KVPY Sample Paper KVPY Stream-SX Model Paper-26

Let the harmonic mean of two positive real numbers a and b be 4. If q is positive real number such that a,5,q,b is an arithmetic progression, then the values of q is equal to

A) 3

B) 4

C) 5

D) 6

Correct Answer: B

Solution :

Given, harmonic mean of a and b is 4
\[\therefore \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,4=\frac{2ab}{a+b}\,\,\]	?. (i)
And a, 5, q, b are in AP
\[\therefore \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,q=10-a,q=\frac{b+5}{2}\]\[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,a=10-q,b=2q-5\]
On putting, the values of a and b in Eq. (i), we get
\[4=\frac{2\left( 10-q \right)\left( 2q-5 \right)}{10-q+2q-5}\]\[\Rightarrow \,\,2{{q}^{2}}-23q+60=0\]\[\Rightarrow \,\,2{{q}^{2}}-15q-8q+60=0\]\[\Rightarrow \,\,\left( 2q-15 \right)\left( q-4 \right)=0\]\[\Rightarrow \,\,q=4,\frac{15}{2}\]