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
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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