Practice Paper 1 (Pure 1) (CIE AS Maths: Pure 2)

Practice Paper Questions

1a
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3 marks

Find the first three terms in the expansion of left parenthesis 2 space long dash space 3 x right parenthesis to the power of 7.

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 1 space long dash 2 x close parentheses space open parentheses 2 long dash space 3 x close parentheses to the power of 7 almost equal to 128 space long dash space 1600 x space plus space 8736 x squared

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2a
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2 marks

The equation 2 k space long dash space 3 k x minus x squared space equals 0 has two distinct real roots.

k is a negative constant.

Find the possible values of k.

2b
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3 marks

In the case space k space equals space long dash 1 spacesketch the graph of y space equals space 2 k space long dash space 3 k x space long dash space x squared, labelling all points where the graph crosses the coordinate axes.

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3a
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2 marks

Show that the equation 2 space sin squared space x space plus space 3 space cos space x space equals space 0 can be written in the form a space cos squared space x space plus space b space cos space x space plus space c space equals space 0 , where ab and c are integers to be found.

3b
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3 marks

Hence, or otherwise, solve the equation 2 space sin squared x space plus space 3 space cos space x space equals 0 spacefor negative 180 degree less or equal than space x space less or equal than 180 degree

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4
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4 marks

The sum of the first three terms in a geometric series is 8.75.
The sum of the first six terms in the same series is 13.23.

Find the common ratio, r, of the series.

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5a
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4 marks

On the same set of axes, sketch the graphs of y space equals space sin 1 half theta space and space y space equals space sin left parenthesis theta space plus space 30 degree right parenthesis in the interval long dash 270 degree space less or equal than theta less or equal than space 270 degree. Label the coordinates of points of intersection with the coordinate axes and of maximum and minimum points where appropriate.

5b
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2 marks

Find the solution to the equation sin 1 half theta space equals space sin open parentheses theta plus 30 degree close parentheses within the interval negative 90 degree less or equal than theta less or equal than 0 degree. Hence, determine the coordinates of the corresponding point of intersection between the two graphs in part (a).

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6a
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5 marks

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

q6a-practicepaper1-seta-cie-alevel-pure

Given that the area of triangle O A B = 5.64 cm2, find the area of the shaded segment.
Give your answer correct to 3 significant figures.

6b
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3 marks

Find the perimeter of the sector O A B, giving your answer correct to 3 significant figures.

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7a
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2 marks

Triangle space A B C space has vertices A(—8, 1), B(12, 16) and C(12, 1). A circle with equation left parenthesis x space long dash space 7 right parenthesis squared space plus space open parentheses y space long dash 6 space close parentheses squared equals space 25 touches Triangle space A B C spaceat the three points P comma space Q and R, as shown in the diagram below:

q7a-practicepaper1-seta-cie-alevel-pure

Write down the coordinates of points R and Q.

7b
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5 marks

Find the coordinates of point P.

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8a
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2 marks

Write the quadratic function y space equals space 4 x squared space plus space 8 x long dash space 5 in the form y space equals space a left parenthesis x space plus space b right parenthesis squared space plus space c where a comma space b and c are integers to be found.

8b
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1 mark

Write down the minimum point on the graph of y space equals space 4 x squared space plus space 8 x space long dash space 5.

8c
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3 marks

Sketch the graph ofspace y space equals space 4 x squared space plus space 8 x space long dash space 5, clearly labelling the minimum point and any point where the graph intersects the coordinate axes.

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9a
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2 marks

The point P left parenthesis long dash 12 comma space long dash 9 right parenthesis lies on the curve with equation y space equals space x squared space plus space 15 x space plus space 27.

The graph is translated so that the point P is mapped to the point left parenthesis long dash 12 comma space 3 right parenthesis.

Write down the equation of the transformed function.

9b
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2 marks

The graph is translated so that the point P is mapped to the point left parenthesis long dash 10 comma space long dash 9 right parenthesis.

Write down the equation of the transformed function in the form y space equals space left parenthesis x space plus space a right parenthesis squared space plus space 15 left parenthesis x space plus space a right parenthesis space plus space 27, where a is a constant to be found.

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10
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5 marks

A function, straight f left parenthesis x right parenthesis, has second derivative given by

straight f space " left parenthesis x right parenthesis space equals space 6 left parenthesis x space long dash space 2 right parenthesis.

Given that space straight f left parenthesis 3 right parenthesis space equals space 20, and straight f space apostrophe left parenthesis 2 right parenthesis equals space 8 , find straight f left parenthesis x right parenthesis.

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11
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5 marks

A curve has the equation y space equals space x left parenthesis x space plus space 6 right parenthesis squared space plus space 4 left parenthesis 3 x space plus space 11 right parenthesis.

The point P left parenthesis x comma space y right parenthesis is the stationary point of the curve.

Find the coordinates of P and determine its nature.

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12
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5 marks

Use calculus to find the value of

      integral subscript 4 superscript 9 space fraction numerator x squared space plus space 1 over denominator square root of x end fraction space straight d x

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13
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4 marks

The diagram below shows part of the curve C defined by the equation y space equals space 1 over a x squared comma where a is a positive constant. The shaded region R is bounded by the curve, the x-axis, and the lines x space equals space 1 and space x space equals space 6.

q13-practicepaper1-seta-cie-alevel-pure

Given that the volume of the solid formed when the region R is rotated 360 degree about the x-axis is  fraction numerator 311 straight pi over denominator 20 end fraction cubic units, find the value of a. 

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14
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5 marks

The curve C has equation space y space equals space 3 x squared space long dash space 6 x space plus space square root of 2 x end root. The point P left parenthesis 2 comma space 2 right parenthesis lies on C.

Find an equation of the tangent to C at P.

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