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Let a1,a2,....an be n nonzero real numbers, of which p are positive and remaining are negative. The number of ordered pairs (j,k), j<k, for which ajak is positive, is 55. Similarly, the number of ordered pairs (j,k), j<k, for which ajak is negative is 50. Then the value of p2+(np)2 is


Your Answer
A
629
Your Answer
B
325
Correct Answer
C
125
Your Answer
D
221

Solution

The correct option is D 125
There are p positive numbers and (np) negative numbers.
The number of pairs for ajak to be negative,
pC1×(np)C1=50
p(np)=50..........(i)
The number of pairs for ajak to be positive,
pC2+(np)C2=55
p(p1)+(np)(np1)=110
using equation (i),
p(p1)+50p(50p1)=110p2p+(50p)2(50p)=110(p+50p)2(p+50p)100=110
Assuming the (p+50p)=y
y2y210=0(y15)(y+14)=0
y=15(p+50p)=15
[y cannot be negative]

p2+(np)2=p2+(50p)2=(p+50p)2100=125

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