Question : If the expression px3 – qx2 – 7x – 6 is completely divisible by x2 – x – 6, then what are the values of p and q respectively?
Option 1: 0 and 1
Option 2: 1 and 0
Option 3: 2 and 1
Option 4: 1 and 2
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Correct Answer: 1 and 0
Solution : Given: The expression px 3 – qx 2 – 7x – 6 is completely divisible by x 2 – x – 6. So, the roots of x 2 – x – 6 = 0 will also be the roots of px 3 – qx 2 – 7x – 6 = 0 ⇒ x 2 – x – 6 = 0 ⇒ x 2 – 3x + 2x – 6 = 0 ⇒ x(x – 3) + 2(x – 3) = 0 ⇒ (x – 3)(x + 2) = 0 ⇒ x = 3 or x = –2 So, x = – 2 and 3 are the roots of px 3 – qx 2 – 7x – 6 = 0 Putting x = – 2 and x = 3 in the expression, we should get the value equal to 0 because we know that if $\alpha$ is the root of the equation $f(x)=0$, then $f(\alpha)=0$ ⇒ p( – 2) 3 – q( – 2) 2 – 7( – 2) – 6 = 0 and p(3) 3 – q(3) 2 – 7(3) – 6 = 0 ⇒ – 8p – 4q + 14 – 6 = 0 and 27p – 9q – 21 – 6 = 0 ⇒ – 8p – 4q = – 8 and 27p – 9q = 27 Solving these equations, we get ⇒ p = 1 and q = 0 Hence, the correct answer is 1 and 0.
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