Algebra PYQ — Page 59

If the sum of the second, third and fourth terms of a positive term G.P. is $3$ and the sum of its sixth, seventh and eighth terms is $243$, then the sum of the first $50$ terms of this G.P. is :

For which of the following ordered pairs $(\mu ,\delta ),$ the system of linear equations $x+2y+3z=1$ $3x+4y+5z=\mu$ $4x+4y+4z=\delta$ is inconsistent?

In the expansion of ${(\frac{x}{\mathrm{cos}⁡\theta }+\frac{1}{x\mathrm{sin}⁡\theta })}^{16},$ if ${l}_{1}$ is the least value of the term independent of $x$ when $\frac{\pi }{8}\leq \theta \leq \frac{\pi }{4}$ and ${l}_{2}$ is the least value of the term independent of $x$ when $\frac{\pi }{16}\leq \theta \leq \frac{\pi }{8},$ then the ratio ${l}_{2}:{l}_{1}$ is equal to:

Let $a,b,c\in R$ be all non-zero and satisfies ${a}^{3}+{b}^{3}+{c}^{3}=2$. If the matrix $A=[\begin{matrix}a & b & c \\ b & c & a \\ c & a & b\end{matrix}]\begin{matrix} \\ \\ \end{matrix}$ satisfies ${A}^{T}A=I,$ then a value of $abc$ can be

The natural number $m$, for which the coefficient of $x$ in the binomial expansion of ${({x}^{m}+\frac{1}{{x}^{2}})}^{22}$ is 1540, is

Let $S$ be the set of all $\lambda \in R$ for which the system of linear equations $2x-y+2z=2$ $x-2y+\lambda z=-4$ $x+\lambda y+z=4$ has no solution. Then the set $S$

The value of${(\frac{-1+i\sqrt{3}}{1-i})}^{30}$ is :

The number of all $3\times 3$ matrices $A,$ with entries from the set ${-1,0,1}$ such that the sum of the diagonal elements of $A{A}^{T}$ is $3,$ is ___________.

Let $\alpha =\frac{-1+i\sqrt{3}}{2}$. If $a=(1+\alpha )\sum _{k=0}^{100}{\alpha }^{2k}$ and $b=\sum _{k=0}^{100}{\alpha }^{3k}$, then $a$ and $b$, are the roots of the quadratic equation.

The sum of distinct values of $\lambda$ for which the system of equations : $(\lambda -1)x+(3\lambda +1)y+2\lambda z=0$ $(\lambda -1)x+(4\lambda -2)y+(\lambda +3)z=0$ $2x+(3\lambda +1)y+3(\lambda -1)z=0$, Has non-zero solutions, is ....... .

If the sum of the coefficients of all even powers of $x$ in the product $(1+x+{x}^{2}+\ldots +{x}^{2n})(1-x+{x}^{2}-{x}^{3}+\ldots +{x}^{2n})$ is $61,$ then n is equal to

Let$n>2$ be an integer. Suppose that there are $n$ Metro stations in a city located around a circular path. Each pair of the nearest stations is connected by a straight track only. Further, each pair of the nearest station is connected by blue line, whereas all remaining pairs of stations are connected by red line. If number of red lines is $99$ times the number of blue lines, then the value of $n$ is

A survey shows that $63%$ of the people in a city read newspaper $A$ whereas $76%$ read news paper $B$. If $x%$ of the people read both the newspapers, then a possible value of $x$ can be:

Let $z=x+\mathrm{iy}$ be a non-zero complex number such that ${z}^{2}=i{|z|}^{2}$, where $i=\sqrt{-1}$, then $z$ lies on the :

A survey shows that $73%$ of the persons working in an office like coffee, whereas $65%$ like tea. If $x$ denotes the percentage of them, who like both coffee and tea, then $x$ cannot be:

If the system of linear equations $2x+2ay+az=0$ $2x+3by+bz=0$ $2x+4cy+cz=0,$ where $a,b,c\in R$ are non-zero and distinct; has a non-zero solution, then

Let $\alpha$ and $\beta$ be the roots of ${x}^{2}-3x+p=0$ and $\gamma$ and $\delta$ be the roots of ${x}^{2}-6x+q=0.$ If $\alpha ,\beta ,\gamma ,\delta$ from a geometric progression. Then ratio $(2q+p):(2q-p)$ is

An urn contains $5$ red marbles, $4$ black marbles and $3$ white marbles. Then, the number of ways in which $4$ marbles can be drawn so that at the most three of them are red is ___________.

If $m$ arithmetic means (A.Ms) and three geometric means (G.Ms) are inserted between $3$ and $243$ such that ${4}^{th}$ A.M. is equal to ${2}^{nd}$ G.M., then $m$ is equal to:

The value of $\sum _{r=0}^{20}C650-r$ is equal to:

Let m and M be respectively the minimum and maximum value values of $|\begin{matrix}{\mathrm{cos}}^{2}x & 1+{\mathrm{sin}}^{2}x & \mathrm{sin}2x \\ 1+{\mathrm{cos}}^{2}x & {\mathrm{sin}}^{2}x & \mathrm{sin}2x \\ {\mathrm{cos}}^{2}x & {\mathrm{sin}}^{2}x & 1+\mathrm{sin}2x\end{matrix}|$ Then the ordered pair (m, M) is equal to:

Let $\alpha$ and $\beta$ be two real roots of the equation $(k+1){tan}^{2}x-\sqrt{2}\cdot \lambda \mathrm{tan}⁡x=(1-k),$ where $k(\neq -1)$ and $\lambda$ are real numbers. If ${tan}^{2}(\alpha +\beta )=50,$ then a value of $\lambda$ is

If the four complex numbers $z,\bar{z},\bar{z}-2Re(\bar{z})$ and $z-2Re(z)$ represent the vertices of a square of side $4$ units in the Argand plane, then $|z|$ is equal to :

The following system of linear equations $7x+6y-2z=0$ $3x+4y+2z=0$ $x-2y-6z=0,$ has