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

Practice Paper Questions

1a
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2 marks

Solve the equation

4 to the power of 3 x plus 2 end exponent equals 16 to the power of x plus 6 end exponent

1b
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3 marks

Solve the equation

4 to the power of 2 x plus 3 end exponent minus 8 equals 92

giving your answer to 3 significant figures.

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2
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3 marks

Find, in ascending powers of x, the binomial expansion of

            begin mathsize 22px style begin inline style 1 over open parentheses 4 plus 8 x close parentheses squared end style end style

up to and including the term in x cubed.

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3a
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2 marks

Show that the equation  3 tan2 x equals 18 minus 2 sec x  can be written as

3 sec2 x plus 2sec x minus 21 equals 0

3b
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4 marks

Hence, or otherwise, solve the equation

3 tan2 x equals 18 minus 2 sec x comma space space space space space space space space space space space space space space space space space space space space space space minus straight pi less or equal than space x less or equal than straight pi

Give your answers to three significant figures.

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4a
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5 marks

Show that if  y equals cosec 2 x , then 

fraction numerator d y over denominator d x end fraction equals negative 2 cosec space 2 x space cot space 2 x

4b
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1 mark

Hence find the gradient of the tangent to the curve space y equals cos e c space 2 x space at the point with coordinates open parentheses pi over 3 comma fraction numerator 2 square root of 3 over denominator 3 end fraction close parentheses

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5a
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4 marks

Express

fraction numerator x squared minus 4 x plus 7 over denominator open parentheses x minus 1 close parentheses open parentheses x minus 3 close parentheses squared end fraction


as partial fractions.

5b
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3 marks

Hence, or otherwise, find

integral fraction numerator x squared minus 4 x plus 7 over denominator open parentheses x minus 1 close parentheses open parentheses x minus 3 close parentheses squared end fraction space straight d x

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6a
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2 marks

The diagram below shows the graphs of space y equals x space and y equals g left parenthesis x right parenthesis.

QpsJxjGx_q1a-10-1-solving-equations-hard-a-level-maths-pure

Show on the diagram, using the value of x subscript 0 indicated, how an iterative process will lead to a sequence of estimates that converge to the x-coordinate of the point P.
Mark the estimates space x subscript 1and x subscript 2 on your diagram.

6b
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3 marks

By finding a suitable iterative formula, use x subscript 0 equals 2 to estimate a root to the equationx minus sin space 0.8 x equals 2.5 correct to two significant figures.

6c
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2 marks

Confirm that your answer to part (b) is correct to two significant figures.

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7a
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3 marks

Show that

integral tan space k x space straight d x equals 1 over k ln space open vertical bar sec space k x close vertical bar plus c

where k is a constant, and c is the constant of integration.

7b
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5 marks

Use calculus to find the exact value of

integral subscript straight pi over 18 end subscript superscript straight pi over 9 end superscript fraction numerator cosec 3 theta space over denominator 3 c o t space 3 theta end fraction space straight d theta

writing your answer in the form  a ln b , where a and b are rational numbers to be found.

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8
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8 marks

A large container of water is leaking at a rate directly proportional to the square of the volume of water in the container.

(i)
Given that the initial volume of water in the container is 4000 litres and that after 10 minutes the volume of water in the container has dropped by 30%, write down and solve a differential equation connecting the volume, V, of water in the container to the time, t.

 

(ii)
What does your solution predict will happen to the volume of water in the container after a very long time?

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9a
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2 marks

The coordinates of three points are A left parenthesis 2 comma negative 1 comma 5 right parenthesis comma space space B left parenthesis 4 comma 1 comma 1 right parenthesis and  C left parenthesis negative 2 comma negative 5 comma 9 right parenthesis.

Find stack B A with rightwards arrow on top and stack B C with rightwards arrow on top.

9b
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3 marks

By considering the scalar product stack B A with rightwards arrow on top times stack B C with rightwards arrow on top, or otherwise, calculate the angle between stack B A with rightwards arrow on top and  stack B C with rightwards arrow on top.  Give your answer in degrees, accurate to 1 decimal place.

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10a
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3 marks

Given that z subscript 1 equals 5 plus 7 i and z subscript 2 equals 2 minus i:

Work out  z subscript 1 cross times space z subscript 2 superscript asterisk times and z subscript 2 cross times z subscript 2 superscript asterisk times

10b
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2 marks

Hence or otherwise work out z subscript 1 over z subscript 2, giving your answer in the form a plus b i where a and b are real numbers.

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11a
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4 marks

On an Argand diagram, sketch the loci (i.e., sets of points) for which each of the following equations is true:

(i)
arg left parenthesis z plus 2 minus 2 i right parenthesis equals pi over 4
(ii)
vertical line z minus 3 minus 2 i vertical line equals 5
11b
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2 marks

Shade the region of your diagram that satisfies both of the following inequalities:

0 less or equal than arg left parenthesis z plus 2 minus 2 i right parenthesis less or equal than pi over 4 space space space space space and space space space space vertical line z minus 3 minus 2 i vertical line less or equal than 5

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12a
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2 marks

The rare Leaping Unicorn jumps in such a way that the length of a jump is always the same distance.  However, the maximum height a Leaping Unicorn reaches during a jump reduces gradually over time as the unicorn tires.

The way in which Leaping Unicorns jump can be modelled by the function

h left parenthesis x right parenthesis equals vertical line A open parentheses e to the power of negative k x end exponent close parentheses sin x vertical line          x greater or equal than 0

where x is the horizontal distance covered and h is the height, both measured in metres.
A and k are both positive constants.

(i)      Write down the length of a Leaping Unicorn jump.

(ii)     Briefly describe how changing the value of the constant k would affect the

         model.

12b
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5 marks

During its first jump, a Leaping Unicorn reaches a maximum height of 1.288 metres after covering 1.471 metres over the ground.
Find the values of A spaceand k.

12c
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2 marks

What is the total distance of ground covered by a Leaping Unicorn when it is at the maximum height of its third jump?

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