Algebra PYQ — Page 33

The sum of the coefficients of three consecutive terms in the binomial expansion of ${(1+x)}^{n+2}$, which are in the ratio $1:3:5$, is equal to

Let $f:R-{0,1}\rightarrow R$ be a function such that $f(x)+f(\frac{1}{1-x})=1+x$. Then $f(2)$ is equal to :

Let the digits $a,b,c$ be in A.P. Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in A.P. at least once. How many such numbers can be formed?

The range of $f(x)=4{\mathrm{sin}}^{-1}(\frac{{x}^{2}}{{x}^{2}+1})$ is

If $A=\frac{1}{2}[\begin{matrix}1 & \sqrt{3} \\ -\sqrt{3} & 1\end{matrix}]$ then,

The mean of the coefficients of $x,{x}^{2},\ldots \ldots ,{x}^{7}$ in the binomial expression of $(2+x{)}^{9}$ is _________

If the coefficients of $x$ and ${x}^{2}$ in $(1+x{)}^{p}(1-x{)}^{q}$ are $4$ and $-5$ respectively, then $2p+3q$ is equal to

Let $[\begin{matrix}\begin{matrix}2 \\ 1 \\ 0\end{matrix} & \begin{matrix}1 \\ 2 \\ -1\end{matrix} & \begin{matrix}0 \\ -1 \\ 2\end{matrix}\end{matrix}]$. If $|adj(adj(adj2A))|=(16{)}^{n}$, then $n$ is equal to

Eight persons are to be transported from city $A$ to city $B$ in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is

An organization awarded $48$ medals in event $A''$, $25$ in event $B''$ and $18$ in event $C''$. If these medals went to total $60$ men and only five men got medals in all the three events, then, how many received medals in exactly two of three events?

Among the relations $S={(a,b):a,b\in R-{0},2+\frac{a}{b}>0}$ and $T={(a,b):a,b\in R,{a}^{2}-{b}^{2}\in Z}$,

Number of $4$-digit numbers (the repetition of digits is allowed) which are made using the digits $1,2,3$ and $5$ , and are divisible by $15$ , is equal to

Let the determinant of a square matrix $A$ of order $m$ be $m-n$, where m and $n$ satisfy $4m+n=22$ and $17m+4n=93$. If $det(nadj(adj(mA)))={3}^{a}{5}^{b}{6}^{c}$, then $a+b+c$ is equal to

If the coefficients of ${x}^{7}$ in ${(a{x}^{2}+\frac{1}{2bx})}^{11}$ and ${x}^{-7}$ in ${(ax-\frac{1}{3b{x}^{2}})}^{11}$ are equal, then

Let $P=[\begin{matrix}\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\end{matrix}],A=[\begin{matrix}1 & 1 \\ 0 & 1\end{matrix}]$ and $Q=PA{P}^{T}$. If ${P}^{T}{Q}^{2007}P=[\begin{matrix}a & b \\ c & d\end{matrix}]$then $2a+b-3c-4d$ is equal to

Let the number of elements in sets $A$ and $B$ be five and two respectively. Then the number of subsets of $A\times B$ each having at least $3$ and at most $6$ elements is

Let ${a}_{1},{a}_{2},{a}_{3},\ldots$. be a G.P. of increasing positive numbers. Let the sum of its ${6}^{\text{th }}$ and ${8}^{\text{th }}$ terms be $2$ and the product of its ${3}^{\text{rd }}$ and ${5}^{\text{th }}$ terms be $\frac{1}{9}$. Then $6({a}_{2}+{a}_{4})({a}_{4}+{a}_{6})$ is equal to

Let ${s}_{1},{s}_{2},{s}_{3}....,{s}_{10}$ respectively be the sum of $12$ terms of $10A.\mathrm{Ps}$ whose first terms are $1,2,3,....,10$ and the common differences are $1,3,5,...,19$ respectively. Then $\sum _{i=1}^{10}{s}_{i}$ is equal to

Let $A={0,3,4,6,7,8,9,10}$ and $R$ be the relation defined on $A$ such that $R{(x,y)\in A\times A:x-y\text{is odd positive integer}\text{or}x-y=2}$. The minimum number of elements that must be added to the relation $R,$ so that it is a symmetric relation, is equal to $_________$

Let ${f}^{1}(x)=\frac{3x+2}{2x+3},x\in R-{-\frac{3}{2}}$. For $n\geq 2$, define ${f}^{n}(x)={f}^{1}o{f}^{n-1}(x)$. If ${f}^{5}(x)=\frac{ax+b}{bx+a},\mathrm{gcd}(a,b)=1$, then $a+b$ is equal to ________

If the coefficient of ${x}^{15}$ in the expansion of ${(a{x}^{3}+\frac{1}{b{x}^{\frac{1}{3}}})}^{15}$ is equal to the coefficient of ${x}^{-15}$ in the expansion of ${(a{x}^{\frac{1}{3}}-\frac{1}{b{x}^{3}})}^{15}$, where $a$ and $b$ are positive real numbers, then for each such ordered pair $(a,b)$:

Let $\alpha$ be the constant term in the binomial expansion of ${(\sqrt{x}-\frac{6}{{x}^{\frac{3}{2}}})}^{n},n\leq 15.$ If the sum of the coefficients of the remaining terms in the expansion is $649$ and the coefficient of ${x}^{-n}$ is $\lambda \alpha ,$ then $\lambda$ is equal to $________.$

Let $A$ be a $n\times n$ matrix such that $|A|=2$. If the determinant of the matrix $Adj(2.Adj(2{A}^{-1}))$ is ${2}^{84}$, then $n$ is equal to _____ .

For $a\in \mathbb{C}$, let $A={z\in \mathbb{C}:Re(a+\bar{z})>Im(\bar{a}+z)}$ and $B={z\in \mathbb{C}:Re(a+\bar{z})<Im(\bar{a}+z)}$. Then among the two statements: $(S1)$ : If $Re(a),Im(a)>0$, then the set $A$ contains all the real numbers $(S2)$ : If $Re(a),Im(a)<0$, then the set $B$ contains all the real numbers,