JEE Main Mathematics — Algebra previous year questions with solutions.
Let $f:R\rightarrow R$ be defined as $f(x)=2x-1$ and $g:R-{1}\rightarrow R$. be defined as $g(x)=\frac{x-\frac{1}{2}}{x-1}$. Then the composition function $f(g(x))$ is:
The following system of linear equations $2x+3y+2z=9$ $3x+2y+2z=9$ $x-y+4z=8$
The number of pairs $(a,b)$ of real numbers, such that whenever $\alpha$ is a root of the equation ${x}^{2}+ax+b=0,{\alpha }^{2}-2$ is also a root of this equation, is :
If $(\frac{{3}^{6}}{{4}^{4}})k$ is the term, independent of $x,$ in the binomial expansion of ${(\frac{x}{4}-\frac{12}{{x}^{2}})}^{12},$ then $k$ is equal to
If $(2021{)}^{3762}$ is divided by $17,$ then the remainder is _______.
If the fourth term in the expansion of ${(x+{x}^{{\mathrm{log}}_{2}x})}^{7}$ is $4480,$ then the value of $x$ where $x\in N$ is equal to:
A possible value of $x,$ for which the ninth term in the expansion of ${{{3}^{{\mathrm{log}}_{3}\sqrt{{25}^{x-1}+7}}+{3}^{(-\frac{1}{8}){\mathrm{log}}_{3}({5}^{x-1}+1)}}}^{10}$ in the increasing powers of ${3}^{(-\frac{1}{8}){\mathrm{log}}_{3}({5}^{x-1}+1)}$ is equal to $180,$ is :
Let $i=\sqrt{-1}.$ If $\frac{{(-1+i\sqrt{3})}^{21}}{{(1-i)}^{24}}+\frac{{(1+i\sqrt{3})}^{21}}{{(1+i)}^{24}}=k,$ and $n=[|k|]$ be the greatest integral part of $|k|.$ Then $\sum _{j=0}^{n+5}{(j+5)}^{2}-\sum _{j=0}^{n+5}(j+5)$ is equal to ________.
The lowest integer which is greater than ${(1+\frac{1}{{10}^{100}})}^{{10}^{100}}$ is
Let $g:N\rightarrow N$ be defined as $g(3n+1)=3n+2$ $g(3n+2)=3n+3$ $g(3n+3)=3n+1$, for all $n\geq 0$ Then which of the following statements is true ?
Let $n$ denote the number of solutions of the equation ${z}^{2}+3\bar{z}=0,$ where $z$ is a complex number. Then the value of $\sum _{k=0}^{\infty }\frac{1}{{n}^{k}}$ is equal to
Let ${a}_{n}$ be the ${n}^{th}$ term of a G.P. of positive terms. If $\sum _{n=1}^{100}{a}_{2n+1}=200$ and $\sum _{n=1}^{100}{a}_{2n}=100,$ then $\sum _{n=1}^{200}{a}_{n}$ is equal to:
The natural number $m$, for which the coefficient of $x$ in the binomial expansion of ${({x}^{m}+\frac{1}{{x}^{2}})}^{22}$ is 1540, is
Consider the two sets: $A={m\in R:$ both the roots of ${x}^{2}-(m+1)x+m+4=0$ are real $}$ and $B=[-3,5)$ Which of the following is not true?
If the system of equations $x+y+z=2$ $2x+4y-z=6$ $3x+2y+\lambda z=\mu$ has infinitely many solutions, then :
The domain of the function $f(x)={\mathrm{sin}}^{-1}(\frac{|x|+5}{{x}^{2}+1})$ is $(-\infty ,-a]\cup [a,\infty )$, then $a$ is equal to
There are $3$ sections in a question paper and each section contains $5$ questions. A candidate has to answer a total of $5$ questions, choosing at least one question from each section. Then the number of ways, in which the candidate can choose the questions, is:
Suppose the vectors ${x}_{1},{x}_{2}$ and ${x}_{3}$ are the solutions of the system of linear equations, $Ax=b$ when the vector $b$ on the right side is equal to ${b}_{1},{b}_{2}$ and ${b}_{3}$ respectively. If ${x}_{1}=[\begin{matrix}1 \\ 1 \\ 1\end{matrix}],{x}_{2}=[\begin{matrix}0 \\ 2 \\ 1\end{matrix}],{x}_{3}=[\begin{matrix}0 \\ 0 \\ 1\end{matrix}]$; ${b}_{1}=[\begin{matrix}1 \\ 0 \\ 0\end{matrix}],{b}_{2}=[\begin{matrix}0 \\ 2 \\ 0\end{matrix}],{b}_{3}=[\begin{matrix}0 \\ 0 \\ 2\end{matrix}]$, then the determinant of $A$ is equal to
Let $\alpha >0,\beta >0$ be such that ${\alpha }^{3}+{\beta }^{2}=4$. If the maximum value of the term independent of $x$ in the binomial expansion of ${(\alpha {x}^{\frac{1}{9}}+\beta {x}^{-\frac{1}{6}})}^{10}$ is $10k$, then $k$ is equal to
Let ${R}_{1}$ and ${R}_{2}$ be two relations defined as follows :${R}_{1}={(a,b)\in {R}^{2}:{a}^{2}+{b}^{2}\in Q}$ and ${R}_{2}={(a,b)\in {R}^{2}:{a}^{2}+{b}^{2}\notin Q}$, where $Q$ is the set of all rational numbers, then
An urn contains $5$ red marbles, $4$ black marbles and $3$ white marbles. Then, the number of ways in which $4$ marbles can be drawn so that at the most three of them are red is ___________.
If the sum of the first $20$ terms of the series ${\mathrm{log}}_{({7}^{1/2})}x+{\mathrm{log}}_{({7}^{1/3})}x+{\mathrm{log}}_{({7}^{1/4})}x+\ldots$is $460$, then $x$ is equal to:
The value of log₂8 is:
Let $z$ be a complex number such that $|\frac{z-i}{z+2i}|=1$ and $|z|=\frac{5}{2}$ . Then, the value of $|z+3i|$ is