Algebra PYQ — Page 36

The constant term in the expansion of ${(2x+\frac{1}{{x}^{7}}+3{x}^{2})}^{5}$ is _____ .

Let $f(x)=2{x}^{2}-x-1$ and $S={n\in \mathbb{Z}:|f(n)|\leq 800}$. Then, the value of $\underset{n\in S}{\sum }f(n)$ is equal to _______.

Let the minimum value ${v}_{0}$ of $v={|z|}^{2}+{|z-3|}^{2}+{|z-6i|}^{2}$, $z\in \mathbb{C}$ is attained at $z={z}_{0}$. Then ${|2{z}_{0}^{2}-{\bar{z}}_{0}^{3}+3|}^{2}+{v}_{0}^{2}$ is equal to

The positive value of the determinant of the matrix $A$, whose $Adj(Adj(A))=[\begin{matrix}14 & 28 & -14 \\ -14 & 14 & 28 \\ 28 & -14 & 14\end{matrix}]$, is ______.

Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of ${(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}})}^{n}$, in the increasing powers of $\frac{1}{\sqrt[4]{3}}$ be $\sqrt[4]{6}:1$. If the sixth term from the beginning is $\frac{\alpha }{\sqrt[4]{3}}$, then $\alpha$ is equal to _______.

In an examination, there are $5$ multiple choice questions with $3$ choices, out of which exactly one is correct. There are $3$ marks for each correct answer, $-2$ marks for each wrong answer and $0$ mark if the question is not attempted. Then, the number of ways a student appearing in the examination gets $5$ marks is _____

The remainder on dividing $1+3+{3}^{2}+{3}^{3}+\ldots +{3}^{2021}$ by $50$ is _____.

Let $x=[\begin{matrix}1 \\ 1 \\ 1\end{matrix}]$ and $A=[\begin{matrix}-1 & 2 & 3 \\ 0 & 1 & 6 \\ 0 & 0 & -1\end{matrix}]$. For $k\in \mathbb{N}$, if ${X}^{'}{A}^{k}X=33$, then $k$ is equal to

If the minimum value of $f(x)=\frac{5{x}^{2}}{2}+\frac{\alpha }{{x}^{5}},x>0$, is $14$, then the value of $\alpha$ is equal to

Let ${x}_{1},{x}_{2},{x}_{3},\ldots ..,{x}_{20}$ be in geometric progression with ${x}_{1}=3$ and the common ration $\frac{1}{2}$. A new data is constructed replacing each ${x}_{i}$ by ${({x}_{i}-i)}^{2}$. If $x$ is the mean of new data, then the greatest integer less than or equal to $x$ is

The remainder when ${(11)}^{1011}+{(1011)}^{11}$ is divided by $9$ is _____ .

Let $S={z=x+iy:|z-1+i|\geq |z|,|z|<2,|z+i|=|z-1|}$. Then the set of all values of $x$, for which $w=2x+iy\in S$ for some $y\in \mathbb{R}$, is

The number of bijective function $f(1,3,5,7,\cdots ,99)\rightarrow (2,4,6,8,\cdots ,100)$ if $f(3)>f(5)>f(7)\cdots >f(99)$ is

If the system of equations $x+y+z=6$ $2x+5y+\alpha z=\beta$ $x+2y+3z=14$ has infinitely many solutions, then $\alpha +\beta$ is equal to

Let $A$ be a $3\times 3$ matrix having entries from the set ${-1,0,1}$. The number of all such matrices $A$ having sum of all the entries equal to $5$, is _____

The number of $7$-digit numbers which are multiples of $11$ and are formed using all the digits $1,2,3,4,5,7$ and $9$ is _____.

Considering only the principal values of the inverse trigonometric functions, the domain of the function $f(x)={\mathrm{cos}}^{-1}(\frac{{x}^{2}-4x+2}{{x}^{2}+3})$ is

Let $A=[{a}_{ij}]$ be a square matrix of order $3$ such that ${a}_{ij}={2}^{j-i}$, for all $i,j=1,2,3$. Then, the matrix ${A}^{2}+{A}^{3}+\ldots +{A}^{10}$ is equal to

The number of points of intersection $|z-(4+3i)|=2|$ and $|z|+|z-4|=6,z\in C$ is

The number of functions $f$, from the set $A={x\in N:{x}^{2}-10x+9\leq 0}$ to the set $B={{n}^{2}:n\in N}$ such that $f(x)\leq {(x-3)}^{2}+1$, for every $x\in A$, is _______.

Consider the sequence ${a}_{1},{a}_{2},{a}_{3},\ldots \ldots$ such that ${a}_{1}=1,{a}_{2}=2$ and ${a}_{n+2}=\frac{2}{{a}_{n+1}}+{a}_{n}$ for $n=1,2,3,\ldots$ If $(\frac{{a}_{1}+\frac{1}{{a}_{2}}}{{a}_{3}})\cdot (\frac{{a}_{2}+\frac{1}{{a}_{3}}}{{a}_{4}})\cdot (\frac{{a}_{3}+\frac{1}{{a}_{4}}}{{a}_{5}})\ldots (\frac{{a}_{30}+\frac{1}{{a}_{31}}}{{a}_{32}})={2}^{\alpha }(C3161)$ then $\alpha$ is equal to

Let $\alpha ,\beta$ be the roots of the equation ${x}^{2}-\sqrt{2}x+\sqrt{6}=0$ and $\frac{1}{{\alpha }^{2}}+1,\frac{1}{{\beta }^{2}}+1$ be the roots of the equation ${x}^{2}+ax+b=0$. Then the roots of the equation ${x}^{2}-(a+b-2)x+(a+b+2)=0$ are :

The number of $3$-digit odd numbers, whose sum of digits is a multiple of $7$, is _____.

Let $S={z\in \mathbb{C}:{z}^{2}+\bar{z}=0}$. Then $\underset{z\in S}{\sum }(Re(z)+Im(z))$ is equal to _______.