Algebra PYQ — Page 51

The number of solutions of the equation x² - 5|x| + 6 = 0 is:

The sum of all $3$-digit numbers less than or equal to $500,$ that are formed without using the digit $1$ and they all are multiple of $11,$ is ______.

If for $x\in (0,\frac{\pi }{2}),{\mathrm{log}}_{10}\mathrm{sin}x+{\mathrm{log}}_{10}\mathrm{cos}x=-1$ and ${\mathrm{log}}_{10}(\mathrm{sin}x+\mathrm{cos}x)=\frac{1}{2}({\mathrm{log}}_{10}n-1),n>0$, then the value of $n$ is equal to :

If $\alpha +\beta +\gamma =2\pi ,$ then the system of equations $x+(\mathrm{cos}\gamma )y+(\mathrm{cos}\beta )z=0$ $(\mathrm{cos}\gamma )x+y+(\mathrm{cos}\alpha )z=0$ $(\mathrm{cos}\beta )x+(\mathrm{cos}\alpha )y+z=0$ has :

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 _____.

Let ${S}_{1}$ be the sum of first $2n$ terms of an arithmetic progression. Let ${S}_{2}$ be the sum of first $4n$ terms of the same arithmetic progression. If $({S}_{2}-{S}_{1})$ is $1000$, then the sum of the first $6n$ terms of the arithmetic progression is equal to:

For real numbers $\alpha$ and $\beta ,$ consider the following system of linear equations: $x+y-z=2,x+2y+\alpha z=1$ and $2x-y+z=\beta .$ If the system has infinite solutions, then $\alpha +\beta$ is equal to ______.

The number of the real roots of the equation ${(x+1)}^{2}+|x-5|=\frac{27}{4}$ is ________.

If ${e}^{({\mathrm{cos}}^{2}x+{\mathrm{cos}}^{4}x+{\mathrm{cos}}^{6}x+....\infty ){\mathrm{log}}_{e}2}$ satisfies the equation ${t}^{2}-9t+8=0$, then the value of $\frac{2\mathrm{sin}x}{\mathrm{sin}x+\sqrt{3}\mathrm{cos}x}$, where $0<x<\frac{\pi }{2}$, is equal to

If the coefficient of ${a}^{7}{b}^{8}$ in the expansion of $(a+2b+4ab{)}^{10}$ is $K\cdot {2}^{16},$ then $K$ is equal to

The number of solutions of the equation ${32}^{{\mathrm{tan}}^{2}x}+{32}^{{\mathrm{sec}}^{2}x}=81,0\leq x\leq \frac{\pi }{4}$ is :

The sum of first four terms of a geometric progression $(G.P.)$ is $\frac{65}{12}$ and the sum of their respective reciprocals is $\frac{65}{18}.$ If the product of first three terms of the $G.P.$ is $1,$ and the third term is $\alpha ,$ then $2\alpha$ is _________.

If the domain of the function $f(x)=\frac{{\mathrm{cos}}^{-1}\sqrt{{x}^{2}-x+1}}{\sqrt{{\mathrm{sin}}^{-1}(\frac{2x-1}{2})}}$ is the interval $(\alpha ,\beta ],$ then $\alpha +\beta$ is equal to:

The probability of selecting integers $a\in [-5,30]$ such that ${x}^{2}+2(a+4)x-5a+64>0$, for all $x\in R$, is:

The number of pairs $(a,b)$ of real numbers, such that whenever $\alpha$ is a root of the equation ${x}^{2}+ax+b=0,{\alpha }^{2}-2$ is also a root of this equation, is :

$\frac{1}{{3}^{2}-1}+\frac{1}{{5}^{2}-1}+\frac{1}{{7}^{2}-1}+\ldots +\frac{1}{(201{)}^{2}-1}$ is equal to

Let $x$ denote the total number of one-one functions from a set $A$ with $3$ elements to a set $B$ with $5$ elements and $y$ denote the total number of one-one functions from the set $A$ to the set $A\times B$. Then :

Let $A={n\in N\mid {n}^{2}\leq n+10,000},B={3k+1\mid k\in N}$ and $C={2k\mid k\in N},$ then the sum of all the elements of the set $A\cap (B-C)$ is equal to ________.

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

Let $n$ denote the number of solutions of the equation ${z}^{2}+3\bar{z}=0,$ where $z$ is a complex number. Then the value of $\sum _{k=0}^{\infty }\frac{1}{{n}^{k}}$ is equal to

The range of the function $f(x)={\mathrm{log}}_{\sqrt{5}}(3+\mathrm{cos}(\frac{3\pi }{4}+x)+\mathrm{cos}(\frac{\pi }{4}+x)+\mathrm{cos}(\frac{\pi }{4}-x)-\mathrm{cos}(\frac{3\pi }{4}-x))$ is :

Let $M={A=[\begin{matrix}a & b \\ c & d\end{matrix}]:a,b,c,d\in (\pm 3,\pm 2,\pm 1,0)}$. Define $f:M\rightarrow Z,$ as $f(A)=det(A),$ for all $A\in M$ where $Z$ is set of all integers. Then the number of $A\in M$ such that $f(A)=15$ is equal to .

Let ${A}_{1},{A}_{2},{A}_{3},\ldots ..$ be squares such that for each $n\geqslant 1,$ the length of the side of ${A}_{n}$ equals the length of diagonal of ${A}_{n+1}$. If the length of ${A}_{1}$ is $12\mathrm{cm}$, then the smallest value of $n$ for which area of ${A}_{n}$ is less than one, is

The integer $k,$ for which the inequality ${x}^{2}-2(3k-1)x+8{k}^{2}-7>0$ is valid for every $x$ in $R$ is: