Algebra PYQ — Page 77

If $\omega(\neq 1)$ is a cube root of unity, and $(1+\omega)^7=A+B \omega$. Then $(A, B)$ equals

The coefficient of $x^7$ in the expansion of $\left(1-x-x^2+x^3\right)^6$ is

The domain of the function $f(x)=\frac{1}{\sqrt{|x|-x}}$ is

Let $R$ be the set of real numbers This question has Statement $-1$ and Statement $-2$. Of the four choices given after the statements, choose the one that best describes the two statements. Statement-1 : $A=\{(x, y) \in R \times R: y-x$ is an integer $\}$ is an equivalence relation on $R$. Statement-2 : $B=\{(x, y) \in R \times R: x=\alpha y$ for some rational number $\alpha\}$ is an equivalence relation on $\mathrm{R}$.

Let $S$ be a non-empty subset of R. Consider the following statement: $\mathrm{P}$ : There is a rational number $\mathrm{x} \in \mathrm{S}$ such that $\mathrm{x}>0$. Which of the following statements is the negation of the statement $P$ ?

Consider the following relations: $R=\{(x, y) \mid x, y$ are real numbers and $x=$ wy for some rational number w $\}$; $S=\left\{\left(\frac{m}{n}, \frac{p}{q}\right) \mid m, n, p\right.$ and $q$ are integers such that $n, q \neq 0$ and $\left.q m=p n\right\}$. Then

If $\alpha$ and $\beta$ are the roots of the equation $x^2-x+1=0$, then $\alpha^{2009}+\beta^{2009}=$

The number of complex numbers $z$ such that $|z-1|=|z+1|=|z-i|$ equals

Consider the system of linear equations: $$ \begin{aligned} & x_1+2 x_2+x_3=3 \\ & 2 x_1+3 x_2+x_3=3 \\ & 3 x_1+5 x_2+2 x_3=1 \end{aligned} $$ The system has

A person is to count 4500 currency notes. Let $a_n$ denote the number of notes he counts in the $\mathrm{n}^{\text {th }}$ minute. If $\mathrm{a}_1=\mathrm{a}_2=\ldots \ldots=\mathrm{a}_{10}=150$ and $\mathrm{a}_{10}, \mathrm{a}_{11}, \ldots \ldots$ are in A.P. with common difference $-2$, then the time taken by him to count all notes is

The number of $3 \times 3$ non-singular matrices, with four entries as 1 and all other entries as 0 , is

Let $A$ be a $2 \times 2$ matrix with non-zero entries and let $A^2=1$, where 1 is $2 \times 2$ identity matrix. Define $\operatorname{Tr}(\mathrm{A})=$ sum of diagonal elements of $A$ and $|A|=$ determinant of matrix $A$. Statement-1: $\operatorname{Tr}(\mathrm{A})=0$ Statement-2: $|\mathrm{A}|=1$

There are two urns. Urn A has 3 distinct red balls and urn B has 9 distinct blue balls. From each urn two balls are taken out at random and then transferred to the other. The number of ways in which this can be done is

Let $f(x)=(x+1)^2-1, x \geq-1$ Statement-1: The set $\left\{x: f(x)=f^{-1}(x)\right\}=\{0,-1\}$ Statement-2 : $\mathrm{f}$ is a bijection.

If $\left|z-\frac{4}{z}\right|=2$, then the maximum value of $|z|$ is equal to

Let $a, b, c$ be such that $b(a+c) \neq 0$. If $\left|\begin{array}{ccc}a & a+1 & a-1 \\ -b & b+1 & b-1 \\ c & c-1 & c+1\end{array}\right|+\left|\begin{array}{ccc}a+1 & b+1 & c-1 \\ a-1 & b-1 & c+1 \\ (-1)^{n+2} a & (-1)^{n+1} b & (-1)^n c\end{array}\right|=0$, then the value of ' $n$ ' is

Let $A$ and $B$ denote the statements A: $\cos \alpha+\cos \beta+\cos \gamma=0$ B: $\sin \alpha+\sin \beta+\sin \gamma=0$ If $\cos (\beta-\gamma)+\cos (\gamma-\alpha)+\cos (\alpha-\beta)=-\frac{3}{2}$, then

For real $x$, let $f(x)=x^3+5 x+1$, then

The remainder left out when $8^{2 n}-(62)^{2 n+1}$ is divided by 9 is

From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on the shelf so that the dictionary is always in the middle. Then the number of such arrangements is

If the roots of the equation $b x^2+c x+a=0$ be imaginary, then for all real values of $x$, the expression $3 b^2 x^2+6 b c x+2 c^2$ is

Let A be a $2 \times 2$ matrix Statement-1 $: \operatorname{adj}(\operatorname{adj} A)=A$ Statement-2 : $|\operatorname{adj} \mathrm{A}|=|\mathrm{A}|$

If $A, B$ and $C$ are three sets such that $A \cap B=A \cap C$ and $A \cup B=A \cup C$, then

The conjugate of a complex number is $\frac{1}{i-1}$. Then the complex number is