Algebra PYQ — Page 49

Let ${z}_{1},{z}_{2}$ be the roots of the equation ${z}^{2}+az+12=0$ and ${z}_{1},{z}_{2}$ form an equilateral triangle with origin. Then, the value of $|a|$ is

Let $\alpha ,\beta ,\gamma$ be the real roots of the equation, ${x}^{3}+a{x}^{2}+bx+c=0,$ ($a,b,c\in R$ and $a,b\neq 0$). If the system of equations (in, $u,v,w$) given by $\alpha u+\beta v+\gamma w=0,\beta u+\gamma v+\alpha w=0,\gamma u+\alpha v+\beta w=0$ has non-trivial solution, then the value of $\frac{{a}^{2}}{b}$ is

Let $[\lambda ]$ be the greatest integer less than or equal to $\lambda$. The set of all values of $\lambda$ for which the system of linear equations $x+y+z=4,3x+2y+5z=3,9x+4y+(28+[\lambda ])z=[\lambda ]$ has a solution is:

Let ${A}_{1},{A}_{2},{A}_{3},\ldots ..$ be squares such that for each $n\geqslant 1,$ the length of the side of ${A}_{n}$ equals the length of diagonal of ${A}_{n+1}$. If the length of ${A}_{1}$ is $12\mathrm{cm}$, then the smallest value of $n$ for which area of ${A}_{n}$ is less than one, is

Let ${P}_{1},{P}_{2}\ldots ,{P}_{15}$ be $15$ points on a circle. The number of distinct triangles formed by points ${P}_{i},{P}_{j},{P}_{k}$ such that $i+j+k\neq 15,$ is :

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 ________.

Let $a,b,c,d$ be in arithmetic progression with common difference $\lambda$. If $|\begin{matrix}x+a-c & x+b & x+a \\ x-1 & x+c & x+b \\ x-b+d & x+d & x+c\end{matrix}|=2$, then value of ${\lambda }^{2}$ is equal to________.

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 $f:R-{\frac{\alpha }{6}}\rightarrow R$ be defined by $f(x)=(\frac{5x+3}{6x-\alpha })$. Then the value of $\alpha$ for which $(fof)(x)=x$, for all $x\in R-{\frac{\alpha }{6}},$ is

Let $f:R-{3}\rightarrow R-{1}$ be defined by $f(x)=\frac{x-2}{x-3}$. Let $g:R\rightarrow R$ be given as $g(x)=2x-3$. Then, the sum of all the values of $x$ for which ${f}^{-1}(x)+{g}^{-1}(x)=\frac{13}{2}$ is equal to

Let $f:R\rightarrow R$ be defined as $f(x+y)+f(x-y)=2f(x)f(y),f(\frac{1}{2})=-1.$ Then the value of $\sum _{k=1}^{20}\frac{1}{\mathrm{sin}(k)\mathrm{sin}(k+f(k))}$ is equal to :

Let $g:N\rightarrow N$ be defined as $g(3n+1)=3n+2$ $g(3n+2)=3n+3$ $g(3n+3)=3n+1$, for all $n\geq 0$ Then which of the following statements is true ?

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:

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 ${a}_{1},{a}_{2},\ldots ,{a}_{10}$ be an $A.P.$ with common difference $-3$ and ${b}_{1},{b}_{2},\ldots ,{b}_{10}$ be a $G.P.$ with common ratio $2.$ Let ${c}_{k}={a}_{k}+{b}_{k},k=1,2,\ldots ,10.$ If ${c}_{2}=12$ and ${c}_{3}=13,$ then $\sum _{k=1}^{10}{c}_{k}$ is equal to ______.

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 $I$ be an identity matrix of order $2\times 2$ and $P=[\begin{matrix}2 & -1 \\ 5 & -3\end{matrix}]$. Then the value of $n\in N$ for which ${P}^{n}=5I-8P$ is equal to ___ .

Let ${a}_{1},{a}_{2},{a}_{3},\ldots$ be an A.P. If $\frac{{a}_{1}+{a}_{2}+\ldots +{a}_{10}}{{a}_{1}+{a}_{2}+\ldots +{a}_{p}}=\frac{100}{{p}^{2}},p\neq 10,$ then $\frac{{a}_{11}}{{a}_{10}}$ is equal to :

Let $A$ be a symmetric matrix of order $2$ with integer entries. If the sum of the diagonal elements of ${A}^{2}$ is $1,$ then the possible number of such matrices is:

Let ${{{a}_{n}}}_{n=1}^{\infty }$ be a sequence such that ${a}_{1}=1,{a}_{2}=1$ and ${a}_{n+2}=2{a}_{n+1}+{a}_{n}$ for all $n\geq 1$. Then the value of $47\sum _{n=1}^{\infty }(\frac{{a}_{n}}{{2}^{3n}})$ is equal to ________.

Let $A=[{a}_{ij}]$ be a real matrix of order $3\times 3,$ such that ${a}_{i1}+{a}_{i2}+{a}_{i3}=1,$ for $i=1,2,3.$ Then, the sum of all the entries of the matrix ${A}^{3}$ is equal to:

Let $A$ be a $3\times 3$ real matrix. If $det(2Adj(2Adj(Adj(2A))))={2}^{41},$ then the value of $det({A}^{2})$ equals ______.

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 ______ .

Let $f(x)$ be a polynomial of degree $3$ such that $f(k)=-\frac{2}{k}$ for $k=2,3,4,5.$ Then the value of $52-10f(10)$ is equal to _____ .