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Find unknown coefficients so that a 0/0 limit has a given value

This type of question, asked 4 times: May 2022 S1, 4 May 2023 S2, 17 Apr 2026 S1, 19 Apr 2026 S2. All 4 questions below, oldest first.

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  1. Question 1

    May 2022 S1

    134
    If lim⁡x→1x2−ax+bx−1=5\lim_{x \to 1} \frac{x^2 - ax + b}{x - 1} = 5, then (a+b)(a+b) is equal to

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  2. Question 2

    4 May 2023 S2

    132
    If lim⁡x→1x2−ax+bx−1=7\lim_{x\to1}\frac{x^2-ax+b}{x-1}=7, then a+ba+b is equal to

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  3. Question 3

    17 Apr 2026 S1

    60
    If lim⁡x→1sin⁡(3x2−4x+1)−x2+12x3−7x2+ax+b=−2\lim_{x \to 1}\dfrac{\sin(3x^2 - 4x + 1) - x^2 + 1}{2x^3 - 7x^2 + ax + b} = -2, then the quadratic equation having roots aa and bb is

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  4. Question 4

    19 Apr 2026 S2

    59
    If lim⁡x→1x3+ax2+bx+cx2−2x+1=2026\lim_{x \to 1}\dfrac{x^3 + ax^2 + bx + c}{x^2 - 2x + 1} = 2026 then the value of a−ca - c is...

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