CIE A Level Maths: Pure 1

Topic Questions

1.1 Quadratics

1
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3 marks

Expand and simplify

(i)
 left parenthesis x plus 4 right parenthesis left parenthesis 2 x minus 3 right parenthesis

(ii)
 left parenthesis 3 x minus 4 right parenthesis left parenthesis 3 x plus 4 right parenthesis

(iii)
 left parenthesis 2 x plus 1 right parenthesis squared

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2
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3 marks

Factorise

(i)
 x squared plus 5 x minus 14
(ii)
 25 x squared minus 36
(iii)
 2 x squared plus 9 x plus 9

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3
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3 marks

Complete the square for

(i)
 x squared plus 8 x minus 4
(ii)
 2 x squared plus 12 x minus 5
(iii)
 5 x squared minus 3 x plus 2

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4
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3 marks

Solve

(i)
 x squared plus 8 x minus 9 equals 0
(ii)
 3 x squared minus 13 x plus 4 equals 0
(iii)
 4 x squared minus 6 x minus 5 equals 0

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5
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3 marks

Write down the value of the discriminant of

(i)
 x squared minus 3 x plus 4
(ii)
 4 x plus 3 minus 2 x squared
(iii)
 5 minus 8 x plus 2 x squared

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6a
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3 marks
(i)
Write down the y-axis intercept on the graph of y equals 2 x squared plus 5 x minus 3.

(ii)
Find the roots of y equals 2 x squared plus 5 x minus 3.
6b
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3 marks

Sketch the graph of y equals 2 x squared plus 5 x minus 3, labelling all points where the graph crosses the coordinate axes.

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7
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2 marks

The function straight f left parenthesis x right parenthesis equals x squared plus k x plus 3 has no real roots.

Show that k squared less than 12.

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8a
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2 marks

Write x squared plus 10 x plus 24 in the form left parenthesis x plus a right parenthesis squared plus b, where a are b constants to be found.

8b
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1 mark

Hence write down the minimum point on the graph of y equals x squared plus 10 x plus 24.

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9
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2 marks

The function straight f left parenthesis x right parenthesis equals k x squared plus 2 k x minus 3 has two distinct real roots.

Show that 4 k left parenthesis k plus 3 right parenthesis greater than 0.

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10a
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3 marks
(i)
Find the roots of the function straight g left parenthesis x right parenthesis equals 12 plus 4 x minus x squared.

(ii)
Write down the y-axis intercept on the graph of y equals straight g left parenthesis x right parenthesis.
10b
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3 marks
(i)
Write straight g left parenthesis x right parenthesis space in the form a minus left parenthesis x minus b right parenthesis squared, where a are b constants to be found.

(ii)
Hence write down the coordinates of the turning point on the graph of space y equals straight g left parenthesis x right parenthesis.
10c
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2 marks

Sketch the graph of space y equals straight g left parenthesis x right parenthesis, labelling all points where the graph intercepts the coordinate axes and the turning point.

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11
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3 marks

Sketch the graph of y equals left parenthesis 2 x minus 5 right parenthesis squared, labelling any points where the graph intercepts the coordinate axes.

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12
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1 mark

Without showing it algebraically, explain how you know that the function straight f left parenthesis x right parenthesis equals left parenthesis a x minus b right parenthesis squared has a discriminant of zero.

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1a
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2 marks

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

1b
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1 mark

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

1c
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3 marks

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

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2a
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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.

2b
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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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3a
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2 marks

The equation y equals 3 x squared plus 2 p x plus 4 q has no real roots.  Show that p squared less than 12 q.

3b
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3 marks

Given that the curve y equals 3 x squared plus 2 p x plus 4 q passes through left parenthesis negative 2 comma space 6 right parenthesis space and left parenthesis 2 comma space 6 right parenthesis find the values of p and q.

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4a
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2 marks

The equation 2 k minus 3 k x minus x squared equals 0 has two distinct real roots.
k is a negative constant.

Find the possible values of k.

4b
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3 marks

In the case k equals negative 1 space sketch the graph of y equals 2 k minus 3 k x minus x squared, labelling all points where the graph crosses the coordinate axes.

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5a
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2 marks

Find the minimum value of the function straight f left parenthesis x right parenthesis equals x squared plus 4 x plus c, giving your answer in terms of c.

5b
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2 marks

Given that c equals 5, hence, or otherwise, show that the function straight f left parenthesis x right parenthesis equals x squared plus 4 x plus c has no real roots.

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6
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3 marks

Sketch the graph of space y equals 12 x squared minus 5 x minus 72, labelling any points where the graph crosses the coordinate axes.  (You do not need to label the turning point.)

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7
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3 marks

The function straight f left parenthesis x right parenthesis equals k x squared plus 2 k x minus 3 has two distinct real roots.

The function straight g left parenthesis x right parenthesis equals k x squared plus 4 k x minus 16 has no real roots.

Find the possible values of k.

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8a
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1 mark

The graph below shows the curve space y equals straight f left parenthesis x right parenthesis where f left parenthesis x right parenthesis equals 5 minus x squared over 6.

The curve is to be used as the model for the arch on a bridge where the water level under the bridge is represented by the x-axis. All measurements are in meters.

2-2-edexcel-alevel-maths-pure-q8hard

Write down the maximum height of the bridge above the water.

8b
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2 marks

Is the bridge wide enough to span a river of width 11 m?

8c
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2 marks

A second bridge is modelled by the curve space y equals straight f left parenthesis x right parenthesis space where straight f left parenthesis x right parenthesis equals 4 minus x squared over 8. To support the bridge the arch will continue 2 m under the water (ground) level.

Find the distance between the base of the arch on either side of the river.

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9a
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3 marks

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

9b
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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.

9c
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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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10
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3 marks

The diagram below shows the graph of y equals straight f left parenthesis x right parenthesis, where space f left parenthesis x right parenthesis is a quadratic function.  The intercepts with the coordinate axes and the turning point have been labelled.

2-2-edexcel-alevel-maths-pure-q10hard

Sketch the graph of space y equals f left parenthesis x plus 3 right parenthesis, stating the coordinates of any points that intersect the coordinate axes and the turning point.

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11a
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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 24 plus 2 t minus 0.5 t squared, t greater or equal than 0, where his 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.

11b
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2 marks

Find the maximum height the stone reaches above the sea.

11c
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2 marks

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

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1a
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3 marks

The curve C has equation y equals x squared minus 3 x plus 2.

Find the coordinates of any points where C intersects the coordinate axes.

1b
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3 marks

Sketch the graph of C, showing clearly all points of intersection with the coordinate axes.

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2a
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2 marks

Write the quadratic function y equals x squared plus 8 x minus 9 in the form y equals a left parenthesis x plus b right parenthesis squared plus c where a, b and c are integers to be found.

2b
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1 mark

Write down the minimum point on the graph of y equals x squared plus 8 x minus 9.

2c
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3 marks

Sketch the graph of y equals x squared plus 8 x minus 9, clearly labelling the minimum point and any point where the graph intersects the coordinate axes.

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3a
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2 marks

Solve the equation 2 x squared plus x minus 6 equals 0.

3b
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3 marks

Find the coordinates of the turning point on the graph of y equals 2 x squared plus x minus 6.

3c
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2 marks

Sketch the graph of y equals 2 x squared plus x minus 6, labelling the turning point and any points where the graph crosses the coordinate axes.

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4a
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3 marks

Find the minimum value of the function straight f left parenthesis x right parenthesis equals x squared plus 4 x plus 5.

4b
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2 marks

Hence, or otherwise, prove that the function straight f left parenthesis x right parenthesis equals x squared plus 4 x plus 5 has no real roots.

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5
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3 marks

The function straight f left parenthesis x right parenthesis equals k x squared plus 2 k x minus 3 has two distinct real roots.

Show that k less than negative 3 space or space k greater than 0.

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6
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3 marks

The equation 2 x squared minus 4 x plus 3 minus 2 k equals 0 has real roots.

Find the possible values of k.

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7
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2 marks

The equation y equals x squared plus p x plus q has no real roots. Show that p squared less than 4 q.

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8a
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1 mark

The graph below shows the curve straight f left parenthesis x right parenthesis equals 4 minus x squared over 8.

The curve is to be used as the model for the arch on a bridge where the water level under the bridge is represented by the x-axis. All measurements are in meters.

q8a-2-2-quadratics-medium-a-level-maths-pure-2

Write down the maximum height of the bridge above the water.

8b
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3 marks

Is the bridge wide enough to span a river of width 11 m?

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9
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3 marks

The diagram below shows the graph of y equals straight f left parenthesis x right parenthesis, where straight f left parenthesis x right parenthesis is a quadratic function.
The intercepts with the x-axis and the turning point have been labelled.

2-2-edexcel-alevel-maths-pure-q9medium

Sketch the graph of y equals straight f left parenthesis x plus 2 right parenthesis, stating the coordinates of any points that intersect the x-axis and the coordinates of the turning point.

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10
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3 marks

Solve the equation x to the power of 4 minus 13 x squared plus 36 equals 0.

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11
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4 marks

Solve x to the power of 2 over 5 end exponent plus x to the power of 1 fifth end exponent equals 6.

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1a
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2 marks

Write the quadratic function space y equals negative 6 x squared plus 8 x minus 5 in the form space y equals a minus b left parenthesis x plus c right parenthesis squared where a, b and c are constants to be found.

1b
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1 mark

Write down the maximum point on the graph of space y equals negative 6 x squared plus 8 x minus 5.

1c
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3 marks

Sketch the graph of y equals negative 6 x squared plus 8 x minus 5, clearly labelling the maximum point and any point where the graph intersects the coordinate axes.

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2a
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2 marks

The equation y equals x squared plus p x plus q has no real roots. Show that p squared less than 4 q and explain why q must be a positive value.

2b
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2 marks

Given that the minimum point on the graph of y equals x squared plus p x plus q is (3, 1) find the values of p and q.

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3
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2 marks

The equation k squared x squared minus 4 x plus 5 equals k squared has two distinct real roots.

Find the possible values of k.

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4a
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2 marks

The equation 4 k minus 6 k x minus x squared equals 0 has two distinct real roots, alpha space and space beta.
k is a negative constant and 0 less than alpha less than beta.

Sketch the graph of y equals 4 k minus 6 k x minus x squared, labelling the points where the graph crosses the coordinate axes.

4b
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3 marks

Find the possible values of k.

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5a
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2 marks

Find the minimum value of the function straight f left parenthesis x right parenthesis equals x squared plus 8 x plus c, giving your answer in terms of c.

5b
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2 marks

Find the values of c for which the function straight f left parenthesis x right parenthesis equals x squared plus 8 x plus c has no real roots.

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6a
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2 marks

The graph below shows the curve space y equals straight f left parenthesis x right parenthesis where straight f open parentheses x close parentheses equals 4 minus x squared over 8.

The curve is to be used as the model for the arch on a bridge where the water level under the bridge is represented by the x-axis. All measurements are in meters.

2-2-edexcel-alevel-maths-pure-q6vhard

Depending on rainfall throughout the year, the water level can rise by up to 0.5 m, determine whether the bridge is still wide enough to span a river of width 11 m when it is at its peak height.

6b
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2 marks

A barge in the shape of a cuboid (above water level) has a cross-section measuring 6 m wide by 2.5 m tall. The barge regularly travels along the river where the bridge is to be built.  Justifying your answer, determine if the barge will fit underneath the bridge or not.

6c
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2 marks

To support the bridge the arch will continue 2.5 m under the water (ground) level.

Find the exact distance between the base of the arch on either side of the river.

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7a
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2 marks

Show that the equation a x squared plus b x plus c equals 0 can be written in the form

a left parenthesis x plus fraction numerator b over denominator 2 a end fraction right parenthesis squared minus fraction numerator left parenthesis b squared minus 4 a c right parenthesis over denominator 4 a end fraction equals 0

7b
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2 marks

Hence show that x equals fraction numerator negative b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction.

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8a
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3 marks

The function straight f open parentheses x close parentheses is defined by straight f left parenthesis x right parenthesis equals left parenthesis k minus 1 right parenthesis x squared minus left parenthesis k minus 2 right parenthesis x minus 2 k comma space x element of straight real numbers.

The function straight g left parenthesis x right parenthesis is defined by straight g left parenthesis x right parenthesis equals left parenthesis k minus 1 right parenthesis x squared minus 3 k x plus k plus 1 comma space x element of straight real numbers.

 k is a non-zero constant and k not equal to 1.

The graphs of space y equals straight f left parenthesis x right parenthesis space space and space y equals straight g left parenthesis x right parenthesis space intersect once.
Find the x-coordinate of the intersection, giving your answer in terms of k.

8b
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2 marks

In the case when k equals 3, find the coordinates of the point of intersection of space y equals straight f left parenthesis x right parenthesis space space and space y equals straight g left parenthesis x right parenthesis space.

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9a
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3 marks

Solve the equation 8 square root of x equals 48 minus x.

9b
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3 marks

Solve the equation 2 to the power of 4 x end exponent plus 64 equals 20 left parenthesis 2 to the power of 2 x end exponent right parenthesis.

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10
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3 marks

The diagram below shows the graph of space y equals f left parenthesis x right parenthesis. The intercepts with the coordinate axes and the turning point have been labelled.

2-2-edexcel-alevel-maths-pure-q10vhard

The graph is transformed by the function space y equals f left parenthesis x right parenthesis plus 6. One of the new x-axis intercepts is (-2, 0).

Sketch the graph of space y equals f left parenthesis x right parenthesis plus 6, stating the coordinates of any points that intersect the coordinate axes and the turning point.

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11a
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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.

11b
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2 marks

Find the maximum height the stone reaches above the sea.

11c
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2 marks

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

11d
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2 marks

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

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12a
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1 mark

Factorise x squared plus 6 x plus 9

12b
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2 marks

Factorise x squared plus 6 x y plus 9 y squared

12c
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2 marks

Find a relationship between x and y such that x squared plus 6 x y plus 9 y squared equals 0

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