Algebra PYQ — Page 46

Let ${z}_{1}$ and ${z}_{2}$ be two complex numbers such that $\mathrm{arg}({z}_{1}-{z}_{2})=\frac{\pi }{4}$ and ${z}_{1},{z}_{2}$ satisfy the equation $|z-3|=Re(z)$. Then the imaginary part ${z}_{1}+{z}_{2}$ is equal to

If $(\sqrt{3}+i{)}^{100}={2}^{99}(p+iq),$ then $p$ and $q$ are roots of the equation :

Let $z=\frac{1-i\sqrt{3}}{2},i=\sqrt{-1}$. Then the value of $21+{(z+\frac{1}{z})}^{3}+{({z}^{2}+\frac{1}{{z}^{2}})}^{3}+{({z}^{3}+\frac{1}{{z}^{3}})}^{3}+\ldots +{({z}^{21}+\frac{1}{{z}^{21}})}^{3}$ is______.

Let $\mathbb{C}$ be the set of all complex numbers. Let ${S}_{1}={z\in \mathbb{C}:|z-2|\leq 1}$ and ${S}_{2}={z\in \mathbb{C}:z(1+i)+\bar{z}(1-i)\geq 4}.$ Then, the maximum value of ${|z-\frac{5}{2}|}^{2}$ for $z\in {S}_{1}\cap {S}_{2}$ is equal to :

The equation of a circle is $Re({z}^{2})+2{(\mathrm{Im}(z))}^{2}+2Re(z)=0,$ where $z=x+iy.$ A line which passes through the centre of the given circle and the vertex of the parabola, ${x}^{2}-6x-y+13=0,$ has $y$-intercept equal to _________.

Let $C$ be the set of all complex numbers. Let ${S}_{1}={z\in C|{|z–3–2i|}^{2}=8},$ ${S}_{2}=z\in C|\mathrm{Re}(z)\geq 5$ and ${S}_{3}={z\in C||z–\bar{z}|\geq 8}.$ Then the number of elements in ${S}_{1}\cap {S}_{2}\cap {S}_{3}$ is equal to

Let ${z}_{1},{z}_{2}$ be the roots of the equation ${z}^{2}+az+12=0$ and ${z}_{1},{z}_{2}$ form an equilateral triangle with origin. Then, the value of $|a|$ is

Let the lines $(2-i)z=(2+i)\bar{z}$ and $(2+i)z+(i-2)\bar{z}-4i=0,$ (here ${i}^{2}=-1$) be normal to a circle $C$. If the line $iz+\bar{z}+1+i=0$ is tangent to this circle $C$, then its radius is :

If the least and the largest real values of $\alpha ,$ for which the equation $z+\alpha |z-1|+2i=0$ $(z\in C\text{and}i=\sqrt{-1})$ has a solution, are $p$ and $q$ respectively; then $4({p}^{2}+{q}^{2})$ is equal to_______.

The least value of $|z|$ where $z$ is complex number which satisfies the inequality ${e}^{(\frac{(|z|+3)(|z|-1)}{||z|+1|}{\mathrm{log}}_{e}2)}\geq {\mathrm{log}}_{\sqrt{2}}|5\sqrt{7}+9i|$, $i=\sqrt{-1},$ is equal to :

Let $z$ be those complex numbers which satisfy $|z+5|\leq 4$ and $z(1+i)+\bar{z}(1-i)\geqslant -10,i=\sqrt{-1}$. If the maximum value of $|z+1{|}^{2}$ is $\alpha +\beta \sqrt{2}$, then the value of $(\alpha +\beta )$ is

Let $[\lambda ]$ be the greatest integer less than or equal to $\lambda$. The set of all values of $\lambda$ for which the system of linear equations $x+y+z=4,3x+2y+5z=3,9x+4y+(28+[\lambda ])z=[\lambda ]$ has a solution is:

The number of six letter words (with or without meaning), formed using all the letters of the word 'VOWELS', so that all the consonants never come together, is

Let $n$ be a non-negative integer. Then the number of divisors of the form $4n+1$ of the number ${(10)}^{10}\cdot {(11)}^{11}\cdot {(13)}^{13}$ is equal to _____.

The number of three-digit even numbers, formed by the digits $0,1,3,4,6,7$ if the repetition of digits is not allowed, is______.

There are $15$ players in a cricket team, out of which $6$ are bowlers, $7$ are batsmen and $2$ are wicketkeepers. The number of ways, a team of $11$ players be selected from them so as to include at least $4$ bowlers, $5$ batsmen and $1$ wicketkeeper, is

If the sides $AB,BC$ and $CA$ of a triangle $ABC$ have $3,5$ and $6$ interior points respectively, then the total number of triangles that can be constructed using these points as vertices, is equal to:

The total number of $4$-digit numbers whose greatest common divisor with $18$ is $3$ is _____.

A natural number has prime factorization given by $n={2}^{x}{3}^{y}{5}^{z}$, where $y$ and $z$ are such that $y+z=5$ and ${y}^{-1}+{z}^{-1}=\frac{5}{6},y>z$. Then the number of odd divisors of $n$, including $1$, is:

The term independent of $x$ in the expansion of ${(\frac{x+1}{{x}^{2/3}-{x}^{1/3}+1}-\frac{x-1}{x-{x}^{1/2}})}^{10},$ where $x\neq 0,1$ is equal to

If the sum of an infinite $\mathrm{GP}$, $a,ar,a{r}^{2},a{r}^{3},\ldots$ is $15$ and the sum of the squares of its each term is $150,$ then the sum of $ar,2a{r}^{4},a{r}^{6},\ldots$ is:

Let ${{{a}_{n}}}_{n=1}^{\infty }$ be a sequence such that ${a}_{1}=1,{a}_{2}=1$ and ${a}_{n+2}=2{a}_{n+1}+{a}_{n}$ for all $n\geq 1$. Then the value of $47\sum _{n=1}^{\infty }(\frac{{a}_{n}}{{2}^{3n}})$ is equal to ________.

If $1,{\mathrm{log}}_{10}({4}^{x}-2)$ and ${\mathrm{log}}_{10}({4}^{x}+\frac{18}{5})$ are in arithmetic progression for a real number $x$ then the value of the determinant $|\begin{matrix}2(x-\frac{1}{2}) & x-1 & {x}^{2} \\ 1 & 0 & x \\ x & 1 & 0\end{matrix}|$ is equal to:

In an increasing geometric series, the sum of the second and the sixth term is $\frac{25}{2}$ and the product of the third and fifth term is $25.$ Then, the sum of ${4}^{th},{6}^{th}$ and ${8}^{th}$ terms is equal to: