Algebra PYQ — Page 71

If $z$ is a non-real complex number, then the minimum value of $\frac{Im{z}^{5}}{{(Imz)}^{5}}$ is (Where $Imz=$ Imaginary part of $z$)

If $\frac{1}{\sqrt{\alpha }},\frac{1}{\sqrt{\beta }}$ are the roots of the equation $a{x}^{2}+bx+1=0,(a\neq 0,a,b\in R)$, then the equation $x(x+{b}^{3})+({a}^{3}-3abx)=0$ has roots:

If $\mathrm{f}(\theta)=\left|\begin{array}{ccc}1 & \cos \theta & 1 \\ -\sin \theta & 1 & -\cos \theta \\ -1 & \sin \theta & 1\end{array}\right|$ and $\mathrm{A}$ and $\mathrm{B}$ are respectively the maximum and the minimum values of $f(\theta)$, then $(A, B)$ is equal to:

If the coefficients of ${x}^{3}$ and ${x}^{4}$ in the expansion of $(1+ax+b{x}^{2}){(1-2x)}^{18}$ in powers of $x$ are both zero, then $(a,b)$ is equal to

The sum of the digits in the unit's place of all the $4$ - digit numbers formed by using the numbers $3,4,5$ and$6$, without repetition is :

The coefficient of $x^{50}$ in the binomial expansion of $(1+x)^{1000}+x(1+x)^{999}+x^2(1+x)^{998}+\ldots$ $+x^{1000}$ is:

If $A=\left[\begin{array}{ccc}1 & 2 & x \\ 3 & -1 & 2\end{array}\right]$ and $B=\left[\begin{array}{c}y \\ x \\ 1\end{array}\right]$ be such that $\mathrm{AB}=\left[\begin{array}{l}6 \\ 8\end{array}\right]$, then:

The least positive integer $\mathrm{n}$ such that $1-\frac{2}{3}-\frac{2}{3^2}-\ldots .-\frac{2}{3^{n-1}} < \frac{1}{100}$, is:

If $\mathrm{z}_1, \mathrm{z}_2$ and $\mathrm{z}_3, \mathrm{z}_4$ are 2 pairs of complex conjugate numbers, then $\arg \left(\frac{z_1}{z_4}\right)+\arg \left(\frac{z_2}{z_3}\right)$ equals:

Let $f⁡(n)=[\frac{1}{3}+\frac{3n}{100}]n$, where $[n]$ denotes the greatest integer less than or equal to $n$. Then $\sum _{n=1}^{56}f(n)$ is equal to

If $\alpha ,\beta \neq 0$, $f(n)={\alpha }^{n}+{\beta }^{n}$ and $|\begin{matrix}3 & 1+f(1) & 1+f(2) \\ 1+f(1) & 1+f(2) & 1+f(3) \\ 1+f(2) & 1+f(3) & 1+f(4)\end{matrix}|=K{(1-\alpha )}^{2}{(1-\beta )}^{2}{(\alpha -\beta )}^{2}$, then $K$ is equal to

If $$ \begin{aligned} &\left.\mid \begin{array}{ccc} a^2 & b^2 & c^2 \\ (a+\lambda)^2 & (b+\lambda)^2 & \left(c+\lambda^2\right) \\ (a-\lambda)^2 & \left(b-\lambda^2\right) & \left(-\lambda^2\right. \end{array}\right) \\ &=k \lambda\left|\begin{array}{ccc} a^2 & b^2 & c^2 \\ a & b & c \\ 1 & 1 & 1 \end{array}\right|, \lambda \neq 0 \end{aligned} $$ then $\mathrm{k}$ is equal to:

If $\left(2+\frac{x}{3}\right)^{55}$ is expanded in the ascending powers of $x$ and the coefficients of powers of $x$ in two consecutive terms of the expansion are equal, then these terms are:

If the sum $\frac{3}{{1}^{2}}+\frac{5}{{1}^{2}+{2}^{2}}+\frac{7}{{1}^{2}+{2}^{2}+{3}^{2}}+.....+$ up to $20$ terms is equal to $\frac{k}{21},$ then $k$ is equal to

If $f(x)=x^2-x+5, x>\frac{1}{2}$, and $\mathrm{g}(\mathrm{x})$ is its inverse function, then $\mathrm{g}^{\prime}(7)$ equals:

The number of terms in an $A.P.$ is even, the sum of the odd terms in it is $24$ and that the even terms is $30.$ If the last term exceeds the first term by $10\frac{1}{2},$ then the number of terms in the $A.P.$ is

Let $A$ and $B$ be any two $3 \times 3$ matrices. If $A$ is symmetric and $B$ is skew symmetric, then the matrix $AB-BA$ is

If $B$ is a $3\times 3$ matrix such that ${B}^{2}=0$, then $det.[{(I+B)}^{50}-50B]$ is equal to :

If $a\in R$ and the equation $-3(x-[x]{)}^{2}+2(x-[x])+{a}^{2}=0$ (where $[x]$ denotes the greatest integer $\leq$$x$) has no integral solution, then all possible values of $a$ lie in the interval

Given an A.P. whose terms are all positive integers. The sum of its first nine terms is greater than $200$ and less than $220.$ If the second term in it is $12$, then its ${4}^{\mathrm{th}}$ term is :

If $1+x^4+x^5=\sum_{i=0}^5 a_i\left(1+x^i\right)$, for all $x$ in $R$, then $a_2$ is:

If ${\Delta }_{r}=|\begin{matrix}r & 2r-1 & 3r-2 \\ \frac{n}{2} & n-1 & a \\ \frac{1}{2}n(n-1) & {(n-1)}^{2} & \frac{1}{2}(n-1)(3n+4)\end{matrix}|$, then the value of $\sum _{r=1}^{n-1}{\Delta }_{r}$

The sum of the first 20 terms common between the series $3+7+11+15+$ and $1+6+11+$ $16+\ldots .$. is

8-digit numbers are formed using the digits 1, 1, 2, $2,2,3,4,4$. The number of such numbers in which the odd digits do no occupy odd places, is: