Algebra PYQ — Page 58

If $Re(\frac{z-1}{2z+i})=1,$ where $z=x+iy,$ then the point $(x,y)$ lies on a

Two families with three members each and one family with four members are to be seated in a row. In how many ways can they be seated so that the same family members are not separated ?

The number of $4$ letter words (with or without meaning) that can be formed from the eleven letters of the word $\mathrm{EXAMINATION}$ is

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

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 ${(\frac{1+i}{1-i})}^{\frac{m}{2}}={(\frac{1+i}{i-1})}^{\frac{n}{3}}=1,(m,n\in N)$ then the greatest common divisor of the least values of $m$ and $n$ is

If the term independent of $x$ in the expansion of ${(\frac{3}{2}{x}^{2}-\frac{1}{3x})}^{9}$ is $k$, then $18k$ is equal to:

If $f(x+y)=f(x)f(y)$ and $\Sigma _{x=1}^{\infty }f(x)=2,x,y\in N$, where $N$ is the set of all natural numbers, then the value of $\frac{f(4)}{f(2)}$ is

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?

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 number of ordered pairs $(r,k)$ for which $6.{C}_{r}35=({k}^{2}-3).{C}_{r+1}36,$ where $k$ is an integer is

There are $3$ sections in a question paper and each section contains $5$ questions. A candidate has to answer a total of $5$ questions, choosing at least one question from each section. Then the number of ways, in which the candidate can choose the questions, is:

If $z$ is a complex number satisfying $|Re(z)|+|Im(z)|=4,$ then $|z|$ cannot be

If the system of linear equations, $x+y+z=6$ $x+2y+3z=10$ $3x+2y+\lambda z=\mu$ has more than two solutions, then $\mu -{\lambda }^{2}$, is equal to.

Let $a-2b+c=1.$ If $f(x)=|\begin{matrix}x+a & x+2 & x+1 \\ x+b & x+3 & x+2 \\ x+c & x+4 & x+3\end{matrix}|,$ then:

Let $\cup _{i=1}^{50}{X}_{i}=\cup _{i=1}^{n}{Y}_{i}=T$, where each ${X}_{i}$ contains $10$ elements and each ${Y}_{i}$ contains $5$ elements. If each element of the set $T$ is an element of exactly $20$ of sets ${X}_{i}$'s and exactly $6$ of sets ${Y}_{i}$'s then $n$ is equal to :

A test consists of $6$ multiple choice questions, each having $4$ alternative answers of which only one is correct. The number of ways, in which a candidate answers all six questions such that exactly four of the answers are correct, is ___________

If $a+x=b+y=c+z+1,$ where $a,b,c,x,y,z$ are non-zero distinct real numbers, then$|\begin{matrix}x & a+y & x+a \\ y & b+y & y+b \\ z & c+y & z+c\end{matrix}|$ is equal to :

If $1+(1-{2}^{2}\cdot 1)+(1-{4}^{2}\cdot 3)+(1-{6}^{2}\cdot 5)+\ldots \ldots +(1-{20}^{2}\cdot 19)=\alpha -220\beta$, then an ordered pair $(\alpha ,\beta )$ is equal to:

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

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:

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

Let $A=[{a}_{ij}]$ and $B=[{b}_{ij}]$ be two $3\times 3$ real matrices such that ${b}_{ij}={(3)}^{(i+j-2)}{a}_{ij}$ , where $i,j=1,2,3$ . If the determinant of $B$ is $81$ , then determinant of $A$ i s

If the minimum and the maximum values of the function $f:[\frac{\pi }{4},\frac{\pi }{2}]\rightarrow R,$ defined by $f(\theta )=|\begin{matrix}-{\mathrm{sin}}^{2}\theta & -1-{\mathrm{sin}}^{2}\theta & 1 \\ -{\mathrm{cos}}^{2}\theta & -1-{\mathrm{cos}}^{2}\theta & 1 \\ 12 & 10 & -2\end{matrix}|$ are $m$ and $M$respectively, then the ordered pair $(m,M)$ is equal to :