Algebra PYQ — Page 46

The total number of numbers, lying between $100$ and $1000$ that can be formed with the digits $1,2,3,4,5,$ if the repetition of digits is not allowed and numbers are divisible by either $3$ or $5,$ is

If ${x}^{2}+9{y}^{2}-4x+3=0,x,y\in R,$ then $x$ and $y$ respectively lie in the intervals

If $(2021{)}^{3762}$ is divided by $17,$ then the remainder is _______.

The sum of all integral values of $k(k\neq 0)$ for which the equation $\frac{2}{x-1}-\frac{1}{x-2}=\frac{2}{k}$ in $x$ has no real roots, is_____.

If $A={x\in R:|x-2|>1},B={x\in R:\sqrt{{x}^{2}-3}>1},C={x\in R:|x-4|\geqslant 2}$ and $Z$ is the set of all integers, then the number of subsets of the set $(A\cap B\cap C{)}^{c}\cap Z$ is _________.

The set of all values of $k>-1$, for which the equation ${(3{x}^{2}+4x+3)}^{2}-(k+1)(3{x}^{2}+4x+3)(3{x}^{2}+4x+2)+$ $k{(3{x}^{2}+4x+2)}^{2}=0$ has real roots, is:

Let $\alpha ,\beta$ be two roots of the equation ${x}^{2}+{(20)}^{1/4}x+{(5)}^{1/2}=0$. Then ${\alpha }^{8}+{\beta }^{8}$ is equal to

Let $\alpha$ and $\beta$ be two real numbers such that $\alpha +\beta =1$ and $\alpha \beta =-1$. Let ${p}_{n}={(\alpha )}^{n}+{(\beta )}^{n}$, ${p}_{n-1}=11$ and ${p}_{n+1}=29$ for some integer $n\geqslant 1$. Then, the value of ${p}_{n}^{2}$ is______.

Let $\alpha$ and $\beta$ be the roots of ${x}^{2}-6x-2=0$. If ${a}_{n}={\alpha }^{n}-{\beta }^{n}$ for $n\geqslant 1$, then the value of $\frac{{a}_{10}-2{a}_{8}}{3{a}_{9}}$ is:

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______.

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

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 :

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

The minimum value of $f(x)={a}^{{a}^{x}}+{a}^{1-{a}^{x}}$, where $a,x\in R$ and $a>0$, is equal to:

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

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______.

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:

If ${a}_{r}=\mathrm{cos}\frac{2r\pi }{9}+i\mathrm{sin}\frac{2r\pi }{9},r=1,2,3,\ldots ,i=\sqrt{-1}$, then the determinant $|\begin{matrix}{a}_{1} & {a}_{2} & {a}_{3} \\ {a}_{4} & {a}_{5} & {a}_{6} \\ {a}_{7} & {a}_{8} & {a}_{9}\end{matrix}|$ is equal to :

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 co-efficient of ${x}^{7}$ and ${x}^{8}$ in the expansion of ${(2+\frac{x}{3})}^{n}$ are equal, then the value of $n$ is equal to :

The value of $3+\frac{1}{4+\frac{1}{3+\frac{1}{4+\frac{1}{3+\ldots \infty }}}}$ is equal to