is p^3 + q^5 > r^3?

Asked: Is \(P^3\) + \(q^5\)>\(r^4\)?

1) \(p <q < r\)
p, q, & r can take multiple values giving different results.
Case 1: p=1;q=2;r=3 => 1+32 =33 < 81
Case 2: p=98; q=99;r=100 => 98^3 + 99^5 > 100^4
NOT SUFFICIENT

2) \(q < 0\)
No relation with p & r is provided.
NOT SUFFICIENT

Combining (1) & (2)
1) \(p <q < r\)
2) \(q < 0\)
If q<0 => p<0 since p<q
\(P^3\) + \(q^5\)>\(r^4\)
\(p^3<0 \ &\ q^5<0\)
\(p^3+q^5<0 \ but\ r^4>0\)
\(p^3+q^5<r^4 \ and\ NOT\ >r^4\)
SUFFICIENT

IMO C

As question does not specify anything about P q & r whether integer or simply a number we can test any number.

Statement 1) \(p <q < r\)
We don't know anything about whether all of them(P, q a& r) are negative or positive. So various scenarios may follow:

A: \(0 < P < q < r\) i.e. \(1^3 + 2^5 < 3^4\) (Y)
B: \(P < 0 < q < r\) i.e. \((-1)^3 + 2^5 < 3^4\) (Y) OR \((-1)^3 + (3)^5 < 4^4\) (N)
C: \(P < q < 0 < r\)
D: \(P < q < r < 0\)

From A and B we have two cases - a 'yes' and a 'no' which prove statement 1) is INSUFFICIENT.

Statement 2) \(q < 0\)
As nothing is given as far as q's relation with P and r is concerned, various scenarios can follow i.e.
A: \(P < r < q < 0\)
B: \(q < P < r < 0\)
C: \(q < r < P < 0\)
D: \(r < P < q < 0\)

And others when EITHER of P and r is positive OR both of P and r are positive.

Hence Statement 2) is INSUFFICIENT.

Together 1) and 2) we have following scenarios:
A: \(P < q < 0 < r\) i.e. \((-3)^3 + (-2)^5 < 3^4\) (Y)
B: \(P < q < r < 0\) i.e. \((-2)^3 + (-1)^5 < (-1)^4\)

i.e. in any case LHS will be negative because of odd powers and RHS will be positive because of even power.

SUFFICIENT.

Answer (C)