JEE Main Mathematics — Algebra previous year questions with solutions.
Let $D$ be the domain of the function $f(x)={\mathrm{sin}}^{-1}({\mathrm{log}}_{3x}(\frac{6+2{\mathrm{log}}_{3}x}{-5x}))$. If the range of the function $g:D\rightarrow \mathbb{R}$ defined by $g(x)=x-[x],$ ($[x]$ is the greatest integer function), is $(\alpha ,\beta ),$ then ${\alpha }^{2}+\frac{5}{\beta }$ is equal to
Let $\lambda \in \mathbb{R}$ and let the equation $E$ be $|x{|}^{2}-2|x|+|\lambda -3|=0$. Then the largest element in the set $S=$ ${x+\lambda :x$ is an integer solution of $E}$ is ______
Let ${x}_{1},{x}_{2},\ldots ,{x}_{100}$ be in an arithmetic progression, with ${x}_{1}=2$ and their mean equal to $200$ . If ${y}_{i}=i({x}_{i}-i),1\leq i\leq 100$, then the mean of ${y}_{1},{y}_{2},\ldots$, ${y}_{100}$ is
Let $A={1,2,3,4,5}$ and $B={1,2,3,4,5,6}$. Then the number of functions $f:A\rightarrow B$ satisfying $f(1)+f(2)=f(4)-1$ is equal to........
If the domain of the function $f(x)=\frac{[x]}{1+{x}^{2}}$, where $[x]$ is greatest integer $\leq x$, is $[2,6)$, then its range is
The absolute difference of the coefficients of ${x}^{10}$ and ${x}^{7}$ in the expansion of ${(2{x}^{2}+\frac{1}{2x})}^{11}$ is equal to
The number of real solutions of the equation $3({x}^{2}+\frac{1}{{x}^{2}})-2(x+\frac{1}{x})+5=0$, is
Let $[t]$ denote the greatest integer $\leq t$. if the constant term in the expansion of ${(3{x}^{2}-\frac{1}{2{x}^{5}})}^{7}$ is $\alpha$ then $[\alpha ]$ is equal to $_____$
The number of elements in the set ${n\in \mathbb{N}:10\leq n\leq 100$ and ${3}^{n}-3$ is a multiple of $7}$ is _______.
Let $\alpha$ and $\beta$ be real numbers. Consider a $3\times 3$ matrix $A$ such that ${A}^{2}=3A+\alpha I$. If ${A}^{4}=21A+\beta I$, then
For $\alpha ,\beta ,z\in \mathbb{C}$ and $\lambda >1$, if $\sqrt{\lambda -1}$ is the radius of the circle ${|z-\alpha |}^{2}+{|z-\beta |}^{2}=2\lambda ,$ then $|\alpha -\beta |$ is equal to _____.
Suppose $f:R\rightarrow (0,\infty )$ be a differentiable function such that $5f(x+y)=f(x)\cdot f(y),\forall x,y\in R$, If $f(3)=320$, then $\sum _{n=0}^{5}f(n)$ is equal to:
The constant term in the expansion of ${(2x+\frac{1}{{x}^{7}}+3{x}^{2})}^{5}$ is _____ .
The sum ${1}^{2}-2.{3}^{2}+3.{5}^{2}-4.{7}^{2}+5.{9}^{2}-\ldots ..+15.{29}^{2}$ is _____ .
Let $A={1,2,3,4,5,6,7}$. Then the relation $R={(x,y)\in A\times A:x+y=7}$ is
If the constant term in the binomial expansion of ${(\frac{{x}^{\frac{5}{2}}}{2}-\frac{4}{{x}^{l}})}^{9}$ is $-84$ and the coefficient of ${x}^{-3l}$ is ${2}^{\alpha }\beta$ where $\beta <0$ is an odd number, then $|\alpha l-\beta |$ is equal to _____ .
For the system of linear equations $x+y+z=6$ $\alpha x+\beta y+7z=3$ $x+2y+3z=14$ which of the following is NOT true ?
The ${8}^{\text{th }}$ common term of the series ${S}_{1}=3+7+11+15+19+\ldots$ ${S}_{2}=1+6+11+16+21+\ldots$. is
The coefficient of ${x}^{18}$ in the expansion of ${({x}^{4}-\frac{1}{{x}^{3}})}^{15}$ is $____________$
If the system of linear equations $7x+11y+\alpha z=13$ $5x+4y+7z=\beta$ $175x+194y+57z=361$ has infinitely many solutions, then $\alpha +\beta +2$ is equal to
The value of ${(\frac{1+\mathrm{sin}\frac{2\pi }{9}+i\mathrm{cos}\frac{2\pi }{9}}{1+\mathrm{sin}\frac{2\pi }{9}-i\mathrm{cos}\frac{2\pi }{9}})}^{3}$ is
Let ${a}_{1},{a}_{2},{a}_{3},\ldots$. be a GP of increasing positive numbers. If the product of fourth and sixth terms is $9$ and the sum of fifth and seventh terms is $24$ , then ${a}_{1}{a}_{9}+{a}_{2}{a}_{4}{a}_{9}+{a}_{5}+{a}_{7}$ is equal to
If the sum of first n terms of an AP is 3n² + 5n, then the common difference is:
If α and β are roots of x² - 5x + 6 = 0, then α³ + β³ is equal to: