Algebra PYQ — Page 35

A boy needs to select five courses from $12$ available courses, out of which $5$ courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is

The coefficient of ${x}^{7}$ in ${(1-x+2{x}^{3})}^{10}$ is __________ .

If the term without $x$ in the expansion of ${({x}^{\frac{2}{3}}+\frac{\alpha }{{x}^{3}})}^{22}$ is $7315$, then $|\alpha |$ is equal to _____ .

The number of triplets $(x,y,z)$ where $x,y,z$ are distinct non negative integers satisfying $x+y+z=15$, is

The total number of three-digit numbers, divisible by $3$, which can be formed using the digits $1,3,5,8$, if repetition of digits is allowed, is

Let $A={1,2,3,4,..........10}$ and $B={0,1,2,3,4}.$ The number of elements in the relation $R={(a,b)\in A\times A:2{(a-b)}^{2}+3(a-b)\in B}$ is $__________.$

The number of five-digit numbers, greater than $40000$ and divisible by $5$, which can be formed using the digits $0,1,3,5,7$ and $9$ without repetition, is equal to

Let the complex number $z=x+iy$ be such that $\frac{2z-3i}{2z+i}$ is purely imaginary. If $x+{y}^{2}=0$, then ${y}^{4}+{y}^{2}-y$ is equal to

Let $A=[\begin{matrix}1 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 12 & -3\end{matrix}]$. Then the sum of the diagonal elements of the matrix ${(A+I)}^{11}$ is equal to:

Let $5f(x)+4f(\frac{1}{x})=\frac{1}{x}+3,x>0.$ Then $18{\int }_{1}^{2}f(x)dx$ is equal to

Let $a\neq b$ be two non-zero real numbers. Then the number of elements in the set $X={z\in C:\mathrm{Re}(a{z}^{2}+bz)=a\mathrm{and}\mathrm{Re}(b{z}^{2}+az)=b}$ is equal to

Let $A={[{a}_{ij}]}_{2\times 2},$ where $a\neq ij0$ for all $i,j$ and ${A}^{2}=I$, Let $a$ be the sum of all diagonal elements of $A$ and $b=|A|$ Then $3{a}^{2}+4{b}^{2}$ is equal to

The remainder, when ${7}^{103}$ is divided by $17,$ is

If the number of words, with or without meaning. which can be made using all the letters of the word MATHEMATICS in which $C$ and $S$ do not come together, is $(6!)k$ then $k$ is equal to

Let $A$ be a $3\times 3$ matrix such that $|adj(adj(adj.A))|={12}^{4}$. Then $|{A}^{-1}adjA|$ is equal to

Let $S={1,2,3,4,5,6}$. Then the number of oneone functions $f:S\rightarrow P(S)$, where $P(S)$ denote the power set of $S$, such that $f(n)\subset f(m)$ where $n<m$ is

Let $P(S)$ denote the power set of $S={1,2,3,\ldots ,10}$. Define the relations ${R}_{1}$ and ${R}_{2}$ on $P(S)$ as $A{R}_{1}B$ if $(A\cap {B}^{c})\cup (B\cap {A}^{c})=\phi$ and $A{R}_{2}B$ if $A\cup {B}^{c}=B\cup {A}^{c},\forall A,B\in P(S)$ . Then :

The number of real roots of the equation $\sqrt{{x}^{2}-4x+3}+\sqrt{{x}^{2}-9}=\sqrt{4{x}^{2}-14x+6}$, is:

Let $A={-4,-3,-2,0,1,3,4}$ and $R={(a,b)\in A\times A$ : $b=|a|$ or ${b}^{2}=a+1}$ be a relation on $A$. Then the minimum number of elements, that must be added to the relation $R$ so that it becomes reflexive and symmetric, is

The ${8}^{\text{th }}$ common term of the series ${S}_{1}=3+7+11+15+19+\ldots$ ${S}_{2}=1+6+11+16+21+\ldots$. is

For the differentiable function $f:\mathbb{R}-{0}-\mathbb{R},$ let $3f(x)+2f(\frac{1}{x})=\frac{1}{x}-10,$ then $|f(3)+{f}^{'}(\frac{1}{4})|$ is equal to

For the system of linear equations $x+y+z=6$ $\alpha x+\beta y+7z=3$ $x+2y+3z=14$ which of the following is NOT true ?

If the constant term in the binomial expansion of ${(\frac{{x}^{\frac{5}{2}}}{2}-\frac{4}{{x}^{l}})}^{9}$ is $-84$ and the coefficient of ${x}^{-3l}$ is ${2}^{\alpha }\beta$ where $\beta <0$ is an odd number, then $|\alpha l-\beta |$ is equal to _____ .

The sum ${1}^{2}-2.{3}^{2}+3.{5}^{2}-4.{7}^{2}+5.{9}^{2}-\ldots ..+15.{29}^{2}$ is _____ .