Algebra PYQ — Page 20

Let $\lambda, \mu \in \mathbf{R}$. If the system of equations $\begin{aligned} & 3 x+5 y+\lambda z=3 \\ & 7 x+11 y-9 z=2 \\ & 97 x+155 y-189 z=\mu \end{aligned}$ has infinitely many solutions, then $\mu+2 \lambda$ is equal to :

Suppose $28-p,p,70-\alpha ,\alpha$ are the coefficient of four consecutive terms in the expansion of $(1+x{)}^{n}$. Then the value of $2\alpha -3p$ equals

Let $\alpha$ and $\beta$ be the roots of the equation $p{x}^{2}+qx-r=0$, where $p\neq 0$. If $p,q$ and $r$ be the consecutive terms of a non-constant G.P and $\frac{1}{\alpha }+\frac{1}{\beta }=\frac{3}{4}$, then the value of ${(\alpha -\beta )}^{2}$ is:

If $z=\frac{1}{2}-2i$, is such that $|z+1|=\alpha z+\beta (1+i),i=\sqrt{-1}$ and $\alpha ,\beta \in R,$ then $\alpha +\beta$ is equal to

Let $A=\{1,2,3,4,5\}$. Let $\mathrm{R}$ be a relation on $\mathrm{A}$ defined by $x \mathrm{R} y$ if and only if $4 x \leq 5 \mathrm{y}$. Let $\mathrm{m}$ be the number of elements in $\mathrm{R}$ and $\mathrm{n}$ be the minimum number of elements from $\mathrm{A} \times \mathrm{A}$ that are required to be added to $\mathrm{R}$ to make it a symmetric relation. Then $\mathrm{m}+\mathrm{n}$ is equal to :

If the second, third and fourth terms in the expansion of $(x+y)^n$ are 135,30 and $\frac{10}{3}$, respectively, then $6\left(n^3+x^2+y\right)$ is equal to _______

If all the words with or without meaning made using all the letters of the word "NAGPUR" are arranged as in a dictionary, then the word at $315^{\text {th }}$ position in this arrangement is :

If the constant term in the expansion of $\left(\frac{\sqrt[5]{3}}{x}+\frac{2 x}{\sqrt[3]{5}}\right)^{12}, x \neq 0$, is $\alpha \times 2^8 \times \sqrt[5]{3}$, then $25 \alpha$ is equal to :

Let $A=\{(x, y): 2 x+3 y=23, x, y \in \mathbb{N}\}$ and $B=\{x:(x, y) \in A\}$. Then the number of one-one functions from $A$ to $B$ is equal to _______

Let $A=\{2,3,6,7\}$ and $B=\{4,5,6,8\}$. Let $R$ be a relation defined on $A \times B$ by $\left(a_1, b_1\right) R\left(a_2, b_2\right)$ if and only if $a_1+a_2=b_1+b_2$. Then the number of elements in $R$ is _________

The function $\text { f: R->R, }$ $f(x)=\frac{x^2+2 x-15}{x^2-4 x+9}, x \in \mathbb{R}$ is

Let $z$ be a complex number such that the real part of $\frac{z-2 i}{z+2 i}$ is zero. Then, the maximum value of $|z-(6+8 i)|$ is equal to

The number of distinct real roots of the equation $|x+1||x+3|-4|x+2|+5=0$, is

Let $R=(\begin{matrix}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{matrix})$ be a non-zero $3\times 3$ matrix, where $x\mathrm{sin}\theta =y\mathrm{sin}(\theta +\frac{2\pi }{3})=z\mathrm{sin}(\theta +\frac{4\pi }{3})$ $\neq 0,\theta \in (0,2\pi )$. For a square matrix $M$, let Trace$(M)$ denote the sum of all the diagonal entries of $M.$ Then, among the statements: $(I)$ Trace$(R)=0$ $(\mathrm{II})$ If Trace$(adj(adj(R))=0$, then $R$ has exactly one non-zero entry.

Let $A$ be a square matrix such that ${\mathrm{AA}}^{T}=I$. Then $\frac{1}{2}A[{(A+{A}^{T})}^{2}+{(A-{A}^{T})}^{2}]$ is equal to

If $n$ is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then $n$ is equal to:

If three successive terms of a G.P. with common ratio $r(r>1)$ are the length of the sides of a triangle and $[r]$ denotes the greatest integer less than or equal to r, then $3[r]+[-r]$ is equal to:

The number of common terms in the progressions $4,9,14,19,\ldots \ldots$, up to ${25}^{\text{th }}$ term and $3,6,9,12$,.... up to ${37}^{\text{th }}$ term is :

Let $S$ be the set of positive integral values of $a$ for which $\frac{a{x}^{2}+2(a+1)x+9a+4}{{x}^{2}-8x+32}<0,\forall x\in \mathbb{R}$. Then, the number of elements in $S$ is:

Let $\alpha, \beta$ be roots of $x^2+\sqrt{2} x-8=0$. If $\mathrm{U}_{\mathrm{n}}=\alpha^{\mathrm{n}}+\beta^n$, then $\frac{\mathrm{U}_{10}+\sqrt{2} \mathrm{U}_9}{2 \mathrm{U}_8}$ is equal to______

Let $x_1, x_2, x_3, x_4$ be the solution of the equation $4 x^4+8 x^3-17 x^2-12 x+9=0$ and $\left(4+x_1^2\right)\left(4+x_2^2\right)\left(4+x_3^2\right)\left(4+x_4^2\right)=\frac{125}{16} m$. Then the value of $m$ is

For $0<c<b<a$, let $(a+b–2c){x}^{2}+(b+c–2a)x+(c+a–2b)=0$ and $\alpha \neq 1$ be one of its root. Then, among the two statements (I) If $\alpha \in (-1,0)$, then $b$ cannot be the geometric mean of $a$ and $c$. (II) If $\alpha \in (0,1)$, then $b$ may be the geometric mean of $a$ and $c$.

If $z_1, z_2$ are two distinct complex number such that $\left|\frac{z_1-2 z_2}{\frac{1}{2}-z_1 \bar{z}_2}\right|=2$, then

If $z$ is a complex number such that $|z|\leq 1$, then the minimum value of $|z+\frac{1}{2}(3+4i)|$ is: