Algebra PYQ — Page 2

Let $\mathrm{A}=\{0,1,2, \ldots, 9\}$. Let R be a relation on A defined by $(x, y) \in \mathrm{R}$ if and only if $|x-y|$ is a multiple of 3. Given below are two statements: Statement I: $n(\mathrm{R})=36$. Statement II: $R$ is an equivalence relation. In the light of the above statements, choose the correct answer from the options given below

Let $\mathrm{S}=\left\{x^{3}+a x^{2}+b x+c: a, b, c \in \mathrm{~N}\right.$ and $\left.a, b, c \leq 20\right\}$ be a set of polynomials. Then the number of polynomials in S, which are divisible by $x^{2}+2$, is

Let $A$ be a $3 \times 3$ matrix such that $A^T \begin{bmatrix}1\\0\\1\end{bmatrix} = \begin{bmatrix}5\\2\\2\end{bmatrix}$, $A^T \begin{bmatrix}0\\0\\1\end{bmatrix} = \begin{bmatrix}3\\1\\1\end{bmatrix}$, $A \begin{bmatrix}1\\0\\1\end{bmatrix} = \begin{bmatrix}3\\4\\4\end{bmatrix}$ and $A \begin{bmatrix}0\\0\\1\end{bmatrix} = \begin{bmatrix}1\\3\\1\end{bmatrix}$. If $\det(A) = 1$, then $\det(\operatorname{adj}(A^2 + A))$ is equal to:

The common difference of the A.P.: $a_{1}, a_{2}, \ldots, a_{\mathrm{m}}$ is 13 more than the common difference of the A.P.: $b_{1}, b_{2}, \ldots, b_{n}$. If $b_{31}=-277, b_{43}=-385$ and $a_{78}=327$, then $a_{1}$ is equal to

If the system of equations $x + 5y + 6z = 4$, $2x + 3y + 4z = 7$, $x + 6y + az = b$ has infinitely many solutions, then the point $(a, b)$ lies on the line

Let $A=\left[\begin{array}{ll}3 & -4 \\ 1 & -1\end{array}\right]$ and $B$ be two matrices such that $A^{100}=100 B+I$. Then the sum of all the elements of $\mathrm{B}^{100}$ is $\_\_\_\_$

Let $\alpha, \beta$ be the roots of the equation $x^2 - 3x + r = 0$, and $\dfrac{\alpha}{2}, 2\beta$ be the roots of the equation $x^2 + 3x + r = 0$. If the roots of the equation $x^2 + 6x = m$ are $2\alpha + \beta + 2r$ and $\alpha - 2\beta - \dfrac{r}{2}$, then $m$ is equal to:

If the roots of x² - 5x + k = 0 are in the ratio 2:3, then k equals:

For the matrices $\mathrm{A}=\left[\begin{array}{ll}3 & -4 \\ 1 & -1\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ll}-29 & 49 \\ -13 & 18\end{array}\right]$, if $\left(\mathrm{A}^{15}+\mathrm{B}\right)\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{l}0 \\ 0\end{array}\right]$, then among the following which one is true?

The largest value of $n$, for which $40^{n}$ divides $60!$, is

If for $3 \leq r \leq 30$, $\binom{30}{30-r} + 3\binom{30}{31-r} + 3\binom{30}{32-r} + \binom{30}{33-r} = \binom{m}{r}$, then $m$ equals:

Let the smallest value of $k \in \mathbb{N}$, for which the coefficient of $x^3$ in $(1+x)^3 + (1+x)^4 + (1+x)^5 + \ldots + (1+x)^{99} + (1+kx)^{100}$, $x \neq 0$, is $\left(43n + \dfrac{101}{4}\right)\left(^{100}C_3\right)$ for some $n \in \mathbb{N}$, be $p$. Then the value of $p + n$ is:

The sum of squares of all the real solutions of the equation $\log_{(x+1)}(2x^2+5x+3) = 4 - \log_{(2x+3)}(x^2+2x+1)$ is equal to ________.

The smallest positive integral value of $a$, for which all the roots of $x^{4}-a x^{2}+9=0$ are real and distinct, is equal to

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined as $f(x) = \dfrac{2x^2 - 3x + 2}{3x^2 + x + 3}$. Then $f$ is :

Let $f$ be a function such that $3 f(x)+2 f\left(\frac{m}{19 x}\right)=5 x, x \neq 0$, where $m=\sum_{i=1}^{9}(i)^{2}$. Then $f(5)-f(2)$ is equal to

The sum of all the elements in the range of $f(x)=\operatorname{Sgn}(\sin x)+\operatorname{Sgn}(\cos x)+\operatorname{Sgn}(\tan x)+\operatorname{Sgn}(\cot x)$, $x \neq \frac{\mathrm{n} \pi}{2}, \mathrm{n} \in \mathbf{Z}$, where $\operatorname{Sgn}(\mathrm{t})=\left\{\begin{aligned} 1, & \text { if } \mathrm{t}>0 \\ -1, & \text { if } \mathrm{t}<0\end{aligned}\right.$, is :

Let the domain of the function $f(x)=\log _{3} \log _{5}\left(7-\log _{2}\left(x^{2}-10 x+85\right)\right)+\sin ^{-1}\left(\left|\frac{3 x-7}{17-x}\right|\right)$ be $(\alpha, \beta]$. Then $\alpha+\beta$ is equal to :

Let A be a $3 \times 3$ matrix such that $\mathrm{A}+\mathrm{A}^{\mathrm{T}}=\mathrm{O}$. If $\mathrm{A}\left[\begin{array}{c}1 \\ -1 \\ 0\end{array}\right]=\left[\begin{array}{l}3 \\ 3 \\ 2\end{array}\right], \mathrm{A}^{2}\left[\begin{array}{c}1 \\ -1 \\ 0\end{array}\right]=\left[\begin{array}{c}-3 \\ 19 \\ -24\end{array}\right]$ and $\operatorname{det}(\operatorname{adj}(2 \operatorname{adj}(\mathrm{~A}+\mathrm{I})))=(2)^{\alpha} \cdot(3)^{\beta} \cdot(11)^{\gamma}, \alpha, \beta, \gamma$ are non-negative integers, then $\alpha+\beta+\gamma$ is equal to $\_\_\_\_$

Consider two sets $\mathrm{A}=\{x \in \mathrm{Z}:|(|x-3|-3)| \leq 1\}$ and $\mathrm{B}=\left\{x \in \mathbb{R}-\{1,2\}: \frac{(x-2)(x-4)}{x-1} \log _{e}(|x-2|)=0\right\}$. Then the number of onto functions $f: \mathrm{A} \rightarrow \mathrm{B}$ is equal to

The sum of squares of all the real solutions of the equation $\log_{(x+1)}(2x^2+5x+3) = 4 - \log_{(2x+3)}(x^2+2x+1)$ is equal to ________.

Let $A = \begin{bmatrix} 1 & 1 & 2 \\ -2 & 0 & 1 \\ 1 & 3 & 5 \end{bmatrix}$. Then the sum of all elements of the matrix $\text{adj}(\text{adj}(2(\text{adj}A)^{-1}))$ is equal to:

If $\alpha$ and $\beta(\alpha<\beta)$ are the roots of the equation $(-2+\sqrt{3})(|\sqrt{x}-3|)+(x-6 \sqrt{x})+(9-2 \sqrt{3})=0, x \geqslant 0$, then $\sqrt{\frac{\beta}{\alpha}}+\sqrt{\alpha \beta}$ is equal to :

If $\sum_{r=1}^{25}\left(\frac{r}{r^{4}+r^{2}+1}\right)=\frac{p}{q}$, where $p$ and $q$ are positive integers such that $\operatorname{gcd}(p, q)=1$, then $p+q$ is equal to $\_\_\_\_$。