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

Practice Paper Questions

1
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6 marks

The curve C has equation y equals space fraction numerator 9 over denominator square root of 3 x end root end fraction minus 3 over x.  The point P open parentheses 3 comma space 2 close parentheses lies on C.

The normal to C at P intersects the x-axis at the point Q.

Find the coordinates of Q.

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2
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2 marks

Factorise completely 15 x cubed plus 19 x squared minus 10 x.

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3a
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3 marks

Solve the equation 5 square root of x plus 3 equals 2 x.

3b
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3 marks

Solve x to the power of 2 over 3 end exponent plus 2 x to the power of 1 third end exponent equals 8.

3c
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3 marks

Solve the equation 2 to the power of 2 x end exponent plus 64 equals 20 left parenthesis 2 to the power of x right parenthesis.

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4a
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3 marks

A triangle ABC has sides A B equals 2 x space c m, A C equals open parentheses x plus 6 close parentheses space c m and angle B A C equals 30 to the power of degree.

The area of the triangle is  5 begin inline style 3 over 4 end style cm2.

Show that x satisfies the equation 2 x squared plus 12 x minus 23 equals 0.

4b
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3 marks

Hence, or otherwise, find the perimeter of the triangle. Giving your answer to 3 significant figures.

4c
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3 marks

Given angle ABC is obtuse, find the ratio of the angles of the triangle, to the nearest degree.

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5
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4 marks

Find the equation of the curve passing through the point (4, -8) and given by

y equals integral open parentheses fraction numerator 2 over denominator square root of x end fraction minus x minus 3 close parentheses space straight d x

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6a
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1 mark

A stone is thrown vertically upwards from the top of a cliff.  The path of the stone is modelled by the quadratic function h left parenthesis t right parenthesis equals 52 plus 3 t minus 0.5 t squared, t greater or equal than 0, where h is the height, in meters, of the stone above the sea and t is the time in seconds since the stone was thrown.

Write down the height of the cliff from which the stone was thrown.

6b
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2 marks

Find the maximum height the stone reaches above the sea.

6c
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2 marks

How long does it take for the stone to hit the sea?

6d
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2 marks

How long does the stone stay above it’s starting height for?

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7
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5 marks

A firework is launched inside a large shed with a sloping roof.  In relation to the horizontal distance from the point it was launched, the height of the firework, h m, can be modelled by the quadratic equation

 h equals 0.84 x minus 0.07 x squared

The sloping roof of the shed can be modelled with the equation

 h equals 2 plus 0.1 x

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Determine whether, according to the model, the firework will hit the roof of the shed before escaping out the open end of the shed on the right of the diagram.

                      

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8
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5 marks

The curve C has equation y equals 3 x squared minus 6 x plus square root of 2 x end root.  The point P(2,  2) lies on C.

Find an equation of the tangent to C at P.

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9a
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2 marks

Show that fraction numerator open parentheses 5 plus 2 square root of x close parentheses squared over denominator square root of x end fraction  can be written as 25 x to the power of negative 1 half end exponent plus 20 plus 4 x to the power of 1 half end exponent.

9b
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3 marks

Show that fraction numerator 729 over denominator 4 square root of 3 minus square root of 27 end fraction equals 3 to the power of a, where a is a rational number to be found.

9c
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2 marks

Show that  fraction numerator 2 square root of x over denominator square root of 2 end fraction open parentheses square root of 8 x plus 6 x close parentheses  can be written as a y squared plus b y cubed, where space y equals square root of 2 x end root and a and b are integers to be found.

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10
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5 marks

The point P open parentheses negative 5 comma negative 2 close parentheses lies on the line l subscript 1, l subscript 1 crosses the x-axis at the point R.

Another line, l subscript 2, is perpendicular to l subscript 1 at the point P and crosses the x-axis at the point Q(-1, 0).

Find the area of the triangle P Q R.

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11
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5 marks

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The diagram shows a prism with cross-section in the shape of a sector of a circle. 

The radius of the sector is open parentheses 2 a minus 3 close parentheses cm, and the angle at the centre is 1.7 radians.

The height of the prism is 4 cm.

Given that the volume of the prism is 7.65 cm3, find the possible value(s) of a.

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12a
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4 marks

On the same diagram, sketch the graphs of space y equals x cubed minus 3 x squared minus 6 x plus 8 and space y equals 3 over x squared.

12b
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1 mark

Write down the number of solutions to the equation

x to the power of 5 minus 3 x to the power of 4 minus 6 x cubed plus 8 x squared equals 3.

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13a
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4 marks

The graph below shows two curves with equations y equals p space sin space x and y equals cos open parentheses x plus q degree close parentheses, in the interval negative 180 degree less or equal than space x less or equal than 180 degree, where p and q are integers.

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Using the graph above, find the values of p and q and label the points of intersection each graph has with the coordinate axes.

13b
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2 marks

Within the stated interval, the curves intersect at the two points R and S as shown in the diagram. The coordinates of point R are (9.90°, 0.34), accurate to 2 decimal places. By considering the graph, as well as the properties of the sine and cosine functions, state the coordinates of Point S, to two decimal places.

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