Algebra PYQ — Page 41

Let a function $f:\mathbb{N}\rightarrow \mathbb{N}$ be defined by $f(n)=[\begin{matrix}2n, & n=2,4,6,8,\ldots .. \\ n-1, & n=3,7,11,15,\ldots .. \\ \frac{n+1}{2}, & n=1,5,9,13,\ldots ..\end{matrix}$ then, $f$ is

If the sum of the co-efficients of all the positive even powers of $x$ in the binomial expansion of ${(2{x}^{3}+\frac{3}{x})}^{10}$ is ${5}^{10}-\beta \cdot {3}^{9}$, then $\beta$ is equal to _____.

Let $M=[\begin{matrix}0 & -\alpha \\ \alpha & 0\end{matrix}]$, where $\alpha$ is a non-zero real number and $N=\sum _{k=1}^{49}{M}^{2k}$. If $(I-{M}^{2})N=-2I$, then the positive integral value of $\alpha$ is ______.

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 _____

If $x=\sum _{n=0}^{\infty }{a}^{n},y=\sum _{n=0}^{\infty }{b}^{n},z=\sum _{n=0}^{\infty }{c}^{n}$, where $a,b,c$ are in A.P. and $|a|<1,|b|<1,|c|<1$, $abc\neq 0$, then

Let $A$ be a $3\times 3$ real matrix such that $A(\begin{matrix}1 \\ 1 \\ 0\end{matrix})=(\begin{matrix}1 \\ 1 \\ 0\end{matrix});A(\begin{matrix}1 \\ 0 \\ 1\end{matrix})=(\begin{matrix}-1 \\ 0 \\ 1\end{matrix})$ and $A(\begin{matrix}0 \\ 0 \\ 1\end{matrix})=(\begin{matrix}1 \\ 1 \\ 2\end{matrix})$. If $X={[{x}_{1}{x}_{2}{x}_{3}]}^{T}$ and $I$ is an identity matrix of order $3$, then the system $(A-2I)X=(\begin{matrix}4 \\ 1 \\ 1\end{matrix})$ has

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

The area of the polygon, whose vertices are the non-real roots of the equation $\bar{z}=i{z}^{2}$ is

Let $f:N\rightarrow R$ be a function such that $f(x+y)=2f(x)f(y)$ for natural numbers $x$ and $y$. If $f(1)=2$, then the value of $\alpha$ for which $\sum _{k=1}^{10}f(\alpha +k)=\frac{512}{3}({2}^{20}-1)$ holds, is

Suppose ${a}_{1},{a}_{2},\ldots ,{a}_{n},\ldots$ be an arithmetic progression of natural numbers. If the ratio of the sum of the first five terms to the sum of first nine terms of the progression is $5:17$ and $110<{a}_{15}<120$, then the sum of the first ten terms of the progression is equal to

Let $A={n\in N:H.C.F.(n,45)=1}$ and let $B={2k:k\in {1,2,\ldots ,100}}$. Then the sum of all the elements of $A\cap B$ is _____.

If ${a}_{1}(>0),{a}_{2},{a}_{3},{a}_{4},{a}_{5}$ are in a G.P. , ${a}_{2}+{a}_{4}=2{a}_{3}+1$ and $3{a}_{2}+{a}_{3}=2{a}_{4}$, then ${a}_{2}+{a}_{4}+2{a}_{5}$ is equal to _____.

Let $f,g:\mathbb{N}-{1}\rightarrow \mathbb{N}$ be functions defined by $f(a)=\alpha$, where $\alpha$ is the maximum of the powers of those primes $p$ such that ${p}^{\alpha }$ divides $a$, and $g(a)=a+1$, for all $a\in \mathbb{N}-{1}$. Then, the function $f+g$ is

The number of real values of $\lambda$, such that the system of linear equations $2x-3y+5z=9$ $x+3y-z=-18$ $3x-y+({\lambda }^{2}-|\lambda |)z=16$ has no solutions, is

Let $A$ be a matrix of order $3\times 3$ and $\mathrm{det}(A)=2$. Then $\mathrm{det}(det(A)adj(5adj({A}^{3}))$ is equal to _____.

Let $3,6,9,12,\ldots$ upto $78$ terms and $5,9,13,17,\ldots$ upto $59$ terms be two series. Then, the sum of the terms common to both the series is equal to ______.

Let $S={x\in [-6,3]-{-2,2}:\frac{|x+3|-1}{|x|-2}\geq 0}$ and $T={x\in Z:{x}^{2}-7|x|+9\leq 0}$. Then the number of elements in $S\cap T$ is

Let $f(x)$ be a quadratic polynomial with leading coefficient $1$ such that $f(0)=p,p\neq 0$, and $f(1)=\frac{1}{3}$. If the equations $f(x)=0$ and $fofofof(x)=0$ have a common real root, then $f(-3)$ is equal to ______.

Let the sum of an infinite $G.P.$, whose first term is $a$ and the common ratio is $r$, be $5$. Let the sum of its first five terms be $\frac{98}{25}$. Then the sum of the first $21$ terms of an $\mathrm{AP}$, whose first term is $10ar,{n}^{\mathrm{th}}$ term is ${a}_{n}$ and the common difference is $10{ar}^{2}$, is equal to

The value of $i^{100}$ where $i = \\sqrt{-1}$ is

If ${a}_{1},{a}_{2},{a}_{3}\ldots$ and ${b}_{1},{b}_{2},{b}_{3}\ldots .$ are A.P. and ${a}_{1}=2,{a}_{10}=3,{a}_{1}{b}_{1}=1={a}_{10}{b}_{10}$ then ${a}_{4}{b}_{4}$ is equal to

If the coefficient of ${x}^{10}$ in the binomial expansion of ${(\frac{\sqrt{x}}{{5}^{\frac{1}{4}}}+\frac{\sqrt{5}}{{x}^{\frac{1}{3}}})}^{60}$ is ${5}^{k}l$, where $l,k\in N$ and $l$ is coprime to $5$, then $k$ is equal to ______.

Let $f(x)$ be a quadratic polynomial such that $f(-2)$ $+f(3)=0$. If one of the roots of $f(x)=0$ is $-1$, then the sum of the roots of $f(x)=0$ is equal to

Let $S={1,2,3,4}$. Then the number of elements in the set {$f:S\times S\rightarrow S:f$ is onto and $f(a,b)=f(b,a)$ $\geq a\forall (a,b)\in S\times S$} is