Algebra PYQ — Page 41

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 $\alpha$ and $\beta$ be the roots of the equation ${x}^{2}+(2i-1)=0$. Then, the value of $|{\alpha }^{8}+{\beta }^{8}|$ is equal to

Let $c,k\in R$. If $f(x)=(c+1){x}^{2}+(1-{c}^{2})x+2k$ and $f(x+y)=f(x)+f(y)-xy$, for all $x,y\in R$, then the value of $|2(f(1)+f(2)+f(3)+\ldots \ldots +f(20))|$ is equal to ______.

The number of values of $\alpha$ for which the system of equations $x+y+z=\alpha$ $\alpha x+2\alpha y+3z=-1$ $x+3\alpha y+5z=4$ is inconsistent, is

Let $S={z\in C:|z-2|\leq 1,z(1+i)+\bar{z}(1-i)\leq 2}$. Let $|z-4i|$ attains minimum and maximum values, respectively, at ${z}_{1}\in S$ and ${z}_{2}\in S$. If $5({|{z}_{1}|}^{2}+{|{z}_{2}|}^{2})=\alpha +\beta \sqrt{5}$, where $\alpha$ and $\beta$ are integers, then the value of $\alpha +\beta$ is equal to ______.

Different A.P.'s are constructed with the first term $100$, the last term $199$, And integral common differences. The sum of the common differences of all such, A.P's having at least $3$ terms and at most $33$ terms is.

The greatest integer less than or equal to the sum of first $100$ terms of the sequence $\frac{1}{3},\frac{5}{9},\frac{19}{27},\frac{65}{81},\ldots$ is equal to ______

If ${z}^{2}+z+1=0,z\in C$, then $|\sum _{n=1}^{15}{({z}^{n}+(-1{)}^{a}\frac{1}{{z}^{n}})}^{2}|$ is equal to _____.

Let $A={1,2,3,4,5,6,7}$. Define $B=${$T\subseteq A$: either $1\notin T$ or $2\in T$} and $C=${$T\subseteq A:T$ the sum of all the elements of $T$ is a prime number.} Then the number of elements in the set $B\cup C$ is _______.

The system of equations $-kx+3y-14z=25$ $-15x+4y-kz=3$ $-4x+y+3z=4$ Question: is consistent for all $k$ in the set

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

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 for some $p,q,r\in R$, all have positive sign, one of the roots of the equation $({p}^{2}+{q}^{2}){x}^{2}-2q(p+r)x+{q}^{2}+{r}^{2}=0$ is also a root of the equation ${x}^{2}+2x-8=0$, then $\frac{{q}^{2}+{r}^{2}}{{p}^{2}}$ is equal to-

Let $\alpha ,\beta$ be the roots of the equation ${x}^{2}-4\lambda x+5=0$ and $\alpha ,\gamma$ be the roots of the equation ${x}^{2}-(3\sqrt{2}+2\sqrt{3})x+7+3\lambda \sqrt{3}=0$. If $\beta +\gamma =3\sqrt{2}$, then ${(\alpha +2\beta +\gamma )}^{2}$ is equal to

Let ${S}_{1}={{z}_{1}\in C:|{z}_{1}-3|=\frac{1}{2}}$ and ${S}_{2}={{z}_{2}\in C:|{z}_{2}-|{z}_{2}+1||=|{z}_{2}+|{z}_{2}-1||}.$ Then, for ${z}_{1}\in {S}_{1}$ and ${z}_{2}\in {S}_{2}$, the least value of $|{z}_{2}-{z}_{1}|$ is

Let $(z)$ represent the principal argument of the complex number $z$. The, $|z|=3$ and $\mathrm{arg}(z-1)-\mathrm{arg}(z+1)=\frac{\pi }{4}$ intersect:

Let $A={z\in C:|\frac{z+1}{z-1}|<1}$ and $B={z\in C:\mathrm{arg}(\frac{z-1}{z+1})=\frac{2\pi }{3}}$. Then $A\cap B$ is

Let ${z}_{1}$ and ${z}_{2}$ be two complex numbers such that ${\bar{z}}_{1}=i{\bar{z}}_{2}$ and $\mathrm{arg}(\frac{{z}_{1}}{{\bar{z}}_{2}})=\pi$, then the argument of ${z}_{1}$ is

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 minimum value of the sum of the squares of the roots of ${x}^{2}+(3-a)x=2a-1$ is

Let $S$ be the set of all passwords which are six to eight characters long, where each character is either an alphabet from ${A,B,C,D,E}$ or a number from ${1,2,3,4,5}$ with the repetition of characters allowed. If the number of passwords in $S$ whose at least one character is a number from ${1,2,3,4,5}$ is $\alpha \times {5}^{6}$, then $\alpha$ is equal to

The total number of four digit numbers such that each of the first three digits is divisible by the last digit, is equal to ______.

The number of ways, $16$ identical cubes, of which $11$ are blue and rest are red, can be placed in a row so that between any two red cubes there should be at least $2$ blue cubes, is ______.

The probability that a randomly chosen $2\times 2$ matrix with all the entries from the set of first $10$ primes, is singular, is equal to