Algebra PYQ — Page 29

The relation $R={(a,b):gcd(a,b)=1,2a\neq b,a,b\in \mathbb{Z}}$ is:

Let the system of linear equations $–x+2y-9z=7$ $-x+3y+7z=9$ $-2x+y+5z=8$ $-3x+y+13z=\lambda$ has a unique solution $x=\alpha ,y=\beta ,z=\gamma$. Then the distance of the point $(\alpha ,\beta ,\gamma )$ from the plane $2x-2y+z=\lambda$ is

Number of integral solutions to the equation $x+y+z=21$, where $x\geq 1,y\geq 3,z\geq 4$, is equal to _____ .

Let ${s}_{1},{s}_{2},{s}_{3}....,{s}_{10}$ respectively be the sum of $12$ terms of $10A.\mathrm{Ps}$ whose first terms are $1,2,3,....,10$ and the common differences are $1,3,5,...,19$ respectively. Then $\sum _{i=1}^{10}{s}_{i}$ is equal to

Let $A={0,3,4,6,7,8,9,10}$ and $R$ be the relation defined on $A$ such that $R{(x,y)\in A\times A:x-y\text{is odd positive integer}\text{or}x-y=2}$. The minimum number of elements that must be added to the relation $R,$ so that it is a symmetric relation, is equal to $_________$

Let the sixth term in the binomial expansion of ${(\sqrt{{2}^{{\mathrm{log}}_{2}(10-{3}^{x})}}+\sqrt[5]{{2}^{(x-2){\mathrm{log}}_{2}3}})}^{m}$ powers of ${2}^{(x-2){\mathrm{log}}_{2}3}$, be $21$ . If the binomial coefficients of the second, third and fourth terms in the expansion are respectively the first, third and fifth terms of an A.P., then the sum of the squares of all possible values of $x$ is _____ .

Let $R$ be a relation on $N\times N$ defined by $(a,b)R(c,d)$ if and only if $ad(b-c)=bc(a-d)$. Then $R$ is

In an examination, $5$ students have been allotted their seats as per their roll numbers. The number of ways, in which none of the students sits on the allotted seat, is

Let $A,B,C$ be $3\times 3$ matrices such that $A$ is symmetric and $B$ and $C$ are skew-symmetric. Consider the statements $(S1){A}^{13}{B}^{26}-{B}^{26}{A}^{13}$ is symmetric $(S2){A}^{26}{C}^{13}-{C}^{13}{A}^{26}$ is symmetric Then,

Five digit numbers are formed using the digits $1,2,3,5,7$ with repetitions and are written in descending order with serial numbers. For example, the number $77777$ has serial number $1$. Then the serial number of $35337$ is

If domain of the function ${\mathrm{log}}_{e}(\frac{6{x}^{2}+5x+1}{2x-1})+{\mathrm{cos}}^{-1}(\frac{2{x}^{2}-3x+4}{3x-5})$ is $(\alpha ,\beta )\cup (\gamma ,\delta )$, then $18({\alpha }^{2}+{\beta }^{2}+{\gamma }^{2}+{\delta }^{2})$ is equal to

The equation ${x}^{2}–4x+[x]+3=x[x]$, where $[x]$ denotes the greatest integer function, has:

Let ${S}_{1}$ and ${S}_{2}$ be respectively the sets of all $a\in R-{0}$ for which the system of linear equations $ax+2ay-3az=1$ $(2a+1)x+(2a+3)y+(a+1)z=2$ $(3a+5)x+(a+5)y+(a+2)z=3$ has unique solution and infinitely many solutions. Then

The number of integral terms in the expansion of ${({3}^{\frac{1}{2}}+{5}^{\frac{1}{4}})}^{680}$ is equal to

Let $f(x)=2{x}^{n}+\lambda ,\lambda \in \mathbb{R},n\in \mathbb{N}$, and $f(4)=133$, $f(5)=255$. Then the sum of all the positive integer divisors of $(f(3)-f(2))$ is

The number of square matrices of order $5$ with entries from the set ${0,1}$, such that the sum of all the elements in each row is $1$ and the sum of all the elements in each column is also $1$, is

Let ${a}_{1},{a}_{2},{a}_{3},\ldots$. be a GP of increasing positive numbers. If the product of fourth and sixth terms is $9$ and the sum of fifth and seventh terms is $24$ , then ${a}_{1}{a}_{9}+{a}_{2}{a}_{4}{a}_{9}+{a}_{5}+{a}_{7}$ is equal to

Let ${A}_{1},{A}_{2},{A}_{3}$ be the three A.P. with the same common difference $d$ and having their first terms as $A,A+1,A+2$, respectively. Let $a,b,c$ be the ${7}^{\text{th }},{9}^{\text{th }},{17}^{\text{th }}$ terms of ${A}_{1},{A}_{2},{A}_{3}$, respectively such that $|\begin{matrix}a & 7 & 1 \\ 2b & 17 & 1 \\ c & 17 & 1\end{matrix}|+70=0$. If $a=29$, then the sum of first $20$ terms of an AP whose first term is $c-a-b$ and common difference is $\frac{d}{12}$, is equal to _____ .

Let $a,b,c>1,{a}^{3},{b}^{3}$ and ${c}^{3}$ be in $A.P.$ and ${\mathrm{log}}_{a}b$, ${\mathrm{log}}_{c}a$ and ${\mathrm{log}}_{b}c$ be in $G.P.$ If the sum of first $20$ terms of an $A.P.$, whose first term is $\frac{a+4b+c}{3}$ and the common difference is $\frac{a-8b+c}{10}$ is $-444$, then $abc$ is equal to

Let $x$ and $y$ be distinct integers where $1\leq x\leq 25$ and $1\leq y\leq 25$. Then, the number of ways of choosing $x$ and $y$, such that $x+y$ is divisible by $5$ , is _____ .

If the center and radius of the circle $|\frac{z-2}{z-3}|=2$ are respectively $(\alpha ,\beta )$ and $\gamma$, then $3(\alpha +\beta +\gamma )$ is equal to

The number of $4$-letter words, with or without meaning, each consisting of $2$ vowels and $2$ consonants, which can be formed from the letters of the word UNIVERSE without repetition is _____.

Let $x,y,z>1$ and $A=[\begin{matrix}1 & {\mathrm{log}}_{x}y & {\mathrm{log}}_{x}z \\ {\mathrm{log}}_{y}x & 2 & {\mathrm{log}}_{y}z \\ {\mathrm{log}}_{z}x & {\mathrm{log}}_{z}y & 3\end{matrix}]$. Then $|adj(adj{A}^{2})|$ is equal to

The sum of all the four-digit numbers that can be formed using all the digits $2,1,2,3$ is equal to ____.