Algebra PYQ — Page 24

Consider the system of linear equation $x+y+z=$ $4\mu ,x+2y+2\lambda z=10\mu ,x+3y+4{\lambda }^{2}z={\mu }^{2}+15$, where $\lambda ,\mu \in R$. Which one of the following statements is NOT correct?

Let $f(x)={2}^{x}-{x}^{2},x\in R$. If $m$ and $n$ are respectively the number of points at which the curves $y=f(x)$ and $y={f}^{'}(x)$ intersects the $x-$axis, then the value of $m+n$ is

The function $f:N-{1}\rightarrow N$; defined by $f(n)=$ the highest prime factor of $n$, is :

If $\left(\frac{1}{\alpha+1}+\frac{1}{\alpha+2}+\ldots \ldots+\frac{1}{\alpha+1012}\right)-\left(\frac{1}{2 \cdot 1}+\frac{1}{4 \cdot 3}+\frac{1}{6 \cdot 5}+\ldots . .+\frac{1}{2024 \cdot 2023}\right)=\frac{1}{2024}$, then $\alpha$ is equal to________

Let $3,a,b,c$ be in $A.P.$ and $3,a-1,b+1,c+9$ be in $G.P.$ Then, the arithmetic mean of $a,b$ and $c$ is:

For $x \geqslant 0$, the least value of $\mathrm{K}$, for which $4^{1+x}+4^{1-x}, \frac{\mathrm{K}}{2}, 16^x+16^{-x}$ are three consecutive terms of an A.P., is equal to :

If $\mathrm{S}(x)=(1+x)+2(1+x)^2+3(1+x)^3+\cdots+60(1+x)^{60}, x \neq 0$, and $(60)^2 \mathrm{~S}(60)=\mathrm{a}(\mathrm{b})^{\mathrm{b}}+\mathrm{b}$, where $a, b \in N$, then $(a+b)$ equal to ______

If $S=\{a \in \mathbf{R}:|2 a-1|=3[a]+2\{a\}\}$, where $[t]$ denotes the greatest integer less than or equal to $t$ and $\{t\}$ represents the fractional part of $t$, then $72 \sum_{a \in S} a$ is equal to ______

Let the relations $R_1$ and $R_2$ on the set $X=\{1,2,3, \ldots, 20\}$ be given by $R_1=\{(x, y): 2 x-3 y=2\}$ and $R_2=\{(x, y):-5 x+4 y=0\}$. If $M$ and $N$ be the minimum number of elements required to be added in $R_1$ and $R_2$, respectively, in order to make the relations symmetric, then $M+N$ equals

Let the sum of the maximum and the minimum values of the function $f(x)=\frac{2 x^2-3 x+8}{2 x^2+3 x+8}$ be $\frac{\mathrm{m}}{\mathrm{n}}$, where $\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$. Then $\mathrm{m}+\mathrm{n}$ is equal to :

Let $\alpha, \beta$ be the distinct roots of the equation $x^2-\left(t^2-5 t+6\right) x+1=0, t \in \mathbb{R}$ and $a_n=\alpha^n+\beta^n$. Then the minimum value of $\frac{a_{2023}+a_{2025}}{a_{2024}}$ is

The sum of the coefficient of $x^{2 / 3}$ and $x^{-2 / 5}$ in the binomial expansion of $\left(x^{2 / 3}+\frac{1}{2} x^{-2 / 5}\right)^9$ is

In an examination of Mathematics paper, there are $20$ questions of equal marks and the question paper is divided into three sections : $A,B$ and $C$. A student is required to attempt total $15$ questions taking at least $4$ questions from each section. If section $A$ has $8$ questions, section $B$ has $6$ questions and section $C$ has $6$ questions, then the total number of ways a student can select $15$ questions is _________.

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 three real numbers $a, b, c$ be in arithmetic progression and $a+1, b, c+3$ be in geometric progression. If $a>10$ and the arithmetic mean of $a, b$ and $c$ is 8, then the cube of the geometric mean of $a, b$ and $c$ is

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

Let $A={1,2,3,...20}$. Let ${R}_{1}$ and ${R}_{2}$ two relation on $A$ such that ${R}_{1}={(a,b):b$ is divisible by $a$} ${R}_{2}={(a,b):a$ is an integral multiple of $b$} Then, number of elements in ${R}_{1}-{R}_{2}$ is equal to __________.

Let $\alpha =\frac{(4!)!}{(4!{)}^{3!}}$ and $\beta =\frac{(5!)!}{(5!{)}^{4!}}$. Then :

Let $S={z\in \mathbb{C}-{i,2i}:\frac{{z}^{2}+8iz-15}{{z}^{2}-3iz-2}\in \mathbb{R}}$. $\alpha -\frac{13}{11}i\in S,\alpha \in \mathbb{R}-{0}$, then $242{\alpha }^{2}$ is equal to

If the ${1011}^{\mathrm{th}}$ term from the end in the binomial expansion of ${(\frac{4x}{5}-\frac{5}{2x})}^{2022}$ is $1024$ times ${1011}^{\mathrm{th}}$ term from the beginning, then $32|x|$ is equal to

Let $N$ denote the number that turns up when a fair die is rolled. If the probability that the system of equations $x+y+z=1 2x+Ny+2z=2 3x+3y+Nz=3$ has unique solution is $\frac{k}{6},$ then the sum of value of $k$ and all possible values of $N$ is

Let $A={1,2,3,5,8,9}$. Then the number of possible functions $f:A\rightarrow A$ such that $f(m\cdot n)=f(m)\cdot f(n)$ for every $m,n\in A$ with $m\cdot n\in A$ is equal to

Let the sets $A$ and $B$ denote the domain and range respectively of the function $f(x)=\frac{1}{\sqrt{[x]-x}},$ where $[x]$ denotes the smallest integer greater than or equal to $x$. Then among the statements $(S1):A\cap B=(1,\infty )-\mathbb{N}$ and $(S2):A\cup B=(1,\infty )$

Let $A={\theta \in (0,2\pi ):\frac{1+2i\mathrm{sin}\theta }{1-i\mathrm{sin}\theta }\mathrm{is}\mathrm{purely}\mathrm{imaginary}}$ Then the sum of the elements is in $A$ is