10 free IGCSE Additional Mathematics Vectors & Calculus questions written in the style of Cambridge Additional Mathematics papers, each with its answer and mark scheme. 4 of them are A* challenge questions, built to catch the mistakes strong students still make.
This topic covers Vectors in two dimensions, Differentiation, Integration, Kinematics. Work through the questions below, then generate a full Vectors & Calculus paper with a fresh set every time.
Generate a full Vectors & Calculus paper →The radius of a circle increases at 0.5 cm/s. Find the rate of increase of the area when the radius is 10 cm.
| Part | Answer and marking points | Marks |
|---|---|---|
| 31.4 cm²/s dA/dr = 2πr B1 chain rule M1 31.4 A1 | 3 |
A boat travels at 8 m/s in still water. It heads due north across a river flowing at 6 m/s due east.
Find the resultant speed and the angle between the boat’s path and the bank.
| Part | Answer and marking points | Marks |
|---|---|---|
| 10 m/s; 53.1° Pythagoras M1 10 m/s A1 tan θ = 8/6 M1 53.1° A1 | 4 |
Find the area enclosed between the curve y = x2 − 7x + 10 and the x-axis.
| Part | Answer and marking points | Marks |
|---|---|---|
| 92 roots x = 2 and x = 5 B1 integrate M1 substitute limits M1 92 A1 | 4 |
A particle moves so that its displacement from O after t seconds is s = 4e−t + 4t m.
Find the velocity and the acceleration when t = ln 2, and explain whether the particle is speeding up or slowing down at this time.
| Part | Answer and marking points | Marks |
|---|---|---|
| v = 2; a = 2 v = −4e−t + 4 M1 v = 2 m/s A1 a = 4e−t = 2 m/s² A1 speeding up (v and a same sign) B1 | 4 |
The curve y = (x)(5 − x) meets the x-axis at two points. Find the exact area enclosed by the curve and the x-axis.
| Part | Answer and marking points | Marks |
|---|---|---|
| 20.8333 oe Expand to quadratic M1 Integrate correctly M1 Limits 0 and 5 M1 Correct substitution M1 1256 A1 | 5 |
p = −5i + 3j and q = i − j.
Find the unit vector in the direction of p + q.
| Part | Answer and marking points | Marks |
|---|---|---|
| 1√20(−4i + 2j) p + q = −4i + 2j B1 |p + q| = √20 M1 unit vector A1 | 3 |
A curve has equation y = 2x3 − 2x2 − x.
Find dydx.
Find the equation of the normal to the curve at the point where x = 1.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | 6x2 − 4x − 1 Two terms correct B1 All correct B1 | 2 |
| (b) | y + 1 = −(x − 1) gradient of tangent = 1 M1 gradient of normal = −1 M1 point (1, −1) B1 Correct equation in any form A1 | 4 |
Differentiate with respect to x: (a) (3x + 1)5; (b) x2 ln x; (c) e3xx.
| Part | Answer and marking points | Marks |
|---|---|---|
| chain/product/quotient 5(3x + 1)4 × 3 M1 15(3x + 1)4 A1 product rule M1 2x ln x + x A1 quotient rule M1 3xe3x − e3xx2 A1 | 6 |
A particle moves in a straight line so that its velocity t seconds after passing a fixed point O is v = 3t2 − 6t m/s.
Find the acceleration when t = 1.
Find the time when the particle is instantaneously at rest (t > 0).
Find the displacement from O when t = 6.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | 0 m/s² differentiate M1 0 A1 | 2 |
| (b) | t = 2 v = 0 M1 t = 2 A1 | 2 |
| (c) | 108 m s = t3 − 3t2 M1 108 m A1 | 2 |
A curve has equation y = x3 − 3x + 4.
Find the coordinates of the stationary points.
Determine the nature of each stationary point.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | (1, 2) and (−1, 6) dydx = 3x2 − 3 M1 x = ±1 A1 substitute M1 (1, 2) and (−1, 6) A1 | 4 |
| (b) | minimum at x = 1, maximum at x = −1 d2ydx2 = 6x M1 both correct A1 | 2 |
In the 0606 syllabus, Vectors & Calculus covers Vectors in two dimensions, Differentiation, Integration, Kinematics.
This page has 10 Vectors & Calculus questions with answers. For more, the free generator on mentorriyaz.in builds a new Vectors & Calculus paper with a full mark scheme each time you click.
Riyaz Shaikh has mentored IGCSE, IB and A Level students for 15+ years, one-on-one, online and in Mumbai. The hardest questions on this site are the ones he uses to find a student's weak points.
Meet Riyaz →Original Cambridge-style practice questions written for mentorriyaz.in. These are not Cambridge past paper questions, and this site is not affiliated with or endorsed by Cambridge University Press & Assessment. Updated 2026-11-04. Paper generator · mentorriyaz.in