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 1Option 2: 1 and 0Option 3: 2 and 1Option 4: 1 and 2 (2024)

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Team Careers360 17th Jan, 2024

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 and3 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 theseequations, we get
⇒ p = 1 andq = 0
Hence, the correct answer is 1 and0.

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