Let $\lambda$ be a real number for which the system of linear equations $$ \begin{aligned} & x+y+z=6 \\ & 4 x+\lambda y-\lambda z=\lambda-2 \\ &...

If for some $\alpha \in \mathrm{R}$, the lines $L_1: \frac{x+1}{2}=\frac{y-2}{-1}=\frac{z-1}{1}$ and $\mathrm{L}_2: \frac{\mathrm{x}+2}{\alpha}=\frac{\mathrm{y}+1}{5-\alpha}=\frac{\mathrm{z}+1}{1}$ are coplanar, then the line $L_2$ passes through the point :

The sum of the real roots of the equation $...

Suppose the vectors $x_1, x_2$ and $x_3$ are the solutions of the system of linear equations, $A x=b$ when the vector $b$ on the right...

Let [ $\lambda$ ] be the greatest integer less than or equal to $\lambda$. The set of all values of $\lambda$ for which the system...

If the system of equations $$ \begin{aligned} & x-2 y+3 z=9 \\ & 2 x+y+z=b \end{aligned} $$ $x-7 y+a z=24$, has infinitely many solutions, then...

Let $d \in R$, and $$ A\left[\begin{array}{ccc} -2 & 4+d & (\sin \theta)-2 \\ 1 & (\sin \theta)+2 & d \\ 5 & (2 \sin...

The following system of linear equations
7x + 6y – 2z = 0
3x + 4y + 2z = 0
X – 2y – 6z...

The solution of the differential equation, $\frac{d y}{d x}=(x-y)^2$, when $y(1)=1$, is

If the system of equations
$$ \begin{aligned} & x+y+z=2 \\ & 2 x+4 y-z=6 \\ & 3 x+2 y+\lambda z=\mu \end{aligned} $$ has infinitely many...