Algebra PYQ — Page 68

If $x=a, y=b, z=c$ is a solution of the system of linear equations $x+8y+7z=0$ $9x+2y+3z=0$ $x+y+z=0$ Such that the point $(a,b,c)$ lies on the plane $x+2y+z=6$ , then $2a+b+c$ equals:

For two $3\times 3$ matrices $A$ and $B$, let $A+B=2{B}^{'}$ and $3A+2B={I}_{3},$ where ${B}^{'}$ is the transpose of $B$ and ${I}_{3}$ is $3\times 3$ identity matrix. Then :

Let $A$ be any $3\times 3$ invertible matrix. Then which one of the following is not always true?

Let $f(x)={2}^{10}x+1$ and $g(x)={3}^{10}x-1.$ If $(fog)(x)=x,$ then $x$ is equal to:

Let ${S}_{n}=\frac{1}{{1}^{3}}+\frac{1+2}{{1}^{3}+{2}^{3}}+\frac{1+2+3}{{1}^{3}+{2}^{3}+{3}^{3}}+\ldots +\frac{1+2+\ldots ,+n}{{1}^{3}+{2}^{3}+\ldots {n}^{3}}$ . If $100 {S}_{n}=n,$ then $n$ is equal to:

Let $\omega$ be a complex number such that $2\omega +1=z$ where $z=\sqrt{-3}$ . If $|\begin{matrix}1 & 1 & 1 \\ 1 & -{\omega }^{2}-1 & {\omega }^{2} \\ 1 & {\omega }^{2} & {\omega }^{7}\end{matrix}|=3k,$ Then $k$ can be equal to:

The coefficient of ${x}^{-5}$ in the binomial expansion of ${(\frac{x+1}{{x}^{\frac{2}{3}}-{x}^{\frac{1}{3}}+1 }-\frac{x-1}{x-{x}^{\frac{1}{2}}})}^{10}$ where $x\neq 0,1$ is

The equation $Im(\frac{iz-2}{z-i})+1=0, z\in C, z\neq i$ represents a part of a circle having radius equal to :

If the sum of the first $n$ terms of the series $\sqrt{3}+ \sqrt{75}+ \sqrt{243}+ \sqrt{507}+\ldots$ is $435\sqrt{3},$ then $n$ equals:

For any three positive real numbers $a,b$ and $c$. If $9(25{a}^{2}+{b}^{2})+25({c}^{2}-3ac)=15b(3a+c).$ Then

If $S={x\in [0, 2\pi ] :|\begin{matrix}0 & \mathrm{cos}⁡x & -\mathrm{sin}⁡x \\ \mathrm{sin}⁡x & 0 & \mathrm{cos}⁡x \\ \mathrm{cos}⁡x & \mathrm{sin}⁡x & 0\end{matrix}|=0},$ then $\underset{x \in S}{\sum }\mathrm{tan}(\frac{\pi }{3}+x)$ is equal to:

The number of real values of $\lambda$ for which the system of linear equations, $2x+4y-\lambda z=0$, $4x+\lambda y+2z=0$ and $\lambda x+2y+2z=0$, has infinitely many solutions, is:

The function $f :R\rightarrow [-\frac{1}{2},\frac{1}{2}]$ defined as $f(x)=\frac{x}{1+{x}^{2}},$ is:

The function $f :N\rightarrow I$ defined by $f(x)=x-5[\frac{x}{5}]$ , where $N$ is the set of natural numbers and $[x]$ denotes the greatest integer less than or equal to $x$, is:

If three positive numbers $a$, $b$ and $c$ are in A.P. such that $abc=8$, then the minimum possible value of $b$ is:

The number of ways in which$5$ boys and $3$ girls can be seated on a round table if a particular boy ${B}_{1}$ and a particular girl ${G}_{1}$ never sit adjacent to each other, is:

If $S$ is the set of distinct values of $b$ for which the following system of linear equations $x+y+z=1$ $x+ay+z=1$ $ax+by+z=0$ has no solution, then $S$ is:

Let $a, b, c\in R$ . If $f(x)=a{x}^{2}+bx+c$ is such that $a+b+c=3$ and $f(x+y)=f(x)+f(y)+xy, \forall x, y\in R$ , then $\sum _{ n=1 }^{ 10 } f(n)$ is equal to:

If, for a positive integer $n$, the quadratic equation, $x(x+1)+(x+1)(x+2)+...+(x+\bar{n-1})(x+n)=10n$ has two consecutive integral solutions, then $n$ is equal to:

Let $p(x)$ be a quadratic polynomial such that $p(0)=1.$ If $p(x)$ leaves remainder $4$ when divided by $x-1$ and it leaves remainder $6$ when divided by $x+1$ then:

If ${(27)}^{999}$ is divided by $7$, then the remainder is

The sum of all the real values of $x$ Satisfying the equation ${2}^{(x-1)({x}^{2}+5x-50)}=1$ is:

Let $z\in C,$ the set of complex numbers. Then the equation, $2|z+3i|-|z-i|=0$ represents:

If the arithmetic mean of two numbers $a$ and $b, a>b>0$ , is five times their geometric mean, then $\frac{a+b}{a-b}$ is equal to: