Practice Paper 2 (Pure) (Edexcel A Level Maths: Mechanics)

Practice Paper Questions

1
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6 marks

The diagram below shows part of the graph with equation y equals 5 minus 3 e to the power of negative x end exponent.

The trapezium rule is to be used to estimate the shaded area of the graph which is given by the integral

integral subscript 1 superscript 2 5 minus 3 e to the power of negative x end exponent space space straight d x

(i)
Given that 4 strips are to be used, calculate the width of each strip, h.

(ii)
Complete the table of values below, giving each entry correct to three significant figures.

x 

1

1.25

1.5

1.75

2

y 

3.90

   

4.48

 

(iii)
Use the trapezium rule with the values from the table in part (ii) to find an estimate of the shaded area.

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2
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3 marks

In triangle ABC, stack A B with rightwards arrow on top equals bold a bold space and  stack A C with rightwards arrow on top equals bold b.  Point P divides BC in the ratio 3:2.

q9-11-1-vectors-in-2-dimensions-easy-a-level-maths-pure

i)
Write down vectorstack space B C space with rightwards arrow on top in terms of a and b.

 

ii)
Find stack B P with rightwards arrow on top in terms of a and b.

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3
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3 marks

Solve the equation

2 space log subscript x open parentheses x plus 2 close parentheses equals 3

giving your answer correct to 3 significant figures.

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4
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3 marks

In the expansion of left parenthesis 4 minus p x right parenthesis to the power of 6, the coefficient of the x to the power of 4 term is 19 440.
Given that p is a positive integer find the value of p.

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5a
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3 marks

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

5b
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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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6
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4 marks

Write  fraction numerator x squared plus 3 x plus 10 over denominator x squared plus 8 x plus 15 end fraction  in the form  A plus fraction numerator B over denominator x plus 3 end fraction plus fraction numerator C over denominator x plus 5 end fraction , where A comma B spaceand C are integers to be found.

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7
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6 marks

The diagram below shows the graph of y equals straight f open parentheses x close parentheses, where straight f open parentheses x close parentheses is the function defined by

space straight f open parentheses x close parentheses equals fraction numerator sin space 3 x over denominator e to the power of 2 x minus 3 end exponent end fraction space comma space   space space space space space space space space space 0 less or equal than x less or equal than fraction numerator 2 pi over denominator 3 end fraction

q8a-7-3-further-differentiation-vh-a-level-maths-pure-screenshots

The points A and B are maximum and minimum points, respectively.

Find the difference between the y-values of the coordinates of A and B, giving your answer correct to 3 decimal places.

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8a
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1 mark

The diagram below shows the graphs of y equals x subscript blank and space y equals ln space open parentheses x minus 1 close parentheses plus 3 space.

q6-10-1-solving-equations-medium-a-level-maths-pure

The iterative formula

x subscript n plus 1 end subscript equals ln open parentheses space x subscript n minus 1 close parentheses plus 3

is to be used to find an estimate for a root, alpha, of the functionspace straight f left parenthesis x right parenthesis.

Write down an expression for straight f left parenthesis x right parenthesis.

8b
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2 marks

Using an initial estimate, x subscript 0 space equals space 2, show, by adding to the diagram above, which of the two points (S or T) the sequence of estimates x subscript 1 comma x subscript 2 comma x subscript 3 comma horizontal ellipsis will converge to.
Hence deduce whether alpha  is the x-coordinate of point S or point T.

8c
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2 marks

Find the estimates x subscript 1 comma x subscript 2 comma space x subscript 3 and x subscript 4, giving each to three decimal places.

8d
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2 marks

Confirm that α = 4.146 correct to three decimal places.

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9a
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6 marks

straight f left parenthesis x right parenthesis equals x cubed plus r x squared plus s x minus 30. Given that straight f left parenthesis 2 right parenthesis equals 0 space and straight f left parenthesis negative 3 right parenthesis equals negative 240

find the values of r and s.

9b
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3 marks

Factorise straight f left parenthesis x right parenthesis completely.

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10a
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1 mark

A simple model for the acceleration of a rocket,A space m s to the power of negative 2 end exponent , is given as

 A equals A subscript 0 e to the power of 0.2 t end exponent

where t is the time in seconds after lift-off.  A subscript 0 is a constant.

What does the constant A subscript 0 represent?

10b
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2 marks

After 10 seconds, the acceleration is 20 space m s to the power of negative 2 end exponent.
Find the value of A subscript 0.

10c
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2 marks

Find how long it takes for the acceleration of the rocket to reach 100 space ms to the power of negative 2 end exponent

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11a
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5 marks

By letting B equals 2 A, use the identity for tan open parentheses A plus B close parentheses to derive an expression for tan space 3 A in terms of tan A.

 

11b
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3 marks

Hence, or otherwise, solve the equation

         fraction numerator 6 space tan space x minus 2 space tan cubed space x over denominator 1 minus 3 space tan squared space x end fraction equals 2 space space space space space space space space space 0 less or equal than space x less or equal than straight pi

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12a
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3 marks

On the same axes, sketch the graphs of y equals vertical line straight f left parenthesis x right parenthesis vertical line  and y equals vertical line g left parenthesis x right parenthesis vertical line   where

straight f left parenthesis x right parenthesis space equals space 3 x space minus space 1 space space space space space space space space space space x element of straight real numbers

g left parenthesis x right parenthesis space equals space 2 x space plus space 2 space space space space space space space space space space x element of straight real numbers

Label the points at which the graphs intersect the coordinate axes.

12b
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3 marks

Solve the equation vertical line straight f left parenthesis x right parenthesis vertical line space equals space vertical line g left parenthesis x right parenthesis vertical line.

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13a
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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

q6-9-2-further-parametric-equations-very-hard-a-level-maths-pure

(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.

13b
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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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14a
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2 marks

It is given

straight f left parenthesis x right parenthesis space equals space 4 x cubed space plus space 4 x to the power of 2 space end exponent space minus space 7 x space plus space 2              

Write down the domain and range of the function f(x).

14b
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3 marks

 Sketch the graph of  y = f(x), stating the coordinates of any intersections with the coordinate axes. (You do not need to give the coordinates of any turning points.)

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15
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6 marks

The points A (2, -21) and B (-5, 3) are the two endpoints of the diameter AB of a circle. Find the equation of the circle in the form a x squared plus a y squared plus b x plus c y plus d equals 0, where a, b, c and d are integers to be found.

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16a
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3 marks

A geometric series has first term 9, and the sum of the first three terms of the series is 19.  The common ratio of the series is r.

Show that 9 r squared plus 9 r minus 10 equals 0.

16b
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2 marks

Find the two possible values of r.

16c
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3 marks

Given that the series converges, find the sum to infinity of the series.

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17a
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2 marks

Fully factorise n cubed plus 6 n squared plus 8 n.

17b
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3 marks

Prove that, if n is odd, n cubed plus 6 n squared plus 8 n is odd and that if n is even, n cubed plus 6 n squared plus 8 n is even.

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