Practice Paper Pure 2 (Edexcel International A Level Maths: Pure 2)

Practice Paper Questions

1a
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3 marks

Find the first three terms in the expansion of left parenthesis 4 space long dash space 3 x right parenthesis to the power of 9.

1b
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3 marks

Given that x is small such that x cubed and higher powers of x can be ignored show that 

         open parentheses 3 space long dash space 2 x squared space close parentheses open parentheses 4 minus space 3 x close parentheses to the power of 9 almost equal to space 786432 space minus space 5308416 x space plus space 15400960 x squared

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2a
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4 marks

The trapezium rule is to be used to find an estimate for the integral

 

integral subscript 4 superscript 8 space straight f left parenthesis x right parenthesis space space straight d x

The table below shows values for x and f(x), rounded to three significant figures where appropriate.

x 

4

4.5

5

5.5

6

6.5

7

7.5

8

 f(x)

3.16

3.39

3.61

3.81

4

4.18

4.36

4.53

4.69

Using the values in the table find 

(i)      an estimate for the integral using 2 strips,
(ii)     an estimate for the integral using 4 strips,
(iii)    an estimate for the integral using 8 strips.

2b
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2 marks

Justify which of the estimates from part (a) will be the most accurate estimate for the integral.

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3a
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5 marks

The function f left parenthesis x right parenthesis equals 2 x cubed minus 5 x squared plus a x plus b  has left parenthesis 2 x plus 3 right parenthesis as a factor, and when f left parenthesis x right parenthesis is divided by left parenthesis x minus 2 right parenthesis the remainder is 7. 

Show that a and b must satisfy the simultaneous equations:  

2 a plus b equals 11

3 a minus 2 b equals negative 36

3b
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2 marks

Hence find a and b.

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4a
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3 marks

A curve has the equation y equals x cubed minus 12 x plus 7.

Find expressions for fraction numerator d y over denominator d x end fraction and  fraction numerator d squared y over denominator d x squared end fraction.

4b
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3 marks

Determine the coordinates of the local minimum of the curve.

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5a
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4 marks

A circle C has centre (-2, 3) and passes through the point P(6, -3)

Find an equation for the circle C.

5b
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3 marks

Find an equation for the tangent to the circle at P.

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6a
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2 marks

A training track for cyclists is in the shape of a circle made up of several laneF9HMZxJf_image

One lap of the inner lane is 10 pi m, with each lane working outwards having a lap distance of 5 pi m more than the lane immediately inside it.

During a training session, a cyclist is expected to complete one lap of each lane, starting with the inner lane, before moving onto the next one.

A cyclist trains until they have completed the first five lanes.
Find the total distance travelled by the cyclist.

6b
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2 marks

There are 8 lanes in total on the training track.

Find the lap distance of the outside lane.

6c
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3 marks

During a particular training session, a cyclist completes a lap of each lane but in addition, also completes a further 10 laps of the outside lane.
Find the total distance the cyclist travels during this training session.

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7
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8 marks

The diagram below shows the graphs of  y equals 12 x minus x squared minus 27 space and space y equals x squared minus 14 x plus 45.

screenshot-2023-08-09-at-12-10-41-pm

Find the area of the shaded region, R.

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8
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2 marks

Show that

                                             fraction numerator 1 minus cos to the power of 2 space end exponent x over denominator tan squared space x end fraction identical to cos to the power of 2 space end exponent x

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9
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5 marks

Solve the equation tan  2 x equals fraction numerator 3 over denominator tan space 2 x end fraction  for negative 180 degree less or equal than space x less or equal than 180 degree .

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10a
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2 marks

A convergent geometric series has first term 64, and the sum to infinity of the series is 384.

Show that the common ratio, r, of the series is 5 over 6.  

10b
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2 marks

Find the difference between the ninth and tenth terms of the series, giving your answer correct to 3 significant figures.

10c
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2 marks

Calculate the sum of the first eight terms in the series, giving your answer correct to 3 significant figures.

10d
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4 marks

Given that the sum of the first k terms of the series is greater than 380, find the smallest possible value of k.

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11
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3 marks

Solve the equation

log subscript 3 open parentheses x plus 4 close parentheses equals 4 plus 2 space log subscript 3 space x

giving your answers correct to 3 significant figures.

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12
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8 marks

Solve the equation

2 cross times 5 to the power of 2 x plus 1 end exponent plus 21 equals 41 cross times 5 to the power of x

giving your answers in the form  log subscript a space b, where a and b are rational numbers to be found.

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