Question

Given below are Matching Type Questions, with two columns(each having some items) each.Each item of column $I$ has to be matched with the items of column $II$, by encircling the correct matches:
Note:An item of Column $I$ can be matched with more than one items in Column $II$.All the items of Column $II$ have to be matched.

Answer 1

$\left(A\right)$Let slopes be $m,m^2$ of the equation $a{x}^2-6xy+y^2=0$
Given:Slope of one line is square of slope of the other line.
Then $m+m^2=6$ and $m{m}^2=a$
$\Rightarrow m^2+m-6=0$
$\Rightarrow (m+3)(m-2)=0$
$\therefore m=-3,2$
$\Rightarrow a=m.m^2=-27,8$ when $m=(-3,2)$ respectively.
$\left(B\right)a{x}^3-9x^{2}y-x{y}^2+4y^3=(x^2+pxy-y^2)(ax-4y)$
On comparing , we get
$-9=-4+ap$
$\Rightarrow p=-\frac{5}{a}$
and $-1=-4p-a$

$\Rightarrow p=\frac{1-a}{4}$

$\Rightarrow -\frac{5}{a}=\frac{1-a}{4}$

$\Rightarrow -20=a-a^2$

$\Rightarrow a^2-a-20=0$

$\Rightarrow (a-5)(a+4)=0$
$\therefore a=5,-4$
$\left(C\right)$ By hypothesis $x^4+x^{3}y+c{x}^2{y}^2-x{y}^3+y^4=(a{x}^2+2hxy+b{y}^2)(h{x}^2-2axy-h{y}^2)$
$=ah(x^4+y^4)+2(h^2-a^2)(x^{3}y-x{y}^3)-6ah{x}^2{y}^2$
Comparing with the coefficients, then we get
$ah=1,2(h^2-a^2)=1,c=-6ah$
$\therefore c=-6\times 1=-6$