Algebra PYQ — Page 72

Two women and some men participated in a chess tournament in which every participant played two games with each of the other participants. If the number of games that the men played between them-selves exceeds the number of games that the men played with the women by$66$, then the number of men who participated in the tournament lies in the interval

Let A be a $3 \times 3$ matrix such that $$ \mathrm{A}\left[\begin{array}{lll} 1 & 2 & 3 \\ 0 & 2 & 3 \\ 0 & 1 & 1 \end{array}\right]=\left[\begin{array}{lll} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right] $$ Then $\mathrm{A}^{-1}$ is:

The function $f(x)=|sin4x|+|cos2x|,$ is a periodic function with a fundamental period

Let $f:R\rightarrow R$ be defined by $f(x)=\frac{|x|-1}{|x|+1},$ then $f$ is

If $\alpha$ and $\beta$ are roots of the equation, $x^2-4 \sqrt{2} k x+2 e^{4 \ln k}-1=0$ for some $k$, and $\alpha^2+\beta^2=66$, then $\alpha^3+\beta^3$ is equal to:

The sum of the roots of the equation, $\mathrm{x}^2+|2 \mathrm{x}-3|-4=0$, is:

For all complex numbers $z$ of the form $1+i\alpha ,\alpha \in R,$ if ${z}^{2}=x+iy,$ then

If $A$is a $3\times 3$ non-singular matrix such that $A{A}^{'}={A}^{'}A$ and $B={A}^{-1}{A}^{'},$ then $B{B}^{'}$ equals, where ${X}^{'}$ denotes the transpose of the matrix $X$.

Let $\mathrm{f}$ be an odd function defined on the set of real numbers such that for $\mathrm{x} \geq 0$, $f(x)=3 \sin x+4 \cos x$. Then $\mathrm{f}(\mathrm{x})$ at $\mathrm{x}=-\frac{11 \pi}{6}$ is equal to:

Let $P$ be the relation defined on the set of all real numbers such that $P={(a,b):{\mathrm{sec}}^{2}a-{\mathrm{tan}}^{2}b=1}$. Then, $P$ is

A relation on the set $\mathrm{A}=\{\mathrm{x}:|\mathrm{x}| < 3, \mathrm{x} \in \mathrm{Z}\}$, where $Z$ is the set of integers is defined by $\mathrm{R}=\{(\mathrm{x}, \mathrm{y}): \mathrm{y}=|\mathrm{x}|, \mathrm{x} \neq-1\}$. Then the number of elements in the power set of $R$ is:

If $g$ is the inverse of a function $f$ and ${f}^{'}(x)=\frac{1}{1+{x}^{5}},$ then ${g}^{'}(x)$ is equal to

Let $\alpha \text{ and } \beta$ be the roots of equation $p{x}^{2}+qx+r=0$, $p\neq 0$. If $p,q,r$are in A.P. and $\frac{ 1 }{ \alpha } + \frac{ 1 }{ \beta } = 4$, then the value of $| \alpha - \beta |$ is

Three positive numbers form an increasing $G.P.$ If the middle term in this $G.P.$ is doubled, the new numbers are in $A.P.$ Then the common ratio of the $G.P.$ is :

If equations $a{x}^{2}+bx+c=0,(a,b,c\in R,a\neq 0)$ and $2{x}^{2}+3x+4=0$ have a common root, then $a:b:c$ equals :

The equation $\sqrt{3{x}^{2}+x+5}=x-3$, where $x⁡$ is real, has

If $z$ is a complex number such that $|z|\geq 2,$ then the minimum value of $|z+\frac{1}{2}|$ :

Let $w(Imw\neq 0)$ be a complex number. Then, the set of all complex numbers $z$ satisfying the equation $w-\bar{w}z=k(1-z)$, for some real number $k$, is

The number of terms in the expansion of ${(1+x)}^{101}{(1-x+{x}^{2})}^{100}$ in powers of $x$ is

If $X={{4}^{n}-3n-1:n\in N}$ and $Y={9(n-1):n\in N}$, where $N$ is the set of natural numbers, then $X\cup Y$ is equal to

An eight digit number divisible by 9 is to be formed using digits from 0 to 9 without repeating the digits. The number of ways in which this can be done is:

The coefficient of ${ x⁡ }^{ 1 0 1 2 }$ in the expansion of ${(1+{x}^{n}+{x}^{253})}^{10},$ (where $n\leq 22$ is any positive integer), is

In a geometric progression, if the ratio of the sum of first 5 terms to the sum of their reciprocals is 49 , and the sum of the first and the third term is 35. Then the first term of this geometric progression is:

Let for $\mathrm{i}=1,2,3, \mathrm{p}_{\mathrm{i}}(\mathrm{x})$ be a polynomial of degree 2 in $x, p^{\prime}{ }_i(x)$ and $p^{\prime \prime}{ }_i(x)$ be the first and second order derivatives of $p_i(x)$ respectively. Let, $$ \left.\mathrm{A}(\mathrm{x})=\left[\begin{array}{lll} \mathrm{p}_1(\mathrm{x}) & \mathrm{p}_1^{\prime}(\mathrm{x}) & \mathrm{p}_1^{\prime \prime} \mathrm{x}( \\ \mathrm{p}_2(\mathrm{x}) & \mathrm{p}_2^{\prime}(\mathrm{x}) & \mathrm{p}_2^{\prime \prime}( \\ \mathrm{p}_3(\mathrm{x}) & \mathrm{p}_3^{\prime}(\mathrm{x}) & \mathrm{p}_3^{\prime \prime}(\mathrm{x} \end{array}\right]\right) $$ and $\mathrm{B}(\mathrm{x})=[\mathrm{A}(\mathrm{x})]^{\mathrm{T}} \mathrm{A}(\mathrm{x})$. Then determinant of $\mathrm{B}(\mathrm{x})$ :