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Mathematics

Assertion (A) : The value of 'a' if the division of ax3 + 9x2 + 4x - 10 by x + 3 leaves remainder 5 is -2.

Reason (R) : If f(x), a polynomial in x, is divided by x - a. Then f(x) = (x - a) Quotient + Remainder. Putting x = a on both sides we obtained, Remainder = f(a).

  1. A is true, R is false.

  2. A is false, R is true.

  3. Both A and R are true.

  4. Both A and R are false.

Factorisation

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Answer

By remainder theorem,

When a polynomial p(x) is divided by a linear polynomial (x - a), then the remainder is equal to p(a).

Given,

Division of ax3 + 9x2 + 4x - 10 by x + 3 leaves remainder 5.

⇒ x + 3 = 0

⇒ x = -3.

Substituting value of x = -3 in ax3 + 9x2 + 4x - 10, remainder is 5.

⇒ a(-3)3 + 9(-3)2 + 4(-3) - 10 = 5

⇒ -27a + 9 × 9 - 12 - 10 = 5

⇒ -27a + 81 - 22 = 5

⇒ -27a + 59 = 5

⇒ 27a = 59 - 5

⇒ 27a = 54

⇒ a = 5427\dfrac{54}{27} = 2.

∴ Assertion (A) is false.

By remainder theorem,

On dividing f(x) by x - a, we get remainder = f(a).

∴ f(x) = (x - a) Quotient + Remainder.

∴ Reason (R) is true.

Hence, Option 2 is the correct option.

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