Algebra PYQ — Page 21

Let $A=\{2,3,6,7\}$ and $B=\{4,5,6,8\}$. Let $R$ be a relation defined on $A \times B$ by $\left(a_1, b_1\right) R\left(a_2, b_2\right)$ if and only if $a_1+a_2=b_1+b_2$. Then the number of elements in $R$ is _________

Let $\alpha \in(0, \infty)$ and $A=\left[\begin{array}{lll}1 & 2 & \alpha \\ 1 & 0 & 1 \\ 0 & 1 & 2\end{array}\right]$. If $\operatorname{det}\left(\operatorname{adj}\left(2 A-A^T\right) \cdot \operatorname{adj}\left(A-2 A^T\right)\right)=2^8$, then $(\operatorname{det}(A))^2$ is equal to:

Let $\alpha \beta \neq 0$ and $A=\left[\begin{array}{rrr}\beta & \alpha & 3 \\ \alpha & \alpha & \beta \\ -\beta & \alpha & 2 \alpha\end{array}\right]$. If $B=\left[\begin{array}{rrr}3 \alpha & -9 & 3 \alpha \\ -\alpha & 7 & -2 \alpha \\ -2 \alpha & 5 & -2 \beta\end{array}\right]$ is the matrix of cofactors of the elements of $A$, then $\operatorname{det}(A B)$ is equal to :

Let $A$ and $B$ be two square matrices of order 3 such that $|A|=3$ and $|B|=2$. Then $\left|\mathrm{A}^{\mathrm{T}} \mathrm{A}(\operatorname{adj}(2 \mathrm{~A}))^{-1}(\operatorname{adj}(4 \mathrm{~B}))(\operatorname{adj}(\mathrm{AB}))^{-1} \mathrm{AA}^{\mathrm{T}}\right|$ is equal to :

Let $a$ and $b$ be two distinct positive real numbers. Let ${11}^{\text{th }}$ term of a GP, whose first term is $a$ and third term is $b$, is equal to ${p}^{\text{th }}$ term of another GP, whose first term is $a$ and fifth term is $b$. Then $p$ is equal to

Let ${z}_{1}$ and ${z}_{2}$ be two complex number such that ${z}_{1}+{z}_{2}=5$ and ${z}_{1}^{3}+{z}_{2}^{3}=20+15i$. Then $|{z}_{1}^{4}+{z}_{2}^{4}|$ equals-

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 $\alpha$ and $\beta$ be the sum and the product of all the non-zero solutions of the equation $(\bar{z})^2+|z|=0$, $z \in$ C. Then $4\left(\alpha^2+\beta^2\right)$ is equal to :

Let $\alpha$ and $\beta$ be the roots of the equation $p{x}^{2}+qx-r=0$, where $p\neq 0$. If $p,q$ and $r$ be the consecutive terms of a non-constant G.P and $\frac{1}{\alpha }+\frac{1}{\beta }=\frac{3}{4}$, then the value of ${(\alpha -\beta )}^{2}$ 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 :

Let $f:R\rightarrow R$ and $g:R\rightarrow R$ be defined as $f(x)={\begin{matrix}{\mathrm{log}}_{e}x, & x>0 \\ {e}^{-x}, & x\leq 0\end{matrix}$ and $g(x)={\begin{matrix}x, & x\geq 0 \\ {e}^{x}, & x<0\end{matrix}$. Then, $gof:R\rightarrow R$ is:

Let $A={1,2,3,4}$ and $R={(1,2),(2,3),(1,4)}$ be a relation on $A$. Let $S$ be the equivalence relation on $A$ such that $R\subset S$ and the number of elements in $S$ is $n$. Then, the minimum value of $n$ is _______

Let $B=\left[\begin{array}{ll}1 & 3 \\ 1 & 5\end{array}\right]$ and $A$ be a $2 \times 2$ matrix such that $A B^{-1}=A^{-1}$. If $B C B^{-1}=A$ and $C^4+\alpha C^2+\beta I=O$, then $2 \beta-\alpha$ is equal to

Let $f(x)=x^5+2 x^3+3 x+1, x \in \mathbf{R}$, and $g(x)$ be a function such that $g(f(x))=x$ for all $x \in \mathbf{R}$. Then $\frac{g(7)}{g^{\prime}(7)}$ 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 a relation $\mathrm{R}$ on $\mathrm{N} \times N$ be defined as: $\left(x_1, y_1\right) \mathrm{R}\left(x_2, y_2\right)$ if and only if $x_1 \leq x_2$ or $y_1 \leq y_2$. Consider the two statements: (I) $\mathrm{R}$ is reflexive but not symmetric. (II) $R$ is transitive Then which one of the following is true?

Let A be a $3\times 3$ real matrix such that $A(\begin{matrix}1 \\ 0 \\ 1\end{matrix})=2(\begin{matrix}1 \\ 0 \\ 1\end{matrix}),A(\begin{matrix}-1 \\ 0 \\ 1\end{matrix})=4(\begin{matrix}-1 \\ 0 \\ 1\end{matrix}),A(\begin{matrix}0 \\ 1 \\ 0\end{matrix})=2(\begin{matrix}0 \\ 1 \\ 0\end{matrix})$. Then, the system $(A-3I)(\begin{matrix}x \\ y \\ z\end{matrix})=(\begin{matrix}1 \\ 2 \\ 3\end{matrix})$ has

In the expansion of $(1+x)(1-{x}^{2}){(1+\frac{3}{x}+\frac{3}{{x}^{2}}+\frac{1}{{x}^{3}})}^{5},x\neq 0$, the sum of the coefficient of ${x}^{3}$ and ${x}^{-13}$ is equal to ______

In an increasing geometric progression of positive terms, the sum of the second and sixth terms is $\frac{70}{3}$ and the product of the third and fifth terms is 49 . Then the sum of the $4^{\text {th }}, 6^{\text {th }}$ and $8^{\text {th }}$ terms is equal to :

In an examination of Mathematics paper, there are $20$ questions of equal marks and the question paper is divided into three sections : $A,B$ and $C$. A student is required to attempt total $15$ questions taking at least $4$ questions from each section. If section $A$ has $8$ questions, section $B$ has $6$ questions and section $C$ has $6$ questions, then the total number of ways a student can select $15$ questions is _________.

In an A.P., the sixth term ${a}_{6}=2$. If the ${a}_{1}{a}_{4}{a}_{5}$ is the greatest, then the common difference of the A.P., is equal to

In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let $\mathrm{m}$ and $\mathrm{n}$ respectively be the least and the most number of students who studied all the three subjects. Then $\mathrm{m}+\mathrm{n}$ is equal to ______

If $S=\{a \in \mathbf{R}:|2 a-1|=3[a]+2\{a\}\}$, where $[t]$ denotes the greatest integer less than or equal to $t$ and $\{t\}$ represents the fractional part of $t$, then $72 \sum_{a \in S} a$ is equal to ______

If $1+\frac{\sqrt{3}-\sqrt{2}}{2 \sqrt{3}}+\frac{5-2 \sqrt{6}}{18}+\frac{9 \sqrt{3}-11 \sqrt{2}}{36 \sqrt{3}}+\frac{49-20 \sqrt{6}}{180}+\ldots$ upto $\infty=2+\left(\sqrt{\frac{b}{a}}+1\right) \log _e\left(\frac{a}{b}\right)$, where $\mathrm{a}$ and $\mathrm{b}$ are integers with $\operatorname{gcd}(\mathrm{a}, \mathrm{b})=1$, then $11 \mathrm{a}+18 \mathrm{~b}$ is equal to ______