Question : If $a + b + c = 5, a^{3} + b^{3} + c^{3} = 85$ , and $a b c = 25$ , then find the value of $a^{2} + b^{2} + c^{2} - a b - b c - c a$ .
Option 1: 2
Option 2: 4
Option 3: 6
Option 4: 8

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Question : If a : b = 4 : 5 and b : c = 3 : 10, then a : c is:

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Answer 1

Correct Answer: 2

Solution : $a + b + c = 5$
Squaring both sides,
⇒ $(a + b + c)^{2} = 25$
⇒ $a^{2} + b^{2} + c^{2} + 2 (a b + b c + c a) = 25$
⇒ $a^{2} + b^{2} + c^{2} - a b - b c - c a + 3 (a b + b c + c a) = 25$ ........(1)
Now we have to find the value of $(a b + b c + c a)$
$a + b + c = 5$
Cubing both sides,
⇒ $a^{3} + b^{3} + c^{3} + 3 [(a + b + c) (a b + a c + b c) - a b c] = 125$
⇒ $85 + 3 [5 (a b + b c + c a) - 25] = 125$
⇒ $15 (a b + b c + c a) - 75 = 40$
⇒ $3 (a b + b c + c a) = \frac{115}{5}$
⇒ $3 (a b + b c + c a) = 23$
Putting in (1),
⇒ $a^{2} + b^{2} + c^{2} - a b - b c - c a = 25 - 23$
⇒ $a^{2} + b^{2} + c^{2} - a b - b c - c a = 2$
Hence, the correct answer is 2.

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Question : If a+b+c=5,a3+b3+c3=85, and abc=25, then find the value of a2+b2+c2−ab−bc−ca.Option 1: 2Option 2: 4Option 3: 6Option 4: 8

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Question : If $a + b + c = 5, a^{3} + b^{3} + c^{3} = 85$ , and $a b c = 25$ , then find the value of $a^{2} + b^{2} + c^{2} - a b - b c - c a$ .
Option 1: 2
Option 2: 4
Option 3: 6
Option 4: 8