Algebra PYQ — Page 44

If $\alpha ,\beta$ are the roots of the equation ${x}^{2}-(5+{3}^{\sqrt{{\mathrm{log}}_{3}5}}-{5}^{\sqrt{{\mathrm{log}}_{5}3}})x+3({3}^{{({\mathrm{log}}_{3}5)}^{\frac{1}{3}}}-{5}^{{({\mathrm{log}}_{5}3)}^{\frac{2}{3}}}-1)=0$ then the equation, whose roots are $\alpha +\frac{1}{\beta }$ and $\beta +\frac{1}{\alpha }$,

If ${a}_{1},{a}_{2},{a}_{3}\ldots$ and ${b}_{1},{b}_{2},{b}_{3}\ldots .$ are A.P. and ${a}_{1}=2,{a}_{10}=3,{a}_{1}{b}_{1}=1={a}_{10}{b}_{10}$ then ${a}_{4}{b}_{4}$ is equal to

The sum of all real roots of equation $({e}^{2x}-4)(6{e}^{2x}-5{e}^{x}+1)=0$ is

Let $S=${$z\in \mathbb{C}:|z-3|\leq 1$ and$z(4+3i)+\bar{z}(4-3i)\leq 24$}. If $\alpha +i\beta$ is the point in $S$ which is closest to $4i$, then $25(\alpha +\beta )$ is equal to ______.

If the sum of the squares of the reciprocals of the roots $\alpha$ and $\beta$ of the equation $3{x}^{2}+\lambda x-1=0$ is $15$, then $6{({\alpha }^{3}+{\beta }^{3})}^{2}$ is equal to

Let for some real numbers $\alpha$ and $\beta ,a=\alpha -i\beta$. If the system of equations $4ix+(1+i)y=0$ and $8(\mathrm{cos}\frac{2\pi }{3}+i\mathrm{sin}\frac{2\pi }{3})x+\bar{a}y=0$ has more than one solution then $\frac{\alpha }{\beta }$ is equal to

If the coefficient of ${x}^{10}$ in the binomial expansion of ${(\frac{\sqrt{x}}{{5}^{\frac{1}{4}}}+\frac{\sqrt{5}}{{x}^{\frac{1}{3}}})}^{60}$ is ${5}^{k}l$, where $l,k\in N$ and $l$ is coprime to $5$, then $k$ is equal to ______.

The number of one-one functions $f:{a,b,c,d}\rightarrow {0,1,2,\ldots ,10}$ such that $2f(a)-f(b)+3f(c)+f(d)=0$ is _____

If $z=x+iy$ satisfies $|z|-2=0$ and $|z-i|-|z+5i|=0$, then

Which of the following matrices can NOT be obtained from the matrix $[\begin{matrix}-1 & 2 \\ 1 & -1\end{matrix}]$ by a single elementary row operation?

Let $\alpha ,\beta$ and $\gamma$ be three positive real numbers. Let $f(x)=\alpha {x}^{5}+\beta {x}^{3}+\gamma x,x\in R$ and $g:R\rightarrow R$ be such that $g(f(x))=x$ for all $x\in R$. If ${a}_{1},{a}_{2},{a}_{3},\ldots ,{a}_{n}$ be in arithmetic progression with mean zero, then the value of $f(g(\frac{1}{n}\sum _{i=1}^{n}f({a}_{i})))$ is equal to

Let${A}_{1},{A}_{2},{A}_{3},\ldots \ldots$ be an increasing geometric progression of positive real numbers. If ${A}_{1}{A}_{3}{A}_{5}{A}_{7}=\frac{1}{1296}$ and ${A}_{2}+{A}_{4}=\frac{7}{36}$, then, the value of ${A}_{6}+{A}_{8}+{A}_{10}$ is equal to

The number of real solutions of the equation, ${x}^{2}-|x|-12=0$ is:

If $n$ is the number of irrational terms in the expansion of ${({3}^{1/4}+{5}^{1/8})}^{60}$, then $(n-1)$ is divisible by :

Let the coefficients of third, fourth and fifth terms in the expansion of ${(x+\frac{a}{{x}^{2}})}^{n},x\neq 0,$ be in the ratio $12:8:3.$ Then the term independent of $x$ in the expansion, is equal to _______.

Let $M$ be any $3\times 3$ matrix with entries from the set ${0,1,2}$. The maximum number of such matrices, for which the sum of diagonal elements of ${M}^{T}M$ is seven, is______.

Let $a,b,c$ be in arithmetic progression. Let the centroid of the triangle with vertices $(a,c),(2,b)$ and $(a,b)$ be $(\frac{10}{3},\frac{7}{3}).$ If $\alpha ,\beta$ are the roots of the equation $a{x}^{2}+bx+1=0,$ then the value of ${\alpha }^{2}+{\beta }^{2}-\alpha \beta$ is:

Let $f,g:N\rightarrow N$ such that $f(n+1)=f(n)+f(1)\forall n\in N$ and $g$ be any arbitrary function. Which of the following statements is NOT true?

A point $z$ moves in the complex plane such that $\mathrm{arg}(\frac{z-2}{z+2})=\frac{\pi }{4},$ then the minimum value of $|z-9\sqrt{2}-2i{|}^{2}$ is equal to

The number of solutions of the equation ${\mathrm{log}}_{4}(x-1)={\mathrm{log}}_{2}(x-3)$ is ______.

There are $5$ students in class $10,6$ students in class $11$ and $8$ students in class $12.$ If the number of ways, in which $10$ students can be selected from them so as to include at least $2$ students from each class and at most $5$ students from the total $11$ students of class $10$ and $11$ is $100k,$ then $k$ is equal to

Let $\lambda \neq 0$ be in $R.$ If $\alpha$ and $\beta$ are the roots of the equation ${x}^{2}-x+2\lambda =0,$ and $\alpha$ and $\gamma$ are the roots of the equation $3{x}^{2}-10x+27\lambda =0,$ then $\frac{\beta \gamma }{\lambda }$ is equal to ________.

The lowest integer which is greater than ${(1+\frac{1}{{10}^{100}})}^{{10}^{100}}$ is

A function $f(x)$ is given by $f(x)=\frac{{5}^{x}}{{5}^{x}+5}$, then the sum of the series $f(\frac{1}{20})+f(\frac{2}{20})+f(\frac{3}{20})+\ldots +f(\frac{39}{20})$ is equal to: