Question

If a,b,c,d are in H.P., then [RPET 1991]

• A. a + d > b + c
• B. ad > bc
• C. Both (a) and (b)
• D. None of these

Answer 1

The correct option is C

Both (a) and (b)

As a,b,c,d are in H.P. So b is H.M. between a and c. Also G.M. between a and c = $\sqrt{ac}.$
Now, G.M. > H.M. so that $\sqrt{ac}>b\Rightarrow ac>b^2$ …..(i)
Again a,b,c,d are in H.P. So c is H.M. between b and d.
Therefore bd > $c^2$ …..(ii)
Multiplying (i) and (ii), we get

$abcd>b^2{c}^2$ or ad > bc. Hence answer (b) is true.

Now A.M. between a and c = $\frac{1}{2}(a+c)$

Now as A.M. > H.M. so here $\Rightarrow$ a + c > 2b ....(iii)

And c is H.M. between b and d $\Rightarrow$ b + d > 2c .....(iv)

Adding (iii) and (iv), we get
(a+c)+(b+d) > 2(b+c) $\Rightarrow$ a + d > b + c
Hence answer (a) is true. So both (a) and (b) are correct.