Algebra PYQ — Page 19

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

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 __________.

In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let $\mathrm{m}$ and $\mathrm{n}$ respectively be the least and the most number of students who studied all the three subjects. Then $\mathrm{m}+\mathrm{n}$ is equal to ______

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

The number of solutions, of the equation ${e}^{\mathrm{sin}x}-2{e}^{-\mathrm{sin}x}=2$ is

Number of integral terms in the expansion of ${{{7}^{(\frac{1}{2})}+{11}^{(\frac{1}{6})}}}^{824}$ is equal to ______.

Let \(\alpha, \beta \in\) be roots of equation \(x^2-70 x+\lambda=0\), where \(\frac{\lambda}{2}, \frac{\lambda}{3} \notin\). If \(\lambda\) assumes the minimum possible value, then \(\frac{(\sqrt{\alpha-1}+\sqrt{\beta-1})(\lambda+35)}{\boldsymbol{|\alpha-\beta|}}\) is equal to :

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 $R$ 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 sum of all possible values of $\theta \in[-\pi, 2 \pi]$, for which $\frac{1+i \cos \theta}{1-2 i \cos \theta}$ is purely imaginary, is equal

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_______

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:

If $A$ is a square matrix of order 3 such that $\operatorname{det}(A)=3$ and $\operatorname{det}\left(\operatorname{adj}\left(-4 \operatorname{adj}\left(-3 \operatorname{adj}\left(3 \operatorname{adj}\left((2 \mathrm{~A})^{-1}\right)\right)\right)\right)\right)=2^{\mathrm{m}} 3^{\mathrm{n}}$, then $\mathrm{m}+2 \mathrm{n}$ is equal to :

Let $0 \leq \mathrm{r} \leq \mathrm{n}$. If ${ }^{\mathrm{n}+1} \mathrm{C}_{\mathrm{r}+1}:{ }^n \mathrm{C}_{\mathrm{r}}:{ }^{\mathrm{n}-1} \mathrm{C}_{\mathrm{r}-1}=55: 35: 21$, then $2 \mathrm{n}+5 \mathrm{r}$ is equal to:

In the expansion of $(1+x)(1-{x}^{2}){(1+\frac{3}{x}+\frac{3}{{x}^{2}}+\frac{1}{{x}^{3}})}^{5},x\neq 0$, the sum of the coefficient of ${x}^{3}$ and ${x}^{-13}$ is equal to ______

Let $\alpha ,\beta$ be the roots of the equation ${x}^{2}-x+2=0$ with $Im(\alpha )>Im(\beta )$. Then ${\alpha }^{6}+{\alpha }^{4}+{\beta }^{4}-5{\alpha }^{2}$ is equal to

Let $\alpha \in(0, \infty)$ and $A=\left[\begin{array}{lll}1 & 2 & \alpha \\ 1 & 0 & 1 \\ 0 & 1 & 2\end{array}\right]$. If $\operatorname{det}\left(\operatorname{adj}\left(2 A-A^T\right) \cdot \operatorname{adj}\left(A-2 A^T\right)\right)=2^8$, then $(\operatorname{det}(A))^2$ is equal to: