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

Practice Paper Questions

1
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4 marks

Prove by contradiction that there are an infinite number of powers of 2.

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2a
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3 marks

The functions straight f left parenthesis x right parenthesis and g left parenthesis x right parenthesis are given as follows

         f left parenthesis x right parenthesis equals open parentheses 1 minus 1 half x close parentheses to the power of 1 half end exponent space space space space space space space space g left parenthesis x right parenthesis equals left parenthesis 2 plus x right parenthesis to the power of negative 2 end exponent

(i)
Expand straight f left parenthesis x right parenthesis, in ascending powers of x up to and including the term in x squared.
(ii)
Find the values for x for which the expansion is valid.
2b
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3 marks
(i)
Expand g left parenthesis x right parenthesis comma in ascending powers of x up to and including the term in x squared.
(ii)
Find the values for x for which the expansion is valid.
2c
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2 marks
(i)
Find the expansion of fraction numerator square root of 1 minus 1 half x end root over denominator open parentheses 2 plus x close parentheses squared end fraction in ascending powers of x, up to and including the term in x squared.
(ii)
Find the values for x for which the expansion is valid.

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3
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4 marks

The diagram below shows part of the curve C defined by the equation y space equals space 1 over a x squared comma where a is a positive constant.  The shaded region R is bounded by the curve, the  x-axis, and the lines x space equals space 1 and x space equals space 6. 

screenshot-2023-08-09-at-10-19-59-am

Given that the volume of the solid formed when the region R  is rotated 360 degree about the x-axis is fraction numerator 311 straight pi over denominator 20 end fraction  cubic units, find the value of a.

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4a
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5 marks

A sketch of the graph with parametric equations

  x equals t squared plus 4 t    y equals t squared minus 1

is shown below.

q5a-9-1-parametric-equations-hard-a-level-maths-pure-screenshot

Find the coordinates of all points where the graph intersects the coordinate axes.

4b
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4 marks

Find the coordinates of the points where the graph intersects the line with equation x minus 5 y plus 19 equals 0.

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5a
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3 marks

Find an expression for fraction numerator d y over denominator d x end fraction in terms of t for the parametric equations

 space x equals e to the power of 2 t space end exponent    y equals 3 t squared plus 1

5b
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3 marks

The graph of  y  against  x  passes through the point P (1 , 1).

(i)
Find the value of t at the point P.

(ii)
Find the gradient at the point P.

(iii)
What does the value of the gradient tell you about point P?

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6
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5 marks

Use the substitution  u equals 2 plus ln space x space space to show that

integral fraction numerator 1 over denominator x open parentheses 2 plus ln space x close parentheses cubed end fraction space straight d x equals fraction numerator negative 1 over denominator 2 open parentheses ln space x space plus 2 close parentheses squared end fraction plus c

where c is the constant of integration.

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7a
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4 marks

The curve C is described by the equation

3 x squared plus 2 x y cubed plus 16 equals 0.

Show that the normal to C at the point where x equals negative 4 spaceis parallel to the normal to C at the point where x equals 4.

7b
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4 marks

Find the distance between the y-axis intercepts of these two normals.

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8
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6 marks

Find the integral

integral fraction numerator 8 x squared minus 8 x minus 1 over denominator left parenthesis 4 x squared minus 1 right parenthesis left parenthesis x minus 2 right parenthesis end fraction space straight d x

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9a
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3 marks

Find an expression for y given that

y equals integral 6 x squared e to the power of x cubed end exponent space straight d x

9b
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3 marks

Integrate

integral open parentheses 16 minus 32 x close parentheses sin open parentheses space 4 x minus 2 close parentheses squared space straight d x

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10
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10 marks

Determine whether each of the following pairs of lines intersect, are parallel, or are skew.  If the lines intersect, find the coordinates of the point of intersection.

(i)
bold italic r equals negative 7 bold italic i minus bold italic j plus 6 bold italic k plus s left parenthesis bold italic i plus bold italic j minus bold italic k right parenthesis   and    bold italic r equals 2 bold italic i plus 2 bold italic j plus 11 bold italic k plus t left parenthesis 2 bold italic i minus bold italic j plus 5 bold italic k right parenthesis
(ii)
bold italic r equals 3 bold italic i minus 3 bold italic j plus bold italic k plus s left parenthesis bold italic i minus 2 bold italic j plus bold italic k right parenthesis     and    bold italic r equals negative bold italic i minus bold italic j plus 5 bold italic k plus t left parenthesis 2 bold italic i plus 2 bold italic j minus 3 bold italic k right parenthesis.

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11a
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7 marks

A bar of soap in the shape of a cuboid is placed in a bowl of warm water and its volume is recorded at regular intervals.  The water is maintained at a constant temperature.

Before being placed in the water the soap measures 3 cm by 6 cm by 10 cm.

Two minutes later the bar of soap measures 2.85 cm by 5.7 cm by 9.5 cm.

The rate of decrease in volume of the bar of soap is modelled as being directly proportional to its volume.

Defining any variables you use, find and solve a differential equation linking the volume of the bar of soap and time.

11b
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2 marks

What happens to the volume of the bar of soap for large values of t?
Briefly explain why this could be considered a criticism of the model.

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