Question : If a + b + c = 15 and a2 + b2 + c2 = 83 then the value of a3 + b3 + c3 – 3abc:
Option 1: 200
Option 2: 180
Option 3: 190
Option 4: 210
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Correct Answer: 180
Solution : Given: a2+b2+c2 = 83, a+b+c = 15 We know that, (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca) ⇒ (15)2 = 83 + 2(ab + bc + ca) ⇒ 225 – 83 = 2(ab + bc + ca) ⇒$\frac{142}{2}$ = ab + bc + ca $\therefore$ ab + bc + ca = 71 Now, a3 + b3 + c3–3abc = (a + b + c)(a2 + b2 + c2 – ab – bc – ca) Putting the values, we get: = (15)(83 – 71) = 15 × 12 = 180 Hence, the correct answer is 180.
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Question : If a + b + c = 0, then the value of (a + b – c)2 + ( b + c – a)2 + ( c + a – b)2 is:
Option 1: 0
Option 2: 8abc
Option 3: 4(a2 + b2 + c2)
Option 4: 4(ab + bc + ca)
Question : If a + b + c = 1, ab + bc + ca = –1, and abc = –1, then what is the value of a3 + b3 + c3?
Option 1: 1
Option 2: 5
Option 3: 3
Option 4: 2
Question : What is the value of (a + b + c) {(a - b)2 + (b - c)2 + (c - a)2}?
Option 1: $2 a^3+2 b^3+2 c^3$
Option 2: $2a^3+2b^3+2c^3-6abc$
Option 3: $3abc$
Option 4: $6abc$
Question : What is the value of (a + b)3 – a3 – b3?
Option 1: – 3ab(a – b)
Option 2: 3ab(a + b)
Option 3: – 3ab(a + b)
Option 4: 3ab(a – b)
Question : If A + B = – 5 and AB = 6, then find the value of A3 + B3.
Option 1: 35
Option 2: 45
Option 3: –35
Option 4: 215
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