Algebra PYQ — Page 60

Let $A={a,b,c}$ and $B={1,2,3,4}.$ Then the number of elements in the set$C={f:A\rightarrow B\mid 2\in f(A)$ and $f$ is not one-one$}$ is $\ldots$

The common difference of the $A.P.{b}_{1},{b}_{2},....,{b}_{m}$ is $2$ more than common difference of $A.P.{a}_{1},{a}_{2},.....,{a}_{n}$. If ${a}_{40}=-159,{a}_{100}=-399$ and ${b}_{100}={a}_{70}$, then ${b}_{1}$ is equal to :

If ${2}^{10}+{2}^{9}\cdot {3}^{1}+{2}^{8}\cdot {3}^{2}+\ldots \ldots +2\cdot {3}^{9}+{3}^{10}=S-{2}^{11}$, then $S$ is equal to

Let $\alpha$ be a root of the equation ${x}^{2}+x+1=0$ and the matrix $A=\frac{1}{\sqrt{3}}[\begin{matrix}1 & 1 & 1 \\ 1 & \alpha & {\alpha }^{2} \\ 1 & {\alpha }^{2} & {\alpha }^{4}\end{matrix}],$ then the matrix ${A}^{31}$ is equal to

If $A=[\begin{matrix}1 & 1 & 2 \\ 1 & 3 & 4 \\ 1 & -1 & 3\end{matrix}],B=adjA$ and $C=3A,$ then $\frac{|adjB|}{|C|}$ is equal to

The total number of $3-$digit numbers whose sum of digits is $10$, is ..........

Let $f:R\rightarrow R$ be such that for all $x\in R({2}^{1+x}+{2}^{1-x}),f(x)$ and $({3}^{x}+{3}^{-x})$ are in A.P., then the minimum value of $f(x)$ is

If one real root of the quadratic equation $81 x^{2}+k x+256=0$ is cube of the other root, then a value of $\mathrm{k}$ is :

For $x\in R,$ Let $[x]$ denotes the greatest integer $\leq x,$then the sum of the series $[-\frac{1}{3}]+[-\frac{1}{3}-\frac{1}{100}]+[-\frac{1}{3}-\frac{2}{100}]+.....+[-\frac{1}{3}-\frac{99}{100}]$ is

The greatest value of $c\in R$ for which the system of linear equations $x-cy-cz=0$, $cx-y+cz=0$, $cx+cy-z=0$ has a non-trivial solution, is

The number of$6$ digit number that can be formed using the digits $0, 1, 2, 5, 7$ and $9$ which are divisible by $11$ and no digit is repeated is:

Let $f:[0,1]\rightarrow R$ be such that $f(xy)=f(x).f(y),$ for all $x,y\in [0,1],$ and $f(0)\neq 0.$ If $y=y(x)$ satisfies the differential equation, $\frac{dy}{dx}=f(x)$ with $y(0)=1$ then $y(\frac{1}{4})+y(\frac{3}{4})$ is equal to:

If $\frac{z - \alpha }{z + \alpha }(\alpha \in R)$ is a purely imaginary number and $|z|=2$, then a value of $\alpha$ is :

The sum of all natural numbers $n$ such that $100<n<200$ and $H.C.F.$ $(91,n)>1$ is

Let ${a}_{1}, {a}_{2}, \ldots , {a}_{30}$ be an A.P., $S= \sum _{i=1}^{30}{a}_{i}$ and $T= \sum _{i=1}^{15}{a}_{(2i-1)}.$ If ${a}_{5}=27$ and $S-2T=75,$ then ${a}_{10}$ is equal to:

If ${a}_{1},{a}_{2},{a}_{3},....$ are in A.P. such that ${a}_{1}+{a}_{7}+{a}_{16}=40,$ then the sum of the first $15$ terms of this A.P is:

The number of integral values of $m$ for which the quadratic expression $(1+2m) {x}^{2}-2(1+3m)x+4(1+m), x\in R$ is always positive, is

Let $\alpha$ and $\beta$ be the roots of the equation ${x}^{2}+x+1=0.$ Then for $y\neq 0$ in $R,|\begin{matrix}y+1 & \alpha & \beta \\ \alpha & y+\beta & 1 \\ \beta & 1 & y+\alpha \end{matrix}|$ is equal to

The total number of matrices $A=(\begin{matrix}0 & 2y & 1 \\ 2x & y & -1 \\ 2x & -y & 1\end{matrix}),(x,y\in R,x\neq y)$ for which ${A}^{T}A=3{I}_{3}$ is:

If $A=[\begin{matrix}cos\theta & -sin\theta \\ sin\theta & cos\theta \end{matrix}]$ , then the matrix ${A}^{-50}$ when $\theta =\frac{\pi }{12},$ is equal to:

If three distinct numbers $a, b, c$ are in G.P. and the equations $a{x}^{2}+2bx+c=0$ and $d{x}^{2}+2ex+f=0$ have a common root, then which one of the following statements is correct?

If $B=[\begin{matrix}5 & 2\alpha & 1 \\ 0 & 2 & 1 \\ \alpha & 3 & -1\end{matrix}]$ is the inverse of a $3\times 3$ matix $A,$ then the sum of all values of $\alpha$ for which $det(A)+1=0,$ is:

The smallest natural number $n$ , such that the coefficient of $x$ in the expansion of ${({x}^{2}+\frac{1}{{x}^{3}})}^{n}$ is $C23 n$ , is

If the fourth term in the binomial expansion of ${(\sqrt{{x}^{\frac{1}{1+{log}_{10}x}}}+{x}^{\frac{1}{12}})}^{6}$ is equal to $200,$ and $x>1,$ then the value of $x$ is