JEE Main Mathematics — Algebra previous year questions with solutions.
Let ${S}_{1},{S}_{2}$ and ${S}_{3}$ be three sets defined as ${S}_{1}={z\in \mathbb{C}:|z-1|\leq \sqrt{2}},$ ${S}_{2}={z\in \mathbb{C}:Re((1-i)z)\geq 1}$ and ${S}_{3}={z\in \mathbb{C}:Im(z)\leq 1}.$ Then, the set ${S}_{1}\cap {S}_{2}\cap {S}_{3}$
If $\alpha$ and $\beta$ are the distinct roots of the equation ${x}^{2}+{(3)}^{\frac{1}{4}}x+{3}^{\frac{1}{2}}=0$, then the value of ${\alpha }^{96}({\alpha }^{12}-1)+{\beta }^{96}({\beta }^{12}-1)$ is equal to:
The range of the function $f(x)={\mathrm{log}}_{\sqrt{5}}(3+\mathrm{cos}(\frac{3\pi }{4}+x)+\mathrm{cos}(\frac{\pi }{4}+x)+\mathrm{cos}(\frac{\pi }{4}-x)-\mathrm{cos}(\frac{3\pi }{4}-x))$ is :
If $\alpha ,\beta$ are natural numbers such that ${100}^{\alpha }-199\beta =(100)(100)+(99)(101)+(98)(102)+\ldots ..+(1)(199),$ then the slope of the line passing through $(\alpha ,\beta )$ and origin is:
The number of solutions of the equation ${\mathrm{log}}_{(x+1)}(2{x}^{2}+7x+5)+{\mathrm{log}}_{(2x+5)}(x+1{)}^{2}-4=0,x>0,$ is
Let$A=[\begin{matrix}1 & 2 \\ -1 & 4\end{matrix}].$ If ${A}^{-1}=\alpha I+\beta A,\alpha ,\beta \in R,I$ is a $2\times 2$ identity matrix, then $4(\alpha -\beta )$ is equal to :
The term independent of $x$ in the expansion of ${[\frac{x+1}{{x}^{2/3}-{x}^{1/3}+1}-\frac{x-1}{x-{x}^{1/2}}]}^{10},x\neq 1$, is equal to ___.
Let the domain of the function $f(x)={\mathrm{log}}_{4}({\mathrm{log}}_{5}({\mathrm{log}}_{3}(18x-{x}^{2}-77)))$ be $(a,b)$. Then the value of the integral ${\int }_{a}^{b}\frac{{\mathrm{sin}}^{3}x}{({\mathrm{sin}}^{3}x+{\mathrm{sin}}^{3}(a+b-x))}$ is equal to _____.
Let $z$ and $w$ be two complex numbers such that $w=z\bar{z}-2z+2,|\frac{z+i}{z-3i}|=1$ and $Re(w)$ has minimum value. Then, the minimum value of $n\in N$ for which ${w}^{n}$ is real, is equal to _______.
Let $[x]$ denote greatest integer less than or equal to $x$. If for $n\in N,{(1-x+{x}^{3})}^{n}=\sum _{j=0}^{3n}{a}_{j}{x}^{j}$, then $\sum _{j=0}^{[\frac{3n}{2}]}{a}_{2j}+4\sum _{j=0}^{[\frac{3n-1}{2}]}{a}_{2j+1}$ is equal to :
Consider an arithmetic series and a geometric series having four initial terms from the set ${11,8,21,16,26,32,4}$. If the last terms of these series are the maximum possible four digit numbers, then the number of common terms in these two series is equal to _______.
The domain of the function, $f(x)={\mathrm{sin}}^{-1}(\frac{3{x}^{2}+x-1}{(x-1{)}^{2}})+{\mathrm{cos}}^{-1}(\frac{x-1}{x+1})$ is:
The ratio of the coefficient of the middle term in the expansion of $(1+x{)}^{20}$ and the sum of the coefficients of two middle terms in expansion of $(1+x{)}^{19}$ is .
The sum of the roots of the equation, $x+1-2{\mathrm{log}}_{2}(3+{2}^{x})+2{\mathrm{log}}_{4}(10-{2}^{-x})=0,$ is :
The sum of all the $4$-digit distinct numbers that can be formed with the digits $1,2,2$ and $3$ is:
If $A={x\in R:|x-2|>1},B={x\in R:\sqrt{{x}^{2}-3}>1},C={x\in R:|x-4|\geqslant 2}$ and $Z$ is the set of all integers, then the number of subsets of the set $(A\cap B\cap C{)}^{c}\cap Z$ is _________.
If for $x\in (0,\frac{\pi }{2}),{\mathrm{log}}_{10}\mathrm{sin}x+{\mathrm{log}}_{10}\mathrm{cos}x=-1$ and ${\mathrm{log}}_{10}(\mathrm{sin}x+\mathrm{cos}x)=\frac{1}{2}({\mathrm{log}}_{10}n-1),n>0$, then the value of $n$ is equal to :
The set of all values of $k>-1$, for which the equation ${(3{x}^{2}+4x+3)}^{2}-(k+1)(3{x}^{2}+4x+3)(3{x}^{2}+4x+2)+$ $k{(3{x}^{2}+4x+2)}^{2}=0$ has real roots, is:
The sum of all integral values of $k(k\neq 0)$ for which the equation $\frac{2}{x-1}-\frac{1}{x-2}=\frac{2}{k}$ in $x$ has no real roots, is_____.
Let $\alpha =\underset{x\in R}{\mathrm{max}}{{8}^{2\mathrm{sin}3x}\cdot {4}^{4\mathrm{cos}3x}}$ and $\beta =\underset{x\in R}{\mathrm{min}}{{8}^{2\mathrm{sin}3x}\cdot {4}^{4\mathrm{cos}3x}}.$ If $8{x}^{2}+bx+c=0$ is a quadratic equation whose roots are ${\alpha }^{1/5}$ and ${\beta }^{1/5},$ then the value of $c-b$ is equal to :
Let $\alpha ,\beta$ be two roots of the equation ${x}^{2}+{(20)}^{1/4}x+{(5)}^{1/2}=0$. Then ${\alpha }^{8}+{\beta }^{8}$ is equal to
Let $\alpha$ and $\beta$ be two real numbers such that $\alpha +\beta =1$ and $\alpha \beta =-1$. Let ${p}_{n}={(\alpha )}^{n}+{(\beta )}^{n}$, ${p}_{n-1}=11$ and ${p}_{n+1}=29$ for some integer $n\geqslant 1$. Then, the value of ${p}_{n}^{2}$ is______.
The integer $k,$ for which the inequality ${x}^{2}-2(3k-1)x+8{k}^{2}-7>0$ is valid for every $x$ in $R$ is:
Let $\alpha$ and $\beta$ be the roots of ${x}^{2}-6x-2=0$. If ${a}_{n}={\alpha }^{n}-{\beta }^{n}$ for $n\geqslant 1$, then the value of $\frac{{a}_{10}-2{a}_{8}}{3{a}_{9}}$ is: