Algebra PYQ — Page 42

The total number of functions, $f:{1,2,3,4}\rightarrow {1,2,3,4,5,6}$ such that $f(1)+f(2)=f(3)$, is equal to

If $\frac{1}{2\times 3\times 4}+\frac{1}{3\times 4\times 5}+\frac{1}{4\times 5\times 6}+\ldots +$ $\frac{1}{100\times 101\times 102}=\frac{k}{101}$, then $34k$ is equal to _______.

If $\frac{1}{(20-a)(40-a)}+\frac{1}{(40-a)(60-a)}+\ldots \ldots +$ $\frac{1}{(180-a)(200-a)}=\frac{1}{256}$, then the maximum value of $a$ is

Let ${a}_{1}={b}_{1}=1,{a}_{n}={a}_{n-1}+2$ and ${b}_{n}={a}_{n}+{b}_{n-1}$ for every natural number $n\geq 2$. Then $\sum _{n=1}^{15}{a}_{n}\cdot {b}_{n}$ is equal to _____ .

If the sum of the coefficients of all the positive powers of $x$, in the binomial expansion of ${({x}^{n}+\frac{2}{{x}^{5}})}^{7}$ is $939$, then the sum of all the possible integral values of $n$ is

Let $A$ and $B$ be two $3\times 3$ matrices such that $AB=I$ and $|A|=\frac{1}{8}$ then $|adj(Badj(2A))|$ is equal to

Let ${R}_{1}={(a,b)\in N\times N:|a-b|\leq 13}$ and ${R}_{2}={(a,b)\in N\times N:|a-b|\neq 13}$ Then on $N$:

Let $f(x)=|\begin{matrix}a & -1 & 0 \\ ax & a & -1 \\ a{x}^{2} & ax & a\end{matrix}|,a\in R$. Then the sum of the squares of all the values of a for $2{f}^{'}(10)-{f}^{'}(5)+100=0$ is

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 $X=[\begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0\end{matrix}],Y=\alpha l+\beta X+\gamma {X}^{2}$ and $Z={\alpha }^{2}I-\alpha \beta X+({\beta }^{2}-\alpha \gamma ){X}^{2},\alpha ,\beta ,\gamma \in \mathbb{R}$. If ${Y}^{-1}=[\begin{matrix}\frac{1}{5} & \frac{-2}{5} & \frac{1}{5} \\ 0 & \frac{1}{5} & \frac{-2}{5} \\ 0 & 0 & \frac{1}{5}\end{matrix}]$, then ${(\alpha -\beta +\gamma )}^{2}$ is equal to ______.

Let $A$ be a $2 \times 2$ matrix with $\operatorname{det}(A)=-1$ and $\operatorname{det}((A+I)(\operatorname{Adj}(A)+I))=4$. Then the sum of the diagonal elements of $A$ can be:

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

For $p,q\in R$, consider the real valued function $f(x)={(x-p)}^{2}-q,x\in R$ and $q>0$. Let ${a}_{1},{a}_{2},{a}_{3}$ and ${a}_{4}$ be in an arithmetic progression with mean $p$ and positive common difference. If $|f({a}_{i})|=500$ for all $i=1,2,3,4$, then the absolute difference between the roots of $f(x)=0$ is

Let for $n=1,2,\ldots \ldots ,50,{S}_{n}$ be the sum of the infinite geometric progression whose first term is ${n}^{2}$ and whose common ratio is $\frac{1}{{(n+1)}^{2}}$. Then the value of $\frac{1}{26}+\sum _{n=1}^{50}({S}_{n}+\frac{2}{n+1}-n-1)$ is equal to

Let $R$ be a relation from the set ${1,2,3\ldots \ldots \ldots ,60}$ to itself such that $R=${$(a,b):b=pq$, where $p,q\geq 3$ are prime numbers}. Then, the number of elements in $R$ is

For $n\in N$, let ${S}_{n}={z\in C:|z-3+2i|=\frac{n}{4}}$ and ${T}_{n}={z\in C:|z-2+3i|=\frac{1}{n}}$. Then the number of elements in the set ${n\in N:{S}_{n}\cap {T}_{n}=\phi }$ is

Let $A$ be a $3\times 3$ invertible matrix. If $|\mathrm{adj}(24A)|=$$adj(3adj(2A))|$, then ${|A|}^{2}$ is equal to

Let $p$ and $p+2$ be prime numbers and let $\Delta =|\begin{matrix}p! & (p+1)! & (p+2)! \\ (p+1)! & (p+2)! & (p+3)! \\ (p+2)! & (p+3)! & (p+4)!\end{matrix}|$ Then the sum of the maximum values of $\alpha$ and $\beta$, such that ${p}^{\alpha }$ and ${(p+2)}^{\beta }$ divide $\Delta$, is _______.

Let $A={z\in C:1\leqslant |z-(1+i)|\leqslant 2}$ and $B={z\in A:|z-(1-i)|=1}$. Then, $B$

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

Let $A=[\begin{matrix}1 & -1 \\ 2 & \alpha \end{matrix}]$ and $B=[\begin{matrix}\beta & 1 \\ 1 & 0\end{matrix}],\alpha ,\beta \in R$. Let ${\alpha }_{1}$ be the value of $\alpha$ which satisfies ${(A+B)}^{2}={A}^{2}+[\begin{matrix}2 & 2 \\ 2 & 2\end{matrix}]$ and ${\alpha }_{2}$ be the value of $\alpha$ which satisfies ${(A+B)}^{2}={B}^{2}$. Then $|{\alpha }_{1}-{\alpha }_{2}|$ is equal to

The number of positive integers $k$ such that the constant term in the binomial expansion of ${(2{x}^{3}+\frac{3}{{x}^{k}})}^{12},x\neq 0$ is ${2}^{8}\cdot l$, where $l$ is an odd integer, is ______.

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be defined as $f(x)=x-1$ and $g:R\rightarrow {1,-1}\rightarrow \mathbb{R}$ be defined as $g(x)=\frac{{x}^{2}}{{x}^{2}-1}$. Then the function $fog$ is:

Let $a,b\in R$ be such that the equation $a{x}^{2}-2bx+15=0$ has repeated root $\alpha$ and if $\alpha$ and $\beta$ are the roots of the equation ${x}^{2}-2bx+21=0$, then ${\alpha }^{2}+{\beta }^{2}$ is equal to: