Question : What is the value of $(a^2+b^2+4c^2-ab-2bc-2ca)(a+b+2c)$?
Option 1: $a^3 + b^3 + 8c^3 – 8abc$
Option 2: $a^3 + b^3 + 6c^3 – 4abc$
Option 3: $4a^3 + b^3 + 8c^3 – 9abc$
Option 4: $a^3 + b^3 + 8c^3 – 6abc$
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Correct Answer: $a^3 + b^3 + 8c^3 – 6abc$
Solution : $(a^2+b^2+4c^2-ab-2bc-2ca)(a+b+2c)$ $= a^3+b^2a+4c^2a-a^2b-2abc-2ca^2 + a^2b+b^3+4c^2b-ab^2-2b^2c-2abc + 2a^2c+2b^2c+8c^3-2abc-4bc^2-4c^2a$ $= a^3+b^3+8c^3-2abc-2abc-2abc+b^2a-ab^2+4c^2a-4c^2a-a^2b+ a^2b-2ca^2 + 2a^2c-2b^2c+2b^2c-4bc^2+4c^2b$ $= a^3+b^3+8c^3-6abc$ Hence, the correct answer is $ a^3+b^3+8c^3-6abc$.
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Question : If $\frac{(a+b+c)}{2}=16$ and $2ab+2 bc+2ca=120$ ,then what is the value of $4a^2+4b^2+4c^2$?
Option 1: 2828
Option 2: 3368
Option 3: 3616
Option 4: 3056
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 : If $a =\cot A+\cos A$ and $b =\cot A-\cos A$, then find the value of $a^2-b^2-4 \sqrt{ab}$.
Option 1: 0
Option 2: –1
Option 3: 1
Option 4: –4
Question : A slope of line AB is $-\frac{2}{3}$. Co-ordinates of points A and B are $(x,–3)$ and $(5, 2)$, respectively. What is the value of $x$?
Option 1: 4
Option 2: –14
Option 3: 12.5
Question : If a + b + c = 0, then the value of (a + b – c)2 + ( b + c – a)2 + ( c + a – b)2 is:
Option 2: 8abc
Option 3: 4(a2 + b2 + c2)
Option 4: 4(ab + bc + ca)
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