Selangor Trial 2025 Add Maths Paper 2, Question 8Differentiation

Solved on video in English · Form 5, Differentiation · 7:06 video

The question

(a)(i)

(a) It is given that the gradient function of a curve is 8 − x³ and the curve has a turning point at (p, 3). (i) Find the value of p.

(a)(ii)

(a)(ii) Determine whether the turning point is a maximum point or a minimum point.

(b)(i)

(b) The volume of air, V cm³, in a spherical balloon is given by V = (4/3)πr³, where r cm is the radius of the balloon. (i) The balloon is filled with air at the rate of 189 cm³ s⁻¹. Calculate the rate of change of radius, in cm s⁻¹, when the radius is 6 cm.

(b)(ii)

(b)(ii) It is found that the balloon is leaking and the radius decreases from 20 cm to 19.9 cm. Determine the approximate change in volume of the balloon.

Selangor Trial 2025 Add Maths Paper 2, Question 8: the question as drawn in Eduly's video7:06

Part (a)(i)

Step 1

At a turning point the gradient is zero, so set the gradient function to 0: 8 − x³ = 0.

Why this step

A turning point is where the tangent is flat, so the gradient is 0. Set the gradient function to 0, not 3, because 3 is the y-value.

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

Final answer

p = 2

Where students lose marks

Watch out: set the gradient function to 0, not 3. The 3 is the y-value, not the gradient.

Part (a)(ii)

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

Final answer

d²y/dx² = −12 < 0, so it is a maximum point

Where students lose marks

Watch out: differentiate the gradient function, not y, and keep the sign: −3(2)² = −12. Negative means maximum.

Part (b)(i)

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

Final answer

dr/dt = 21/(16π) ≈ 0.4178 cm s⁻¹

Where students lose marks

Watch out: substitute r = 6 first, then divide both sides by 144π. Multiplying by 144π instead gives a wrong rate.

Part (b)(ii)

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

Final answer

δV ≈ −160π cm³ (a decrease of 160π cm³)

Where students lose marks

Watch out: the radius decreases, so δr = −0.1, and use r = 20 (the starting radius), not 19.9.

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