JEE Main Mathematics — Algebra previous year questions with solutions.
Let $A={-4,-3,-2,0,1,3,4}$ and $R={(a,b)\in A\times A$ : $b=|a|$ or ${b}^{2}=a+1}$ be a relation on $A$. Then the minimum number of elements, that must be added to the relation $R$ so that it becomes reflexive and symmetric, is
Let $z=a+ib,b\neq 0$ be complex numbers satisfying ${z}^{2}=\bar{z}\cdot {2}^{1-|z|}$. Then the least value of $n\in N$, such that ${z}^{n}={(z+1)}^{n}$, is equal to _____ .
Let for some real numbers $\alpha$ and $\beta ,a=\alpha -i\beta$. If the system of equations $4ix+(1+i)y=0$ and $8(\mathrm{cos}\frac{2\pi }{3}+i\mathrm{sin}\frac{2\pi }{3})x+\bar{a}y=0$ has more than one solution then $\frac{\alpha }{\beta }$ is equal to
Sum of squares of modulus of all the complex numbers $z$ satisfying $\bar{z}=i{z}^{2}+{z}^{2}-z$ is equal to
Let $f:R\rightarrow R$ be a continuous function such that $f(3x)-f(x)=x$. If $f(8)=7$, then $f(14)$ is equal to:
The number of one-one functions $f:{a,b,c,d}\rightarrow {0,1,2,\ldots ,10}$ such that $2f(a)-f(b)+3f(c)+f(d)=0$ is _____
There are ten boys ${B}_{1},{B}_{2},\ldots .,{B}_{10}$ and five girls ${G}_{1},{G}_{2},\ldots .{G}_{5}$ in a class. Then the number of ways of forming a group consisting of three boys and three girls, if both ${B}_{1}$ and ${B}_{2}$ together should not be the members of a group, is _____.
Let $\alpha ,\beta$ and $\gamma$ be three positive real numbers. Let $f(x)=\alpha {x}^{5}+\beta {x}^{3}+\gamma x,x\in R$ and $g:R\rightarrow R$ be such that $g(f(x))=x$ for all $x\in R$. If ${a}_{1},{a}_{2},{a}_{3},\ldots ,{a}_{n}$ be in arithmetic progression with mean zero, then the value of $f(g(\frac{1}{n}\sum _{i=1}^{n}f({a}_{i})))$ is equal to
The remainder when ${3}^{2022}$ is divided by $5$ is
Let $A={x\in R:|x+1|<2}$ and $B={x\in R:|x-1|\geq 2}$. Then which one the following statements is NOT true?
Let $f(x)=2{x}^{2}-x-1$ and $S={n\in \mathbb{Z}:|f(n)|\leq 800}$. Then, the value of $\underset{n\in S}{\sum }f(n)$ is equal to _______.
Let $A$ be a $2 \times 2$ matrix with $\operatorname{det}(A)=-1$ and $\operatorname{det}((A+I)(\operatorname{Adj}(A)+I))=4$. Then the sum of the diagonal elements of $A$ can be:
Different A.P.'s are constructed with the first term $100$, the last term $199$, And integral common differences. The sum of the common differences of all such, A.P's having at least $3$ terms and at most $33$ terms is.
Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of ${(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}})}^{n}$, in the increasing powers of $\frac{1}{\sqrt[4]{3}}$ be $\sqrt[4]{6}:1$. If the sixth term from the beginning is $\frac{\alpha }{\sqrt[4]{3}}$, then $\alpha$ is equal to _______.
In an examination, there are $5$ multiple choice questions with $3$ choices, out of which exactly one is correct. There are $3$ marks for each correct answer, $-2$ marks for each wrong answer and $0$ mark if the question is not attempted. Then, the number of ways a student appearing in the examination gets $5$ marks is _____
The total number of functions, $f:{1,2,3,4}\rightarrow {1,2,3,4,5,6}$ such that $f(1)+f(2)=f(3)$, is equal to
If $\alpha ,\beta ,\gamma ,\delta$ are the roots of the equation ${x}^{4}+{x}^{3}+{x}^{2}+x+1=0$, then ${\alpha }^{2021}+{\beta }^{2021}+{\gamma }^{2021}+{\delta }^{2021}$ is equal to
Let $f(x)=a{x}^{2}+bx+c$ be such that $f(1)=3,f(-2)=\lambda$ and $f(3)=4$. If $f(0)+f(1)+f(-2)+f(3)=14$, then $\lambda$ is equal to
Let $\alpha ,\beta$ be the roots of the equation ${x}^{2}-\sqrt{2}x+\sqrt{6}=0$ and $\frac{1}{{\alpha }^{2}}+1,\frac{1}{{\beta }^{2}}+1$ be the roots of the equation ${x}^{2}+ax+b=0$. Then the roots of the equation ${x}^{2}-(a+b-2)x+(a+b+2)=0$ are :
The number of $3$-digit odd numbers, whose sum of digits is a multiple of $7$, is _____.
Let $S={z\in \mathbb{C}:{z}^{2}+\bar{z}=0}$. Then $\underset{z\in S}{\sum }(Re(z)+Im(z))$ is equal to _______.
Let $\alpha$ be a root of the equation $1+{x}^{2}+{x}^{4}=0$. Then the value of ${\alpha }^{1011}+{\alpha }^{2022}-{\alpha }^{3033}$ is equal to:
The total number of three-digit numbers, with one digit repeated exactly two times, is ______.
Let $f(x)=\frac{x-1}{x+1},x\in R-{0,-1,1)$ . If ${f}^{n+1}(x)=f({f}^{n}(x))$ for all $n\in N$, then ${f}^{6}(6)+{f}^{7}(7)$ is equal to