The number of values of the pair $(a, b)$ for which $a (x + 1)^{2} + b (x^{2} - 3 x - 2) + x + 1 = 0$

0

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infinite

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Solution

The correct option is C $0$
$a (x^{2} + 2 x + 1) + b (x^{2} - 3 x - 2) + x + 1 = 0$
$(a + b) x^{2} + x (2 a - 3 b + 1) - 2 b = 0$
$- 2 b = 0, b = 0$ ....(1)
$a + b = 0, a = - b$ .....(2)
But $2 a - 3 b + 1$ can not be zero
$1 \neq 0$
Therefore, no solution exits.
Hence, option 'A' is correct.

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The number of values of the pair (a,b) for which a(x+1)2+b(x2−3x−2)+x+1=0

The correct option is C 0a(x2+2x+1)+b(x2−3x−2)+x+1=0(a+b)x2+x(2a−3b+1)−2b=0−2b=0,b=0....(1)a+b=0,a=−b.....(2)But 2a−3b+1 can not be zero1≠0Therefore, no solution exits.Hence, option 'A' is correct.

The number of values of the pair $(a, b)$ for which $a (x + 1)^{2} + b (x^{2} - 3 x - 2) + x + 1 = 0$

The correct option is C $0$
$a (x^{2} + 2 x + 1) + b (x^{2} - 3 x - 2) + x + 1 = 0$
$(a + b) x^{2} + x (2 a - 3 b + 1) - 2 b = 0$
$- 2 b = 0, b = 0$ ....(1)
$a + b = 0, a = - b$ .....(2)
But $2 a - 3 b + 1$ can not be zero
$1 \neq 0$
Therefore, no solution exits.
Hence, option 'A' is correct.