Practice Paper 1 (Pure) (AQA A Level Maths: Mechanics)

Practice Paper Questions

1
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1 mark

Given that a space greater than space 0, determine which of these expressions is not equivalent to the others.

Circle your answer.

log subscript 2 open parentheses fraction numerator 1 over denominator square root of a end fraction close parentheses log subscript 2 a to the power of begin inline style 1 half end style end exponent 1 half log subscript 2 open parentheses a to the power of negative 1 end exponent close parentheses negative 1 half log subscript 2 a

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2
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1 mark

Given y space equals space e to the power of 6 x end exponent, find fraction numerator straight d y over denominator straight d x end fraction.

Circle your answer.

fraction numerator straight d y over denominator straight d x end fraction equals 6 e to the power of 6 x end exponent fraction numerator straight d y over denominator straight d x end fraction equals 6 x e to the power of 6 x minus 1 end exponent fraction numerator straight d y over denominator straight d x end fraction equals e to the power of 6 x end exponent fraction numerator straight d y over denominator straight d x end fraction equals e to the power of 6 x end exponent over 6

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3
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1 mark

The diagram below shows a sector of a circle.

q4-aqa-a-level-maths-practice-paper-pure

The radius of the circle is 7 cm and theta space equals space 0.9 radians.

Find the area of the sector.

Circle your answer.

6.3 cm2 44.1 cm2 220.5 cm2 22.05 cm2

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4a
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3 marks

The line l passes through the points (p, 2p) and (3p, 9p).

Find an equation for the line l

4b
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2 marks

l intercepts the y-axis at (0, 3).

Find the value of p.

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5a
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2 marks

The sum of the first n terms of an arithmetic series is

         S subscript n equals left parenthesis k plus 7 right parenthesis plus left parenthesis 2 k plus 18 right parenthesis plus left parenthesis 3 k plus 29 right parenthesis plus horizontal ellipsis plus 36

Show that  n equals fraction numerator 40 over denominator k plus 11 end fraction.

5b
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3 marks

Hence show that the sum of the first n terms is  fraction numerator 20 k plus 860 over denominator k plus 11 end fraction.

5c
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1 mark

Given that S subscript n equals 180, find the value of k.

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6a
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2 marks

The function straight f left parenthesis x right parenthesis is defined as

straight f colon x rightwards arrow from bar fraction numerator x squared plus 1 over denominator x squared end fraction space space space space space space space space space space space x element of straight real numbers comma space x not equal to 0

Show that straight f left parenthesis x right parenthesis can be written in the form

straight f colon x rightwards arrow from bar 1 plus 1 over x squared

6b
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2 marks

Explain why the inverse of straight f left parenthesis x right parenthesis does not exist and suggest an adaption to its domain so the inverse does exist.

6c
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4 marks

The domain of straight f left parenthesis x right parenthesis is changed to x space greater than space 0.

Find an expression for straight f to the power of negative 1 end exponent left parenthesis x right parenthesis and state its domain and range.

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7a
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3 marks

The diagram below shows part of the graph space y equals straight f left parenthesis x right parenthesis spacewhere space straight f left parenthesis x right parenthesis equals 2 x space cos space left parenthesis 3 x right parenthesis minus 1.

q1a-10-1-solving-equations-medium-a-level-maths-pure

(i)
Find straight f open parentheses 1.6 close parentheses and straight f open parentheses 1.7 close parentheses, giving your answers to three significant figures.

(ii)
Briefly explain the significance of your results from part (i).
7b
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3 marks

One of the solutions to the equation straight f open parentheses x close parentheses equals 0 is x equals 2.55, correct to three significant figures.

(i)
Write down the upper and lower bound of 2.55.

(ii)
Hence, use the sign change rule to confirm that this is a solution
(to three significant figures) to the equation straight f open parentheses x close parentheses equals 0.

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8a
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2 marks

Show that the equation  x cubed plus 3 equals 5 x can be rewritten as

x equals cube root of 5 x minus 3 end root

8b
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3 marks

Starting with x subscript 0 equals 1.8, use the iterative formula

x subscript n plus 1 end subscript equals cube root of 5 x subscript n minus 3 end root

to find a root of the equation x cubed plus 3 equals 5 x, correct to two decimal places.

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9
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4 marks

A geometric series is given by 1 plus 2 x plus 4 x squared plus horizontal ellipsis

(i)
Write down the common ratio, r, of the series.
(ii)
Given that the series is convergent, and that sum from n equals 1 to infinity of space 2 x to the power of n minus 1 end exponent equals 19, calculate the value of x.

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10
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6 marks

Prove by contradiction that square root of 11 is an irrational number.  You may use without proof the fact that if n squared is a multiple of 11, then n is a multiple of 11.

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11a
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3 marks

In a computer animation, the side length, s mm, of a square is increasing at a constant rate of 2 millimetres per second.

(i)
Write down the value of fraction numerator d s over denominator d t end fraction, where t is time and measured in seconds.

(ii)
Write down a formula for the area, A mm2, of the square and hence find  fraction numerator d A over denominator d s end fraction.
11b
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3 marks

Use the chain rule to find an expression for  fraction numerator d A over denominator d t end fraction in terms of s and hence find the rate at which the area is increasing when  s=10.

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12a
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4 marks

Given that straight f open parentheses x close parentheses equals sin space x

Show that

straight f apostrophe open parentheses x close parentheses equals limit as h rightwards arrow 0 of open parentheses sin space x space open parentheses fraction numerator cos space h space minus 1 over denominator h end fraction close parentheses plus cos space x space open parentheses fraction numerator sin space h over denominator h end fraction space close parentheses close parentheses

12b
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3 marks

Hence prove that straight f apostrophe open parentheses x close parentheses equals cos space x.

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13a
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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

13b
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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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14a
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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

14b
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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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15
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5 marks

The diagram below shows part of the graph with equation y equals 3 x minus e to the power of x squared end exponent.

q3a-10-1-solving-equations-hard-a-level-maths-pure

Use the trapezium rule with 5 strips to find an estimate for the shaded area, giving your answer to three significant figures.

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16
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5 marks

Use a suitable substitution to find the following

integral 8 x space sin space left parenthesis 3 x squared plus 1 right parenthesis space space straight d x

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17a
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1 mark

A soft ball is thrown upwards from the top of a building.
The height, h m of the ball above the ground after t seconds is modelled by the function

h left parenthesis t right parenthesis equals 15 plus 8.4 t minus 4.9 t squared space space space space space space t greater than 0

What is the significance of the constant 15 in the function?

17b
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2 marks

At what time is the ball at the same height as when it was thrown?

17c
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3 marks

Find the time at which the ball is at its maximum height and what this maximum height is.

17d
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2 marks

How long does it take for the ball to first hit the ground?

17e
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2 marks

Given that the ball first hits the ground at a distance 20 m from the base of the building find the shortest distance between this point and where the ball was thrown from.

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18a
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3 marks

Differentiate with respect to x, simplifying your answers as far as possible:

open parentheses 4 cos space x space minus 3 sin space x close parentheses space e to the power of 3 x space minus space 5 space end exponent

18b
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3 marks

open parentheses x cubed minus 4 x squared plus 7 close parentheses ln space x

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19a
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2 marks

Integrate

integral 2 sin space x space cos space x space space straight d x

19b
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4 marks

Use calculus to find the value of

integral subscript 1 superscript 3 open parentheses 4 x plus 1 close parentheses to the power of 5 space straight d x

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