Which 2012 DSE M2 questions are explained here?

Featured questions on curve sketching, tangents, areas, volumes of revolution, rates of change and 3D vectors. Each one has a step-by-step solution and a short GeoGebra animation.

Are these the official HKEAA answers?

No. They are SketchMath’s own worked solutions. The final answers have been checked numerically, but the layout of the working is ours, not the marking scheme’s.

Can I change the numbers and see the graph update?

Yes. Click any video to open that construction in SketchMath Chat, then ask for a variation — a different curve, limit or vector — and the graph is redrawn with the new working.

2012 DSE M2 Past Paper: Solutions with Video Explanations

We are featuring 4 questions from the 2012 DSE M2 paper, each explained with a video. Take a look.

Ask AI about a 2012 DSE M2 question

2012 Q5 — Minimum point and asymptotes of a rational curve

For the curve y = (x² + x + 1)/(x + 1), find its minimum point and its asymptotes.

A tangent slides along the right-hand branch of y = (x² + x + 1)/(x + 1). It is horizontal at the minimum point (0, 1), and the curve hugs the asymptotes x = −1 and y = x. Try it yourself →

Divide: x² + x + 1 = x(x + 1) + 1, so y = x + 1/(x + 1)
Asymptotes: vertical x = −1 (denominator zero) and oblique y = x (since 1/(x + 1) → 0)
dy/dx = 1 − 1/(x + 1)² = x(x + 2)/(x + 1)²
dy/dx = 0 at x = 0 or x = −2. The sign of dy/dx changes from − to + at x = 0, so it is a minimum (at x = −2 it is a maximum)
At x = 0: y = 1

Answer: Minimum point (0, 1); asymptotes x = −1 and y = x

2012 Q6 — Water poured into a frustum-shaped glass

A glass is an inverted frustum 10 cm high, with base radius 3 cm and rim radius 4 cm. When the water is h cm deep its volume is V = (π/300)(h³ + 90h² + 2700h) cm³. Water is poured in at 7π cm³/s. Find the rate at which the depth is increasing when h = 5.

Side view of the glass as it fills. The water surface widens as it rises, so the same inflow of 7π cm³/s raises the level more and more slowly: 4/7 cm/s at a depth of 5 cm. Try it yourself →

Why the formula holds: at depth h the surface radius is 3 + h/10, and the frustum formula gives V = (π/3)h[3² + 3(3 + h/10) + (3 + h/10)²], which expands to the given expression
dV/dh = (π/300)(3h² + 180h + 2700) = (π/100)(h + 30)²
Chain rule: dV/dt = (dV/dh)(dh/dt)
At h = 5: 7π = (π/100)(35)² · dh/dt = 12.25π · dh/dt
dh/dt = 7/12.25 = 4/7

Answer: The depth increases at 4/7 cm/s

2012 Q7 — Area and height of a parallelepiped

A parallelepiped has edges OA = 6i + 2j − k, OB = 2i + j and OC = 5i − j + 2k; OADB is the face spanned by OA and OB. Find(a) the area of the parallelogram OADB,(b) the distance from C to the plane OADB.

The camera circles the parallelepiped. The blue face is OADB (area 3) and the red segment is the perpendicular from C to that face, of length 11/3. Try it yourself →

OA × OB = (2·0 − (−1)·1)i − (6·0 − (−1)·2)j + (6·1 − 2·2)k = i − 2j + 2k
(a) Area = |OA × OB| = √(1 + 4 + 4) = 3
(b) The distance is the length of the projection of OC on the normal n = i − 2j + 2k
OC · n = 5 + 2 + 4 = 11
Distance = |OC · n| / |n| = 11/3

Answer: (a) 3; (b) 11/3

2012 Q9 — Volume of revolution of y = √(x sin x)

The region bounded by the curve y = √(x sin x) (0 ≤ x ≤ π) and the x-axis is revolved about the x-axis. Find the volume of the solid, after first finding ∫ x sin x dx.

The red curve y = √(x sin x) sweeps out the blue solid as it turns about the x-axis. Each cross-section is a disc of area πy² = πx sin x, which is what gets integrated. Try it yourself →

By parts with u = x, dv = sin x dx: ∫ x sin x dx = −x cos x + ∫ cos x dx = −x cos x + sin x + C
Disc method: V = π∫₀^π y² dx = π∫₀^π x sin x dx
V = π[−x cos x + sin x]₀^π = π[(π + 0) − 0]

Answer: V = π²

How to use these 2012 DSE M2 solutions

Try each question on paper first, then watch its animation: every video is a real GeoGebra construction of that question, so you can see why the answer is what it is before reading the steps. Click a video to open the same graph in SketchMath and drag or rotate it yourself. For the theory behind a question, see the DSE M2 notes; for other years, see all M2 past paper solutions.

Each question is described in our own words with the same numbers as the 2012 HKDSE Mathematics Extended Part Module 2 paper, and the solutions and animations are written by SketchMath.