Practice Paper Pure 1 (Edexcel International AS Maths: Pure 2)

Practice Paper Questions

1a
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2 marks

Use calculus to work out the following integrals:

integral open parentheses 1 third x squared space space plus space 3 x to the power of negative 1 half end exponent close parentheses space d x

1b
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3 marks

integral fraction numerator open parentheses x space plus space 3 close parentheses squared over denominator square root of x end fraction d x

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2a
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2 marks

A curve has the equation y equals 3 x minus 4 x to the power of negative 2 end exponent comma space space x not equal to 0.

Find fraction numerator d y over denominator d x end fraction.

2b
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2 marks

Find the coordinates of the point on the curve where the gradient is 2.

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3
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4 marks

Solve the inequality fraction numerator 5 over denominator 3 x squared plus 2 end fraction less or equal than 2.

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4
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3 marks

Write down the inequalities that define the region R shown in the diagram below.

2-4-q8-inequalities-a-level-maths

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5
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5 marks
(i)

On the same set of axes, sketch the graphs of y equals cos space theta and y equals cos space 3 theta in the interval negative 180 degree less or equal than theta less or equal than 180 degree, giving the coordinates of all points of intersection with the coordinate axes.

(ii)
State the coordinates of any points where the graphs intersect.

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6a
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4 marks

Two perpendicular lines l subscript 1 and l subscript 2 intersect at point P(2, 5).

l subscript 2 crosses the x-axis at point Q(-3, 0).

Find an equation for the line l subscript 1.

6b
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4 marks

l subscript 1 crosses the x-axis at the point R.
Find the area of the triangle P Q R.

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7
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3 marks

Sketch the graph of y equals 2 x squared left parenthesis x plus 3 right parenthesis labelling any points where the graph intersects the coordinate axes.

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8a
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2 marks

The curve C has equation y equals 3 x squared minus 6 plus 4 over x.  The point Popen parentheses 1 comma space 1 close parentheses lies on C.

Find an expression for fraction numerator d y over denominator d x end fraction.

8b
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3 marks

Show that an equation of the normal to C at point is x plus 2 y equals 3.

8c
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2 marks

This normal cuts the x-axis at the point Q.

Find the length of PQ, giving your answer as an exact value.

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9a
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5 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-hard

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

9b
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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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10a
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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.

10b
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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.

10c
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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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11
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4 marks

The line l passes through the points (4, 2) and (8, 5).

Find the equation of the line l, giving your answer in the form a x plus b y plus c equals 0
where a,b and c are integers to be found.

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12a
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6 marks

Given that straight f left parenthesis x right parenthesis equals x cubed minus left parenthesis 2 square root of 3 right parenthesis x squared plus 3 x

Sketch the graph of y equals straight f left parenthesis x right parenthesis , showing clearly the coordinates of the points where the curve crosses or touches the coordinate axes.

12b
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3 marks

The functions straight g left parenthesis x right parenthesis and straight h left parenthesis x right parenthesis are defined by the equations

straight g left parenthesis x right parenthesis equals straight f left parenthesis negative x right parenthesis space
straight h left parenthesis x right parenthesis equals straight g left parenthesis x plus a right parenthesis

The graph of straight h left parenthesis x right parenthesis touches the x-axis at the point left parenthesis 5 comma 0 right parenthesis.  Find the value of a, giving your answer as an exact value.

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13
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5 marks

A function,space straight f open parentheses x close parentheses, has second derivative given by

space space straight f apostrophe apostrophe open parentheses x close parentheses equals 6 left parenthesis x minus 2 right parenthesis.

Given that straight f open parentheses 3 close parentheses equals 20, and straight f apostrophe open parentheses 2 close parentheses equals 8 comma find straight f open parentheses x close parentheses.

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14a
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2 marks

Given that a to the power of 1 third end exponent equals negative 4, find the value of a.

14b
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2 marks

Simplify x to the power of 11 over 6 end exponent divided by x to the power of 4 over 3 end exponent

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