Algebra PYQ — Page 26

Let ${\alpha }_{1},{\alpha }_{2},\ldots ,{\alpha }_{7}$${\alpha }_{1},{\alpha }_{2},\ldots ,{\alpha }_{7}$ be the roots of the equation ${x}^{7}+3{x}^{5}-13{x}^{3}-15x=0$ and $|{\alpha }_{1}|\geq |{\alpha }_{2}|\geq \ldots \geq |{\alpha }_{7}|$. Then, ${\alpha }_{1}{\alpha }_{2}-{\alpha }_{3}{\alpha }_{4}+{\alpha }_{5}{\alpha }_{6}$ is equal to _______

Let $R={a,b,c,d,e}$ and $S={1,2,3,4}$. Total number of onto functions $f:R\rightarrow S$ such that $f(a)\neq 1$, is equal to ________.

Let $w=z\bar{z}+{k}_{1}z+{k}_{2}iz+\lambda (1+i),{k}_{1},{k}_{2}\in \mathbb{R}.$ . Let $Re(w)=0$ be the circle $C$ of radius $1$ in the first quadrant touching the line $y=1$ and the $y-$axis. If the curve $Im(w)=0$ intersects $C$ at $A$ and $B,$ then $30(AB{)}^{2}$ is equal to $_______.$

Let ${w}_{1}$ be the point obtained by the rotation of ${z}_{1}=5+4i$ about the origin through a right angle in the anticlockwise direction, and ${w}_{2}$ be the point obtained by the rotation of ${z}_{2}=3+5i$ about the origin through a right angle in the clockwise direction. Then the principal argument ${w}_{1}-{w}_{2}$ is equal to

Let $S={z=x+iy:\frac{2z-3i}{4z+2i}$ is a real number $}$. Then which of the following is NOT correct?

Let $a,b$ be two real numbers such that $ab<0$. If the complex number $\frac{1+ai}{b+i}$ is of unit modulus and $a+ib$ lies on the circle $|z-1|=|2z|$, then a possible value of $\frac{1+[a]}{4b}$, where $[t]$ is greatest integer function, is :

Let $z$ be a complex number such that $|\frac{z-2i}{z+i}|=2,z\neq -i$. Then $z$ lies on the circle of radius $2$ and centre

For two non-zero complex number ${z}_{1}$ and ${z}_{2}$, if $Re({z}_{1}{z}_{2})=0$ and $Re({z}_{1}+{z}_{2})=0$, then which of the following are possible? (A) $Im({z}_{1})>0$ and $Im({z}_{2})>0$ (B) $Im({z}_{1})<0$ and $Im({z}_{2})>0$ (C) $Im({z}_{1})>0$ and $Im({z}_{2})<0$ (D) $Im({z}_{1})<0$ and $Im({z}_{2})<0$ Choose the correct answer from the options given below:

Let ${z}_{1}=2+3i$ and ${z}_{2}=3+4i$. The set $S={z\in C:{|z-{z}_{1}|}^{2}-{|z-{z}_{2}|}^{2}={|{z}_{1}-{z}_{2}|}^{2}}$ represents a

A person forgets his $4$-digit ATM pin code. But he remembers that in the code all the digits are different, the greatest digit is $7$ and the sum of the first two digits is equal to the sum of the last two digits. Then the maximum number of trials necessary to obtain the correct code is________.

Total numbers of $3$-digit numbers that are divisible by $6$ and can be formed by using the digits $1,2,3,4,5$ with repetition, is ________

If the letters of the word MATHS are permuted and all possible words so formed are arranged as in a dictionary with serial numbers, then the serial number of the word THAMS is

In an examination, $5$ students have been allotted their seats as per their roll numbers. The number of ways, in which none of the students sits on the allotted seat, is

All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary with serial numbers. Then the serial number of the word PUBLIC is

The number of words, with or without meaning, that can be formed using all the letters of the word ASSASSINATION so that the vowels occur together, is _____ .

Number of integral solutions to the equation $x+y+z=21$, where $x\geq 1,y\geq 3,z\geq 4$, is equal to _____ .

The total number of six digit numbers, formed using the digits $4,5,9$ only and divisible by $6$, is _____ .

The number of seven digits odd numbers, that can be formed using all the seven digits $1,2,2,2,3,3,5$ is

Five digit numbers are formed using the digits $1,2,3,5,7$ with repetitions and are written in descending order with serial numbers. For example, the number $77777$ has serial number $1$. Then the serial number of $35337$ is

Let $[\alpha ]$ denote the greatest integer $\leq \alpha$. Then $[\sqrt{1}]+[\sqrt{2}]+[\sqrt{3}]+.............+[\sqrt{120}]$ is equal to

Let $0<z<y<x$ be three real numbers such that $\frac{1}{x},\frac{1}{y},\frac{1}{z}$ are in an arithmetic progression and $x,\sqrt{2}y,z$ are in a geometric progression. If $xy+yz+zx=\frac{3}{\sqrt{2}}xyz$, then $3(x+y+z{)}^{2}$ is equal to

Let ${a}_{1},{a}_{2},{a}_{3},....,{a}_{n}$ be $n$ positive consecutive terms of an arithmetic progression. If $d>0$ is its common difference, then $\underset{n\rightarrow \infty }{\mathrm{lim}}\sqrt{\frac{d}{n}}(\frac{1}{\sqrt{{a}_{1}}+\sqrt{{a}_{2}}}+\frac{1}{\sqrt{{a}_{2}}+\sqrt{{a}_{3}}}+\ldots +\frac{1}{\sqrt{{a}_{n-1}}+\sqrt{{a}_{n}}})$ is

The number of $3$-digit numbers, that are divisible by either $2$ or $3$ but not divisible by $7$ is _____ .

Let ${a}_{1},{a}_{2},\ldots \ldots ,{a}_{n}$ be in A.P. If ${a}_{5}=2{a}_{7}$ and ${a}_{11}=18$, then $12(\frac{1}{\sqrt{{a}_{10}}+\sqrt{{a}_{11}}}+\frac{1}{\sqrt{{a}_{11}}+\sqrt{{a}_{12}}}+\ldots ..\frac{1}{\sqrt{{a}_{17}}+\sqrt{{a}_{18}}})$ is equal to _____ .