JEE Main Mathematics — Algebra previous year questions with solutions.
Let $x={(8\sqrt{3}+13)}^{13}$ and $y={(7\sqrt{2}+9)}^{9}$. If $[t]$ denotes the greatest integer $\leq t$, then
The remainder on dividing ${5}^{99}$ by $11$ is _____ .
In a group of $100$ persons $75$ speak English and $40$ speak Hindi. Each person speaks at least one of the two languages. If the number of persons who speak only English is $\alpha$ and the number of persons who speaks only Hindi is $\beta$, then the eccentricity of the ellipse $25({\beta }^{2}{x}^{2}+{\alpha }^{2}{y}^{2})={\alpha }^{2}{\beta }^{2}$ is
The sum of all those terms, of the arithmetic progression $3,8,13,...,373$, which are not divisible by $3$, is equal to ________.
The relation $R={(a,b):gcd(a,b)=1,2a\neq b,a,b\in \mathbb{Z}}$ is:
For the system of linear equations $2x-y+3z=5$ $3x+2y-z=7$ $4x+5y+\alpha z=\beta$, which of the following is NOT correct?
Let $\left\{a_k\right\}$ and $\left\{b_k\right\}, k \in \mathbb{N}$, be two G.P.s with common ratio $r_1$ and $r_2$ respectively such that $\mathrm{a}_1=\mathrm{b}_1=4$ and $\mathrm{r}_1<\mathrm{r}_2$. Let $\mathrm{c}_{\mathrm{k}}=\mathrm{a}_{\mathrm{k}}+\mathrm{b}_{\mathrm{k}}, \mathrm{k} \in \mathbb{N}$. If $\mathrm{c}_2=5$ and $\mathrm{c}_3=\frac{13}{4}$ then $\sum_{\mathrm{k}=1}^{\infty} \mathrm{c}_{\mathrm{k}}-\left(12 \mathrm{a}_6+8 \mathrm{~b}_4\right)$ is equal to
Let $f:(0,1)\rightarrow \mathbb{R}$ be a function defined by $f(x)=\frac{1}{1-{e}^{-x}}$, and $g(x)=(f(-x)-f(x))$. Consider two statements (I) $g$ is an increasing function in $(0,1)$ (II) $g$ is one-one in $(0,1)$ Then,
Fractional part of the number $\frac{{4}^{2022}}{15}$ is equal to
Let $S={z\in \mathbb{C}:\bar{z}=i({z}^{2}+Re(\bar{z}))}$. Then $\underset{z\in S}{\sum }|z{|}^{2}$ is equal to
Let $\alpha =8-14i,A={z\in \mathbb{C}:\frac{\alpha z-\bar{\alpha }\bar{z}}{{z}^{2}-(\bar{z}{)}^{2}-112i}=1}$ and $B={z\in \mathbb{C}:|z+3i|=4}$ Then, $\underset{z\in A\cap B}{\sum }(Rez-Imz)$ is equal to ________
The remainder when ${19}^{200}+{23}^{200}$ is divided by $49$, is _____ .
The sum to $20$ terms of the series $2\cdot {2}^{2}-{3}^{2}+2\cdot {4}^{2}-{5}^{2}+2\cdot {6}^{2}-............$ is equal to$__________.$
For the system of equations $x+y+z=6$ $x+2y+\alpha z=10$ $x+3y+5z=\beta$, which one of the following is NOT true?
Suppose Anil's mother wants to give $5$ whole fruits to Anil from a basket of $7$ red apples, $5$ white apples and $8$ oranges. If in the selected $5$ fruits, at least $2$ orange, at least one red apple and at least one white apple must be given, then the number of ways, Anil's mother can offer $5$ fruits to Anil is _____ .
The largest natural number $n$ such that $3n$ divides $66!$ is $_______$
${25}^{190}-{19}^{190}-{8}^{190}+{2}^{190}$ is divisible by
Let $A={1,3,4,6,9}$ and $B={2,4,5,8,10}$. Let $R$ be a relation defined on $A\times B$ such that $R={({a}_{1},{b}_{1}),({a}_{2},{b}_{2}):{a}_{1}\leq {b}_{2}\text{ and}{b}_{1}\leq {a}_{2}}$. Then the number of elements in the set $R$ is
If the system of equations $2x+y-z=5$ $2x-5y+\lambda z=\mu$ $x+2y-5z=7$ has infinitely many solutions, then$(\lambda +\mu {)}^{2}+(\lambda -\mu {)}^{2}$ is equal to
Let $R$ be a relation on $N\times N$ defined by $(a,b)R(c,d)$ if and only if $ad(b-c)=bc(a-d)$. Then $R$ is
Let $m$ and $n$ be the numbers of real roots of the quadratic equations ${x}^{2}-12x+[x]+31=0$ and ${x}^{2}-5|x+2|-4=0$ respectively, where $[x]$ denotes the greatest integer $\leq x$. Then ${m}^{2}+mn+{n}^{2}$ is equal to
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a function defined by $f(x)={\mathrm{log}}_{\sqrt{m}}{\sqrt{2}(\mathrm{sin}x-\mathrm{cos}x)+m-2}$, for some $m$, such that the range of $f$ is $[0,2]$. Then the value of $m$ is _____ .
Let $\lambda \neq 0$ be a real number. Let $\alpha ,\beta$ be the roots of the equation $14{x}^{2}-31x+3\lambda =0$ and $\alpha ,\gamma$ be the roots of the equation $35{x}^{2}-53x+4\lambda =0$. Then $\frac{3\alpha }{\beta }$ and $\frac{4\alpha }{\gamma }$ are the roots of the equation :
Consider the following system of questions $\alpha x+2y+z=1$ $2\alpha x+3y+z=1$ $3x+\alpha y+2z=\beta$ For some $\alpha ,\beta \in \mathbb{R}$. Then which of the following is NOT correct.