KVPY Sample Paper KVPY Stream-SX Model Paper-7

If 5, 5r, \[5{{r}^{2}}\]are the lengths of the sides of a triangle, be equal to:

A) \[\frac{3}{4}\]

B) \[\frac{5}{4}\]

C) \[\frac{7}{4}\]

D) \[\frac{3}{2}\]

Correct Answer: C

Solution :

5, 5r, \[5{{r}^{2}}\]sides of triangle,
\[5+5r>5{{r}^{2}}\]	? (1)
\[5+5{{r}^{2}}>5r\]	? (2)
From \[{{r}^{2}}-r-1<0\]
\[\left[ r-\left( \frac{1-\sqrt{5}}{2},\frac{1+\sqrt{5}}{2} \right) \right]\]	? (4)
From (2), \[{{r}^{2}}-r+1>0\] \[\Rightarrow \]\[r\in \,R\]	? (5)

From (3), \[{{r}^{2}}+r-1>0\]
So, \[\left( r+\frac{\sqrt{1+\sqrt{5}}}{2} \right)\left( r+\frac{1-\sqrt{5}}{2} \right)>0\]
\[r\in \left( -\infty ,\frac{1+\sqrt{5}}{2} \right)\cup \left( -\frac{1-\sqrt{5}}{2},\infty \right)\]	? (6)
From (4), (5), (6), \[r\in \left( \frac{-1+\sqrt{5}}{2},\frac{1+\sqrt{5}}{2} \right)\]
Now check options.