Algebra PYQ — Page 33

If the co-efficient of ${x}^{9}$ in ${(\alpha {x}^{3}+\frac{1}{\beta x})}^{11}$and the co-efficient of ${x}^{-9}$ in ${(\alpha x-\frac{1}{\beta {x}^{3}})}^{11}$ are equal, then $(\alpha \beta {)}^{2}$ 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 $A=[\begin{matrix}m & n \\ p & q\end{matrix}],d=|A|\neq 0$ and $|A-d(AdjA)|=0$. Then

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

If for $z=\alpha +i\beta ,|z+2|=z+4(1+i),$ then $\alpha +\beta$ and $\alpha \beta$ are the roots of the equation

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 ________

The total number of $4$-digit numbers whose greatest common divisor with $54$ is $2$ , is

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 _____ .

The number of $9$ digit numbers, that can be formed using all the digits of the number $123412341$ so that the even digits occupy only even places, is ______

Some couples participated in a mixed doubles badminton tournament. If the number of matches played, so that no couple played in a match, is $840$, then the total numbers of persons, who participated in the tournament, is ________.

For some $a,b,c\in \mathbb{N}$, let $f(x)=ax-3$ and $g(x)={x}^{b}+c,x\in \mathbb{R}$. If ${(fog)}^{-1}(x)={(\frac{x-7}{2})}^{\frac{1}{3}}$, then $(f\circ g)(ac)+(g\circ f)(b)$ is equal to _____ .

Let $\alpha$ be a root of the equation $(a-c){x}^{2}+(b-a)x+(c-b)=0$ where $a,b,c$ are distinct real numbers such that the matrix $[\begin{matrix}{\alpha }^{2} & \alpha & 1 \\ 1 & 1 & 1 \\ a & b & c\end{matrix}]$ is singular. Then the value of $\frac{{(a-c)}^{2}}{(b-a)(c-b)}+\frac{{(b-a)}^{2}}{(a-c)(c-b)}+\frac{{(c-b)}^{2}}{(a-c)(b-a)}$ is

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

For the system of equations $x+y+z=6$ $x+2y+\alpha z=10$ $x+3y+5z=\beta$, which one of the following is NOT true?

Let $x={(8\sqrt{3}+13)}^{13}$ and $y={(7\sqrt{2}+9)}^{9}$. If $[t]$ denotes the greatest integer $\leq t$, then

In a group of $100$ persons $75$ speak English and $40$ speak Hindi. Each person speaks at least one of the two languages. If the number of persons who speak only English is $\alpha$ and the number of persons who speaks only Hindi is $\beta$, then the eccentricity of the ellipse $25({\beta }^{2}{x}^{2}+{\alpha }^{2}{y}^{2})={\alpha }^{2}{\beta }^{2}$ is

For the system of linear equations $2x+4y+2az=b$ $x+2y+3z=4$ $2x+5y+2z=8$ which of the following is $\mathrm{NOT}$ correct?

If the system of equations $x+y+az=b$ $2x+5y+2z=6$ $x+2y+3z=3$ has infinitely many solutions, then $2a+3b$ is equal to

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 :

The coefficient of ${x}^{18}$ in the expansion of ${({x}^{4}-\frac{1}{{x}^{3}})}^{15}$ is $____________$

The number of real roots of the equation $x|x|-5|x+2|+6=0$, is

Let $a,b,c$ be the three distinct positive real numbers such that ${(2a)}^{{\mathrm{log}}_{e}a}={(bc)}^{{\mathrm{log}}_{e}b}$ and ${b}^{{\mathrm{log}}_{e}2}={a}^{{\mathrm{log}}_{e}c}$ Then $6a+5bc$ is equal to ______.