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1. If \( x-a \) is a factor of \( x^{3}-a^{2} x+x+2 \), then find the value of \( a \) 2. For any real numbers \( a, b, c \) find the smallest value of the expression \( 3 a^{2}+27 b^{2}+5 c^{2}-18 a b-30 c+237 \) : 3. When a polynomial \( P(x) \) is divided by \( (x-2) \) and \( (x-3) \), remainders are \( 3 \& 2 \) respectively. What is the |
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Answer» 1. Let f(x) = x3 - a2x + x + 2 ∵ x - a is a factor ∴ f(a) = 0 ⇒ a3 - a2 + a + 2 = 0 ⇒ a = -2 2. 3a2 + 27b2 + 5c2 - 18ab - 30c + 237 = 3(a2 + 9b2 - 6ab) + 5(c2 - 6c + 9) + 237 - 45 = 3(a - 3b)2 + 5(c - 3)2 + 192 ∴ Smallest value of expression is 192 which occurs when a = 3b & c = 3. 3. P(2) = 3 & P(3) = 2 Let P(x) = x2 + bx + c ∴ P(2) = 4 + 2b + c P(3) = 9 + 3b + c P(3) - P(2) = b + 5 ⇒ b + 5 = 2 - 3 = -1 ∴ b = -6 Also 4 - 12 + c = P(2) c = 8 + P(2) = 8 + 3 = 11 ∴ P(x) = x2 - 6x + 11 |
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