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Remainder on division by a linear divisor (x - a) is f(a): compute it or find k

This type of question, asked 3 times: CDS 2 2017, CDS 1 2018, CDS 2 2018. All 5 questions below, oldest first.

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  1. Question 1

    CDS 2 2017

    16
    Let f(x)=a0xn+a1xn−1+a2xn−2+…+an−1x+anf(x) = a_0 x^n + a_1 x^{n-1} + a_2 x^{n-2} + \ldots + a_{n-1} x + a_n, where a0,a1,a2,…,ana_0, a_1, a_2, \ldots, a_n are real numbers. If f(x)f(x) is divided by (ax−b)(ax - b), then the remainder is

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  2. Question 2

    CDS 2 2017

    7
    If 5x3+5x2−6x+95x^3 + 5x^2 - 6x + 9 is divided by (x+3)(x + 3), then the remainder is

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  3. Question 3

    CDS 1 2018

    11
    If (x+3)(x + 3) is a factor of x3+3x2+4x+kx^3 + 3x^2 + 4x + k, then what is the value of kk?

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  4. Question 4

    CDS 1 2018

    16
    The remainder when 3x3+kx2+5x−63x^3 + kx^2 + 5x - 6 is divided by (x+1)(x + 1) is −7-7. What is the value of kk?

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  5. Question 5

    CDS 2 2018

    24
    The remainder when 3x3−2x2y−13xy2+10y33x^3 - 2x^2y - 13xy^2 + 10y^3 is divided by (x−2y)(x - 2y) is equal to

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