Edexcel IGCSE Maths
Topic questions Pure mathematics Differentiation
12 questions · 106 marks · about 2.5 hours
A curve with equation y 
9x2 
16x 
5 is drawn on a grid where 1 unit 
1cm on each axis.
P and Q are two points on the curve.
P is the point with coordinates 
1,
2
The gradient of the tangent to the curve at Q is 6.5
Find the length, in cm to 3 significant figures, of PQ
You must show clear algebraic working.
A particle is moving along a straight line. At time t seconds, the displacement, x metres, of the particle from a fixed point O on the line is given by
At time t seconds, the velocity of the particle is v m/s
Find an expression for v in terms of t
Find the least speed, in m/s, of the particle.
The curve C has equation
Find, to one decimal place, the coordinates of the stationary point of C
Complete the table of values for y
Give your values of y to one decimal place where necessary.
| x | 0.4 | 0.6 | 0.8 | 1 | 2 | 3 | 4 | 5 |
| y | 35.2 | 0 | –8 | –3 | 4.8 |
On the grid opposite, plot the stationary point and plot the points from your completed table. Join these to form a smooth curve.
By drawing a suitable straight line on the grid, find estimates, to one decimal place, of the solutions to the equation
within the range 0 < x 
5
Turn over for a spare grid if you need to redraw your curve.
Only use this grid if you need to redraw your curve.
The equation of a curve C is y 
1 where k is a constant. A minimum point on C occurs when x = –1.5
Show that the value of k is 12
Complete the table of values for y 
1
Give your values of y to 2 decimal places.
| x | –6 | –5 | –4 | –3 | –2 | –1 | –0.8 |
| y | –1.44 | –2.00 | –2.75 | –2.00 |
On the grid opposite, plot the minimum point and the points from your completed table.
Join all the points to form a smooth curve.
By drawing a suitable straight line on the grid, find the range of values, to one decimal place, of x in 
6 
x 
0 8., for which
Turn over for a spare grid if you need to redraw your graph.
Only use this grid if you need to redraw your graph.
Also involves Coordinate geometry · Tangents and normals
A curve C and a straight line L are drawn on a grid.
C has equation y = 5x2 – 16x – 5
L has equation y + 5x = 7
Find the coordinates of the points of intersection of C and L
Show clear algebraic working.
P is the point on the curve with equation y = 5x2 – 16x – 5 with x coordinate 2
The line Q is the tangent to the curve at the point P
The line Q crosses the x‑axis at the point X and the y‑axis at the point Y The point M lies on Q and is such that XM = MY
Calculate the coordinates of the point M
Give your coordinates as exact values.
A particle P is moving along a straight line through the fixed point O
The speed, v m/s, of P at time t seconds is given by
Complete the table of values for v 
2 
Give your values of v to 2 decimal places.
| t | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 | 4.5 | 5 |
| v | 5.04 | 2.90 | 2.71 | 2.70 | 2.78 | 2.90 |
On the grid opposite, plot the points from your completed table and join them to form a smooth curve.
Using your curve, find an estimate, to one decimal place, for the speed of P when t = 1.75
The acceleration of P at time t seconds where 1 
t 
5 is a m/s2
Find an expression for a in terms of t
Using your answer to part (d), find the value of t when P has its minimum speed in
Turn over for a spare grid if you need to redraw your curve.
Only use this grid if you need to redraw your curve.
Also involves Gradient and intercept
Also involves Tangents and normals
f(x) = 3x3 – 7x2 + 5x – 1
Use the factor theorem to show that (3x – 1) is a factor of f(x).
Hence solve the equation f(x) = 0
Show clear algebraic working.
The curve C has equation y = f(x)
Find the exact coordinates of the turning points on C.
The curve C crosses the y-axis at the point P.
Find the gradient of C at P.
Hence find an equation of the tangent to C at P.
f (x) = 4x3 – 13x – 6
Use the factor theorem to show that (2x + 1) is a factor of f (x).
Hence factorise f (x) fully.
The curve C has equation y = f (x)
Find the coordinates of the points of intersection of C with the x‑axis.
Find the coordinates, to 2 decimal places, of the turning points of C.
The table below gives the coordinates of three points on C.
| x | –2 | 0.5 | 1.5 |
| y | –12 | –12 | –12 |
On the grid opposite, draw the curve C for –2 
x 
2
Clearly label the coordinates of the turning points of C and the coordinates of the points of intersection with the x ‑axis and the y ‑axis.
Turn over for a spare grid if you need to redraw your curve.
Only use this grid if you need to redraw your curve.
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