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Question

# Let a,b,c,d be distinct real numbers and a and b are the roots of quadratic equation x2−2cx−5d=0. If c and d are the roots of the quadratic equation x2−2ax−5b=0 .then find the numerical values of a+b+c+d.

A
30
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B
15
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C
10
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D
60
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Solution

## The correct option is A 30a,b are roots of x2−2cx−5d=0⇒a+b=2cand c,d are roots of x2−2ax−5b=0 ⇒c+d=2a∴(a−c)+(b−d)=2(c−a)⇒(b−d)=3(c−a)…(1) Since c is the root of x2−2ax−5b=0⇒c2−2ac−5b=0…(2)Similarly a is root of x2−2cx−5d=0⇒a2−2ac−5d=0…(3)On subtracting (3) from (2)c2−a2=5(b−d)⇒(c−a)(c+a)=5×3(c−a)……(from(1))⇒c+a=15∴a+b+c+d=2c+2a=2(c+a)=2×15=30

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