Algebra PYQ — Page 18

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?

If all the words with or without meaning made using all the letters of the word "NAGPUR" are arranged as in a dictionary, then the word at $315^{\text {th }}$ position in this arrangement is :

Let $0 \leq \mathrm{r} \leq \mathrm{n}$. If ${ }^{\mathrm{n}+1} \mathrm{C}_{\mathrm{r}+1}:{ }^n \mathrm{C}_{\mathrm{r}}:{ }^{\mathrm{n}-1} \mathrm{C}_{\mathrm{r}-1}=55: 35: 21$, then $2 \mathrm{n}+5 \mathrm{r}$ is equal to:

If the term independent of $x$ in the expansion of $\left(\sqrt{\mathrm{a}} x^2+\frac{1}{2 x^3}\right)^{10}$ is 105 , then $\mathrm{a}^2$ is equal to :

If the coefficients of $x^4, x^5$ and $x^6$ in the expansion of $(1+x)^n$ are in the arithmetic progression, then the maximum value of $n$ is:

If $\alpha$ denotes the number of solutions of ${|1-i|}^{x}={2}^{x}$ and $\beta =(\frac{|z|}{\mathrm{arg}(z)})$, where $z=\frac{\pi }{4}{(1+i)}^{4}(\frac{1-\sqrt{\pi }\cdot i}{\sqrt{\pi }+i}+\frac{\sqrt{\pi }-i}{1+\sqrt{\pi }\cdot i})$, $i=\sqrt{-1}$, then the distance of the point $(\alpha ,\beta )$ from the line $4x-3y=7$ is ______

If $f(x)=|\begin{matrix}2{\mathrm{cos}}^{4}x & 2{\mathrm{sin}}^{4}x & 3+{\mathrm{sin}}^{2}2x \\ 3+2{\mathrm{cos}}^{4}x & 2{\mathrm{sin}}^{4}x & {\mathrm{sin}}^{2}2x \\ 2{\mathrm{cos}}^{4}x & 3+2{\mathrm{sin}}^{4}x & {\mathrm{sin}}^{2}2x\end{matrix}|$ then $\frac{1}{5}{f}^{'}(0)$ is equal to ________.

The values of $\alpha$, for which $|\begin{matrix}1 & \frac{3}{2} & \alpha +\frac{3}{2} \\ 1 & \frac{1}{3} & \alpha +\frac{1}{3} \\ 2\alpha +3 & 3\alpha +1 & 0\end{matrix}|=0$, lie in the interval

For $\alpha, \beta \in \mathbb{R}$ and a natural number $n$, let $A_r=\left|\begin{array}{ccc}r & 1 & \frac{n^2}{2}+\alpha \\ 2 r & 2 & n^2-\beta \\ 3 r-2 & 3 & \frac{n(3 n-1)}{2}\end{array}\right|$. Then

Let ${S}_{n}$ denote the sum of the first n terms of an arithmetic progression. If ${S}_{10}=390$ and the ratio of the tenth and the fifth terms is $15:7$, then ${S}_{15}-{S}_{5}$ is equal to:

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

Consider the matrix $f(x)=[\begin{matrix}\mathrm{cos}x & -\mathrm{sin}x & 0 \\ \mathrm{sin}x & \mathrm{cos}x & 0 \\ 0 & 0 & 1\end{matrix}]$. Given below are two statements : Statement I:$f(-x)$ is the inverse of the matrix $f(x)$. Statement II: $f(x)f(y)=f(x+y)$. In the light of the above statements, choose the correct answer from the options given below

Consider the matrices : $A=\left[\begin{array}{ll}2 & -5 \\ 3 & m\end{array}\right], B=\left[\begin{array}{l}20 \\ m\end{array}\right]$ and $X=\left[\begin{array}{l}x \\ y\end{array}\right]$. Let the set of all $m$, for which the system of equations $A X=B$ has a negative solution (i.e., $x < 0$ and $y < 0$ ), be the interval $(a, b)$. Then $8 \int_a^b|A| d m$ is equal to_________

If $f(x)={\begin{matrix}2+2x,-1\leq x<0 \\ 1-\frac{x}{3},0\leq x\leq 3\end{matrix};g(x)={\begin{matrix}-x,-3\leq x\leq 0 \\ x,0<x\leq 1\end{matrix}$, then range of $(f\circ g(x))$ is

Let $A=[\begin{matrix}2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2\end{matrix}]$ and $P=[\begin{matrix}1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 & 1 & 5\end{matrix}]$. The sum of the prime factors of $|{P}^{-1}\mathrm{AP}-2I|$ is equal to

Let $A=\left[\begin{array}{lll}2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b\end{array}\right]$. If $A^3=4 A^2-A-21 I$, where $I$ is the identity matrix of order $3 \times 3$, then $2 a+3 b$ is equal to

If $\alpha ,\beta$ are the roots of the equation, ${x}^{2}-x-1=0$ and ${S}_{n}=2023{\alpha }^{n}+2024{\beta }^{n}$, then

Let $A=\{2,3,6,8,9,11\}$ and $B=\{1,4,5,10,15\}$. Let $R$ be a relation on $A \times B$ defined by $(a, b) R(c, d)$ if and only if $3 a d-7 b c$ is an even integer. Then the relation $R$ is

Let the relations $R_1$ and $R_2$ on the set $X=\{1,2,3, \ldots, 20\}$ be given by $R_1=\{(x, y): 2 x-3 y=2\}$ and $R_2=\{(x, y):-5 x+4 y=0\}$. If $M$ and $N$ be the minimum number of elements required to be added in $R_1$ and $R_2$, respectively, in order to make the relations symmetric, then $M+N$ equals

Let $A={1,2,3,...20}$. Let ${R}_{1}$ and ${R}_{2}$ two relation on $A$ such that ${R}_{1}={(a,b):b$ is divisible by $a$} ${R}_{2}={(a,b):a$ is an integral multiple of $b$} Then, number of elements in ${R}_{1}-{R}_{2}$ is equal to __________.

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 $A={1,2,3,....100}$. Let $R$ be a relation on $A$ defined by $(x,y)\in R$ if and only if $2x=3y$. Let ${R}_{1}$ be a symmetric relation on $A$ such that $R\subset {R}_{1}$ and the number of elements in ${R}_{1}$ is $n$. Then the minimum value of $n$ is _______.

Let $S={1,2,3,\ldots ,10}$. Suppose $M$ is the set of all the subsets of $S$, then the relation $R={(A,B):A\cap B\neq \phi ;A,B\in M}$ 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