Practice Paper Pure 4 (Edexcel International A Level Maths: Statistics 1)

Practice Paper Questions

1a
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3 marks

Find the first four terms in the binomial expansion of  fraction numerator 1 over denominator 2 minus 3 x end fraction.

1b
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2 marks

Find the values of x for which the expansion is valid.

1c
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2 marks

The expansion is to be used in a computer program to estimate the value of 5 over 7.
Check that the expansion is valid for this purpose and use the first four terms of the expansion to estimate the value of 5 over 7.

1d
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2 marks

Find the percentage error the computer program will introduce by using the expansion as an approximation to 5 over 7.

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2
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5 marks

Starting with the equation of a semicircle of radius ry space equals square root of r squared minus x squared end root (where r space greater than space 0), use calculus to prove that the general formula for the volume, V, of a sphere of radius r is

V equals 4 over 3 straight pi r cubed.

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3a
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5 marks

The curve C has equation 2 x y squared minus x squared equals 16,

Line L has equation x equals 4.

Show that the two points where C intersects L have equal gradients.

3b
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1 mark

What else can you deduce about the two points where C and L intercept?

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4a
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4 marks

Express

fraction numerator x squared minus 4 x plus 7 over denominator open parentheses x minus 1 close parentheses open parentheses x minus 3 close parentheses squared end fraction


as partial fractions.

4b
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3 marks

Hence, or otherwise, find

integral fraction numerator x squared minus 4 x plus 7 over denominator open parentheses x minus 1 close parentheses open parentheses x minus 3 close parentheses squared end fraction space straight d x

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5a
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1 mark

Verify that the point A(1 , 1) lies on the curve with equation

ln open parentheses space x y close parentheses space plus x y squared equals 1.

5b
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8 marks

The tangent at point A intercepts the x-axis at point B and the y-axis at point C.
Find the area of the triangle OBC.

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6a
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2 marks

The diagram below shows a sketch of the curve defined by the parametric equations

x equals 3 cos space t space   space y equals 5 sin space 2 t space    0 less or equal than t less or equal than 2 pi

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(i)
Write down the equations of the two horizontal tangents to the curve.

(ii)

Write down the equations of the two vertical tangents to the curve.

6b
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8 marks

The four tangents from part (a) create a rectangle around the curve as shown below.

Find the percentage of the area of the rectangle enclosed by the curve

(the shaded area on the diagram).

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7
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6 marks

In the enchanted kingdom of Vectoria, a magical flying unicorn takes off from the wizard’s palace at the point known as O and travels 30 km on a bearing of 300°.  Chased by an evil dragon, it then travels an unknown distance of  k km due north before reaching the enchanted grove at point P.  Given that the position vector of P relative to O is open parentheses x bold i plus y bold j close parentheses space km, and that the straight-line distance between the grove and the palace is known to be 30 square root of 3 space km, find the exact values of x and y.

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8a
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6 marks

A tree disease is spreading throughout a large forested area.

The rate of increase in the number of infected trees is modelled by the differential equation

fraction numerator straight d N over denominator straight d t end fraction equals k N open parentheses N minus 1 close parentheses comma space space space N greater than 1

where N is the number of infected trees, t is the time in days since the disease was first identified and k is a positive constant.

Solve the differential equation above, and show that the general solution can be written in the form

N equals fraction numerator 1 over denominator 1 minus A e to the power of k t end exponent end fraction

where A is a positive constant.

8b
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3 marks

Initially two trees were identified as diseased.
A fortnight later, 4 trees were infected.
Using this information, find the values of the constants A and k.

8c
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3 marks

By considering the solution to the differential equation along with the values of A and k found in part (b), suggest a range of values of  t  for which the model might be considered reliable.

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9a
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1 mark

In triangle ABC, point F lies on AB and point G lies on BC.  

F divides AB in the ratio m:n.  

The line segment FG is parallel to AC.

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Explain why stack B G with rightwards arrow on top equals lambda stack B C with rightwards arrow on top for some constant lambda, where 0 less or equal than lambda less or equal than 1.

9b
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4 marks

Given that space stack A B with rightwards arrow on top equals bold a  and space stack A C with rightwards arrow on top equals bold b,   show that

stack F G with rightwards arrow on top equals open parentheses fraction numerator n over denominator m plus n end fraction minus lambda close parentheses bold a plus lambda bold b

9c
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3 marks

Using your result from (b), prove that G divides BC in the ratio n:m.

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10
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3 marks

Prove by contradiction that if x squared is odd, then x must be odd.

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