Given that 𝑟 < 0 and 0 <𝑝𝑞/𝑟< 1

Considering 0 <𝑝𝑞/𝑟, since 𝑟 < 0 𝑖𝑠 𝑔𝑖𝑣𝑒𝑛, thus 0 <𝑝𝑞/𝑟 will be true only when 𝑝𝑞 < 0

Therefore, 𝑝𝑞 is negative. → 𝑝 𝑎𝑛𝑑 𝑞 will have opposite signs. Which implies:

i) If 𝑝 > 0 then 𝑞 < 0

OR

ii) If 𝑝 < 0 then 𝑞 > 0

Now, considering 𝑝𝑞/𝑟 < 1, since 𝑟 < 0 and when we multiply by negative value on both the sides of the inequality, the inequality signs reverses due to the reversal of the magnitude.

Hence, in 𝑝𝑞/𝑟< 1 multiplying with 𝑟 on both the sides we get, 𝑝𝑞 > r

Hence, we can definitely say that,
𝒑𝒒 < 𝟎 and 𝒑𝒒 > r

(A)𝑝 < 0
This is not always true.
We know, 𝑝𝑞 < 0, which implies that 𝑝 will only be negative if 𝑞 is positive and
nothing is given about the sign of 𝑞, so 𝑝 < 0, is not always true.
(B)𝑞 < 0
This is not always true.
We know, 𝑝𝑞 < 0, which implies that 𝑞 will only be negative if 𝑝 will be positive
and nothing is given about the sign of 𝑝, so 𝑞 < 0, is not always true.
(C) 𝑝 > 𝑞
This is not always true.
We know, 𝑝𝑞 < 0, which implies both 𝑝 and 𝑞 are of opposite signs. So, 𝑝 > 𝑞 will
only be true, when 𝑝 is positive, and 𝑞 is negative which cannot be definitely
concluded.
(D)𝑝 < 𝑞
This is not always true.
We know, 𝑝𝑞 < 0, which implies both 𝑝 and 𝑞 are of opposite signs. So, 𝑝 < 𝑞 will
only be true, when 𝑝 is negative and 𝑞 is positive which cannot be definitely
concluded.
(E) 𝑝𝑞 > 0
As we know, 𝑝𝑞 < 0,
So, 𝑝𝑞 > 0 can never be true.
(F) 𝑝𝑞 < 0
We have already concluded that 𝑝𝑞 < 0.
So, this option will always be true.

(G) 𝑝𝑞 < 𝑟
As 𝑝𝑞 < 0 and r < 0
𝑝𝑞/𝑟 < 1
→ 𝑝𝑞 > 𝑟
So, we can say that 𝑝𝑞 < 𝑟 cannot be true.
(H) 𝑝𝑞 > 𝑟
As 𝑝𝑞 < 0 and r < 0 ,
𝑝𝑞/𝑟< 1
→ 𝑝𝑞 > 𝑟
So, we can say that 𝑝𝑞 > 𝑟 is always true.