Practice Paper 1 (Pure) (OCR A Level Maths: Pure)

Practice Paper Questions

1
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3 marks

The function straight f left parenthesis x right parenthesis equals k x squared plus 2 k x minus 3 has two distinct real roots.

Show that k less than negative 3 space or space k greater than 0.

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2a
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2 marks

A company selling books models the number of books sold per year,N, using the formula

N equals 10 space 000 minus 200 c

where c is the price per book in pounds sterling.

(i)

Find the number of books the company can expect to sell if they are priced at £18 each?

(ii)
Work out the total income the company will receive if they sell all books at £18 each.
2b
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2 marks

Find the number of books the company can expect to sell if they are priced at £16 each and work out the total income the company will receive if they sell all books at this price.

2c
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1 mark

What do your answers to part (a) and (b) suggest about the relationship between the price of a book, the number sold and the total income received?

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3a
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2 marks

The leakage rate of water from a pipe, L space 1 space straight s to the power of negative 1 end exponent (litres per second), is directly proportional to the square root of the flow rate,straight s space straight m space straight s to the power of negative 1 end exponent  (meters per second), which is the speed of the water flowing through the pipe.
It was observed that the leaking rate was 0.72 space straight l space straight s to the power of negative 1 end exponent when the flow rate was 0.64 space straight m space straight s to the power of negative 1 end exponent.

Write down an equation connecting L spaceand S.

3b
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2 marks

Find the flow rate when the leakage rate is 0.49 space straight l space straight s to the power of negative 1 end exponent.

3c
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2 marks

An alternative model for the leakage rate is L equals 0.5 straight s.
Apart from when there is no leak find a flow rate and a leakage rate for when both models predict the same result.

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4a
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2 marks

straight f left parenthesis x right parenthesis equals 4 x cubed minus 7 x minus 3

Use the factor theorem to show that left parenthesis 2 x plus 1 right parenthesis is a factor of straight f left parenthesis x right parenthesis.

4b
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4 marks

Factorise straight f left parenthesis x right parenthesis completely.

4c
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2 marks

Write down all the real roots of the equation straight f left parenthesis x right parenthesis equals 0.

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5
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4 marks

The point space P left parenthesis negative 3 comma negative 2 right parenthesis lies on the curve with equation y equals straight f left parenthesis x right parenthesis

State the coordinates of the image of point P on the curves with the following equations:

(i)

y minus 2 equals straight f left parenthesis x right parenthesis minus 6

(ii)

y equals straight f left parenthesis x minus 3 right parenthesis

(iii)

2 y equals straight f left parenthesis x right parenthesis

(iv)
y equals straight f left parenthesis 1 half space x right parenthesis

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6
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4 marks

The diagram below shows a sketch of the graph with equation y equals straight f open parentheses x close parentheses.

i_Q65bb0_q8a-7-3-further-differentiation-vh-a-level-maths-pure-screenshots

On the sketch, mark the approximate locations of the following ...

(i)
... intercepts with the coordinate axes using the letter C
(ii)
... stationary points using the letter S
(iii)
... points of inflection using the letter I

Also highlight sections of the curve where the graph is convex.

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7a
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2 marks

The functions straight f left parenthesis x right parenthesis and  straight g left parenthesis x right parenthesis are given as follows

               straight f left parenthesis x right parenthesis equals open parentheses 4 plus 3 x close parentheses to the power of 1 half end exponent space space space space space space space space space space space space space space space space space straight g left parenthesis x right parenthesis equals open parentheses 9 minus 2 x close parentheses to the power of negative 1 half end exponent

Expand straight f left parenthesis x right parenthesis, in ascending powers of x up to and including the term in x squared.

7b
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2 marks

Expand straight g open parentheses x close parentheses, in ascending powers of x up to and including the term in x squared.

7c
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2 marks

Find the expansion of square root of fraction numerator 4 plus 3 x over denominator 9 minus 2 x end fraction end root  in ascending powers of x, up to and including the term in x squared.

7d
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2 marks

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

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8a
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3 marks

The function f(x) is defined as

  straight f open parentheses x close parentheses equals 5 cos space x sin space 2 x space space minus 3 space space space space space space space space space space space space space space space space x element of straight real numbers

Show thatspace straight f apostrophe open parentheses x close parentheses equals 10 cos space x space left parenthesis 1 minus 3 sin squared x space right parenthesis.

8b
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3 marks

Use the Newton-Raphson method with x subscript 0 equals 0.3 to find a root of the equation space straight f open parentheses x close parentheses equals 0 spacecorrect to five significant figures.

8c
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1 mark

Write down the exact value of a root to the equation straight f open parentheses x close parentheses equals negative 3.

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9a
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2 marks

The functionspace straight f left parenthesis x right parenthesis spaceis defined as

straight f left parenthesis x right parenthesis space equals space x squared space long dash space 4 space space space space space space space space space space x greater or equal than 0

Work out the range of straight f open parentheses x close parentheses.

9b
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1 mark

If the domain of straight f left parenthesis x right parenthesis is changed to x space less or equal than space 0, what is the range of straight f left parenthesis x right parenthesis?

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10a
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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

10b
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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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11a
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2 marks

The graph of the curve C shown below is defined by the parametric equations

x equals 3 sin space 3 theta     y equals 6 cos space 2 theta   space minus space straight pi over 2 space less or equal than theta less or equal than space straight pi over 2

q4a-9-2-further-parametric-equations-medium-a-level-maths-pure

(i)
Write down the value of  fraction numerator straight d y over denominator straight d theta end fraction  at the point (0 , 6).

(ii)
Write down the value of  fraction numerator straight d x over denominator straight d theta end fraction at the points (-3 , 3) and (3 , 3).
11b
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3 marks

Find an expression for  fraction numerator straight d y over denominator straight d x end fraction  in terms of theta.

11c
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4 marks
(i)
Find the values of x, y and  fraction numerator straight d y over denominator straight d x end fraction  at the point where  theta equals space pi over 12.

(ii)
Hence show the equation of the tangent to C at the point where space theta equals space pi over 12 space spaceis
2 square root of 2 x plus 3 y minus open parentheses 9 square root of 3 plus 6 close parentheses equals 0

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12
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7 marks

The diagram below shows two right-angled triangles.
Angles A and B have been labelled.

q9-5-6-compund-and-double-angle-formulae-a-level-only-edexcel-a-level-pure-maths-veryhard

Given that   alpha equals A plus B, find the exact values of sin alpha,cos alpha and tan alpha.

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13
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5 marks

Using the identities

sin(A + B) sin A cos B + sin B cos A and

cos 2A identical to1 –  2 sin2 A

show that sin 3A identical to 3 sin A – 4 sin3 A

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14a
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1 mark

The diagram below shows a sketch of the curves with equations

y equals square root of 25 minus x squared space space end root     and      space y equals 6 x minus x squared minus 5

q9-8-2-further-integration-veryhard-a-level-maths-pure-screenshot

Show that the two curves intersect at the points (3, 4), (4, 3) and (5, 0).

14b
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7 marks

By using the substitution  x equals 5 sin space u, show that

integral square root of 25 minus x squared end root space straight d x equals fraction numerator 25 arcsin space open parentheses x over 5 close parentheses space plus x square root of 25 minus x squared end root over denominator 2 end fraction plus c

where c is the constant of integration.

14c
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5 marks

Using calculus, and your results from parts (a) and (b), show that the total shaded area enclosed by the two curves in the diagram is equal to

fraction numerator 25 pi over denominator 4 end fraction minus 4 minus 25 over 2 open parentheses 2 arcsin space open parentheses 4 over 5 close parentheses minus arcsin space open parentheses 3 over 5 close parentheses close parentheses space units squared

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15a
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6 marks

A tree disease is spreading throughout a large forested area.

The rate of increase in the number of infected trees is modelled by the differential equation

fraction numerator straight d N over denominator straight d t end fraction equals k N open parentheses N minus 1 close parentheses comma space space space N greater than 1

where N is the number of infected trees, t is the time in days since the disease was first identified and k is a positive constant.

Solve the differential equation above, and show that the general solution can be written in the form

N equals fraction numerator 1 over denominator 1 minus A e to the power of k t end exponent end fraction

where A is a positive constant.

15b
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3 marks

Initially two trees were identified as diseased.
A fortnight later, 4 trees were infected.
Using this information, find the values of the constants A and k.

15c
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3 marks

By considering the solution to the differential equation along with the values of A and k found in part (b), suggest a range of values of  t  for which the model might be considered reliable.

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