Practice Paper Pure 2 (Edexcel International A Level Maths: Pure 4)

Practice Paper Questions

1a
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2 marks

Lloyd is to start training in order to run a marathon.

For the first week of training he will run a total of 2 miles.

Each subsequent week he’ll increase the total number of miles run by 3 miles.

He intends training for 15 weeks.

Calculate how far Lloyd will run during his eighth week of training

1b
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2 marks

Work out how much further Lloyd will run in his last week of training compared to his first.

1c
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3 marks

Find the total number of miles Lloyd will run across all 15 weeks of his training schedule.

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2a
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4 marks

On the same axes, sketch the graphs of y equals 4 to the power of x and y equals 5 to the power of x.

Label any points of intersection with the coordinate axes.
Write down the equations of any asymptotes.

2b
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1 mark

Write down an equation for the graph that is a reflection of y equals 4 to the power of x in the y-axis.

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3
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3 marks

Solve 2 to the power of 2 x end exponent minus 24 open parentheses 2 to the power of x close parentheses plus 128 equals 0.

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4
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3 marks

Prove that the square of an even number is a multiple of 4.

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5
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3 marks

Prove that the (positive) difference between an integer and its cube is the product of three consecutive integers.

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6
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3 marks

Find the coefficient of the term in x to the power of 4 in the expansion of left parenthesis 3 plus 2 x right parenthesis to the power of 9.

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7a
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3 marks

Find the first three terms, in ascending powers of x, in the expansion of left parenthesis 5 minus 2 x right parenthesis to the power of 4.

7b
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2 marks

Use your answer to part (a) to estimate left parenthesis 4.5 right parenthesis to the power of 4.

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8
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5 marks

The diagram below shows part of the graph with equation  y equals 3 plus 2 x minus x squared.

8-1-sq-9a-hard-edexelmodelling-with-sequences-and-series-

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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9
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5 marks

Given

integral subscript 1 superscript p left parenthesis 1 plus 1 over x squared right parenthesis space straight d x equals 15 over 4

find the value of the constant p, where p greater than 0.

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10a
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2 marks

Triangle A B C has vertices A(-8, 1), B(12, 16) and C(12, 1).  A circle with equation open parentheses x minus 7 close parentheses squared plus open parentheses y minus 6 close parentheses squared equals 25 touches Triangle A B C at the three points P comma space Qand R, as shown in the diagram below:

q7-3-2-circles-hard-a-level-maths-pure-screenshot

Write down the coordinates of points R and Q.

10b
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5 marks

Find the coordinates of point P.

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11
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3 marks

Show that

2 space log subscript 3 space x plus log subscript 3 open parentheses x squared minus 1 close parentheses minus 2 space log subscript 3 open parentheses x plus 1 close parentheses identical to log subscript 3 fraction numerator x squared open parentheses x minus 1 close parentheses over denominator open parentheses x plus 1 close parentheses end fraction

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12a
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2 marks

straight f left parenthesis x right parenthesis equals 6 x to the power of 4 plus 7 x cubed minus 27 x squared minus 28 x plus 12

Find the remainder when straight f left parenthesis x right parenthesis is divided by left parenthesis 2 x plus 3 right parenthesis.

12b
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5 marks

Given that left parenthesis x plus 2 right parenthesis is a factor of straight f left parenthesis x right parenthesis, factorise straight f left parenthesis x right parenthesis completely.

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13
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3 marks

Solve the equation 3 sin 3 theta space equals space 4 space cos space 3 theta in the interval 0 space less or equal than space theta space less or equal than space pi, giving your answers to 3 significant figures.

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14
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5 marks

Solve the equation 6 spacecos2 2 thetaequalssin 2 theta plus 5  for negative 180 degree less or equal than theta less or equal than 180 degree, giving your answers to 1 decimal place where appropriate.

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15a
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3 marks

The diagram below shows the part of the curve with equation y equals 3 minus 1 fourth x squared for which y greater than 0. The marked point P open parentheses x comma space y close parentheses lies on the curve. O is the origin.

mao-shtQ_q7a-7-2-applications-of-differentiation-medium-a-level-maths-pure

Show thatspace O P squared equals 9 minus 1 half x squared plus 1 over 16 x to the power of 4.

15b
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8 marks

Find the minimum distance from O to the curve, using calculus to prove that your answer is indeed a minimum.

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