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

Practice Paper Questions

1a
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2 marks

Find the first three terms, in ascending powers of x, in the binomial expansion of

               square root of 1 plus 2 x end root 

1b
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1 mark

State the values of x for which your expansion in part (a) is valid.

1c
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1 mark

Using a suitable value of x, use your expansion from part (a) to estimate square root of 1.06 end root, giving your answer to 3 significant figures.

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2
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5 marks

Solve the equation

6 cross times 3 to the power of x minus 1 end exponent equals 6 to the power of 2 x end exponent

giving your answer in the form fraction numerator ln space a over denominator ln space b end fraction, where a and b are integers to be found.

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3
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5 marks

In the triangle ABCstack A B with rightwards arrow on top equals 7 bold i plus bold j minus bold k space and  stack A C with rightwards arrow on top equals negative 2 bold i plus 5 bold k.

 

7R8LyPS9_q1-11-1-vectors-in-2-dimensions-hard-a-level-maths-pure

Show that angle B A C equals 119.6 degree to 1 d.p.

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4a
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2 marks

The function f(x) is defined as

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

 Show that f(x) can be written in the form

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

                             

4b
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2 marks

Explain why the inverse of f(x) does not exist and suggest an adaption to its domain so the inverse does exist.

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5a
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2 marks

Two sequences are being used to model the value of a car, n years after it was new.  At new, the car’s value is £25 000.

Model 1 is an arithmetic sequence.

Model 2 is a geometric sequence.

Both models predict the same value for the car, £7 500, when it is exactly 8 years old.

Find the common difference for Model 1 and the common ratio for Model 2, giving answers to three significant figures where appropriate.

5b
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3 marks

Find the value of the car according to Model 2 in the year Model 1 predicts its value to be £5000.

5c
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1 mark

State one benefit of Model 2 over Model 1 for estimating the value of older cars.

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6a
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3 marks

The alternating voltage,V , in an electrical circuit, t seconds after it is switched on, is modelled by the function

V equals 55 square root of 3 sin πt over 30 plus 55 space cos space πt over 30

Show that

55 square root of 3 sin πt over 30 plus 55 space cos space πt over 30

can be written as

R spacesinopen parentheses πt over 30 plus alpha close parentheses

where R equals 110 and alpha equals straight pi over 6.

6b
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4 marks
(i)
Find the voltage at time t equals 0.

(ii)
Find the voltage after one minute.

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7a
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2 marks

The diagram below shows a sketch of the curves with equations

space space y equals x squared minus 3 x plus 4 and y equals 4 minus x squared plus 2 x

q11-8-2-further-integration-medium-a-level-maths-pure-screenshot

Find the x-coordinates of the intersections of the two graphs.

7b
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2 marks

Show that the area of the shaded region labelled R is given by

integral subscript 0 superscript 5 over 2 end superscript open parentheses 5 x minus 2 x squared close parentheses space d x

7c
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5 marks

Use calculus to find the area of the shaded region labelled R.

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8
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2 marks

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

Using the variables V, for the volume of water in the container, and t, for time, write down a differential equation involving the term fraction numerator d V over denominator d t end fraction, for the volume of water in the container.

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9
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3 marks

Differentiate with respect to x, simplifying your answer:

open parentheses 5 plus sin to the power of 2 space end exponent 3 x close parentheses space e to the power of x squared minus 3 x plus 2 end exponent

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10
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3 marks

The function straight f left parenthesis x right parenthesis is to be transformed by a sequence of functions, in the order detailed below:

  1. A reflection in the y-axis.
  2. A vertical stretch of scale factor 4.
  3. A translation by open parentheses table row 0 row cell negative 2 end cell end table close parentheses.

Write down an expression for the combined transformation in terms of straight f left parenthesis x right parenthesis.

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11
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5 marks

Use calculus and the substitution space x equals cos space theta to find the exact value of

integral subscript fraction numerator 1 over denominator square root of 2 end fraction end subscript superscript fraction numerator square root of 3 over denominator 2 end fraction end superscript fraction numerator 1 over denominator square root of 1 minus x squared end root space end fraction straight d x

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12
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11 marks

The diagram below shows circles C1 and C2 which intersect at the two points A and B. Circle C1 has equation  x squared plus y squared minus 16 x minus 10 y plus 39 equals 0, and points A and B lie along the line with equation  3 x minus y equals negative 1. Circle space C subscript 2 spacealso passes through the point (-13, 2)

HFlQzKQw_q8-3-2-circles-medium-a-level-maths-pure-screenshot

Find an equation of circle C2.

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13a
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3 marks

Show that 6 cos theta minus 8 sin theta can be written in the form  R cosopen parentheses theta plus straight alpha close parentheses, where R greater than 0 and alpha is an acute angle measured in radians.

13b
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3 marks

Hence, or otherwise, solve the equation 3 cos theta minus 4 sin theta minus 2 equals 0,for 0 less or equal than space x less or equal than 2 straight pi.
Give your answers to three significant figures.

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14a
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3 marks

A sequence is defined for k greater or equal than 1 by the recurrence relation

 u subscript k plus 2 end subscript equals u subscript k plus 1 end subscript over u subscript k comma space space space space space space u subscript 1 equals a comma space space space space space space space u subscript 2 equals b

where a and b are real numbers.

Show that the sequence is periodic, and determine its order.

14b
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5 marks

Given that sum from r equals 1 to 44 of space u subscript r space equals negative 50 and   sum from r equals 1 to 84 of space u subscript r space equals negative 92

determine the possible values of a and b.

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15a
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2 marks

A large weather balloon is being inflated at a rate that is inversely proportional to the square of its volume.

Defining variables for the volume of the balloon (m3) and time (seconds) write down a differential equation to describe the relationship between volume and time as the weather balloon is inflated.

15b
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6 marks

Given that initially the balloon may be considered to have a volume of zero, and that after 400 seconds of inflating its volume is 600 m3, find the particular solution to your differential equation.

15c
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2 marks

Although it can be inflated further, the balloon is considered ready for release when its volume reaches 1250 m3.  If the balloon needs to be ready for a midday release, what is the latest time that it can start being inflated?

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16
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8 marks

The curve C is defined by

e to the power of sin space x y end exponent space equals 1 space space space space space space space space space space space space space space open curly brackets space y greater than 0 close curly brackets

Points A and B have coordinates open parentheses pi over 2 space comma space 2 close parentheses and open parentheses negative pi over 2 space comma space 2 close parentheses respectively.

The tangents to C at points A and B intersect at the point P.
The tangent to C at point A intersects the x-axis at point Q.
The tangent to C at point B intersects the x-axis at point R.

Find the area of triangle PQR.

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17
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4 marks

Below is a proof by contradiction that there is no largest multiple of 7.

Line 1: Assume there is a number, S, say, that is the largest multiple of 7.
Line 2: S equals 7 k
Line 3: Consider the number S plus 7.
Line 4: S plus 7 equals 7 k plus 7
Line 5: therefore S plus 7 equals 7 left parenthesis k plus 1 right parenthesis
Line 6: So S plus 7 is a multiple of 7.
Line 7: This is a contradiction to the assumption that S is the largest multiple of 7.
Line 8: Therefore, there is no largest multiple of 7.

The proof contains two omissions in its argument.

Identify both omissions and correct them.

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