Algebra PYQ — Page 44

Considering only the principal values of the inverse trigonometric functions, the domain of the function $f(x)={\mathrm{cos}}^{-1}(\frac{{x}^{2}-4x+2}{{x}^{2}+3})$ is

The number of positive integers $k$ such that the constant term in the binomial expansion of ${(2{x}^{3}+\frac{3}{{x}^{k}})}^{12},x\neq 0$ is ${2}^{8}\cdot l$, where $l$ is an odd integer, is ______.

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be defined as $f(x)=x-1$ and $g:R\rightarrow {1,-1}\rightarrow \mathbb{R}$ be defined as $g(x)=\frac{{x}^{2}}{{x}^{2}-1}$. Then the function $fog$ is:

Let $A=(\begin{matrix}1 & 2 \\ -2 & -5\end{matrix})$. Let $\alpha ,\beta \in \mathbb{R}$ be such that $\alpha {A}^{2}+\beta A=2I$. Then $\alpha +\beta$ is equal to

The number of $\theta \in (0,4\pi )$ for which the system of linear equations $3(\mathrm{sin}3\theta )x-y+z=2$ $3(\mathrm{cos}2\theta )x+4y+3z=3$ $6x+7y+7z=9$ has no solution is

Let $A=(\begin{matrix}2 & -1 \\ 0 & 2\end{matrix})$. If $B=I-C15(adjA)+C25(adjA{)}^{2}-...-C55{(\mathrm{adj}A)}^{5}$, then the sum of all elements of the matrix $B$ is:

The sum of the cubes of all the roots of the equation ${x}^{4}-3{x}^{3}-2{x}^{2}+3x+1=0$ is _____.

The coefficient of ${x}^{101}$ in the expression ${(5+x)}^{500}+x{(5+x)}^{499}+{x}^{2}{(5+x)}^{498}+\ldots \ldots +{x}^{500},x>0$ is

Let $n\geq 5$ be an integer. If ${9}^{n}-8n-1=64\alpha$ and ${6}^{n}-5n-1=25\beta$, then $\alpha -\beta$ is equal to:

Numbers are to be formed between $1000$ and $3000$, which are divisible by $4$, using the digits $1,2,3,4,5$ and $6$ without repetition of digits. Then the total number of such numbers is _______.

The ordered pair $(a,b)$, for which the system of linear equations $3x-2y+z=b$ $5x-8y+9z=3$ $2x+y+az=-1$ has no solution, is

If the system of equations $\alpha x+y+z=5,x+2y+3z=4,x+3y+5z=\beta$. Has infinitely many solutions, then the ordered pair $(\alpha ,\beta )$ is equal to

The system of linear equations$3x-2y-kz=10$ $2x-4y-2z=6$ $x+2y-z=5m$ is inconsistent if :

The number of real solutions of the equation, ${x}^{2}-|x|-12=0$ is:

The number of seven digit integers with sum of the digits equal to $10$ and formed by using the digits $1,2$ and $3$ only is:

If $n$ is the number of irrational terms in the expansion of ${({3}^{1/4}+{5}^{1/8})}^{60}$, then $(n-1)$ is divisible by :

Let the coefficients of third, fourth and fifth terms in the expansion of ${(x+\frac{a}{{x}^{2}})}^{n},x\neq 0,$ be in the ratio $12:8:3.$ Then the term independent of $x$ in the expansion, is equal to _______.

Let $M$ be any $3\times 3$ matrix with entries from the set ${0,1,2}$. The maximum number of such matrices, for which the sum of diagonal elements of ${M}^{T}M$ is seven, is______.

Let $a,b,c$ be in arithmetic progression. Let the centroid of the triangle with vertices $(a,c),(2,b)$ and $(a,b)$ be $(\frac{10}{3},\frac{7}{3}).$ If $\alpha ,\beta$ are the roots of the equation $a{x}^{2}+bx+1=0,$ then the value of ${\alpha }^{2}+{\beta }^{2}-\alpha \beta$ is:

Let $f,g:N\rightarrow N$ such that $f(n+1)=f(n)+f(1)\forall n\in N$ and $g$ be any arbitrary function. Which of the following statements is NOT true?

The total number of positive integral solutions $(x,y,z)$ such that $xyz=24$ is :

If $\alpha ,\beta \in R$ are such that $1-2i$ (here ${i}^{2}=-1$) is a root of ${z}^{2}+\alpha z+\beta =0$, then $(\alpha -\beta )$ is equal to:

Let $A$ be a $3\times 3$ matrix with $det(A)=4$. Let ${R}_{i}$ denote the ${i}^{th}$ row of $A$. If a matrix $B$ is obtained by performing the operation ${R}_{2}\rightarrow 2{R}_{2}+5{R}_{3}$ on $2A$, then $det(B)$ is equal to :

If sum of the first $21$ terms of the series ${\mathrm{log}}_{{9}^{1/2}}x+{\mathrm{log}}_{{9}^{1/3}}x+{\mathrm{log}}_{{9}^{1/4}}x+\ldots ..$ where $x>0$ is $504,$ then $x$ is equal to