Algebra PYQ — Page 63

If $\alpha$ and $\beta$ are the roots of the quadratic equation ${x}^{2}+xsin\theta -2sin\theta =0, \theta \in (0,\frac{\pi }{2})$ , then $\frac{{\alpha }^{12}+{\beta }^{12}}{({\alpha }^{-12}+{\beta }^{-12}){.(\alpha -\beta )}^{24}}$ is equal to :

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

Let a function $f:(0, \infty) \rightarrow(0, \infty)$ be defined by $f(x)=\left|1-\frac{1}{x}\right| .$ Then $f$ is :

If ${}^{n}{C}_{4},{ }^{n}{C}_{5}$ and ${}^{n}{C}_{6}$ are in A.P., then $n$ can be

The product of three consecutive terms of a $G.P.$ is $512$. If $4$ is added to each of the first and the second of these terms, the three terms now form an $A.P.$, then the sum of the original three terms of the given $G.P.$ is :

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:

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

Let $d\in R,$ and $A=[\begin{matrix}-2 & 4+d & (\mathrm{sin}\theta )-2 \\ 1 & (\mathrm{sin}\theta )+2 & d \\ 5 & (2\mathrm{sin}\theta )-d & (-\mathrm{sin}\theta )+2+2d\end{matrix}],$ $\theta \in [0, 2\pi ].$ If the minimum value of $det(A)$ is $8,$ then a value of $d$ is:

The number of integral values of $m$ for which the equation, $(1+{m}^{2}){x}^{2}-2(1+3m)x+(1+8m)=0$ has no real root, is

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 $m$ is chosen in the quadratic equation $({m}^{2}+1){x}^{2}-3x+{({m}^{2}+1)}^{2}=0$ such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is:

If three distinct numbers $a, b, c$ are in G.P. and the equations $a{x}^{2}+2bx+c=0$ and $d{x}^{2}+2ex+f=0$ have a common root, then which one of the following statements is correct?

The value of $\lambda$ such that sum of the squares of the roots of the quadratic equation, ${x}^{2}+(3-\lambda ) x+2=\lambda$ has the least value is:

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

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

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

Let $z$ be a complex number such that $|z|+z=3+i$ $($ where $i=\sqrt{-1})$ Then $|\mathrm{z}|$ is equal to :

If $\alpha$ and $\beta$ be the roots of the equation ${x}^{2}-2x+2=0,$ then the least value of $n$ for which ${(\frac{\alpha }{\beta })}^{n}=1$ is

Let $A={\theta \in (-\frac{\pi }{2},\pi ):\frac{3+2isin\theta }{1-2i sin\theta } is purely imaginary }.$ Then the sum of the elements in $A$ is:

If $z=\frac{\sqrt{3}}{2}+\frac{i}{2} (i=\sqrt{-1}),$ then ${(1+iz+{z}^{5}+i{z}^{8})}^{9}$ is equal to:

Suppose that $20$ pillars of the same height have been erected along the boundary of circular stadium. If the top of each pillar has been connected by beams with the top of all its non-adjacent pillars, then the total number of beams is:

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 three boxes, each containing $10$ balls labelled $1, 2, \ldots ., 10$. Suppose one ball is randomly drawn from each of the boxes. Denote by ${n}_{i}$, the label of the ball drawn from the ${i}^{th}$ box, $(i=1, 2, 3)$. Then, the number of ways in which the balls can be chosen such that ${n}_{1}<{n}_{2}<{n}_{3}$ is :