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If $a, b \in [0, 2 π]$ and the equation $x^{2} + 4 + 3 \sin (ax + b) - 2 x = 0$ has atleast one solution, then the value of $a + b$ is

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$9 π / 2$

$7 π / 2$

$3 π / 2$

$5 π / 2$

answer is A.

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Detailed Solution

$x^{2} - 2 x + 4 + 3 \sin (ax + b) = 0 {(x - 1)}^{2} + 3 + 3 \sin (ax + b) = 0 x - 1 = 0, \sin (ax + b) = - 1 x = 1 \Rightarrow \sin (a + b) = - 1 a + b = \frac{3 π}{2}, \frac{7 π}{2}, \frac{11 π}{2}, \dots a + b = \frac{7 π}{2}$

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