Practice Paper 3 (Pure 3)

1

3 marks

Solve the equation $|x^{2} — 4| = 3$ , giving your answers in exact form.

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2a

4 marks

Show that $2 \sin θ + 4 \cos θ$ can be written as $2 \sqrt{5} \cos (θ — α),$ where $α = 0.464$ to three significant figures.

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2b

3 marks

Hence solve the equation

$2 \sin θ + 4 \cos θ = 3 - π \leq θ \leq π$

giving your answers correct to 3 significant figures.

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3

4 marks

Differentiate $\frac{5 x^{7}}{\sin 2 x}$ with respect to $x$ .

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4

5 marks

Use calculus and the substitution $x = \cos θ$ to find the exact value of

$\int_{\frac{1}{\sqrt{2}}}^{\frac{\sqrt{3}}{2}} \frac{1}{\sqrt{1 - x^{2}}} d x$

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5

5 marks

Solve the equation

$8 \cos^{4} θ - 5 \cos 2 θ - 2 = 0 0 \leq θ \leq π$

tate your answers as multiples of $π$ .

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6a

2 marks

The functions $f (x)$ and $g (x)$ are given as follows

$f (x) = {(4 + 3 x)}^{\frac{1}{2}} g (x) = {(9 — 2 x)}^{- \frac{1}{2}}$

Expand $f (x)$ , in ascending powers of $x$ up to and including the term in $x^{2}$ .

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6b

2 marks

Expand $g (x)$ , in ascending powers of $x$ up to and including the term in $x^{2}$ .

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6c

2 marks

Find the expansion of $\sqrt{\frac{4 + 3 x}{9 - 2 x}}$ in ascending powers of $x$ , up to and including the term in $x^{2}$ .

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6d

2 marks

Find the values of $x$ for which your expansion in part (c) is valid.

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7a

7 marks

A tree disease is spreading throughout a large forested area.

When the disease was first discovered, three trees were infected.

Ten days later ten trees were infected.

The differential equation

$\frac{1}{t} \frac{d N}{d t} = k N$

where k is a positive constant, is used to model the number of infected trees, N, at a time t days after the disease was first discovered.

Find the particular solution to the differential equation.

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7b

4 marks

Scientists believe the majority of the forest can be saved from infection if action is taken before 30 trees are infected.

Measured from the time when the disease was first discovered, how many days does the model predict the scientists have to take action in order to save the majority of the forest from infection?

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8

5 marks

Use integration by parts to find

$\int (2 x^{2} - 1) e^{x} d x$

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9a

2 marks

Show that the equation $x^{3} + 3 = 5 x$ can be rewritten as

$x = \sqrt[3]{5 x - 3}$

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9b

3 marks

Starting with $x_{0} = 1.8,$ use the iterative formula

$x_{n + 1} = \sqrt[3]{5 x_{n} - 3}$

to find a root of the equation $x^{3} + 3 = 5 x$ , correct to two decimal places.

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10

5 marks

Determine whether the following pairs of lines intersect, are parallel, or are skew.

If the lines intersect, find the coordinates of the point of intersection.

$r = — 7 i — j + 6 k + s (i + j - k)$ and $r = 2 i + 2 j + 11 k + t (2 i — j + 5 k) .$

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11

5 marks

Find the perpendicular distance of the point $P (6, 0, 3)$ to the line

$r = — i + 5 j + 2 k + s (3 i - 6 j — 2 k)$

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12a

4 marks

$f (z) = z^{4} - 2 z^{3} + 6 z^{2} + p z + 125$ , where $p$ is a real constant.

Given that $3 - 4 i$ is a root of the equation $f (z) = 0$ , show that $z^{2} + 4 z + 5$ is a factor of $f (z)$ .

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12b

3 marks

Hence find the value of $p$ and solve completely the equation $f (z) = 0$ .

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13

5 marks

On an Argand diagram, shade the region which satisfies both of the following inequalities:

$| z - 4 i | > | z | and | z - 4 i | < | z + 2 |$

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Practice Paper 3 (Pure 3) (CIE A Level Maths: Pure 3)

Practice Paper Questions

Set A

Practice Paper 1 (Pure 1)

Practice Paper 4 (Mechanics)

Practice Paper 5 (Probability & Statistics 1)

Practice Paper 6 (Probability & Statistics 2)

Set B

Practice Paper 1 (Pure 1)

Practice Paper 3 (Pure 3)

Practice Paper 4 (Mechanics)

Practice Paper 5 (Probability & Statistics 1)

Practice Paper 6 (Probability & Statistics 2)

Set C

Practice Paper 1 (Pure 1)

Practice Paper 3 (Pure 3)

Practice Paper 4 (Mechanics)

Practice Paper 5 (Probability & Statistics 1)

Practice Paper 6 (Probability & Statistics 2)