Algebra PYQ — Page 54

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

The following system of linear equations $2x+3y+2z=9$ $3x+2y+2z=9$ $x-y+4z=8$

If the fourth term in the expansion of ${(x+{x}^{{\mathrm{log}}_{2}x})}^{7}$ is $4480,$ then the value of $x$ where $x\in N$ is equal to:

If the coefficients of ${x}^{7}$ in ${({x}^{2}+\frac{1}{bx})}^{11}$ and ${x}^{-7}$ in ${(x-\frac{1}{b{x}^{2}})}^{11},b\neq 0,$ are equal, then the value of $b$ is equal to:

Let $A={1,2,3,\ldots ,10}$ and $f:A\rightarrow A$ be defined as $f(k)={\begin{matrix}k+1 & \mathrm{if}k\mathrm{is}odd \\ k & \mathrm{if}k\mathrm{is}even\end{matrix}$ Then the number of possible functions $g:A\rightarrow A$ such that $gof=f$ is:

If $\alpha ,\beta$ are roots of the equation ${x}^{2}+5(\sqrt{2})x+10=0,\alpha >\beta$ and ${P}_{n}={\alpha }^{n}-{\beta }^{n}$ for each positive integer $n,$ then the value of $(\frac{{P}_{17}{P}_{20}+5\sqrt{2}{P}_{17}{P}_{19}}{{P}_{18}{P}_{19}+5\sqrt{2}{P}_{18}^{2}})$ is equal to

The number of rational terms in the binomial expansion of ${({4}^{\frac{1}{4}}+{5}^{\frac{1}{6}})}^{120}$is_______.

Let $n$ be a positive integer. Let $A=\sum _{k=0}^{n}{(-1)}^{k}\times Ckn[{(\frac{1}{2})}^{k}+{(\frac{3}{4})}^{k}+{(\frac{7}{8})}^{k}+{(\frac{15}{16})}^{k}+{(\frac{31}{32})}^{k}]$. If $63A=1-\frac{1}{{2}^{30}},$ then $n$ is equal to ______ .

The sum of all those terms which are rational numbers in the expansion of ${({2}^{\frac{1}{3}}+{3}^{\frac{1}{4}})}^{12}$ is:

If for the matrix, $A=[\begin{matrix}1 & -\alpha \\ \alpha & \beta \end{matrix}],A{A}^{T}={I}_{2}$, then the value of ${\alpha }^{4}+{\beta }^{4}$ is :

Let ${a}_{1},{a}_{2},\ldots ,{a}_{21}$ be an $A.P.$ such that $\sum _{n=1}^{20}\frac{1}{{a}_{n}{a}_{n+1}}=\frac{4}{9}.$ If the sum of this $A.P.$ is $189,$ then ${a}_{6}{a}_{16}$ is equal to :

Let ${a}_{n}$ be the ${n}^{th}$ term of a G.P. of positive terms. If $\sum _{n=1}^{100}{a}_{2n+1}=200$ and $\sum _{n=1}^{100}{a}_{2n}=100,$ then $\sum _{n=1}^{200}{a}_{n}$ is equal to:

Consider the two sets: $A={m\in R:$ both the roots of ${x}^{2}-(m+1)x+m+4=0$ are real $}$ and $B=[-3,5)$ Which of the following is not true?

If the system of equations $x+y+z=2$ $2x+4y-z=6$ $3x+2y+\lambda z=\mu$ has infinitely many solutions, then :

The domain of the function $f(x)={\mathrm{sin}}^{-1}(\frac{|x|+5}{{x}^{2}+1})$ is $(-\infty ,-a]\cup [a,\infty )$, then $a$ is equal to

Suppose the vectors ${x}_{1},{x}_{2}$ and ${x}_{3}$ are the solutions of the system of linear equations, $Ax=b$ when the vector $b$ on the right side is equal to ${b}_{1},{b}_{2}$ and ${b}_{3}$ respectively. If ${x}_{1}=[\begin{matrix}1 \\ 1 \\ 1\end{matrix}],{x}_{2}=[\begin{matrix}0 \\ 2 \\ 1\end{matrix}],{x}_{3}=[\begin{matrix}0 \\ 0 \\ 1\end{matrix}]$; ${b}_{1}=[\begin{matrix}1 \\ 0 \\ 0\end{matrix}],{b}_{2}=[\begin{matrix}0 \\ 2 \\ 0\end{matrix}],{b}_{3}=[\begin{matrix}0 \\ 0 \\ 2\end{matrix}]$, then the determinant of $A$ is equal to

Let $\alpha >0,\beta >0$ be such that ${\alpha }^{3}+{\beta }^{2}=4$. If the maximum value of the term independent of $x$ in the binomial expansion of ${(\alpha {x}^{\frac{1}{9}}+\beta {x}^{-\frac{1}{6}})}^{10}$ is $10k$, then $k$ is equal to

Let $a,b\in R,a\neq 0$ be such that the equation, $a{x}^{2}-2bx+5=0$ has a repeated root $\alpha ,$ which is also a root of the equation, ${x}^{2}-2bx-10=0.$ If $\beta$ is the other root of this equation, then ${\alpha }^{2}+{\beta }^{2}$ is equal to:

Let $\lambda \in R$. The system of linear equations $2{x}_{1}-4{x}_{2}+\lambda {x}_{3}=1$ ${x}_{1}-6{x}_{2}+{x}_{3}=2$ $\lambda {x}_{1}-10{x}_{2}+4{x}_{3}=3$ is inconsistent for :

The set of all real values of $\lambda$ for which the quadratic equation $({\lambda }^{2}+1){x}^{2}-4\lambda x+2=0$ always have exactly one root in the interval $(0,1)$ is :

The value of ${(\frac{1+\mathrm{sin}\frac{2\pi }{9}+i\mathrm{cos}\frac{2\pi }{9}}{1+\mathrm{sin}\frac{2\pi }{9}-i\mathrm{cos}\frac{2\pi }{9}})}^{3}$ is

If $A={x\in R:|x|<2}$ and $B={x\in R:|x-2|\geq 3};$ then

The sum, $\sum _{n=1}^{7}\frac{n(n+1)(2n+1)}{4}$, is equal to

If the sum of the first $20$ terms of the series ${\mathrm{log}}_{({7}^{1/2})}x+{\mathrm{log}}_{({7}^{1/3})}x+{\mathrm{log}}_{({7}^{1/4})}x+\ldots$is $460$, then $x$ is equal to: