{2} & 2x \\ \mathrm{tan}x & x & 1\end{matrix}|$, then $\underset{x\rightarrow 0}{\mathrm{lim}}\frac{{f}^{'}(x)}{x}$
Let $A$ be a matrix such that $A$. $\left[\begin{array}{ll}1 & 2 \\ 0 & 3\end{array}\right]$ is a scalar matrix and $|3 A|=108$. Then $A^2$ equals
The coefficient of $x^{10}$ in the expansion of $(1+x)^2$ $\left(1+x^2\right)^3\left(1+x^3\right)^4$ is equal to
If the system of linear equations $$ \begin{aligned} &x+a y+z=3 \\ &x+2 y+2 z=6 \\ &x+5 y+3 z=b \end{aligned} $$ has no solution, then