Algebra PYQ — Page 63

The number of all possible positive integral value of $\alpha$ for which the roots of the quadratic equation $6{x}^{2}-11x+\alpha =0$ are rational numbers is:

Let $\alpha$ and $\beta$ be the roots of the equation ${x}^{2}+2x+2=0,$ then ${\alpha }^{15}+{\beta }^{15}$ is equal to

If $z$ and $\omega$ are two complex numbers such that $|z\omega |=1$ and $arg(z)-arg(\omega )=\frac{\pi }{2}$, then:

All the points in the set $S={\frac{\alpha +i}{\alpha -i},\alpha \in R},i=\sqrt{-1}$ lie on a

Let the sum of the first $n$ terms of a non-constant $A.P.,{a}_{1},{a}_{2},{a}_{3},....,{a}_{n}$ be $50n+\frac{n(n-7)}{2}A,$ where $A$ is a constant. If $d$ is the common difference of this $A.P.$, then the ordered pair $(d,{a}_{50})$ is equal to

The number of four-digit numbers strictly greater than $4321$ that can be formed using the digit $0,1,2,3,4,5$ (repetition of digits is allowed) is:

Consider a class of $5$ girls and $7$ boys. The number of different teams consisting of $2$ girls and $3$ boys that can be formed from this class, if there are two specific boys $A$ and $B,$ who refuse to be the members of the same team, is:

Let $f(x)={\mathrm{log}}_{e}(sinx), (0<x<\pi )$ and $g(x)={\mathrm{sin}}^{-1}({e}^{-x}), (x\geq 0).$ If $\alpha$ is a positive real number such that $a={(fog)}^{'}(\alpha )$ and $b=(fog)(\alpha ),$ then

Let ${S}_{n}$ denote the sum of the first $n$ terms of an $A.P.$. If ${S}_{4}=16$ and ${S}_{6}=-48$ , then ${S}_{10}$ is equal to:

If $si{n}^{4}\alpha +4co{s}^{4}\beta +2=4\sqrt{2}sin\alpha cos\beta$, $\alpha ,\beta \in [0,\pi ]$, then $cos(\alpha +\beta )-cos(\alpha -\beta )$ is equal to

The sum of the following series $1+6+\frac{9({1}^{2}+{2}^{2}+{3}^{2})}{7}+\frac{12({1}^{2}+{2}^{2}+{3}^{2}+{4}^{2})}{9}+\frac{15({1}^{2}+{2}^{2}+\ldots +{5}^{2})}{11}+....$ up to $15$ terms, is:

A group of students comprises of $5$ boys and n girls. If the number of ways, in which a team of $3$ students can randomly be selected from this group such that there is at least one boy and at least one girl in each team, is $1750,$ then n is equal to

If the coefficients of ${x}^{2}$ and ${x}^{3}$, are both zero, in the expansion of the expression $(1+ax+b{x}^{2}){(1-3x)}^{15}$, in powers of $x$ , then the ordered pair $(a,b)$ is equal to

The total number of irrational terms in the binomial expansion of ${({7}^{\frac{1}{5}}-{3}^{\frac{1}{10}})}^{60}$ is

The sum of the series $1+2\times 3+3\times 5+4\times 7+\ldots$ upto ${11}^{th}$ term is:

In a class of $140$ students numbered $1$ to$140$ , all even numbered students opted Mathematics course, those whose number is divisible by $3$ opted Physics course and those whose number is divisible by $5$ opted Chemistry course. Then the number of students who did not opt for any of the three courses is:

If $a, b$ and $c$ be three distinct real numbers in G.P. and $a+b+c=xb,$ then $x$ cannot be:

If the system of linear equations $x-4y+7z=g$; $3y-5z=h$; $-2x+5y-9z=k$ is consistent, then:

If the sum and product of the first three terms in an $A.P.$ are $33$ and $1155$, respectively, then a value of its ${11}^{th}$ term is:

Let $P=[\begin{matrix}1 & 0 & 0 \\ 3 & 1 & 0 \\ 9 & 3 & 1\end{matrix}]$ and $Q=[{q}_{ij}]$ be two $3\times 3$ matrices such that $Q-{P}^{5}={I}_{3}$. Then $\frac{{q}_{21}+{q}_{31}}{{q}_{32}}$ is equal to :

Let $\lambda$ be a real number for which the system of linear equations $x+y+z=6,$ $4x+\lambda y-\lambda z=\lambda -2$ and $3x+2y-4z=-5$ has infinitely many solutions. Then $\lambda$ is a root of the quadratic equation:

If $A$ is a symmetric matrix and $B$ is skew- symmetric matrix such that $A+B=[\begin{matrix}2 & 3 \\ 5 & -1\end{matrix}]$ , then $AB$ is equal to:

Let $a_{1}, a_{2}, \ldots, a_{10}$ be a G.P. If $\frac{a_{3}}{a_{1}}=25,$ then $\frac{a_{9}}{a_{5}}$ equals :

The coefficient of ${t}^{4}$ in the expansion of ${(\frac{1-{t}^{6}}{1-t})}^{3}$ is