Algebra PYQ — Page 64

Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^{2} \sin \theta-x(\sin \theta \cos \theta+1)+\cos \theta=0\left(0 < \theta < 45^{\circ}\right),$ and $\alpha < \beta .$ Then $\sum_{n=0}^{\infty}\left(\alpha^{n}+\frac{(-1)^{n}}{\beta^{n}}\right)$ is equal to :

The sum of the real values of $x$ for which the middle term in the binomial expansion of $\left(\frac{x^{3}}{3}+\frac{3}{x}\right)^{8}$ equals 5670 is :

If three of the six vertices of a regular hexagon are chosen at random, then the probability that the triangle formed with these chosen vertices is equilateral 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:

Let ${z}_{0}$ be a root of quadratic equation, ${x}^{2}+x+1=0.$ If $z=3+6i{z}_{0}^{81}-3i{z}_{0}^{93}$ , then $arg$ $(z)$ is equal to:

The term independent of $x$ in the expansion of $(\frac{1}{60}-\frac{{x}^{8}}{81}).{(2{x}^{2}-\frac{3}{{x}^{2}})}^{6}$ is equal to

There are $m$ men and two women participating in a chess tournament. Each participant plays two games with every other participant. If the number of games played by the men between themselves exceeds the number of games played between the men and the women by $84$, then the value of $m$ is :

If $a>0$ and $z=\frac{{(1+i)}^{2}}{a-i}$ , has magnitude $\sqrt{\frac{2}{5}}$ , then $\overset{-}{z}$ is equal to:

Let $S={1, 2, 3,\ldots .,100}$, then number of non-empty subsets $A$ of $S$ such that the product of elements in $A$ is even is :

The sum of an infinite geometric series with positive terms is 3 and the sum of the cubes of its terms is $\frac{27}{19}$. Then the common ratio of this series is:

The Number of ways of choosing $10$ objects out of $31$ objects of which $10$ are identical and the remaining $21$ are distinct, is:

The equation $|z-i|=|z-1|,i=\sqrt{-1},$ represents:

If the fractional part of the number $\frac{{2}^{403}}{15}$ is $\frac{k}{15},$ then $k$ is equal to

If $\alpha$and $\beta$ are the roots of the equation $375 {x}^{2}-25x-2=0,$ then $\underset{n\rightarrow \infty }{lim}\sum _{r=1}^{n}{\alpha }^{r}+\underset{n\rightarrow \infty }{lim}\sum _{r=1}^{n}{\beta }^{r}$ is equal to:

The sum of all two digit positive numbers which when divided by $7$ yield $2$ or $5$ as remainder is

A committee of $11$ member is to be formed from $8$ males and $5$ females. If $m$ is the number of ways the committee is formed with at least $6$ males and $n$ is the number of ways the committee is formed with at least $3$ females, then:

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 $f: R \rightarrow R$ be defined by $f(x)=\frac{x}{1+x^{2}}, x \in R .$ Then the range of $f$ is

The domain of the definition of the function $f(x)=\frac{1}{4-{x}^{2}}+{log}_{10}({x}^{3}-x)$ is:

Let $\left(-2-\frac{1}{3} i\right)^{3}=\frac{x+i y}{27}(i=\sqrt{-1}),$ where $x$ and $y$ are real numbers then $\mathrm{y}-\mathrm{x}$ equals

If the sum of the first $15$ terms of the series ${(\frac{3}{4})}^{3}+{(1\frac{1}{2})}^{3}+{(2\frac{1}{4})}^{3}+{3}^{3}+{(3\frac{3}{4})}_{ }^{3}+\ldots$ is equal to $225K$, then $K$ is equal to :

The sum of the real roots of the equation $|\begin{matrix}x & -6 & -1 \\ 2 & -3x & x-3 \\ -3 & 2x & x+2\end{matrix}|=0,$ is equal to:

Let $z={(\frac{\sqrt{3}}{2}+\frac{i}{2})}^{5}+{(\frac{\sqrt{3}}{2}-\frac{i}{2})}^{5}.$ If $R(z)$ and $I(z)$ respectively denote the real and imaginary parts of $z,$ then

If $A=[\begin{matrix}1 & \mathrm{sin}\theta & 1 \\ -\mathrm{sin}\theta & 1 & \mathrm{sin}\theta \\ -1 & -\mathrm{sin}\theta & 1\end{matrix}]$, then for all $\theta \in (\frac{3\pi }{4},\frac{5\pi }{4})$, $\mathrm{det}(A)$ lies in the interval :