Manipal Entrance Test 2020 Math Paper

Geometric mean of a and b=√ab
⇒‌‌√ab=16‌‌ (given)
⇒‌‌ab=256......(i)
And harmonic mean of a and b=‌

2ab

a+b

∴‌‌‌

2ab

a+b

=‌

64

5

‌‌ (given)
⇒‌‌‌

2×256

a+b

=‌

64

5

[from Eq. (i)]
⇒‌‌a+b=40.....(ii)
Now, ‌‌(a−b)=√(a+b)2−4ab
=√(40)2−4×256
=√1600−1024
=√576
⇒‌‌a−b=24.......(jii)
On solving Eqs. (ii) and (iii), we get
a=32 and b=8
a:b=32:8
=4:1