NIMCET 2023 — Mathematics PYQ

Sol.
(a)
$\vec{a}=\hat{\mathbf{i}}-2x\hat{\mathbf{j}}+2y\hat{\mathbf{k}}$ & $\vec{b}=x\hat{\mathbf{i}}+\hat{\mathbf{j}}+y\hat{\mathbf{k}}$

$\cos \theta =\frac{\vec{a}\cdot \vec{b}}{|\vec{a}||\vec{b}|}=\cos {\cos }^{-1}\left(\frac{3}{\sqrt{20}}\right)=\frac{3}{\sqrt{20}}$

$\frac{3}{\sqrt{20}}=\frac{x-2x+2y^2}{\sqrt{1+4x^2+4y^2}\sqrt{x^2+1+y^2}}$

$\Rightarrow 3\sqrt{1+4x^2+4y^2}\sqrt{x^2+1+y^2}=\sqrt{20}\left[2y^2-x\right]$

From Given options (0, 1) satisfy given equations.