JEE Main Mathematics — Algebra previous year questions with solutions.
If the domain of the function $f(x)=\sin ^{-1}\left(\frac{x-1}{2 x+3}\right)$ is $\mathbf{R}-(\alpha, \beta)$, then $12 \alpha \beta$ is equal to :
Let $A$ be a $3 \times 3$ matrix of non-negative real elements such that $A\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=3\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$. Then the maximum value of $\operatorname{det}(\mathrm{A})$ is ______
Let $f:R-{\frac{-1}{2}}\rightarrow R$ and $g:R-{\frac{-5}{2}}\rightarrow R$ be defined as $f(x)=\frac{2x+3}{2x+1}$ and $g(x)=\frac{|x|+1}{2x+5}$. Then the domain of the function $\mathrm{fog}$ is :
A group of $40$ students appeared in an examination of $3$ subjects - Mathematics, Physics & Chemistry. It was found that all students passed in at least one of the subjects, $20$ students passed in Mathematics, $25$ students passed in Physics, $16$ students passed in Chemistry, at most $11$ students passed in both Mathematics and Physics, at most $15$ students passed in both Physics and Chemistry, at most $15$ students passed in both Mathematics and Chemistry. The maximum number of students passed in all the three subjects is _____.
Let $A=[\begin{matrix}2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1\end{matrix}],B=[\begin{matrix}{B}_{1} & {B}_{2} & {B}_{3}\end{matrix}]$, where ${B}_{1}$, ${B}_{2},{B}_{3}$ are column matrices, and ${\mathrm{AB}}_{1}=[\begin{matrix}1 \\ 0 \\ 0\end{matrix}]$, ${\mathrm{AB}}_{2}=[\begin{matrix}2 \\ 3 \\ 0\end{matrix}],{\mathrm{AB}}_{3}=[\begin{matrix}3 \\ 2 \\ 1\end{matrix}]$ If $\alpha =|B|$ and $\beta$ is the sum of all the diagonal elements of $B$, then ${\alpha }^{3}+{\beta }^{3}$ is equal to
The lines ${L}_{1},{L}_{2},...,{L}_{20}$ are distinct. For $n=1,2,3,...,10$ all the lines ${L}_{2n-1}$ are parallel to each other and all the lines ${L}_{2n}$ pass through a given point $P$. The maximum number of points of intersection of pairs of lines from the set ${{L}_{1},{L}_{2},...,{L}_{20}}$ is equal to:
If $\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\ldots+\frac{1}{\sqrt{99}+\sqrt{100}}=m$ and $\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\ldots+\frac{1}{99 \cdot 100}=n$, then the point $(\mathrm{m}, \mathrm{n})$ lies on the line
Let $A=\{1,3,7,9,11\}$ and $B=\{2,4,5,7,8,10,12\}$. Then the total number of one-one maps $f: \mathrm{A} \rightarrow \mathrm{B}$, such that $f(1)+f(3)=14$, is :
Let $f(x)=\left\{\begin{array}{ccc}-\mathrm{a} & \text { if } & -\mathrm{a} \leq x \leq 0 \\ x+\mathrm{a} & \text { if } & 0 < x \leq \mathrm{a}\end{array}\right.$ where $\mathrm{a}>0$ and $\mathrm{g}(x)=(f(x \mid)-|f(x)|) / 2$. Then the function $g:[-a, a] \rightarrow[-a, a]$ is
The area (in sq. units) of the region $S=\{z \in \mathbb{C}:|z-1| \leq 2 ;(z+\bar{z})+i(z-\bar{z}) \leq 2, \operatorname{Im}(z) \geq 0\}$ is
Number of integral terms in the expansion of ${{{7}^{(\frac{1}{2})}+{11}^{(\frac{1}{6})}}}^{824}$ is equal to ______.
Let \(\alpha, \beta \in\) be roots of equation \(x^2-70 x+\lambda=0\), where \(\frac{\lambda}{2}, \frac{\lambda}{3} \notin\). If \(\lambda\) assumes the minimum possible value, then \(\frac{(\sqrt{\alpha-1}+\sqrt{\beta-1})(\lambda+35)}{\boldsymbol{|\alpha-\beta|}}\) is equal to :
Let $3,7,11,15,..,403$ and $2,5,8,11,...,404$ be two arithmetic progressions. Then the sum, of the common terms in them, is equal to_________
Let $[t]$ be the greatest integer less than or equal to $t$. Let $A$ be the set of all prime factors of 2310 and $f: A \rightarrow \mathbb{Z}$ be the function $f(x)=\left[\log _2\left(x^2+\left[\frac{x^3}{5}\right]\right)\right]$. The number of one-to-one functions from $A$ to the range of $f$ is
The number of solutions, of the equation ${e}^{\mathrm{sin}x}-2{e}^{-\mathrm{sin}x}=2$ is
Let $A B C$ be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle $A B C$ and the same process is repeated infinitely many times. If $\mathrm{P}$ is the sum of perimeters and $Q$ is be the sum of areas of all the triangles formed in this process, then :
If $\alpha$ satisfies the equation ${x}^{2}+x+1=0$ and $(1+\alpha {)}^{7}=A+B\alpha +C{\alpha }^{2},A,B,C\geq 0$, then $5(3A-2B-C)$ is equal to
If $\alpha \neq \mathrm{a}, \beta \neq \mathrm{b}, \gamma \neq \mathrm{c}$ and $\left|\begin{array}{lll}\alpha & \mathrm{b} & \mathrm{c} \\ \mathrm{a} & \beta & \mathrm{c} \\ \mathrm{a} & \mathrm{b} & \gamma\end{array}\right|=0$, then $\frac{\mathrm{a}}{\alpha-\mathrm{a}}+\frac{\mathrm{b}}{\beta-\mathrm{b}}+\frac{\gamma}{\gamma-\mathrm{c}}$ is equal to:
Let $S={z\in \mathbb{C}-{i,2i}:\frac{{z}^{2}+8iz-15}{{z}^{2}-3iz-2}\in \mathbb{R}}$. $\alpha -\frac{13}{11}i\in S,\alpha \in \mathbb{R}-{0}$, then $242{\alpha }^{2}$ is equal to
The number of symmetric matrices of order 3, with all the entries from the set ${0,1,2,3,4,5,6,7,8,9}$ is
The number of integers, greater than $7000$ that can be formed, using the digits $3,5,6,7,8$ without repetition is
The number of elements in the set ${n\in \mathbb{Z}:|{n}^{2}-10n+19|<6}$ is _______ .
The remainder when ${(2023)}^{2023}$ is divided by $35$ is
If $\frac{1}{n+1}{}^{n}{C}_{n}+\frac{1}{n}{}^{n}{C}_{n-1}+...+\frac{1}{2}{}^{n}{C}_{1}{+}^{n}{C}_{0}=\frac{1023}{10}$ then $n$ is equal to