Algebra PYQ — Page 73

Let $\mathrm{G}$ be the geometric mean of two positive numbers $\mathrm{a}$ and $\mathrm{b}$, and $\mathrm{M}$ be the arithmetic mean of $\frac{1}{\mathrm{a}}$ and $\frac{1}{\mathrm{~b}}$. If $\frac{1}{\mathrm{M}}: \mathrm{G}$ is $4: 5$, then $\mathrm{a}: \mathrm{b}$ can be:

Let $z \neq-i$ be any complex number such that $\frac{\mathrm{z}-\mathrm{i}}{\mathrm{z}+\mathrm{i}}$ is a purely imaginary number. Then $\mathrm{z}+\frac{1}{\mathrm{z}}$ is:

If $a,b,c$ are non - zero real numbers and if the system of equations $(a-1)x=y+z$ $(b-1)y=x+z$ $(c-1)z=x+y$ has a non - trivial solution, then $ab+bc+ca$ equals :

If a complex number $z$ statisfies the equation $x+\sqrt{2}|z+1|+i=0$, then $|z|$ is equal to :

The sum of the series: $1+\frac{1}{1+2}+\frac{1}{1+2+3}+\ldots \ldots$. upto 10 terms, is:

The ratio of the coefficient of $x^{15}$ to the term independent of $x$ in the expansion of $\left(x^2+\frac{2}{x}\right)^{15}$ is:

If $x,y,z$ are positive numbers in $A.P.$ and ${\mathrm{tan}}^{-1}x$, ${\mathrm{tan}}^{-1}y$ and ${\mathrm{tan}}^{-1}z$ are also in $A.P.$, then which of the following is correct.

If $z$ is a complex number of unit modulus and argument $\theta$, then arg $(\frac{1+z}{1+\overset{-}{z}})$ can be equal to $(\mathrm{given} z\neq -1)$

The term independent of $x$ in the expansion of ${(\frac{x+1}{{x}^{2/3}-{x}^{1/3}+1}-\frac{x-1}{x-{x}^{1/2}})}^{10}$ is

If $a_1, a_2, a_3, \ldots, a_n, \ldots$. are in A.P. such that $a_4-a_7$ $+a_{10}=m$, then the sum of first 13 terms of this A.P., is :

Let $a_1, a_2, a_3, \ldots$ be an A.P, such that $\frac{a_1+a_2+\ldots+a_p}{a_1+a_2+a_3+\ldots+a_q}=\frac{p^3}{q^3} ; p \neq q$. Then $\frac{a_6}{a_{21}}$ is equal to:

If $P=[\begin{matrix}1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4\end{matrix}]$ is the adjoint of a $3\times 3$ matrix $A$ and $|A|=4$, then $\alpha$ is equal to

The matrix $A^2+4 A-5 I$, where $I$ is identity matrix and $A=\left[\begin{array}{cc}1 & 2 \\ 4 & -3\end{array}\right]$, equals :

The number of values of $k$, for which the system of equations : $(k+1)x+8y=4k$ $kx+(k+3)y=3k-1$ has no solution, is :

The sum $\frac{3}{1^2}+\frac{5}{1^2+2^2}+\frac{7}{1^2+2^2+3^2}+\ldots$. upto 11-terms is:

Let ${T}_{n}$ be the number of all possible triangles formed by joining vertices of an $n$-sided regular polygon. If ${T}_{n+1}-{T}_{n}=10$, then the value of $n$ is :

A committee of 4 persons is to be formed from 2 ladies, 2 old men and 4 young men such that it includes at least 1 lady, at least 1 old man and at most 2 young men. Then the total number of ways in which this committee can be formed is :

The values of ' $a$ ' for which one root of the equation $x^2-(a+1) x+a^2+a-8=0$ exceeds 2 and the other is lesser than 2 , are given by :

If the equations ${x}^{2} + 2 x + 3 = 0$ and $a{x}^{2} +bx+c=0,$ $a,b,c\in R,$ have a common root, then $a:b:c$ is:

If the system of linear equations : $$ \begin{aligned} & x_1+2 x_2+3 x_3=6 \\ & x_1+3 x_2+5 x_3=9 \\ & 2 x_1+5 x_2+a x_3=b \end{aligned} $$ is consistent and has infinite number of solutions, then :

If $a, b, c$ are sides of a scalene triangle, then the value of $\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|$ is :

The least integral value $\alpha$ of $x$ such that $\frac{x-5}{x^2+5 x-14}>0$, satisfies :

Let $z$ satisfy $|z|=1$ and $z=1-\bar{z}$. Statement $1: z$ is a real number. Statement 2 : Principal argument of z is $\frac{\pi}{3}$

Let $\mathrm{A}$, other than $\mathrm{I}$ or $-\mathrm{I}$, be a $2 \times 2$ real matrix such that $\mathrm{A}^2=\mathrm{I}$, I being the unit matrix. Let $\operatorname{Tr}(\mathrm{A})$ be the sum of diagonal elements of A. Statement-1: $\operatorname{Tr}(\mathrm{A})=0$ Statement-2: $\operatorname{det}(\mathrm{A})=-1$