# CAT 2018 Quant Questions

Question:
If u2 + (u−2v−1)2 = −4v(u + v), then what is the value of u + 3v?

 1/4 0 1/2 -1/4
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Given that $U^{2}+(U-2V-1)^{2}$= −$4V(U+V)$

$\Rightarrow$ $U^{2}+(U^2-2UV-U-2UV+4V^2+2V-U+2V+1)$ = −$4V(U+V)$

$\Rightarrow$ $U^{2}+(U^2-4UV-2U+4V^2+4V+1)$ = −$4V(U+V)$

$\Rightarrow$ $2U^2-4UV-2U+4V^2+4V+1=−4UV-4V^2$

$\Rightarrow$ $2U^2-2U+8V^2+4V+1=0$

$\Rightarrow$ $2[U^2-U+\frac{1}{4}]+8[V^2+\frac{V}{2}+\frac{1}{16}]=0$

$\Rightarrow$ $2(U-\frac{1}{2})^2+8(V+\frac{1}{4})^2=0$

Sum of two square terms is zero i.e. individual square term is equal to zero.

$U-\frac{1}{2}$ = 0 and $V+\frac{1}{4}$ = 0

U = $\frac{1}{2}$ and V = $-\frac{1}{4}$

Therefore, $U+3V$ = $\frac{1}{2}$+$\frac{-1*3}{4}$ = $\frac{-1}{4}$.

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