Practice Paper 1 (Pure 1) (CIE AS Maths: Pure 1)

Practice Paper Questions

1a
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3 marks

The fifth term of an arithmetic series is k, where k is a constant, and the sum of the first eight terms of the series is 2 k.

Show that the first term, a, of the series is long dash 5 k.

1b
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2 marks

Find an expression for the common difference, d, of the series in terms of k.

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2a
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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.

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

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3a
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2 marks

A ball is dropped and bounces such that the height of each bounce is 80% of the height of the previous bounce.

The first bounce of the ball reaches a height of 1.60 m.

Find the height the ball bounces on its 8th bounce.

3b
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2 marks

Find the total vertical distance travelled by the ball from when it first hits the ground until it hits the ground at the end of its twentieth bounce.

3c
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1 mark

Give one reason why this model may be unrealistic.

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4a
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2 marks

The point P(3, —12) lies on the curve with equation y space equals space x squared minus space 12 x space plus space 15.

The graph is stretched so that the point P is mapped to the point (3, —4). Write down the equation of the transformed function in the form y space equals space a x squared space plus space b x space plus space c, where a, space b spaceand c are constants to be found.

4b
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2 marks

The graph is stretched so that the point P spaceis mapped to the point left parenthesis 1 comma negative 12 right parenthesis. Write down the equation of the transformed function in the form y equals left parenthesis d x right parenthesis squared minus 12 left parenthesis d x right parenthesis plus 15, where d is a constant to be found.

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5a
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3 marks

The curve C has equation y equals x squared minus 3 x plus 2.  The line l has equation space y equals 3 x minus 7.

Find any points of intersection between C and l.

5b
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3 marks

Sketch the graphs of C and l, showing clearly any points of intersection with the coordinate axes for both graphs, the minimum point of C and any points of intersection found between C and l.

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6a
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1 mark

The functions straight f open parentheses straight x close parentheses and straight g open parentheses x close parentheses are defined as follows

            straight f left parenthesis x right parenthesis equals 3 x squared plus 2            x space element of space straight real numbers

            straight g open parentheses x close parentheses space equals space 1 space minus space 3 x         x space element of space straight real numbers

Write down the range of straight f space open parentheses straight x close parentheses.

6b
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4 marks

Find

(i)
fg open parentheses x close parentheses
(ii)
gf open parentheses x close parentheses
6c
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2 marks

Solve the equation straight f open parentheses x close parentheses space equals space straight g open parentheses x close parentheses space plus space 1

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7
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2 marks

The graph below shows the curve with equation y equals 2 space sin space theta, in the interval  0 degree less or equal than theta less or equal than 360 degree. One value of theta has been labelled open parentheses theta equals a degree close parentheses.

q9-5-2-trigonometry-functions-edexcel-a-level-pure-maths-veryhard 

Use the graph, along with the symmetry properties of the sine function, to verify that 

                              2 space sin space a equals 2 space sin open parentheses 180 degree minus a close parentheses equals negative 2 space sin open parentheses 180 degree plus a close parentheses equals negative 2 space sin open parentheses 360 degree minus a close parentheses.

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8
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3 marks

Solve the equation 2 space sin space 2 theta space equals space 1 spacefor 0 space less or equal than theta space less or equal than 2 straight pi.

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9a
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3 marks

The diagram below shows the sector of a circle O A B.

q5a-5-4-radian-measure-a-level-only-edexcel-a-level-pure-maths-veryhard

Show that the area of the shaded segment is given by  1 half r squared open parentheses theta minus sin space theta close parentheses cm2.

9b
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2 marks

Find, in terms of theta, the percentage of the sector that the segment occupies.

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10a
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3 marks

A curve is described by the equation y equals straight f left parenthesis x right parenthesis, wherespace straight f open parentheses x close parentheses equals 7 minus 2 x squared plus square root of x space comma space x greater or equal than 0.

Find straight f apostrophe open parentheses x close parentheses and straight f apostrophe apostrophe open parentheses x close parentheses.

10b
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4 marks

P is the stationary point on the curve.

Find the coordinates of P and determine its nature.

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11
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7 marks

A circle has equation x squared plus y squared plus 14 x minus 6 y equals negative 41.

The lines l subscript 1and l subscript 2 are both tangents to the circle, and they intersect at the point (0, 14).

q8-3-2-circles-hard-a-level-maths-pure-screenshot

Find the equations of l subscript 1 and l subscript 2, giving your answers in the form y equals m x plus c.

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12a
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5 marks

The line with equation  5 y equals 14 minus 3 x  cuts the curve with equation  5 y equals 20 minus 2 x minus x squared at the points P and Q, as shown.

q9-8-1-integration-hard-a-level-maths-pure-screenshot

Find the x- and y-coordinates of the points P and Q.

12b
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6 marks

Find the exact area of the region labelled R, giving your answer in the form  a over b, where a and b are integers to be found.

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13a
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4 marks

A mathematical model for a bowl is obtained by rotating through 360 degree about the y-axis the part of the curve y space equals space 1 over a x squared  which is between y space equals space 8 and  y space equals space 9 and then adding a flat bottom.  a is a constant such that a space greater than space 0.

Find the capacity of the bowl in terms of a.

13b
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3 marks

In the case where the boundaries  y space equals space 3 and y space equals space 6 are used in place of y space equals space 8 and y space space equals space 9,  the capacity of the bowl is increased by 15 straight pi cubic units.
Find the value of a.

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