Algebra PYQ — Page 34

Let $\alpha$ be a root of the equation $(a-c){x}^{2}+(b-a)x+(c-b)=0$ where $a,b,c$ are distinct real numbers such that the matrix $[\begin{matrix}{\alpha }^{2} & \alpha & 1 \\ 1 & 1 & 1 \\ a & b & c\end{matrix}]$ is singular. Then the value of $\frac{{(a-c)}^{2}}{(b-a)(c-b)}+\frac{{(b-a)}^{2}}{(a-c)(c-b)}+\frac{{(c-b)}^{2}}{(a-c)(b-a)}$ is

Let $A=⌊{a}_{\hat{i}\hat{j}}⌋\cdot {a}_{ij}\in Z\cap [0,4],1\leq i,j\leq 2$. The number of matrices $A$ such that the sum of all entries is $a$ prime number $p\in (2,13)$ is _____ .

If $A$ and $B$ are two non-zero $n\times n$ matrices such that ${A}^{2}+B={A}^{2}B,$ then

Let $P$ be a square matrix such that ${P}^{2}=I-P$. For $\alpha ,\beta ,\gamma ,\delta \in \mathbb{N}$, if ${P}^{\alpha }+{P}^{\beta }=\gamma l-29P$ and ${P}^{\alpha }-{P}^{\beta }=\delta l-13P$, then $\alpha +\beta +\gamma -\delta$ is equal to

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a differentiable function that satisfies the relation $f(x+y)=f(x)+f(y)-1,\forall x$, $y\in \mathbb{R}$. If ${f}^{'}(0)=2$, then $|f(-2)|$ is equal to

For the system of linear equations $2x+4y+2az=b$ $x+2y+3z=4$ $2x+5y+2z=8$ which of the following is $\mathrm{NOT}$ correct?

If the system of equations $x+y+az=b$ $2x+5y+2z=6$ $x+2y+3z=3$ has infinitely many solutions, then $2a+3b$ is equal to

For $x\in \mathbb{R},$ two real valued functions $f(x)$ and $g(x)$ are such that, $g(x)=\sqrt{x}+1$ and $fog(x)=x+3-\sqrt{x}.$ Then $f(0)$ is equal to

If $f(x)=\frac{(\mathrm{tan}{1}^{^{\circ}})x+{\mathrm{log}}_{e}(123)}{x{\mathrm{log}}_{e}(1234)-(\mathrm{tan}{1}^{^{\circ}})},x>0$, then the least value of $f(f(x))+f(f(\frac{4}{x}))$ is

For some $a,b,c\in \mathbb{N}$, let $f(x)=ax-3$ and $g(x)={x}^{b}+c,x\in \mathbb{R}$. If ${(fog)}^{-1}(x)={(\frac{x-7}{2})}^{\frac{1}{3}}$, then $(f\circ g)(ac)+(g\circ f)(b)$ is equal to _____ .

The number of real roots of the equation $x|x|-5|x+2|+6=0$, is

Let $a,b,c$ be the three distinct positive real numbers such that ${(2a)}^{{\mathrm{log}}_{e}a}={(bc)}^{{\mathrm{log}}_{e}b}$ and ${b}^{{\mathrm{log}}_{e}2}={a}^{{\mathrm{log}}_{e}c}$ Then $6a+5bc$ is equal to ______.

If the system of equations $x+2y+3z=3$, $4x+3y-4z=4$ and $8x+4y-\lambda z=9+\mu$ has infinitely many solutions, then the ordered pair $(\lambda ,\mu )$ is equal to

If the domain of the function $f(x)={\mathrm{sec}}^{-1}(\frac{2x}{5x+3})$ is $[\alpha ,\beta )\cup (\gamma ,\delta ]$, then $|3\alpha +10(\beta +\gamma )+21\delta |$ is equal to __________

Let $5$ digit numbers be constructed using the digits $0,2,3,4,7,9$ with repetition allowed, and are arranged in ascending order with serial numbers. Then the serial number of the number $42923$ is _____ .

The range of the function $f(x)=\sqrt{3-x}+\sqrt{2+x}$ is

Let $S$ be the set of values of $\lambda$, for which the system of equations $6\lambda x-3y+3z=4{\lambda }^{2}$, $2x+6\lambda y+4z=1$ and $3x+2y+3\lambda z=\lambda$ has no solution. Then,$12\underset{\lambda \in S}{\sum }|\lambda |$ is equal to _______.

For the system of linear equations $ax+y+z=1$, $x+ay+z=1,x+y+az=\beta$, which one of the following statements is NOT correct?

Let $a,b,c$ be the three distinct positive real numbers such that ${(2a)}^{{\mathrm{log}}_{e}a}={(bc)}^{{\mathrm{log}}_{e}b}$ and ${b}^{{\mathrm{log}}_{e}2}={a}^{{\mathrm{log}}_{e}c}$ Then $6a+5bc$ is equal to ______.

Let $A={1,2,3,4}$ and $R$ be a relation on the set $A\times A$ defined by $R={((a,b),(c,d)):2a+3b=4c+5d}$. Then the number of elements in $R$ is _________.

The total number of three-digit numbers, divisible by $3$, which can be formed using the digits $1,3,5,8$, if repetition of digits is allowed, is

Let $A={\theta \in (0,2\pi ):\frac{1+2i\mathrm{sin}\theta }{1-i\mathrm{sin}\theta }\mathrm{is}\mathrm{purely}\mathrm{imaginary}}$ Then the sum of the elements is in $A$ is

Consider a function $f:\mathbb{N}\rightarrow \mathbb{R}$, satisfying $f(1)+2f(2)+3f(3)+\ldots +xf(x)=x(x+1)f(x);x\geq 2$ with $f(1)=1$. Then $\frac{1}{f(2022)}+\frac{1}{f(2028)}$ is equal to

The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word TOUGH is :