Algebra PYQ — Page 29

If $\frac{{1}^{3}+{2}^{3}+{3}^{3}......\text{upto }n\text{terms}}{1\cdot 3+2\cdot 5+3\cdot 7+....\text{upto }n\text{terms}}=\frac{9}{5}$ then the value of $n$ 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,

If $A=[\begin{matrix}1 & 5 \\ \lambda & 10\end{matrix}],{A}^{-1}=\alpha A+\beta I$ and $\alpha +\beta =-2$, then $4{\alpha }^{2}+{\beta }^{2}+{\lambda }^{2}$ is equal to :

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

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 $S={1,2,3,5,7,10,11}$. The number of non-empty subsets of $S$ that have the sum of all elements a multiple of $3$, is _____ .

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

For the differentiable function $f:\mathbb{R}-{0}-\mathbb{R},$ let $3f(x)+2f(\frac{1}{x})=\frac{1}{x}-10,$ then $|f(3)+{f}^{'}(\frac{1}{4})|$ is equal to

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

If $gcd(m,n)=1$ and ${1}^{2}-{2}^{2}+{3}^{2}-{4}^{2}+....+{(2021)}^{2}-{(2022)}^{2}+{(2023)}^{2}=1012{m}^{2}n$ then ${m}^{2}-{n}^{2}$ is equal to

For $\alpha ,\beta \in \mathbb{R}$, suppose the system of linear equations $x-y+z=5$ $2x+2y+\alpha z=8$ $3x-y+4z=\beta$ has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of

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

If $f(x)={x}^{3}-{x}^{2}{f}^{'}(1)+x{f}^{"}(2)-{f}^{'''}(3)$, $x\in R$, then

Let $A=[\begin{matrix}1 & \frac{1}{51} \\ 0 & 1\end{matrix}]$. If $B=[\begin{matrix}1 & 2 \\ -1 & -1\end{matrix}]A[\begin{matrix}-1 & -2 \\ 1 & 1\end{matrix}]$, then the sum of all the elements of the matrix $\sum _{n=1}^{50}{B}^{n}$ is equal to

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 ${A}_{1},{A}_{2},{A}_{3}$ 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 $x$ 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 _____.

If $P$ is a $3\times 3$ real matrix such that ${P}^{T}=aP+(a-1)I$, where $a>1$, then

Let ${A}_{1}$ and ${A}_{2}$ be two arithmetic means and ${G}_{1},{G}_{2}$ and ${G}_{3}$ be three geometric means of two distinct positive numbers. Then ${G}_{1}^{4}+{G}_{2}^{4}+{G}_{3}^{4}+{G}_{1}^{2}{G}_{3}^{2}$ is equal to

If for $z=\alpha +i\beta ,|z+2|=z+4(1+i),$ then $\alpha +\beta$ and $\alpha \beta$ are the roots of the equation

All words, with or without meaning, are made using all the letters of the word $MONDAY$. These words are written as in a dictionary with serial numbers. The serial number of the word $MONDAY$ is