Algebra PYQ — Page 10

Let $\mathrm{T}_{\mathrm{r}}$ be the $\mathrm{r}^{\text {th }}$ term of an A.P. If for some $\mathrm{m}, \mathrm{T}_{\mathrm{m}}=\frac{1}{25}, \mathrm{~T}_{25}=\frac{1}{20}$, and $20 \sum_{\mathrm{r}=1}^{25} \mathrm{~T}_{\mathrm{r}}=13$, then $5 \mathrm{~m} \sum_{\mathrm{r}=\mathrm{m}}^{2 \mathrm{~m}} \mathrm{~T}_{\mathrm{r}}$ is equal to

Let $\mathrm{R}=\{(1,2),(2,3),(3,3)\}$ be a relation defined on the set $\{1,2,3,4\}$. Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is:

Let $I$ be the identity matrix of order $3 \times 3$ and for the matrix $\mathrm{A}=\left[\begin{array}{ccc}\lambda & 2 & 3 \\ 4 & 5 & 6 \\ 7 & -1 & 2\end{array}\right],|\mathrm{A}|=-1$. Let B be the inverse of the matrix $\operatorname{adj}\left(\mathrm{A} \operatorname{adj}\left(\mathrm{A}^2\right)\right)$. Then $|(\lambda B+1)|$ is equal to _____

Let $A=\left[\begin{array}{cc}\alpha & -1 \\ 6 & \beta\end{array}\right], \alpha \gt 0$, such that $\operatorname{det}(A)=0$ and $\alpha+\beta=1$. If I denotes $2 \times 2$ identity matrix, then the matrix $(1+\mathrm{A})^8$ is:

From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is ' M ', is :

Let $A=\left[a_{i j}\right]$ be $3 \times 3$ matrix such that $A\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right]=\left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right], A\left[\begin{array}{l}4 \\ 1 \\ 3\end{array}\right]=\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right]$ and $A\left[\begin{array}{l}2 \\ 1 \\ 2\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$, then $a_{23}$ equals :

Let $A=\{1,2,3\}$. The number of relations on $A$, containing $(1,2)$ and $(2,3)$, which are reflexive and transitive but not symmetric, is ______ -

Let M denote the set of all real matrices of order $3 \times 3$ and let $\mathrm{S}=\{-3,-2,-1,1,2\}$. Let $$ \begin{aligned} & \mathrm{S}_1=\left\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: \mathrm{A}=\mathrm{A}^{\mathrm{T}} \text { and } a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\right\}, \\ & \mathrm{S}_2=\left\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: \mathrm{A}=-\mathrm{A}^{\mathrm{T}} \text { and } a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\right\}, \\ & \mathrm{S}_3=\left\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: a_{11}+a_{22}+a_{33}=0 \text { and } a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\right\} . \end{aligned} $$ If $n\left(\mathrm{~S}_1 \cup_2 \mathrm{US}_3\right)=125 \alpha$, then $\alpha$ equals _______

Let the matrix $A=\left[\begin{array}{lll}1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0\end{array}\right]$ satisfy $A^n=A^{n-2}+A^2-I$ for $\mathrm{n} \geq 3$. Then the sum of all the elements of $\mathrm{A}^{50}$ is :-

The remainder when $\left((64)^{(64)}\right)^{(64)}$ is divided by 7 is equal to

Consider the sets $\mathrm{A}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{R} \times \mathbb{R}: \mathrm{x}^2+\mathrm{y}^2=25\right\}$, $\mathrm{B}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{R} \times \mathbb{R}: \mathrm{x}^2+9 \mathrm{y}^2=144\right\}, \mathrm{C}=\{(\mathrm{x}, \mathrm{y})$ $\left.\in \mathbb{Z} \times \mathbb{Z}: x^2+y^2 \leq 4\right\}$, and $D=A \cap B$. The total number of one-one functions from the set D to the set C is:

Let $X=\mathbf{R} \times \mathbf{R}$. Define a relation $R$ on $X$ as : $\left(a_1, b_1\right) R\left(a_2, b_2\right) \Leftrightarrow b_1=b_2$ Statement I : $\quad \mathrm{R}$ is an equivalence relation. Statement II : For some $(a, b) \in X$, the set $S=\{(x, y) \in X:(x, y) R(a, b)\}$ represents a line parallel to $y=x$. In the light of the above statements, choose the correct answer from the options given below :

The number of sequences of ten terms, whose terms are either 0 or 1 or 2 , that contain exactly five 1 s and exactly three 2 s , is equal to

The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is

Let $\left(1+x+x^2\right)^{10}=a_0+a_1 x+a_2 x^2+\ldots .+a_{20} x^{20}$. If $\left(a_1+a_3+a_5+\ldots .+a_{19}\right)-11 \mathrm{a}_2=121 \mathrm{k}$, then k is equal to $\qquad$ .

For an integer $\mathrm{n} \geq 2$, if the arithmetic mean of all coefficients in the binomial expansion of $(x+y)^{2 n-3}$ is 16 , then the distance of the point $P\left(2 n-1, n^2-4 n\right)$ from the line $x+y=8$ is:

Let $f(x)+2 f\left(\frac{1}{x}\right)=x^2+5$ and $2 g(x)-3 g\left(\frac{1}{2}\right)=x, x \gt 0$. If $\alpha=\int_1^2 f(x) d x$, and $\beta=\int_1^2 g(x) d x$, then the value of $9 \alpha+\beta$ is:

If the domain of the function $f(x)=\log _e\left(\frac{2 x-3}{5+4 x}\right)+\sin ^{-1}\left(\frac{4+3 x}{2-x}\right) \quad \text { is } \quad[\alpha, \beta)$ then $\alpha^2+4 \beta$ is equal to

If the domain of the function $f(x)=\frac{1}{\sqrt{10+3 x-x^2}}+\frac{1}{\sqrt{x+|x|}}$ is $(a, b)$, then $(1+a)^2+b^2$ is equal to :

If the domain of the function $\log _5\left(18 x-x^2-77\right)$ is $(\alpha, \beta)$ and the domain of the function $\log _{(x-1)}\left(\frac{2 x^2+3 x-2}{x^2-3 x-4}\right)$ is $(\gamma, \delta)$, then $\alpha^2+\beta^2+\gamma^2$ is equal to :

$\text { The number of real solution(s) of the equation } x^2+3 x+2=\min \{|x-3|,|x+2|\} \text { is : }$

Let A be the set of all functions $f: \mathbf{Z} \rightarrow \mathbf{Z}$ and R be a relation on A such that $\mathrm{R}=\{(\mathrm{f}, \mathrm{g}): f(0)=\mathrm{g}(1)$ and $f(1)=\mathrm{g}(0)\}$. Then R is:

Let $O$ be the origin, the point $A$ be $z_1=\sqrt{3}+2 \sqrt{2} i$, the point $B\left(z_2\right)$ be such that $\sqrt{3}\left|z_2\right|=\left|z_1\right|$ and $\arg \left(z_2\right)=\arg \left(z_1\right)+\frac{\pi}{6}$. Then

Let $A$ be a matrix of order $3 \times 3$ and $|A|=5$. If $|2 \operatorname{adj}(3 \mathrm{~A} \operatorname{adj}(2 \mathrm{~A}))|=2^\alpha \cdot 3^\beta \cdot 5^\gamma \alpha, \beta, \gamma \in \mathrm{N}$ then $\alpha+\beta+\gamma$ is equal to