Algebra PYQ — Page 20

If the domain of the function $f(x)={\mathrm{cos}}^{-1}(\frac{2-|x|}{4})+{({\mathrm{log}}_{e}(3-x))}^{-1}$ is $[-\alpha ,\beta )-{\gamma }$, then $\alpha +\beta +\gamma$ is equal to :

If the domain of the function $\sin ^{-1}\left(\frac{3 x-22}{2 x-19}\right)+\log _{\mathrm{e}}\left(\frac{3 x^2-8 x+5}{x^2-3 x-10}\right)$ is $(\alpha, \beta]$, then $3 \alpha+10 \beta$ is equal to:

If the system of linear equations $x-2y+z=-4$ $2x+\alpha y+3z=5$ $3x-y+\beta z=3$ has infinitely many solutions, then $12\alpha +13\beta$ is equal to

The value of $\frac{1 \times 2^2+2 \times 3^2+\ldots+100 \times(101)^2}{1^2 \times 2+2^2 \times 3+\ldots .+100^2 \times 101}$ is

If $f(x)=\frac{4x+3}{6x-4},x\neq \frac{2}{3}$ and $(fof)(x)=g(x)$, where $g:R-{\frac{2}{3}}\rightarrow R-{\frac{2}{3}}$, then $(gogog)(4)$ is equal to

The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is

Let $a$ and $b$ be two distinct positive real numbers. Let ${11}^{\text{th }}$ term of a GP, whose first term is $a$ and third term is $b$, is equal to ${p}^{\text{th }}$ term of another GP, whose first term is $a$ and fifth term is $b$. Then $p$ is equal to

If the second, third and fourth terms in the expansion of $(x+y)^n$ are 135,30 and $\frac{10}{3}$, respectively, then $6\left(n^3+x^2+y\right)$ is equal to _______

Let ${z}_{1}$ and ${z}_{2}$ be two complex number such that ${z}_{1}+{z}_{2}=5$ and ${z}_{1}^{3}+{z}_{2}^{3}=20+15i$. Then $|{z}_{1}^{4}+{z}_{2}^{4}|$ equals-

Let $A$ be a $3\times 3$ matrix and $\mathrm{det}(A)=2$. If $n=\mathrm{det}(\underset{2024-\mathrm{times}}{\underset{⏟}{adj(adj(....(adjA))))}})$, then the remainder when $n$ is divided by $9$ is equal to __________.

If the coefficient of ${x}^{30}$ in the expansion of ${(1+\frac{1}{x})}^{6}{(1+{x}^{2})}^{7}{(1-{x}^{3})}^{8};x\neq 0$ is $\alpha$, then $|\alpha |$ equals _________.

Let the range of the function $f(x)=\frac{1}{2+\sin 3 x+\cos 3 x}, x \in \mathbb{R}$ be $[a, b]$. If $\alpha$ and $\beta$ are respectively the A.M. and the G.M. of $a$ and $b$, then $\frac{\alpha}{\beta}$ is equal to

Let $A$ be a $2 \times 2$ symmetric matrix such that $A\left[\begin{array}{l}1 \\ 1\end{array}\right]=\left[\begin{array}{l}3 \\ 7\end{array}\right]$ and the determinant of $A$ be 1 . If $A^{-1}=\alpha A+\beta I$, where $I$ is an identity matrix of order $2 \times 2$, then $\alpha+\beta$ equals _______

The number of 3-digit numbers, formed using the digits $2,3,4,5$ and 7 , when the repetition of digits is not allowed, and which are not divisible by 3 , is equal to__________

The number of distinct real roots of the equation $|x||x+2|-5|x+1|-1=0$ is_______

If three successive terms of a G.P. with common ratio $r(r>1)$ are the length of the sides of a triangle and $[r]$ denotes the greatest integer less than or equal to r, then $3[r]+[-r]$ is equal to:

If $R$ is the smallest equivalence relation on the set ${1,2,3,4}$ such that ${(1,2),(1,3)}\subset R$, then the number of elements in $R$ is ______.

Let the system of equations $x+2y+3z=5,2x+3y+z=9,4x+3y+\lambda z=\mu$ have infinite number of solutions. Then $\lambda +2\mu$ is equal to:

The number of symmetric relations defined on the set ${1,2,3,4}$ which are not reflexive is _______.

An arithmetic progression is written in the following way  The sum of all the terms of the $10^{\text {th }}$ row is_______

The sum of all rational terms in the expansion of $\left(2^{\frac{1}{5}}+5^{\frac{1}{3}}\right)^{15}$ is equal to :

Let $z$ be a complex number such that $|z+2|=1$ and $\operatorname{Im}\left(\frac{z+1}{z+2}\right)=\frac{1}{5}$. Then the value of $|\operatorname{Re}(\overline{z+2})|$ is

If $z$ is a complex number, then the number of common roots of the equation ${z}^{1985}+{z}^{100}+1=0$ and ${z}^{3}+2{z}^{2}+2z+1=0$, is equal to :

A software company sets up $m$ number of computer systems to finish an assignment in 17 days. If 4 computer systems crashed on the start of the second day, 4 more computer systems crashed on the start of the third day and so on, then it took 8 more days to finish the assignment. The value of $\mathrm{m}$ is equal to: