JEE Main Mathematics — Algebra previous year questions with solutions.
If |z - 2| = |z + 2|, then z lies on:
If the real part of the complex number ${(1-\mathrm{cos}\theta +2i\mathrm{sin}\theta )}^{-1}$ is $\frac{1}{5}$ for $\theta \in (0,\pi ),$ then the value of the integral ${\int }_{0}^{\theta }\mathrm{sin}xdx$ is equal to:
Let ${a}_{1},{a}_{2},\ldots ,{a}_{21}$ be an $A.P.$ such that $\sum _{n=1}^{20}\frac{1}{{a}_{n}{a}_{n+1}}=\frac{4}{9}.$ If the sum of this $A.P.$ is $189,$ then ${a}_{6}{a}_{16}$ is equal to :
If for the complex numbers$z$ satisfying $|z-2-2i|\leq 1$, the maximum value of $|3iz+6|$ is attained at $a+ib$, then $a+b$ is equal to _____ .
The sum of all those terms which are rational numbers in the expansion of ${({2}^{\frac{1}{3}}+{3}^{\frac{1}{4}})}^{12}$ is:
If the co-efficient of ${x}^{7}$ and ${x}^{8}$ in the expansion of ${(2+\frac{x}{3})}^{n}$ are equal, then the value of $n$ is equal to :
The value of $3+\frac{1}{4+\frac{1}{3+\frac{1}{4+\frac{1}{3+\ldots \infty }}}}$ is equal to
The total number of numbers, lying between $100$ and $1000$ that can be formed with the digits $1,2,3,4,5,$ if the repetition of digits is not allowed and numbers are divisible by either $3$ or $5,$ is
Let $n$ be a positive integer. Let $A=\sum _{k=0}^{n}{(-1)}^{k}\times Ckn[{(\frac{1}{2})}^{k}+{(\frac{3}{4})}^{k}+{(\frac{7}{8})}^{k}+{(\frac{15}{16})}^{k}+{(\frac{31}{32})}^{k}]$. If $63A=1-\frac{1}{{2}^{30}},$ then $n$ is equal to ______ .
Let $A=[{a}_{ij}]$ be a real matrix of order $3\times 3,$ such that ${a}_{i1}+{a}_{i2}+{a}_{i3}=1,$ for $i=1,2,3.$ Then, the sum of all the entries of the matrix ${A}^{3}$ is equal to:
The number of rational terms in the binomial expansion of ${({4}^{\frac{1}{4}}+{5}^{\frac{1}{6}})}^{120}$is_______.
Let a complex number be $w=1-\sqrt{3}i$. Let another complex number $z$ be such that $|zw|=1$ and $\mathrm{arg}(z)-\mathrm{arg}(w)=\frac{\pi }{2}$. Then the area of the triangle (in sq. units) with vertices origin, $z$ and $w$ is equal to
If $\alpha ,\beta$ are roots of the equation ${x}^{2}+5(\sqrt{2})x+10=0,\alpha >\beta$ and ${P}_{n}={\alpha }^{n}-{\beta }^{n}$ for each positive integer $n,$ then the value of $(\frac{{P}_{17}{P}_{20}+5\sqrt{2}{P}_{17}{P}_{19}}{{P}_{18}{P}_{19}+5\sqrt{2}{P}_{18}^{2}})$ is equal to
If the coefficient of ${a}^{7}{b}^{8}$ in the expansion of $(a+2b+4ab{)}^{10}$ is $K\cdot {2}^{16},$ then $K$ is equal to
Let $Z$ be the set of all integers, $A={(x,y)\in Z\times Z:(x-2{)}^{2}+{y}^{2}\leq 4}$ $B={(x,y)\in Z\times Z:{x}^{2}+{y}^{2}\leq 4}\text{ and }$ $C={(x,y)\in Z\times Z:(x-2{)}^{2}+(y-2{)}^{2}\leq 4}$ If the total number of relations from $A\cap B$ to $A\cap C$ is ${2}^{p}$, then the value of $p$ is:
The number of solutions of the equation x² - 5|x| + 6 = 0 is:
Let $A={2,3,4,5,\ldots .,30}$ and $'\simeq '$ be an equivalence relation on $A\times A,$ defined by $(a,b)\simeq (c,d),$ if and only if $ad=bc$. Then the number of ordered pairs which satisfy this equivalence relation with ordered pair $(4,3)$ is equal to :
For real numbers $\alpha$ and $\beta ,$ consider the following system of linear equations: $x+y-z=2,x+2y+\alpha z=1$ and $2x-y+z=\beta .$ If the system has infinite solutions, then $\alpha +\beta$ is equal to ______.
If $\mathrm{tan}(\frac{\pi }{9}),x,\mathrm{tan}(\frac{7\pi }{18})$ are in arithmetic progression and $\mathrm{tan}(\frac{\pi }{9}),y,\mathrm{tan}(\frac{5\pi }{18})$ are also in arithmetic progression, then $|x-2y|$ is equal to :
If ${e}^{({\mathrm{cos}}^{2}x+{\mathrm{cos}}^{4}x+{\mathrm{cos}}^{6}x+....\infty ){\mathrm{log}}_{e}2}$ satisfies the equation ${t}^{2}-9t+8=0$, then the value of $\frac{2\mathrm{sin}x}{\mathrm{sin}x+\sqrt{3}\mathrm{cos}x}$, where $0<x<\frac{\pi }{2}$, is equal to
The number of solutions of the equation ${32}^{{\mathrm{tan}}^{2}x}+{32}^{{\mathrm{sec}}^{2}x}=81,0\leq x\leq \frac{\pi }{4}$ is :
The sum of first four terms of a geometric progression $(G.P.)$ is $\frac{65}{12}$ and the sum of their respective reciprocals is $\frac{65}{18}.$ If the product of first three terms of the $G.P.$ is $1,$ and the third term is $\alpha ,$ then $2\alpha$ is _________.
$\frac{1}{{3}^{2}-1}+\frac{1}{{5}^{2}-1}+\frac{1}{{7}^{2}-1}+\ldots +\frac{1}{(201{)}^{2}-1}$ is equal to
The students ${S}_{1},{S}_{2},\ldots ,{S}_{10}$ are to be divided into $3$ groups $A,B$ and $C$ such that each group has at least one student and the group $C$ has at most $3$ students. Then the total number of possibilities of forming such groups is __________.