Algebra PYQ — Page 74

If $Z_1 \neq 0$ and $Z_2$ be two complex numbers such that $\frac{Z_2}{Z_1}$ is a purely imaginary number, then $\left|\frac{2 Z_1+3 Z_2}{2 Z_1-3 Z_2}\right|$ is equal to:

Let $R=\{(3,3)(5,5),(9,9),(12,12),(5,12),(3,9)$, $(3,12),(3,5)\}$ be a relation on the $\operatorname{set} A=\{3,5,9,12\}$. Then, $R$ is :

The number of ways in which an examiner can assign 30 marks to 8 questions, giving not less than 2 marks to any question, is :

Let $A$ and $B$ be two sets containing $2$ elements and $4$ elements respectively. The number of subsets of $A\times B$ having $3$ or more elements is :

If for positive integers $r>1, n>2$, the coefficients of the $(3 r)^{\text {th }}$ and $(r+2)^{\text {th }}$ powers of $x$ in the expansion of $(1+x)^{2 n}$ are equal, then $n$ is equal to:

A common tangent to the conics $x^2=6 y$ and $2 x^2-4 y^2=9$ is:

If $p$ and $q$ are non-zero real numbers and $\alpha^3+\beta^3=-p, \alpha \beta=q$, then a quadratic equation whose roots are $\frac{\alpha^2}{\beta}, \frac{\beta^2}{\alpha}$ is :

The sum of the series : $(2)^2+2(4)^2+3(6)^2+\ldots$ upto 10 terms is :

Given a sequence of 4 numbers, first three of which are in G.P. and the last three are in A.P. with common difference six. If first and last terms of this sequence are equal, then the last term is :

The sum of first $20$ terms of the sequence $0.7,0.77,0.777,......,$ is :

Let $\mathrm{R}=\left\{(x, y): x, y \in N\right.$ and $\left.x^2-4 x y+3 y^2=0\right\}$, where $N$ is the set of all natural numbers. Then the relation $R$ is :

Let $a=\operatorname{Im}\left(\frac{1+z^2}{2 i z}\right)$, where $z$ is any non-zero complex number. The set $\mathrm{A}=\{a:|z|=1$ and $z \neq \pm 1\}$ is equal to:

Statement-1: The system of linear equations $$ \begin{aligned} & x+(\sin \alpha) y+(\cos \alpha) z=0 \\ & x+(\cos \alpha) y+(\sin \alpha) z=0 \\ & x-(\sin \alpha) y-(\cos \alpha) z=0 \end{aligned} $$ has a non-trivial solution for only one value of $\alpha$ lying in the interval $\left(0, \frac{\pi}{2}\right)$. Statement-2: The equation in $\alpha$ $$ \left|\begin{array}{ccc} \cos \alpha & \sin \alpha & \cos \alpha \\ \sin \alpha & \cos \alpha & \sin \alpha \\ \cos \alpha & -\sin \alpha & -\cos \alpha \end{array}\right|=0 $$ has only one solution lying in the interval $\left(0, \frac{\pi}{2}\right)$

The real number $k$ for which the equation, $2{x}^{3}+3x+k=0$ has two distinct real roots in $[0,1]$ belongs to

If $p, q, r$ are 3 real numbers satisfying the matrix equation, $[p q r]\left[\begin{array}{lll}3 & 4 & 1 \\ 3 & 2 & 3 \\ 2 & 0 & 2\end{array}\right]=\left[\begin{array}{lll}3 & 0 & 1\end{array}\right]$ then $2 p+q-r$ equals :

Given sum of the first $n$ terms of an A.P. is $2 n+$ $3 n^2$. Another A.P. is formed with the same first term and double of the common difference, the sum of $n$ terms of the new A.P. is :

Let $A=\{1,2,3,4\}$ and $R: A \rightarrow A$ be the relation defined by $R=\{(1,1),(2,3),(3,4),(4,2)\}$. The correct statement is :

If the 7th term in the binomial expansion of $\left(\frac{3}{\sqrt[3]{84}}+\sqrt{3} \ln x\right)^9, x>0$, is equal to 729 , then $x$ can be:

If $\alpha$ and $\beta$ are roots of the equation $x^2+p x+\frac{3 p}{4}=0$, such that $|\alpha-\beta|=\sqrt{10}$, then $p$ belongs to the set :


5 - digit numbers are to be formed using $2,3,5,7$, 9 without repeating the digits. If $p$ be the number of such numbers that exceed 20000 and $q$ be the number of those that lie between 30000 and 90000 , then $p: q$ is:

The difference between the fourth term and the first term of a Geometrical Progresssion is 52. If the sum of its first three terms is 26 , then the sum of the first six terms of the progression is

If $a, b, c \in \mathrm{R}$ and 1 is a root of equation $a x^2+b x$ $+c=0$, then the curve $y=4 a x^2+3 b x+2 c, a \neq 0$ intersect $x$-axis at

If $n={ }^m C_2$, then the value of ${ }^n C_2$ is given by