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Pearson Edexcel IAL Mathematics P4.5 Differentiation Question Bank

Practise differentiating implicit and parametric curves, forming tangent or normal equations, and applying connected rates in geometric models.

Syllabus
First assessment 2019
Course
Mathematics YMA01
Level
A2

Exam points

  • use dy/dx from parametric or implicit curves to form tangent and normal equations
  • apply the chain rule to linked rates in volumes, areas, radii and lengths
  • work with diagrams or given formulae to express one variable before differentiating

Covered syllabus points

P4.5.1 - Differentiation of simple functionsP4.5.2 - Formation of simple differential

P4.5 - Differentiation question 1

[Maximum number: 7]
Figure 3

Figure 3

Figure 3 shows a sketch of the curve C with parametric equations

x=6t−3sin⁡2ty=2cos⁡t0⩽t⩽π2x=6 t-3 \sin 2 t \quad y=2 \cos t \quad 0 \leqslant t \leqslant \frac{\pi}{2}

The curve meets the y-axis at 2 and the x-axis at k, where k is a constant.

Question (a)

(a)

Use parametric differentiation to show that

dy dx=λcosec⁡t\frac{\mathrm{d} y}{\mathrm{~d} x}=\lambda \operatorname{cosec} t

where λ\lambda is a constant to be found.

The point P with parameter t=π4t=\frac{\pi}{4} lies on C.
The tangent to C at the point P cuts the y-axis at the point N.

[ 4 ]

Question (b)

(b)

Find the exact y coordinate of N, giving your answer in simplest form.

The region bounded by the curve, the x-axis and the y-axis is rotated through 2π2 \pi radians about the x-axis to form a solid of revolution.

[ 3 ]

P4.5 - Differentiation question 2

[Maximum number: 3]
Figure 1

Figure 1

A regular icosahedron of side length x cmx \mathrm{~cm}, shown in Figure 1, is expanding uniformly. The icosahedron consists of 20 congruent equilateral triangular faces of side length x cmx \mathrm{~cm}.

Find the rate of change of the volume of the icosahedron when x=2, giving your answer to 2 significant figures.

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