Practice Paper 3 (Pure 3) (CIE A Level Maths: Mechanics)

Practice Paper Questions

1
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4 marks

The functions straight f left parenthesis x right parenthesis comma space straight g left parenthesis x right parenthesis are defined as follows

      straight f open parentheses x close parentheses space equals open vertical bar space x minus 2 close vertical bar space minus 5 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space x space element of straight real numbers
straight g open parentheses x close parentheses space equals space open vertical bar x close vertical bar space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space x space element of straight real numbers space space


Sketch the graph of y space equals space gf left parenthesis x right parenthesis, stating the coordinates of all points where the graph intercepts the coordinate axes.

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2
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4 marks

Given that vertical line z plus 1 minus 2 i vertical line equals vertical line z minus 7 plus 4 i vertical line:

(i)
On an Argand diagram, sketch the locus (i.e., set of points) for which the equation is true.
(ii)
Shade the region of your diagram that satisfies the inequality vertical line z plus 1 minus 2 i vertical line greater than vertical line z minus 7 plus 4 i vertical line                                     .

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3a
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3 marks

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

 space x equals sin space 2 t space   space y equals e to the power of t

3b
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2 marks

Verify that the graph of x against y passes through the point (0 , 1) and find the gradient at that point.

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4
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3 marks

Solve the equation

         2 space log subscript x left parenthesis x space plus space 2 right parenthesis space equals space 3

giving your answer correct to 3 significant figures.

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5
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3 marks

Solve the equation

5 to the power of 2 x end exponent space minus space 8 space cross times 5 to the power of x space plus space 12 space equals space 0 comma


giving your answers in the form log subscript a space b.

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6a
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2 marks

The function,space straight f left parenthesis x right parenthesis space is defined by straight f open parentheses x close parentheses equals 1 over e to the power of x minus x plus 1 space space space space space space space space space space space space space space x element of straight real numbers

Show that the equation straight f open parentheses x close parentheses equals 0 can be written in the form

space space x equals e to the power of negative x end exponent plus 1

6b
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2 marks

On the same diagram sketch the graphs of y equals x and y equals e to the power of negative x end exponent plus 1.

6c
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2 marks

The equation straight f open parentheses x close parentheses equals 0 has a root, α, close to x equals 1.
The iterative formula  xn+1=e-xn+1 with x0=2 is to be used to find correct to three significant figures.

Show, using a diagram and your answer to part (b), that this formula and initial x value will converge to the root alpha .

6d
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3 marks
(i)
Find the values ofspace x subscript 1 comma space x subscript 2 spaceand x subscript 3, giving each correct to three significant figures.
(ii)
How many iterations are required before x subscript n and x subscript n minus 1 end subscript agree to two decimal places?

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7a
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4 marks

Use the identity


Rsinopen parentheses A plus B close parentheses identical to R spacecos B sin A plus Rsin B cos A

to show that

3 sin theta plus 4 cos theta

can be written as

5sinspace open parentheses theta plus straight alpha close parentheses comma space space space space space space space space space space space space where space space alpha equals tan to the power of negative 1 end exponent space open parentheses begin inline style 4 over 3 end style close parentheses

7b
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3 marks

Hence, or otherwise, solve the equation 3 sin theta plus 4 cos theta equals 1 for  0 less or equal than theta less or equal than straight pi.
Give your answers to three significant figures.

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8a
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3 marks

Show that alpha equals negative 1 plus 4 i  is a root of the cubic equation

z cubed plus 5 z squared plus 23 z plus 51 equals 0

8b
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3 marks

Find the other two roots of the cubic equation in part (a), being sure to show clear algebraic working.

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9
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6 marks

Find the particular solution to the following differential equation, using the given boundary conditions.

         sin squared x space fraction numerator straight d y over denominator straight d x end fraction space equals space cos squared space y space space space space space space space space space space space space space space space x space equals space straight pi over 4 comma space y space equals 0 space

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10a
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3 marks

Express fraction numerator 2 open parentheses 2 minus 5 x plus x squared close parentheses over denominator open parentheses x plus 2 close parentheses open parentheses 2 minus x close parentheses squared end fraction in partial fractions.

10b
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5 marks

Express fraction numerator 2 open parentheses 2 minus 5 x plus x squared close parentheses over denominator open parentheses x plus 2 close parentheses open parentheses 2 minus x close parentheses squared end fractionas the first three terms of a binomial expansion in ascending powers of x.

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11a
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2 marks

Write  e to the power of 2 x end exponent open parentheses 1 plus e to the power of 2 x end exponent plus e squared close parentheses  in the form  space e to the power of straight f open parentheses x close parentheses end exponent plus e to the power of g open parentheses x close parentheses end exponent plus e to the power of h open parentheses x close parentheses end exponent comma spacewhere straight f open parentheses x close parentheses, straight g open parentheses x close parentheses and straight h open parentheses x close parenthesesare all linear functions of x.

11b
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4 marks

Hence show that

        integral subscript 0 superscript 1 space e to the power of 2 x end exponent open parentheses 1 plus e to the power of 2 x end exponent plus e squared close parentheses space d x equals 3 over 4 open parentheses e to the power of 4 minus 1 close parentheses

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12
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5 marks

The coordinates of three points are A(—3, 4, 0), B(2, —2, —3) and C(—l, —1, —4).

Calculate the angle between stack space C A space with rightwards arrow on top and stack C B with rightwards arrow on top. Give your answer in degrees, accurate to 1 decimal place.

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13
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6 marks

The coordinates of three points are A(—2, 3, —5), B(2, —5, —9) and C(—10, —1, —1).

The point M is the midpoint of A B, and the point N lies on B C.

Given that open vertical bar stack B N with rightwards arrow on top close vertical bar = 3 open vertical bar stack N C with rightwards arrow on top close vertical bar, find the equation of the line through points M and N in vector form.

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14
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3 marks

Show that

fraction numerator straight d over denominator straight d x end fraction open square brackets a to the power of k x end exponent close square brackets space equals space k a to the power of k x end exponent space In space a

where a and k are constants.

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