Algebra PYQ — Page 28

Let $a,b,c$ and $d$ be positive real numbers such that $a+b+c+d=11$. If the maximum value of ${a}^{5}{b}^{3}{c}^{2}d$ is $3750\beta$, then the value of $\beta$ is

For the system of linear equations $2x-y+3z=5$ $3x+2y-z=7$ $4x+5y+\alpha z=\beta$, which of the following is NOT correct?

Let $B=[\begin{matrix}1 & 3 & \alpha \\ 1 & 2 & 3 \\ \alpha & \alpha & 4\end{matrix}],\alpha >2$ be the adjoint of a matrix $A$ and $|A|=2.$ Then $[\begin{matrix}\alpha & -2\alpha & \alpha \end{matrix}]B[\begin{matrix}\alpha \\ -2\alpha \\ \alpha \end{matrix}]$ is equal to

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?

Let $\alpha$ and $\beta$ be real numbers. Consider a $3\times 3$ matrix $A$ such that ${A}^{2}=3A+\alpha I$. If ${A}^{4}=21A+\beta I$, then

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?

Let $S={z\in \mathbb{C}:\bar{z}=i({z}^{2}+Re(\bar{z}))}$. Then $\underset{z\in S}{\sum }|z{|}^{2}$ is equal to

Let $\alpha =8-14i,A={z\in \mathbb{C}:\frac{\alpha z-\bar{\alpha }\bar{z}}{{z}^{2}-(\bar{z}{)}^{2}-112i}=1}$ and $B={z\in \mathbb{C}:|z+3i|=4}$ Then, $\underset{z\in A\cap B}{\sum }(Rez-Imz)$ is equal to ________

The remainder when ${19}^{200}+{23}^{200}$ is divided by $49$, is _____ .

Suppose Anil's mother wants to give $5$ whole fruits to Anil from a basket of $7$ red apples, $5$ white apples and $8$ oranges. If in the selected $5$ fruits, at least $2$ orange, at least one red apple and at least one white apple must be given, then the number of ways, Anil's mother can offer $5$ fruits to Anil is _____ .

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

The largest natural number $n$ such that $3n$ divides $66!$ is $_______$

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

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

The coefficient of ${x}^{5}$ in the expansion of ${(2{x}^{3}-\frac{1}{3{x}^{2}})}^{5}$ is

Let $A={1,3,4,6,9}$ and $B={2,4,5,8,10}$. Let $R$ be a relation defined on $A\times B$ such that $R={({a}_{1},{b}_{1}),({a}_{2},{b}_{2}):{a}_{1}\leq {b}_{2}\text{ and}{b}_{1}\leq {a}_{2}}$. Then the number of elements in the set $R$ is

For $x\in \mathbb{R},$ two real valued functions $f(x)$ and $g(x)$ are such that, $g(x)=\sqrt{x}+1$ and $fog(x)=x+3-\sqrt{x}.$ Then $f(0)$ is equal to

Let $A={1,2,3,4,5,6,7}$. Then the relation $R={(x,y)\in A\times A:x+y=7}$ is

If the system of equations $2x+y-z=5$ $2x-5y+\lambda z=\mu$ $x+2y-5z=7$ has infinitely many solutions, then$(\lambda +\mu {)}^{2}+(\lambda -\mu {)}^{2}$ is equal to

Let $m$ and $n$ be the numbers of real roots of the quadratic equations ${x}^{2}-12x+[x]+31=0$ and ${x}^{2}-5|x+2|-4=0$ respectively, where $[x]$ denotes the greatest integer $\leq x$. Then ${m}^{2}+mn+{n}^{2}$ is equal to

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a function defined by $f(x)={\mathrm{log}}_{\sqrt{m}}{\sqrt{2}(\mathrm{sin}x-\mathrm{cos}x)+m-2}$, for some $m$, such that the range of $f$ is $[0,2]$. Then the value of $m$ is _____ .

Let $\lambda \neq 0$ be a real number. Let $\alpha ,\beta$ be the roots of the equation $14{x}^{2}-31x+3\lambda =0$ and $\alpha ,\gamma$ be the roots of the equation $35{x}^{2}-53x+4\lambda =0$. Then $\frac{3\alpha }{\beta }$ and $\frac{4\alpha }{\gamma }$ are the roots of the equation :

Consider the following system of questions $\alpha x+2y+z=1$ $2\alpha x+3y+z=1$ $3x+\alpha y+2z=\beta$ For some $\alpha ,\beta \in \mathbb{R}$. Then which of the following is NOT correct.

If $\frac{{1}^{3}+{2}^{3}+{3}^{3}......\text{upto }n\text{terms}}{1\cdot 3+2\cdot 5+3\cdot 7+....\text{upto }n\text{terms}}=\frac{9}{5}$ then the value of $n$ is