Algebra PYQ — Page 23

The lines ${L}_{1},{L}_{2},...,{L}_{20}$ are distinct. For $n=1,2,3,...,10$ all the lines ${L}_{2n-1}$ are parallel to each other and all the lines ${L}_{2n}$ pass through a given point $P$. The maximum number of points of intersection of pairs of lines from the set ${{L}_{1},{L}_{2},...,{L}_{20}}$ is equal to:

Let $a_1, a_2, a_3, \ldots$ be in an arithmetic progression of positive terms. Let $\mathrm{A}_{\mathrm{k}}=\mathrm{a}_1^2-\mathrm{a}_2^2+\mathrm{a}_3^2-\mathrm{a}_4^2+\ldots+\mathrm{a}_{2 \mathrm{k}-1}^2-\mathrm{a}_{2 \mathrm{k}}^2$. If $\mathrm{A}_3=-153, \mathrm{~A}_5=-435$ and $\mathrm{a}_1^2+\mathrm{a}_2^2+\mathrm{a}_3^2=66$, then $\mathrm{a}_{17}-\mathrm{A}_7$ is equal to______

Consider the relations ${R}_{1}$ and ${R}_{2}$ defined as $a{R}_{1}b\Leftrightarrow {a}^{2}+{b}^{2}=1$ for all $a,b,\in R$ and $(a,b){R}_{2}(c,d)\Leftrightarrow a+d=b+c$ for all $(a,b),(c,d)\in N\times N$. Then

If the range of $f(\theta)=\frac{\sin ^4 \theta+3 \cos ^2 \theta}{\sin ^4 \theta+\cos ^2 \theta}, \theta \in \mathbb{R}$ is $[\alpha, \beta]$, then the sum of the infinite G.P., whose first term is 64 and the common ratio is $\frac{\alpha}{\beta}$, is equal to________

Let $A=\{n \in[100,700] \cap \mathbb{N}: n$ is neither a multiple of 3 nor a multiple of 4$\}$. Then the number of elements in $A$ is

Let $f:R\rightarrow R$ be a function defined $f(x)=\frac{x}{{(1+{x}^{4})}^{1/4}}$ and $g(x)=f(f(f(f(x))))$ then $18{\int }_{0}^{\sqrt{2\sqrt{5}}}{x}^{2}g(x)dx$

If $1+\frac{\sqrt{3}-\sqrt{2}}{2 \sqrt{3}}+\frac{5-2 \sqrt{6}}{18}+\frac{9 \sqrt{3}-11 \sqrt{2}}{36 \sqrt{3}}+\frac{49-20 \sqrt{6}}{180}+\ldots$ upto $\infty=2+\left(\sqrt{\frac{b}{a}}+1\right) \log _e\left(\frac{a}{b}\right)$, where $\mathrm{a}$ and $\mathrm{b}$ are integers with $\operatorname{gcd}(\mathrm{a}, \mathrm{b})=1$, then $11 \mathrm{a}+18 \mathrm{~b}$ is equal to ______

Let $f(x)=\left\{\begin{array}{ccc}-\mathrm{a} & \text { if } & -\mathrm{a} \leq x \leq 0 \\ x+\mathrm{a} & \text { if } & 0 < x \leq \mathrm{a}\end{array}\right.$ where $\mathrm{a}>0$ and $\mathrm{g}(x)=(f(x \mid)-|f(x)|) / 2$. Then the function $g:[-a, a] \rightarrow[-a, a]$ is

The area (in sq. units) of the region $S=\{z \in \mathbb{C}:|z-1| \leq 2 ;(z+\bar{z})+i(z-\bar{z}) \leq 2, \operatorname{Im}(z) \geq 0\}$ is

Let $A=[\begin{matrix}2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2\end{matrix}]$ and $P=[\begin{matrix}1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 & 1 & 5\end{matrix}]$. The sum of the prime factors of $|{P}^{-1}\mathrm{AP}-2I|$ is equal to

Let $A=\left[\begin{array}{lll}2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b\end{array}\right]$. If $A^3=4 A^2-A-21 I$, where $I$ is the identity matrix of order $3 \times 3$, then $2 a+3 b$ is equal to

Let $a,b,c$ be the length of three sides of a triangle satisfying the condition $({a}^{2}+{b}^{2}){x}^{2}-2b(a+c)$ $x+({b}^{2}+{c}^{2})=0$. If the set of all possible values of $x$ is in the interval $(\alpha ,\beta ),$ then $12({\alpha }^{2}+{\beta }^{2})$ is equal to _______.

Let $3,7,11,15,..,403$ and $2,5,8,11,...,404$ be two arithmetic progressions. Then the sum, of the common terms in them, is equal to_________

If the system of equations $2x+3y-z=5$ $x+\alpha y+3z=-4$ $3x-y+\beta z=7$ has infinitely many solutions, then $13\alpha \beta$ is equal to

Let $f, g: \mathbf{R} \rightarrow \mathbf{R}$ be defined as : $f(x)=|x-1| \text { and } g(x)= \begin{cases}\mathrm{e}^x, \text { MARA } & x \geq 0 \\ x+1, & x \leq 0\end{cases}$ Then the function $f(g(x))$ is

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 $[t]$ be the greatest integer less than or equal to $t$. Let $A$ be the set of all prime factors of 2310 and $f: A \rightarrow \mathbb{Z}$ be the function $f(x)=\left[\log _2\left(x^2+\left[\frac{x^3}{5}\right]\right)\right]$. The number of one-to-one functions from $A$ to the range of $f$ is

Let $f:R\rightarrow R$ and $g:R\rightarrow R$ be defined as $f(x)={\begin{matrix}{\mathrm{log}}_{e}x, & x>0 \\ {e}^{-x}, & x\leq 0\end{matrix}$ and $g(x)={\begin{matrix}x, & x\geq 0 \\ {e}^{x}, & x<0\end{matrix}$. Then, $gof:R\rightarrow R$ is:

Let $A B C$ be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle $A B C$ and the same process is repeated infinitely many times. If $\mathrm{P}$ is the sum of perimeters and $Q$ is be the sum of areas of all the triangles formed in this process, then :

If $\alpha$ satisfies the equation ${x}^{2}+x+1=0$ and $(1+\alpha {)}^{7}=A+B\alpha +C{\alpha }^{2},A,B,C\geq 0$, then $5(3A-2B-C)$ is equal to

If $\alpha \neq \mathrm{a}, \beta \neq \mathrm{b}, \gamma \neq \mathrm{c}$ and $\left|\begin{array}{lll}\alpha & \mathrm{b} & \mathrm{c} \\ \mathrm{a} & \beta & \mathrm{c} \\ \mathrm{a} & \mathrm{b} & \gamma\end{array}\right|=0$, then $\frac{\mathrm{a}}{\alpha-\mathrm{a}}+\frac{\mathrm{b}}{\beta-\mathrm{b}}+\frac{\gamma}{\gamma-\mathrm{c}}$ is equal to:

Let the positive integers be written in the form :  If the $k^{\text {th }}$ row contains exactly $k$ numbers for every natural number $k$, then the row in which the number 5310 will be, is _______

Let $A$ be a non-singular matrix of order 3 . If $\operatorname{det}(3 \operatorname{adj}(2 \operatorname{adj}((\operatorname{det} A) A)))=3^{-13} \cdot 2^{-10}$ and $\operatorname{det}(3 \operatorname{adj}(2 \mathrm{~A}))=2^{\mathrm{m}} \cdot 3^{\mathrm{n}}$, then $|3 \mathrm{~m}+2 \mathrm{n}|$ is equal to $\qquad$

The number of elements in the set $S={(x,y,z):x,y,z\in Z,x+2y+3z=42,x,y,z\geq 0}$ equals ________