Kuala Lumpur Trial 2025 Add Maths Paper 2, Question 2Quadratic Functions

Solved on video in English · Form 4, Quadratic Functions · 10:03 video

The question

(a)(i)

It is given that the quadratic function f(x) = −mx² − 4x − mn has two different real roots, where m and n are constants.

(a)(i) Find the range of n in terms of m.

(a)(ii)

(a)(ii) By using the method of completing the square, show that f(x) = −m(x + 2/m)² + (4 − nm²)/m.

(a)(iii)

(a)(iii) Given that the equation of the axis of symmetry of f(x) is −3 and the maximum value is 5, calculate the value of m and of n.

(b)

It is given that the quadratic function f(x) = x² + px + p + 3 has imaginary (non-real) roots. By using the number line method, find the range of values of p.

Kuala Lumpur Trial 2025 Add Maths Paper 2, Question 2: the question as drawn in Eduly's video10:03

Part (a)(i)

Step 1

For two different real roots the discriminant b² − 4ac > 0, with a = −m, b = −4, c = −mn.

Why this step

Two different real roots need the discriminant strictly above zero. Use > 0, not ≥ 0, because zero would give one repeated root.

3 more steps, each with the reason for it, in the full solution

Final answer

n < 4/m²

Where students lose marks

Watch out: (−m)(−mn) = +m²n, so −4ac = −4m²n. Two different roots need > 0, not ≥ 0.

Part (a)(ii)

5 more steps, each with the reason for it, in the full solution

Final answer

Shown

Where students lose marks

Watch out: factor −m out of every term, including −mn, so n stays inside. When multiplying −m back in, check the sign of −4/m².

Part (a)(iii)

5 more steps, each with the reason for it, in the full solution

Final answer

m = 2/3, n = 3/2

Where students lose marks

Watch out: the axis is −3, so set −2/m = −3, not 3. Take care with the fraction arithmetic when substituting m = 2/3.

Part (b)

5 more steps, each with the reason for it, in the full solution

Final answer

−2 < p < 6

Where students lose marks

Watch out: imaginary roots need < 0, not ≤ 0. A product < 0 means between the two roots, not outside them.

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