Algebra PYQ — Page 59

Total number of $6-$ digit numbers in which only and all the five digits $1,3,5,7$ and $9$ appears, is

Let A be a $3\times 3$ matrix such that $adjA=[\begin{matrix}2 & -1 & 1 \\ -1 & 0 & 2 \\ 1 & -2 & -1\end{matrix}]$ and $B=adj(adjA)$. If $|A|=\lambda$ and $|{({B}^{-1})}^{⊤}|=\mu$, then the ordered pair $(|\lambda |,\mu )$ is equal to

Let $a,b,c,d\text{and}p$ be non-zero distinct real numbers such that $({a}^{2}+{b}^{2}+{c}^{2}){p}^{2}-2(ab+bc+cd)p+({b}^{2}+{c}^{2}+{d}^{2})=0$. Then

If $g(x)={x}^{2}+x-1$ and $(gof)(x)=4{x}^{2}-10x+5,$ then $f(\frac{5}{4})$ is equal to

Let $\alpha$ and $\beta$ be the roots of the equation ${x}^{2}-x-1=0$ . If ${p}_{k}={(\alpha )}^{k}+{(\beta )}^{k},k\geq 1,$ then which one of the following statements is not true?

Let $\alpha$ and $\beta$ be two real roots of the equation $(k+1){tan}^{2}x-\sqrt{2}\cdot \lambda \mathrm{tan}⁡x=(1-k),$ where $k(\neq -1)$ and $\lambda$ are real numbers. If ${tan}^{2}(\alpha +\beta )=50,$ then a value of $\lambda$ is

If the equation ${x}^{2}+bx+45=0,b\in R$ has conjugate complex roots and they satisfy $|z+1|=2\sqrt{10},$ then

The coefficient of ${x}^{4}$ in the expansion of ${(1+x+{x}^{2}+{x}^{3})}^{6}$ in powers of $x,$ is $\ldots ..$

If $A=[\begin{matrix}1 & 1 & 2 \\ 1 & 3 & 4 \\ 1 & -1 & 3\end{matrix}],B=adjA$ and $C=3A,$ then $\frac{|adjB|}{|C|}$ is equal to

The total number of $3-$digit numbers whose sum of digits is $10$, is ..........

The region represented by ${z=x+iy\in C:|z|-Re(z)\leq 1}$ is also given by the inequality

If $\alpha$ and $\beta$ are the roots of the equation, $7{x}^{2}-3x-2=0,$ then the value of$\frac{\alpha }{1-{\alpha }^{2}}+\frac{\beta }{1-{\beta }^{2}}$ is equal to:

Five numbers are in $A.P.,$ whose sum is $25$ and product is $2520.$ If one of these five numbers is $-\frac{1}{2},$ then the greatest number amongst them is

If the sum of the second, third and fourth terms of a positive term G.P. is $3$ and the sum of its sixth, seventh and eighth terms is $243$, then the sum of the first $50$ terms of this G.P. is :

If ${3}^{2\mathrm{sin}2\alpha -1},14$ and ${3}^{4-2\mathrm{sin}2\alpha }$ are the first three terms of an A.P. for some $\alpha$ , then the sixth term of this A.P. is

Let $A$ be a $2\times 2$ real matrix with entries from ${0,1}$ and $|A|\neq 0$. Consider the following two statements; $(P)$ If $A\neq {l}_{2}$, then $|A|=-1$ $(Q)$ If $|A|=1$, then $tr(A)=2$ Where ${l}_{2}$ denotes $2\times 2$ identity matrix and $tr(A)$ denotes the sum of the diagonal entries of $A$. Then

Let $f:R\rightarrow R$ be such that for all $x\in R({2}^{1+x}+{2}^{1-x}),f(x)$ and $({3}^{x}+{3}^{-x})$ are in A.P., then the minimum value of $f(x)$ is

If $|x|<1,|y|<1$ and $x\neq 1$, then the sum to infinity of the following series $(x+y)+({x}^{2}+xy+{y}^{2})+({x}^{3}+{x}^{2}y+x{y}^{2}+{y}^{3})+.....$ is

The sum, $\sum _{n=1}^{7}\frac{n(n+1)(2n+1)}{4}$, is equal to

Let $S$ be the set of all $\lambda \in R$ for which the system of linear equations $2x-y+2z=2$ $x-2y+\lambda z=-4$ $x+\lambda y+z=4$ has no solution. Then the set $S$

The inverse function of $f(x)=\frac{{8}^{2x}-{8}^{-2x}}{{8}^{2x}+{8}^{-2x}},x\in (-1,1),$ is __________.

The value of ${(\frac{1+\mathrm{sin}\frac{2\pi }{9}+i\mathrm{cos}\frac{2\pi }{9}}{1+\mathrm{sin}\frac{2\pi }{9}-i\mathrm{cos}\frac{2\pi }{9}})}^{3}$ is

The number of all $3\times 3$ matrices $A,$ with entries from the set ${-1,0,1}$ such that the sum of the diagonal elements of $A{A}^{T}$ is $3,$ is ___________.

The sum of distinct values of $\lambda$ for which the system of equations : $(\lambda -1)x+(3\lambda +1)y+2\lambda z=0$ $(\lambda -1)x+(4\lambda -2)y+(\lambda +3)z=0$ $2x+(3\lambda +1)y+3(\lambda -1)z=0$, Has non-zero solutions, is ....... .