Algebra PYQ — Page 16

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 $\alpha_\theta$ and $\beta_\theta$ be the distinct roots of $2 x^2+(\cos \theta) x-1=0, \theta \in(0,2 \pi)$. If m and M are the minimum and the maximum values of $\alpha_\theta^4+\beta_\theta^4$, then $16(M+m)$ equals :

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

The number of complex numbers $z$, satisfying $|z|=1$ and $\left|\frac{z}{\bar{z}}+\frac{\bar{z}}{z}\right|=1$, is :

Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing positive terms. If $a_1 a_5=28$ and $a_2+a_4=29$, then $a_6$ is equal to:

Consider the sets $\mathrm{A}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{R} \times \mathbb{R}: \mathrm{x}^2+\mathrm{y}^2=25\right\}$, $\mathrm{B}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{R} \times \mathbb{R}: \mathrm{x}^2+9 \mathrm{y}^2=144\right\}, \mathrm{C}=\{(\mathrm{x}, \mathrm{y})$ $\left.\in \mathbb{Z} \times \mathbb{Z}: x^2+y^2 \leq 4\right\}$, and $D=A \cap B$. The total number of one-one functions from the set D to the set C is:

If the system of equation $\begin{aligned}<br/>& 2 x+\lambda y+3 z=5 \\ & 3 x+2 y-z=7 \\ & 4 x+5 y+\mu z=9<br/>\end{aligned}$ has infinitely many solutions, then $\left(\lambda^2+\mu^2\right)$ is equal to :

The number of relations on the set $\mathrm{A}=\{1,2,3\}$ containing at most 6 elements including $(1,2)$, which are reflexive and transitive but not symmetric, is ________

If the sum of the second, fourth and sixth terms of a G.P. of positive terms is 21 and the sum of its eighth, tenth and twelfth terms is 15309 , then the sum of its first nine terms is :

Let $\mathrm{a}_{\mathrm{n}}$ be the $\mathrm{n}^{\text {th }}$ term of an A. P. If $S_n=a_1+a_2+a_3+\ldots+a_n=700, a_6=7$ and $S_7=7$, then $\mathrm{a}_{\mathrm{n}}$ 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:

Let $\alpha, \beta$ be the roots of the equation $x^2+2 \sqrt{2} x-1=0$. The quadratic equation, whose roots are $\alpha^4+\beta^4$ and $\frac{1}{10}\left(\alpha^6+\beta^6\right)$, is :

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 :

Consider the function $f:[\frac{1}{2},1]\rightarrow R$ defined by $f(x)=4\sqrt{2}{x}^{3}-3\sqrt{2}x-1$. Consider the statements (I) The curve $y=f(x)$ intersects the $x$-axis exactly at one point (II) The curve $y=f(x)$ intersects the $x$-axis at $x=\mathrm{cos}\frac{\pi }{12}$ Then

Let $S={z\in C:|z-1|=1\mathrm{and}(\sqrt{2}-1)(z+\bar{z})-i(z-\bar{z})=2\sqrt{2}}$. Let ${z}_{1},{z}_{2}\in S$ be such that $|{z}_{1}|=\underset{z\in s}{\mathrm{max}}|z|$ and $|{z}_{2}|=\underset{z\in s}{\mathrm{min}}|z|$. Then ${|\sqrt{2}{z}_{1}-{z}_{2}|}^{2}$ equals:

Let $m\text{and}n$ be the coefficients of seventh and thirteenth terms respectively in the expansion of ${(\frac{1}{3}{x}^{\frac{1}{3}}+\frac{1}{2{x}^{\frac{2}{3}}})}^{18}$. Then ${(\frac{n}{m})}^{\frac{1}{3}}$ is:

Remainder when ${64}^{{32}^{32}}$ is divided by $9$ is equal to _____.

The total number of words (with or without meaning) that can be formed out of the letters of the word "DISTRIBUTION" taken four at a time, is equal to ______.

Let $A={1,2,3,\ldots .7}$ and let $P(A)$denote the power set of $A$. If the number of functions $f:A\rightarrow P(A)$ such that $a\in f(a),\forall a\in A$ is ${m}^{n},m$ and $n\in N$ and $m$ is least, then $m+n$ is equal to ______.

Let $P={z\in \mathbb{C}:|z+2-3i|\leq 1}$ and $Q={z\in \mathbb{C}:z(1+i)+\bar{z}(1-i)\leq -8}$. Let in $P\cap Q,|z-3+2i|$ be maximum and minimum at ${z}_{1}$ and ${z}_{2}$ respectively. If ${|{z}_{1}|}^{2}+2{|z|}^{2}=\alpha +\beta \sqrt{2},$ where $\alpha ,\beta$ are integers, then $\alpha +\beta$ equals __________

Let $f(x)=x^5+2 x^3+3 x+1, x \in \mathbf{R}$, and $g(x)$ be a function such that $g(f(x))=x$ for all $x \in \mathbf{R}$. Then $\frac{g(7)}{g^{\prime}(7)}$ is equal to :

Let $\alpha ,\beta$ be the roots of the equation ${x}^{2}-\sqrt{6}x+3=0$ such that $Im(\alpha )>Im(\beta )$. Let $a,b$ be integers not divisible by $3$and $n$ be a natural number such that $\frac{{\alpha }^{99}}{\beta }+{\alpha }^{98}={3}^{n}(a+ib),i=\sqrt{-1}$. Then $n+a+b$ is equal to ___________.

Let the set $S=\{2,4,8,16, \ldots, 512\}$ be partitioned into 3 sets $A, B, C$ with equal number of elements such that $\mathrm{A} \cup \mathrm{B} \cup \mathrm{C}=\mathrm{S}$ and $\mathrm{A} \cap \mathrm{B}=\mathrm{B} \cap \mathrm{C}=\mathrm{A} \cap \mathrm{C}=\phi$. The maximum number of such possible partitions of $S$ is equal to:

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________