Algebra PYQ — Page 22

For $0<c<b<a$, let $(a+b–2c){x}^{2}+(b+c–2a)x+(c+a–2b)=0$ and $\alpha \neq 1$ be one of its root. Then, among the two statements (I) If $\alpha \in (-1,0)$, then $b$ cannot be the geometric mean of $a$ and $c$. (II) If $\alpha \in (0,1)$, then $b$ may be the geometric mean of $a$ and $c$.

Let $a,b,c$ be the length of three sides of a triangle satisfying the condition $({a}^{2}+{b}^{2}){x}^{2}-2b(a+c)$ $x+({b}^{2}+{c}^{2})=0$. If the set of all possible values of $x$ is in the interval $(\alpha ,\beta ),$ then $12({\alpha }^{2}+{\beta }^{2})$ is equal to _______.

Let $S$ be the set of positive integral values of $a$ for which $\frac{a{x}^{2}+2(a+1)x+9a+4}{{x}^{2}-8x+32}<0,\forall x\in \mathbb{R}$. Then, the number of elements in $S$ is:

Let the set $C={(x,y)\mid {x}^{2}-{2}^{y}=2023,x,y\in \mathbb{N}}$. Then $\underset{(x,y)\in C}{\sum }(x+y)$ is equal to _______.

If $z_1, z_2$ are two distinct complex number such that $\left|\frac{z_1-2 z_2}{\frac{1}{2}-z_1 \bar{z}_2}\right|=2$, then

If $z$ is a complex number such that $|z|\leq 1$, then the minimum value of $|z+\frac{1}{2}(3+4i)|$ is:

Let $\alpha$ and $\beta$ be the sum and the product of all the non-zero solutions of the equation $(\bar{z})^2+|z|=0$, $z \in$ C. Then $4\left(\alpha^2+\beta^2\right)$ is equal to :

If $z=\frac{1}{2}-2i$, is such that $|z+1|=\alpha z+\beta (1+i),i=\sqrt{-1}$ and $\alpha ,\beta \in R,$ then $\alpha +\beta$ is equal to

If $z=x+iy,xy\neq 0$, satisfies the equation ${z}^{2}+i\bar{z}=0$, then $|{z}^{2}|$ is equal to :

If $S=z\in C:|z-i|=|z+i|=|z-1|$, then, $n(S)$ is:

The number of ways five alphabets can be chosen from the alphabets of the word MATHEMATICS, where the chosen alphabets are not necessarily distinct, is equal to :

The number of integers, between 100 and 1000 having the sum of their digits equals to 14 , is _________

60 words can be made using all the letters of the word BHBJO, with or without meaning. If these words are written as in a dictionary, then the $50^{\text {th }}$ word is :

The number of ways of getting a sum 16 on throwing a dice four times is______

There are 5 points $P_1, P_2, P_3, P_4, P_5$ on the side $A B$, excluding $A$ and $B$, of a triangle $A B C$. Similarly there are 6 points $\mathrm{P}_6, \mathrm{P}_7, \ldots, \mathrm{P}_{11}$ on the side $\mathrm{BC}$ and 7 points $\mathrm{P}_{12}, \mathrm{P}_{13}, \ldots, \mathrm{P}_{18}$ on the side $C A$ of the triangle. The number of triangles, that can be formed using the points $\mathrm{P}_1, \mathrm{P}_2, \ldots, \mathrm{P}_{18}$ as vertices, is :

In an examination of Mathematics paper, there are $20$ questions of equal marks and the question paper is divided into three sections : $A,B$ and $C$. A student is required to attempt total $15$ questions taking at least $4$ questions from each section. If section $A$ has $8$ questions, section $B$ has $6$ questions and section $C$ has $6$ questions, then the total number of ways a student can select $15$ questions is _________.

Number of ways of arranging $8$ identical books into $4$ identical shelves where any number of shelves may remain empty is equal to

All the letters of the word $GTWENTY$ are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word $GTWENTY$ IS

Let $r$ and $\theta$ respectively be the modulus and amplitude of the complex number $z=2-i(2\mathrm{tan}\frac{5\pi }{8})$, then $(r,\theta )$ is equal to

Let $a, a r, a r^2$, $\qquad$ be an infinite G.P. If $\sum_{n=0}^{\infty} a r^n=57$ and $\sum_{n=0}^{\infty} a^3 r^{3 n}=9747$, then $a+18 r$ is equal to

If the set $R=\{(a, b): a+5 b=42, a, b \in \mathbb{N}\}$ has $m$ elements and $\sum_{n=1}^m\left(1-i^{n !}\right)=x+i y$, where $i=\sqrt{-1}$, then the value of $m+x+y$ is

Let the positive integers be written in the form :  If the $k^{\text {th }}$ row contains exactly $k$ numbers for every natural number $k$, then the row in which the number 5310 will be, is _______

If the sum of the series $\frac{1}{1 \cdot(1+\mathrm{d})}+\frac{1}{(1+\mathrm{d})(1+2 \mathrm{~d})}+\ldots+\frac{1}{(1+9 \mathrm{~d})(1+10 \mathrm{~d})}$ is equal to 5 , then $50 \mathrm{~d}$ is equal to :

In an increasing geometric progression of positive terms, the sum of the second and sixth terms is $\frac{70}{3}$ and the product of the third and fifth terms is 49 . Then the sum of the $4^{\text {th }}, 6^{\text {th }}$ and $8^{\text {th }}$ terms is equal to :