Algebra PYQ — Page 21

Let $\alpha$ and $\beta$ be the sum and the product of all the non-zero solutions of the equation $(\bar{z})^2+|z|=0$, $z \in$ C. Then $4\left(\alpha^2+\beta^2\right)$ is equal to :

If $z=x+iy,xy\neq 0$, satisfies the equation ${z}^{2}+i\bar{z}=0$, then $|{z}^{2}|$ is equal to :

If $S=z\in C:|z-i|=|z+i|=|z-1|$, then, $n(S)$ is:

The number of ways five alphabets can be chosen from the alphabets of the word MATHEMATICS, where the chosen alphabets are not necessarily distinct, is equal to :

The number of integers, between 100 and 1000 having the sum of their digits equals to 14 , is _________

60 words can be made using all the letters of the word BHBJO, with or without meaning. If these words are written as in a dictionary, then the $50^{\text {th }}$ word is :

The number of ways of getting a sum 16 on throwing a dice four times is______

There are 5 points $P_1, P_2, P_3, P_4, P_5$ on the side $A B$, excluding $A$ and $B$, of a triangle $A B C$. Similarly there are 6 points $\mathrm{P}_6, \mathrm{P}_7, \ldots, \mathrm{P}_{11}$ on the side $\mathrm{BC}$ and 7 points $\mathrm{P}_{12}, \mathrm{P}_{13}, \ldots, \mathrm{P}_{18}$ on the side $C A$ of the triangle. The number of triangles, that can be formed using the points $\mathrm{P}_1, \mathrm{P}_2, \ldots, \mathrm{P}_{18}$ as vertices, is :

Number of ways of arranging $8$ identical books into $4$ identical shelves where any number of shelves may remain empty is equal to

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 $r$ and $\theta$ respectively be the modulus and amplitude of the complex number $z=2-i(2\mathrm{tan}\frac{5\pi }{8})$, then $(r,\theta )$ is equal to

Let $a, a r, a r^2$, $\qquad$ be an infinite G.P. If $\sum_{n=0}^{\infty} a r^n=57$ and $\sum_{n=0}^{\infty} a^3 r^{3 n}=9747$, then $a+18 r$ is equal to

If the sum of the series $\frac{1}{1 \cdot(1+\mathrm{d})}+\frac{1}{(1+\mathrm{d})(1+2 \mathrm{~d})}+\ldots+\frac{1}{(1+9 \mathrm{~d})(1+10 \mathrm{~d})}$ is equal to 5 , then $50 \mathrm{~d}$ is equal to :

In an increasing geometric progression of positive terms, the sum of the second and sixth terms is $\frac{70}{3}$ and the product of the third and fifth terms is 49 . Then the sum of the $4^{\text {th }}, 6^{\text {th }}$ and $8^{\text {th }}$ terms is equal to :

If the set $R=\{(a, b): a+5 b=42, a, b \in \mathbb{N}\}$ has $m$ elements and $\sum_{n=1}^m\left(1-i^{n !}\right)=x+i y$, where $i=\sqrt{-1}$, then the value of $m+x+y$ is

Let the first term of a series be $T_1=6$ and its $r^{\text {th }}$ term $T_r=3 T_{r-1}+6^r, r=2,3$, $\qquad$ $n$. If the sum of the first $n$ terms of this series is $\frac{1}{5}\left(n^2-12 n+39\right)\left(4 \cdot 6^n-5 \cdot 3^n+1\right)$, then $n$ is equal to______

Let the first three terms $2, p$ and $q$, with $q \neq 2$, of a G.P. be respectively the $7^{\text {th }}, 8^{\text {th }}$ and $13^{\text {th }}$ terms of an A.P. If the $5^{\text {th }}$ term of the G.P. is the $n^{\text {th }}$ term of the A.P., then $n$ is equal to:

Let ${S}_{a}$ denote the sum of first $n$ terms an arithmetic progression. If ${S}_{20}=790$ and ${S}_{10}=145$, then ${S}_{15}-$ ${S}_{5}$ is :

Let $S={z\in C:|z-1|=1\mathrm{and}(\sqrt{2}-1)(z+\bar{z})-i(z-\bar{z})=2\sqrt{2}}$. Let ${z}_{1},{z}_{2}\in S$ be such that $|{z}_{1}|=\underset{z\in s}{\mathrm{max}}|z|$ and $|{z}_{2}|=\underset{z\in s}{\mathrm{min}}|z|$. Then ${|\sqrt{2}{z}_{1}-{z}_{2}|}^{2}$ equals:

The sum of the series $\frac{1}{1-3\cdot {1}^{2}+{1}^{4}}+\frac{2}{1-3\cdot {2}^{2}+{2}^{4}}+\frac{3}{1-3\cdot {3}^{2}+{3}^{4}}+....$ up to $10$ terms is

The number of real solutions of the equation $x|x+5|+2|x+7|-2=0$ is_________

The sum of all rational terms in the expansion of $\left(2^{\frac{1}{5}}+5^{\frac{1}{3}}\right)^{15}$ 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 _______

If $f(x)=|\begin{matrix}{x}^{3} & 2{x}^{2}+1 & 1+3x \\ 3{x}^{2}+2 & 2x & {x}^{3}+6 \\ {x}^{3}-x & 4 & {x}^{2}-2\end{matrix}|$ for all $x\in \mathbb{R}$, then $2f(0)+{f}^{'}(0)$ is equal to