Algebra PYQ — Page 9

Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion of $\left(x+\sqrt{x^3-1}\right)^5+\left(x-\sqrt{x^3-1}\right)^5, x\gt1$. If u and v satisfy the equations $\begin{aligned} & \alpha u+\beta v=18 \\ & \gamma u+\delta v=20 \end{aligned}$ then $u+v$ equals :

Let $A=\{1,2,3, \ldots, 10\}$ and $B=\left\{\frac{m}{n}: m, n \in A, m \lt n\right.$ and $\left.\operatorname{gcd}(m, n)=1\right\}$. Then $n(B)$ is equal to :

If the system of linear equations : $\begin{aligned} & x+y+2 z=6 \\ & 2 x+3 y+\mathrm{a} z=\mathrm{a}+1 \\ & -x-3 y+\mathrm{b} z=2 \mathrm{~b} \end{aligned}$ where $a, b \in \mathbf{R}$, has infinitely many solutions, then $7 a+3 b$ is equal to :

$\begin{aligned}<br/>& \text { If } \frac{1}{1^4}+\frac{1}{2^4}+\frac{1}{3^4}+\ldots . . \infty=\frac{\pi^4}{90}, \\ & \frac{1}{1^4}+\frac{1}{3^4}+\frac{1}{5^4}+\ldots . . \infty=\alpha, \\ & \frac{1}{2^4}+\frac{1}{4^4}+\frac{1}{6^4}+\ldots . \infty=\beta,<br/>\end{aligned}$ then $\frac{\alpha}{\beta}$ is equal to

Let the system of equations : $\begin{aligned}<br/>& 2 x+3 y+5 z=9 \\ & 7 x+3 y-2 z=8 \\ & 12 x+3 y-(4+\lambda) z=16-\mu<br/>\end{aligned}$ have infinitely many solutions. Then the radius of the circle centred at $(\lambda, \mu)$ and touching the line $4 x=3 y$ is

The system of equations $\begin{aligned} & x+y+z=6 \\ & x+2 y+5 z=9, \\ & x+5 y+\lambda z=\mu, \end{aligned}$ has no solution if

If the system of equations $\begin{aligned} & 2 x-y+z=4 \\ & 5 x+\lambda y+3 z=12 \\ & 100 x-47 y+\mu z=212 \end{aligned}$ has infinitely many solutions, then $\mu-2 \lambda$ is equal to

If the system of equations $\begin{aligned} & x+2 y-3 z=2 \\ & 2 x+\lambda y+5 z=5 \\ & 14 x+3 y+\mu z=33 \end{aligned}$ has infinitely many solutions, then $\lambda+\mu$ is equal to :

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 :

Let $\mathrm{a} \in \mathbf{R}$ and A be a matrix of order $3 \times 3$ such that $\operatorname{det}(A)=-4$ and $A+I=\left[\begin{array}{lll}1 & a & 1 \\ 2 & 1 & 0 \\ a & 1 & 2\end{array}\right]$, where $I$ is the identity matrix of order $3 \times 3$. If $\operatorname{det}((a+1) \operatorname{adj}((a-1) A))$ is $2^m 3^n, m, n \in$ $\{0,1,2, \ldots .20\}$, then $\mathrm{m}+\mathrm{n}$ is equal to :

Let $A=\left[\begin{array}{cc}\frac{1}{\sqrt{2}} & -2 \\ 0 & 1\end{array}\right]$ and $P=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right], \theta\gt0$. If $\mathrm{B}=\mathrm{PAP}^{\mathrm{T}}, \mathrm{C}=\mathrm{P}^{\mathrm{T}} \mathrm{B}^{10} \mathrm{P}$ and the sum of the diagonal elements of $C$ is $\frac{\mathrm{m}}{\mathrm{n}}$, where $\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$, then $\mathrm{m}+\mathrm{n}$ is :

If $\sum_{\mathrm{r}=1}^9\left(\frac{\mathrm{r}+3}{2^{\mathrm{r}}}\right) .{ }^9 \mathrm{C}_{\mathrm{r}}=\alpha\left(\frac{3}{2}\right)^9-\beta, \quad \alpha, \beta \in \mathrm{N}, \quad$ then $(\alpha+\beta)^2$ is equal to

Let $A$ be a $3 \times 3$ real matrix such that $A^2(A-2 I)-$ $4(\mathrm{~A}-\mathrm{I})=\mathrm{O}$, where I and O are the identity and null matrices, respectively. If $A^5=\alpha A^2+\beta A+\gamma I$, where $\alpha, \beta$ and $\gamma$ are real constants, then $\alpha+\beta+\gamma$ is equal to:

If the domain of the function $f(x)=\log _7\left(1-\log _4\left(x^2-9 x+18\right)\right)$ is $(\alpha, \beta) \cup(\gamma, \delta)$, then $\alpha+\beta+\gamma+\delta$ is equal to

If $f(x)=\frac{2^x}{2^x+\sqrt{2}}, \mathrm{x} \in \mathbb{R}$, then $\sum_{\mathrm{k}=1}^{81} f\left(\frac{\mathrm{k}}{82}\right)$ is equal to

For some $n \neq 10$, let the coefficients of the 5 th, 6 th and 7 th terms in the binomial expansion of $(1+\mathrm{x})^{\mathrm{n}+4}$ be in A.P. Then the largest coefficient in the expansion of $(1+\mathrm{x})^{\mathrm{n}+4}$ is:

The number of natural numbers, between 212 and 999 , such that the sum of their digits is 15 , is

$1+3+5^2+7+9^2+\ldots$ upto 40 terms is equal to

Let $A=\{-2,-1,0,1,2,3\}$. let R be a relation on A defined by $x R y$ if and only if $y=\max \{x, 1\}$. Let $l$ be the number of elements in R. Let m and n be the minimum number of elements required to be added in R to make it reflexive and symmetric relations, respectively. Then $l+\mathrm{m}+\mathrm{n}$ is equal to

If the system of linear equations $\begin{aligned} & 3 x+y+\beta z=3 \\ & 2 x+\alpha y-z=-3 \\ & x+2 y+z=4\end{aligned}$ has infinitely many solutions, then the value of $22 \beta-9 \alpha$ is :

The interior angles of a polygon with n sides, are in an A.P. with common difference $6^{\circ}$. If the largest interior angle of the polygon is $219^{\circ}$, then n is equal to

Let $\alpha, \beta$ be the roots of the equation $x^2-a x-b=0$ with $\operatorname{Im}(\alpha) \lt \operatorname{Im}(\beta)$. Let $P_n=\alpha^n-\beta^n$. If $\mathrm{P}_3=-5 \sqrt{7} i, \mathrm{P}_4=-3 \sqrt{7} i, \mathrm{P}_5=11 \sqrt{7} i$ and $\mathrm{P}_6=45 \sqrt{7} i$, then $\left|\alpha^4+\beta^4\right|$ is equal to $\qquad$ .

Among the statements (S1) : The set $\left\{\mathrm{z} \in \mathbb{C}-\{-\mathrm{i}\}:|\mathrm{z}|=1\right.$ and $\frac{\mathrm{z}-\mathrm{i}}{\mathrm{z}+\mathrm{i}}$ is purely real} contains exactly two elements, and (S2) : The set $\left\{\mathrm{z} \in \mathbb{C}-\{-1\}:|\mathrm{z}|=1\right.$ and $\frac{\mathrm{z}-1}{\mathrm{z}+1}$ is purely imaginary contains infinitely many elements.

The sum $1+3+11+25+45+71+.$. upto 20 terms, is equal to