Algebra PYQ — Page 66

Let $A_n=\left(\frac{3}{4}\right)-\left(\frac{3}{4}\right)^2+\left(\frac{3}{4}\right)^3-\ldots+(-1)^{n-1}\left(\frac{3}{4}\right)^n$ and $B_n=1-A_n$. Then, the least odd natural number $p$, so that $B_n>A_n$, for all $n \geq p$ is

Consider the following two binary relations on the set $A={a,b,c}:{R}_{1}={(c,a),(b,b),(a,c),(c,c),(b,c),(a,a)}$ and ${R}_{2}={(a,b),(b,a),(c,c),(c,a),(a,a),(b,b),(a,c)}$, then :

Let $S={x\in R :x\geq 0 & 2|\sqrt{x}-3|+\sqrt{x} (\sqrt{x}-6)+6=0}$ . Then $S$:

Let $A$ be the sum of the first $20$ terms and $B$ be the sum of the first $40$ terms of the series ${1}^{2}+{2\cdot 2}^{2}+{3}^{2}+{2\cdot 4}^{2}+{5}^{2}+{2\cdot 6}^{2}+\ldots$ If $B-2A=100\lambda ,$ then $\lambda$ is equal to :

Consider the following two binary relations on the set $A=\{a, b, c\}: R_1=\{(\mathrm{c}, a)(b, b),(\mathrm{a}, c),(c$, $c),(b, c),(a, a)\}$ and $\mathrm{R}_2=\{(\mathrm{a}, \mathrm{b}),(\mathrm{b}, \mathrm{a}),(\mathrm{c}, \mathrm{c})$, (c, a), (a, a), (b, b), (a, c). Then

Let $S$ be the set of all real values of $k$ for which the system of linear equations $x+y+z=2$ $2x+y-z=3$ $3x+2y+kz=4$ has a unique solution. Then, $S$is :

Let $S$ be the set of all real values of $k$ for which the system of linear equations $$ \begin{aligned} &x+y+z=2 \\ &2 x+y-z=3 \\ &3 x+2 y+k z=4 \end{aligned} $$ has a unique solution. Then $S$ is

If the system of linear equations $$ \begin{aligned} &x+a y+z=3 \\ &x+2 y+2 z=6 \\ &x+5 y+3 z=b \end{aligned} $$ has no solution, then

Let $p,q$ and $r$ be real numbers $(p\neq q,r\neq 0)$, such that the roots of the equation $\frac{1}{x+p}+\frac{1}{x+q}=\frac{1}{r}$ are equal in magnitude but opposite in sign, then the sum of squares of these roots is equal to

The number of numbers between $2,000$ and $5,000$ that can be formed with the digits $0,1,2,3,4$ (repetition of digits is not allowed) and are multiple of $3$ is

Let $N$ denote the set of all natural numbers. Define two binary relations on $N$ as ${R}_{1}={(x,y)\in N\times N:2x+y=10}$ and ${R}_{2}={(x,y)\in N\times N:x+2y=10}$. Then

If $b$ is the first term of an infinite G. P whose sum is five, then $b$ lies in the interval.

Let $A$ be a matrix such that $A\cdot [\begin{matrix}1 & 2 \\ 0 & 3\end{matrix}]$ is a scalar matrix and $|3A|=108$. Then, ${A}^{2}$ equals :

The number of values of $k$ for which the system of linear equations $(k+2)x+10y=k&$ $kx+(k+3)y=k-1$ has no solution is

If $n$ is the degree of the polynomial, ${[\frac{2}{\sqrt{5{x}^{3}+1}-\sqrt{5{x}^{3}-1}}]}^{8}+ {[\frac{2}{\sqrt{5{x}^{3}+1}+\sqrt{5{x}^{3}-1}}]}^{8}$ and $m$ is the coefficient of ${x}^{n}$ in it, then the ordered pair $(n,m)$ is equal to

The sum of the first $20$ terms of the series $1+\frac{3}{2}+\frac{7}{4}+\frac{15}{8}+\frac{31}{16}+\ldots$ is

If $n$ is the degree of the polynomial, $$ \left[\frac{1}{\sqrt{5 x^3+1}-\sqrt{5 x^3-1}}\right]^8+\left[\frac{1}{\sqrt{5 x^3+1}+\sqrt{5 x^3-1}}\right]^8 $$ and $\mathrm{m}$ is the coefficient of $\mathrm{x}^{\mathrm{n}}$ in it, then the ordered pair $(\mathrm{n}, \mathrm{m})$ is equal to

If $f(x)=|\begin{matrix}\mathrm{cos}x & x & 1 \\ 2\mathrm{sin}x & {x}^{2} & 2x \\ \mathrm{tan}x & x & 1\end{matrix}|$, then $\underset{x\rightarrow 0}{\mathrm{lim}}⁡\frac{{f}^{'}(x)}{x}$

If $\lambda \in \mathrm{R}$ is such that the sum of the cubes of the roots of the equation, $x^2+(2-\lambda) x+(10-\lambda)=0$ is minimum, then the magnitude of the difference of the roots of this equation is

If $a, b, c$ are in A.P. and $a^2, b^2, c^2$ are in G.P. such that $a < b < c$ and $a+b+c=\frac{3}{4}$, then the value of $a$ is

$n$ - digit numbers are formed using only three digits 2,5 and 7 . The smallest value of $n$ for which 900 such distinct numbers can be formed, is

Let $A$ be a matrix such that $A$. $\left[\begin{array}{ll}1 & 2 \\ 0 & 3\end{array}\right]$ is a scalar matrix and $|3 A|=108$. Then $A^2$ equals

Let $A=[\begin{matrix}1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1\end{matrix}]$ and $B={A}^{20}$. Then the sum of the elements of the first column of $B$ is

Suppose $A$ is any $3 \times 3$ non-singular matrix and $(A-3 I)(A-5 I)=O$, where $I=I_3$ and $O=O_3$. If $\alpha A+$ $\beta A^{-1}=4 I$, then $\alpha+\beta$ is equal to