Algebra PYQ — Page 11

Let M denote the set of all real matrices of order $3 \times 3$ and let $\mathrm{S}=\{-3,-2,-1,1,2\}$. Let $$ \begin{aligned} & \mathrm{S}_1=\left\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: \mathrm{A}=\mathrm{A}^{\mathrm{T}} \text { and } a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\right\}, \\ & \mathrm{S}_2=\left\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: \mathrm{A}=-\mathrm{A}^{\mathrm{T}} \text { and } a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\right\}, \\ & \mathrm{S}_3=\left\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: a_{11}+a_{22}+a_{33}=0 \text { and } a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\right\} . \end{aligned} $$ If $n\left(\mathrm{~S}_1 \cup_2 \mathrm{US}_3\right)=125 \alpha$, then $\alpha$ equals _______

Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its \(11^{\text {th }}\) term is :

Define a relation R on the interval \(\left[0, \frac{\pi}{2}\right)\) by \(x \mathrm{R} y\) if and only if \(\sec ^2 x-\tan ^2 y=1\). Then R is :

For an integer $\mathrm{n} \geq 2$, if the arithmetic mean of all coefficients in the binomial expansion of $(x+y)^{2 n-3}$ is 16 , then the distance of the point $P\left(2 n-1, n^2-4 n\right)$ from the line $x+y=8$ is:

Let $f(x)+2 f\left(\frac{1}{x}\right)=x^2+5$ and $2 g(x)-3 g\left(\frac{1}{2}\right)=x, x \gt 0$. If $\alpha=\int_1^2 f(x) d x$, and $\beta=\int_1^2 g(x) d x$, then the value of $9 \alpha+\beta$ is:

If the domain of the function $f(x)=\log _e\left(\frac{2 x-3}{5+4 x}\right)+\sin ^{-1}\left(\frac{4+3 x}{2-x}\right) \quad \text { is } \quad[\alpha, \beta)$ then $\alpha^2+4 \beta$ is equal to

If the domain of the function $f(x)=\frac{1}{\sqrt{10+3 x-x^2}}+\frac{1}{\sqrt{x+|x|}}$ is $(a, b)$, then $(1+a)^2+b^2$ is equal to :

If the domain of the function $\log _5\left(18 x-x^2-77\right)$ is $(\alpha, \beta)$ and the domain of the function $\log _{(x-1)}\left(\frac{2 x^2+3 x-2}{x^2-3 x-4}\right)$ is $(\gamma, \delta)$, then $\alpha^2+\beta^2+\gamma^2$ is equal to :

Let $f: \mathbf{R}-\{0\} \rightarrow(-\infty, 1)$ be a polynomial of degree 2, satisfying $f(x) f\left(\frac{1}{x}\right)=f(x)+f\left(\frac{1}{x}\right)$. If $f(K)=-2 K$, then the sum of squares of all possible values of $K$ is :

$\text { The number of real solution(s) of the equation } x^2+3 x+2=\min \{|x-3|,|x+2|\} \text { is : }$

Let A be the set of all functions $f: \mathbf{Z} \rightarrow \mathbf{Z}$ and R be a relation on A such that $\mathrm{R}=\{(\mathrm{f}, \mathrm{g}): f(0)=\mathrm{g}(1)$ and $f(1)=\mathrm{g}(0)\}$. Then R is:

Let $O$ be the origin, the point $A$ be $z_1=\sqrt{3}+2 \sqrt{2} i$, the point $B\left(z_2\right)$ be such that $\sqrt{3}\left|z_2\right|=\left|z_1\right|$ and $\arg \left(z_2\right)=\arg \left(z_1\right)+\frac{\pi}{6}$. Then

If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to

Let $A$ be a matrix of order $3 \times 3$ and $|A|=5$. If $|2 \operatorname{adj}(3 \mathrm{~A} \operatorname{adj}(2 \mathrm{~A}))|=2^\alpha \cdot 3^\beta \cdot 5^\gamma \alpha, \beta, \gamma \in \mathrm{N}$ then $\alpha+\beta+\gamma$ is equal to

The product of the last two digits of $(1919)^{1919}$ is $\qquad$

Let the domain of the function $f(x)=\cos ^{-1}\left(\frac{4 x+5}{3 x-7}\right)$ be $[\alpha, \beta]$ and the domain of $\mathrm{g}(\mathrm{x})=\log _2\left(2-6 \log _{27}(2 \mathrm{x}+5)\right)$ be $(\gamma, \delta)$. Then $|7(\alpha+\beta)+4(\gamma+\delta)|$ is equal to ________

The sum of all rational terms in the expansion of $\left(1+2^{1 / 3}+3^{1 / 2}\right)^6$ is equal to

For $t \gt -1$, let $\alpha_t$ and $\beta_t$ be the roots of the equation $\left((t+2)^{\frac{1}{7}}-1\right) x^2+\left((t+2)^{\frac{1}{6}}-1\right)$ $x+\left((t+2)^{\frac{1}{21}}-1\right)=0$ If $\lim _{t \rightarrow-1^{+}} \alpha_t=a$ and $\lim _{t \rightarrow-1^{+}} \beta_t=b$, then $72(a+b)^2$ is equal to ________.

Let the range of the function $f(x)=6+16 \cos x \cdot \cos \left(\frac{\pi}{3}-x\right) \cdot \cos \left(\frac{\pi}{3}+x\right) \cdot \sin 3 x \cdot \cos 6 x, x \in \mathbf{R}$ be $[\alpha, \beta]$. Then the distance of the point $(\alpha, \beta)$ from the line $3 x+4 y+12=0$ is :

Let the coefficients of three consecutive terms $T_r, T_{r+1}$ and $T_{r+2}$ in the binomial expansion of $(a+b)^{12}$ be in a G.P. and let $p$ be the number of all possible values of $r$. Let $q$ be the sum of all rational terms in the binomial expansion of $(\sqrt[4]{3}+\sqrt[3]{4})^{12}$. Then $\mathrm{p}+\mathrm{q}$ is equal to :

If $\sum_{r=0}^5 \frac{{ }^{11} C_{2r+1}}{2 r+2}=\frac{\mathrm{m}}{\mathrm{n}}, \operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$, then $\mathrm{m}-\mathrm{n}$ is equal to _______

Consider two sets $A$ and $B$, each containing three numbers in A.P. Let the sum and the product of the elements of A be 36 and p respectively and the sum and the product of the elements of B be 36 and q respectively. Let d and D be the common differences of AP's in A and B respectively such that $D=d+3, d \gt 0$. If $\frac{p+q}{p-q}=\frac{19}{5}$, then $p-q$ is equal to

The product of all solutions of the equation $\mathrm{e}^{5\left(\log _{\mathrm{e}} x\right)^2+3}=x^8, x\gt0$, is :

$\text { The function } f:(-\infty, \infty) \rightarrow(-\infty, 1) \text {, defined by } f(x)=\frac{2^x-2^{-x}}{2^x+2^{-x}} \text { is : }$