Maths

If the system of linear equations
x + y + 3z = 0
x + 3y + k²z = 0
3x + y + 3z = 0
has a non-zero solution (x, y, z) for some k ∈ R, then x + (k/y)z is equal to:

The value of ((-1+i√3)/(1-i))³⁰ is:

The shortest distance between the lines $\frac{x-3}{3}=\frac{y-8}{-1}=\frac{z-3}{1}$ $\frac{}{}\frac{}{}\frac{}{}$ and $\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-6}{4}$ $\frac{}{}\frac{}{}\frac{}{}$ is:

For $a>0$ $$ , let the curves $C_1:y^2=ax$ ${}_{}{}^{}$ and $C_2:x^2=$ ${}_{}{}^{}$ ay intersect at origin $O$ $$ and a point $$ $P$ . Let the line $x=b(0<b<c)$ $$ intersect the chord $OP$ $$ and the $x$ $$ - axis at points $$ $Q$ and $R$ $$ , respectively. If the line $x=b$ $$ bisects the area bounded by the curves, $C_1$ ${}_{}$ and ${}_{}$ , and the area of $\frac{}{}$ $?\mathrm{O}\mathrm{Q}\mathrm{R}=\frac{1}{2}$ , then 'a' satisfies the equation:

If the equation, $x^2+bx+45=0(b\in R)$ ${}^{}$ has conjugate complex roots and they satisfy $\text{exception 25:}$ $|z+1|=2\sqrt[]{10}$ , then:

Let two points be $A(1,-1)$ $$ and $$ $B\left(\mathrm{0,2}\right)$ . If a point $P\left(x^{\mathrm{\text{'}}},y^{\mathrm{\text{'}}}\right)$ $\left({}^{}{}^{}\right)$ be such that the area of $?PAB=5$ $$ sq. units and it lies on the line, $3x+y-4\lambda =0$ $$ , then a value of $\lambda$ $$ is:

${\mathrm{l}\mathrm{i}\mathrm{m}}_{x\to 0}?{\left(\frac{3x^2+2}{7x^2+2}\right)}^{1/x^2}$  is equal to:

${\mathrm{l}\mathrm{i}\mathrm{m}}_{x\to 0}?{\left(\frac{3x^2+2}{7x^2+2}\right)}^{1/x^2}$ is equal to

Let $y=y\left(x\right)$ be a solution of the differential equation, $\sqrt[]{1-x^2}\frac{dy}{dx}+\sqrt[]{1-y^2}=0,\left|x\right|1$ . If $y\left(\frac{1}{2}\right)=\frac{\sqrt[]{3}}{2}$ , then $y\left(\frac{1}{\sqrt[]{2}}\right)$  is equal to:

Let the volume of a parallelepiped whose coterminous edges are given by $\stackrel{}{}\stackrel{}{}\stackrel{}{}\stackrel{}{}\stackrel{}{}\stackrel{}{}\stackrel{}{}\stackrel{}{}$ $\stackrel{?}{u}=\hat{i}+\hat{j}+\lambda \hat{k},\stackrel{?}{v}=\hat{i}+\hat{j}+3\hat{k}$ and $\stackrel{?}{w}=2\hat{i}+\hat{j}+\hat{k}$ $\stackrel{}{}\stackrel{}{}\stackrel{}{}\stackrel{}{}$ be 1 cu. Unit. If $\theta$ $$ be the angle between the edges $\stackrel{?}{\mathrm{u}}$ $\stackrel{}{}$ and $\stackrel{}{}$ $\stackrel{?}{\mathrm{w}}$ , then $\mathrm{c}\mathrm{o}\mathrm{s}?\theta$ $$ can be: