core 2 revision booklet

AQA Mathematics

AQA Mathematics

5361 and 6361

Core 2

Syllabus

&

Past Paper questions

You may use a graphic calculator

in this module

Sequences and Series

A sequence of numbers may be defined by a term to term rule or by a formula.

e.g = un + 2n with gives the sequence: 0, 2, 6, 12, 20, 30

and = n2 ( n also gives 0, 2, 6, 12, 20, 30

note that and so and .

Some sequences converge to a limit, L. The limit can be found by putting L into the rule in place of both and and solving the equation.

A series is the sum of a sequence.

is a sequence, whereas is a series.

A series is often defined using sigma notation, i.e.

e.g.

An arithmetic sequence or series is one where each term can be found by adding a fixed number, d, to the previous term.d is called the common difference.

The nth term is of an arithmetic sequence or series is a + (n 1) d. An arithmetic series is often referred to as an AP (short for

arithmetic progression).

The sum of the first n terms of an arithmetic series is

( first term + last term) ( number of terms ieS = n (a + l )

or S = n (2a + (n 1)d ) if you don't know what the last term is.

The sum of the first n positive integers is n (n + 1)

= 1 + 2 + 3 + ... + n = n (n + 1)

=

A geometric sequence or series is one where each term can be found by multiplying the previous term by a fixed number, r.

r is called the common ratio.

The nth term of a geometric sequence or series is ar n 1.

A geometic series is often referred to as a GP (short for geometric

progression).

= a + ar + ar2 + ar3 + ... + ar n -1 The sum of the first n terms of a geometric series is

As n ( S but only for (1 < r < 1

1.(a)A geometric progression has first term 12 and second term 8. Find the sum to

infinity.

(b)An arithmetic progression has first term 3 and common difference d. The nth term

is 93 and the sum of the first n terms is 768. Find the values of n and d.

SMP 2000 (8 marks)

2.A geometric sequence , , ... has nth term , where = 1000.

(a) Write down the value of the first term, .

(b)Given that

(i) show that the common ratio, r, is equal to 0.2,

(ii) find the sum to infinity of the series + + ...

AQA (6 marks)

3.A tape-measure is rolled up into a tight spiral. The first complete loop requires 3 cm of

tape, the second 3.25 cm, the third 3.5 cm, with the lengths of subsequent loops

continuing the sequence. The fully rolled tape-measure consists of 25 complete loops.

(a)Calculate the length of the final loop.

(b)Calculate the total length of the tape-measure.

SMP 94 (5 marks)4.(a) Find the sum of the three hundred integers from 101 to 400 inclusive.

(b)Find the sum of the geometric series: 2 + 6 + 18 + ... +, giving your

answer in the form ( q, where p and q are integers.

AQA 2001 (6 marks)

5.(a)Find the limit of the sequence given by = .

(b)For each of the sequences below, write down the first four terms and state

whether the sequence is convergent, divergent or oscillatory. For any convergent

sequence, state the value to which it converges.

(i) = + (ii) 3 + 4n (iii) = 10 +

EXEXC (9 marks)

6.A pipeline is to be constructed under a lake. It is calculated that the first mile will take

15 days to construct. Each further mile will take 3 days longer than the one before, so the

1st, 2nd, and 3rd mile will take 15, 18 and 21 days, respectively, and so on.

(a)Find the nth term of the arithmetic sequence 15, 18, 21,

(b)Show that the total time taken to construct the first n miles of the pipeline is

n(n + 9) days.

(c)Calculate the total length of pipeline that can be constructed in 600 days.

AQA 2002 (8 marks)

7.(a)Evaluate (i)

(ii)

(b) Show that

=

8.The third and fourth terms of a geometric series are 6.4 and 5.12 respectively. Find

(a)the common ratio of the series,

(b)the first term of the series,

(c) the sum to infinity of the series.

(d) Calculate the difference between the sum to infinity of the series and the sum of

the first 25 terms of the series.

EDEXC 2001 (10 marks)

9.A young person decides to save 50 at the start of each month to supplement her pension

when she retires. Interest is calculated at the end of each month and is added to her

account. The total in her account after n months can be modelled by the expression

(a)Find the total amount in her account after 3 months. Give your answer to the

nearest 10p.

(b)Calculate the total amount in her account if she continues this method of saving

without a break for 35 years. Give your answer to the nearest 100.

SMP 96 (8 marks)

The Binomial theorem

For a positive integer integer, n, and all values of a and b,

(a + b)n = + + + ... + + ... +

= + + + ... + + ... +

The series has n + 1 terms.

=

(1 + x)n = 1 + + + ... + + ... +

= 1 + nx + + + ... +

1.(a)Use the binomial series to expand in ascending powers of x up to and

including the term in , giving each coefficient as an integer.

(b)Use your series expansion with a suitable value for x to obtain an estimate for

, giving your answer to 2 decimal places.

EDEXC 97 (7 marks)

2.The first three terms in the expansion, in ascending powers of x, of , are

1 ( 18x + .

Given that n is a positive integer, find the value of n and the value of p.


3.(a)Write down the first four terms of the binomial expansion, in ascending powers of x,

of , where n > 2.

Given that the coefficient of in this expansion is ten times the coefficient of ,

(b)find the value of n.

(c) find the coefficient of in the expansion.


4.Obtain, in ascending powers of x as far as the term containing x3, the expansion of

.

EDEXC (4 marks)

Transformations of graphs

y = f(x ( a) + b translates the graph of f(x) through

x is replaced by (x ( a), y is replaced by (y ( b).

y = f(ax) one way stretch, factor from the origin parallel

to the x-axis, y = bf(x) one way stretch, factor b from the origin parallel

to the y-axis,

y = f((x) reflects the graph of f(x) in the y-axis

x is replaced by ((x)

y = (f(x) reflects the graph of f(x) in the x-axis

eg y = 2 x + 3 is a translation of 3 to the left of the graph of y = 2 x.

y = (cos x is a reflection of the graph of y = cos x in the x-axis

y = sin 2x is a stretch factor parallel to the x-axis, so the

graph now has period 180( 1.A sketch graph for the function of g(x) is shown.

Draw carefully, indicating the intercepts with the

axes in each case, separate sketch graphs for the

functions

(a)g(2x),

(b) g((x),

(c) g(x) ( 2

SMP 95 (6 marks)

2.The function f is given by f(x) = and is sketched below.

Write down an expression for f(x+3), and sketch the graph of y = f(x+3).

SMP 94 (3 marks)

3.The diagram below shows the sketch of a curve with equation y = f(x).

In separate diagrams, show, for , a sketch of the curve with equation

(a)y = f((x), (b) y = (f(x), (c) y = f(3x),

marking on each sketch the x-coordinate of any point or points where a curve touches or

crosses the x-axis.

EDEXC (6 marks)

4.(a)On the same set of axes, sketch the graphs of y = and y = + 5

(b)Write down the equation of the image if the graph of y = is

(i)reflected in the y-axis

(ii) translated through

Trigonometry

Cosine rule

c2 = a2 + b2 ( 2ab

Sine rule

= =

The area of the triangle is ab. Angles can be measured in radians, where ( radians = 180(90( = radians, 60( = radians, 45( = radians

arc length of a sector, l = r(

area of sector, A = r 2(

1.For triangle PQR, calculate angle P

and use it to find the area of the triangle.

2.From crossroads C a girl sees that a tower T is 2.3 km away

on a bearing of 056(, and a radio mast M is 4.5 km away on a bearing of 078(.

(a)State the angle MCT.

(b)Calculate the distance MT.

(c)Show that the angle CTM is about 138( and give the bearing of M from T.

(d)She now walks due east. Calculate how far she must walk until:

(i) she is due south of the mast,

(ii) she is in line with the mast and the tower.

SMP (11 marks)

3.Triangle ABC has AB = 9 cm,

BC = 10 cm and CA = 5 cm.

A circle centre A and radius 3 cm ,

intersects AB and AC at P and Q respectively, as shown.

(a) Show that, to 3 decimal places,

(BAC = 1.504 radians.

Calculate

(b)the area, in cm2, of the sector APQ,

(c)the area, in cm2, of the shaded region BPQC,

(d)the perimeter, in cm, of the shaded region BPQC.


(a)Find the area of the curved surface of the prism (shown shaded).

(b)Find the total surface area of the prism.

SMP 2000 (6 marks)5.The diagram shows a circle with centre O and radius 3 cm.

The points A and B on the circle are such that the

angle AOB is 1.5 radians.

(a)Find the length of the minor arc AB.

(b)Find the area of the minor sector OAB.

(c)Show that the area of the minor segment

is approximately 2.3 cm2.

AQA 2001(7 marks)

6.The diagram shows a flat symmetrical ornament, ABC, which is made up of two

isosceles triangles, OAB and OAC, and a sector, OBC, of a circle with centre O

and radius r. The point A lies on this circle and the angle subtended by the arc BC

at the centre O is 2 radians.

Show that the total area of the ornament is approximately 1.84r2.

AQA (5 marks)

7.

The diagrams show cross-sections of two drawer handles.

Shape X is a rectangle ABCD joined to a semicircle with BC as diameter.

The length AB = d cm and BC = 2d cm.

Shape Y is a sector OPQ of a circle with centre O and radius 2d cm.

Angle POQ is ( radians.

Given that the areas of shapes X and Y are equal,

(a)prove that ( = 1 +

Using this value of (, and given that d = 3, find in terms of (,

(b)the perimeter of shape X,

(c) the perimeter of shape Y.

(d)

Hence find the difference, in mm, between the perimeters

of shapes X and Y.

EDEXC 2002 (12 marks)Trig functions

The graphs of y = sin( and y = cos( have a period of 360( or 2( radians and (1 < y < 1

The graph of y = tan( has a period of 180( or ( radians. It has discontinuities at ( = 90(, 270(, etc The graph of y = cos( is symmetrical about the y-axis.

cos 0 = 1 cos 90( = 0cos((( ) = cos( The graph of y = sin( has rotational symmetry about the origin.

sin 0 = 0 sin 90( = 1sin((( ) = (sin( The graph of y = tan( has rotational symmetry about the origin.

tan((( ) = (tan(

tan( = 0 for ( = 0, 180(, 360(....

The same transformations can be applied as for other graphs, eg

the graph of y = 5sin( is a stretch factor 5 of y = sin( parallel to the y-axis, so has a period of 360( and (5 < y < 5.

The graph of y = sin5( is a stretch factor of y = sin( parallel to the x-axis, so has a period of ( and (1 < y < 1.

The graph of y = sin(( ( c) is a translation of y = sin( through .

1. The diagram shows a sketch of the graph of with a line of symmetry L.

(a)Describe the geometrical transformation by which the graph of

can be obtained from that of .

(b)Write down the equation of the line L.

(c)Write down the equation of the image if the graph of y = cos x is

(i)translated through

(ii) reflected in the x-axis.Solving trig equations

tan( =

cos2( + sin2( = 1

Equations such as sin2x = 0.2 have many solutions. Check the required interval and check you are working in the correct mode, degrees or

radians.

Use inverse sin, cos or tan to obtain the first solution. Sketch the appropriate basic graph (eg y = sin( ) and use symmetry to obtain

other solutions.

If the equation is like 2sinx ( cosx = 0, rearrange and use .

If the equation is like 2sin2 x ( 5cos x = 4, replace sin2 x using

sin2x = 1 ( cos2x so that you get a quadratic equation which involves

only one trig function, Then solve it in the usual way, by factorising

or the quadratic formula.1.Find all the values of ( in the interval for which

(a) cos (( + 75)( = 0.5,

(b) sin 2( ( = 0.7, giving your answers to one decimal place.


2.Given that 2 sin 2( = cos 2( ,

(a)show that tan 2( = 0.5

(b)Hence find the values of ( , to one decimal place, in the interval for

which 2 sin 2( = cos 2(EDEXC 2001 (6 marks)

3. (a)Given that = 1, show that = 0.

(b)Hence find all the values of in the interval for which

= 1, giving each answer in terms of .

AQA 2001 (8 marks)

4.The diagram shows part of the graph of .

(a)Find the x-coordinate of the point labelled A, where the graph crosses the x-axis.

(b)Describe a sequence of geometrical transformations by which the graph of

can be obtained from that of .

(c)Find the two solutions in the interval for the equation f(x) = .

AQA (11 marks)

5.

f(x) = 5 sin 3x,

(a)Sketch the graph of f(x), indicating the value of x at each point where the graph

intersects the x-axis.

(b)Write down the coordinates of all the maximum and minimum points of f(x).

(c)Calculate all the values of x for which f(x) = 2.5.


6.The equation of a sine curve is

(a) For this curve, write down the value of:(i) the period,

(ii) the amplitude

(b) Solve the equation = 6 for

SMP 97 (6 marks)

7.Show that

(a)

+

(b)(cos y + sin y)2 = 1 + 2sin y cos y

(c)(1 ( sin t )2 + (1 + sin t )2 4 ( 2 cos2 t

8.(a)Given that (1 is a root of the equation ( ( 4x + 3 = 0, find the two

positive roots.

(b)Hence, by substituting x = cos t, solve the equation

( ( 4cos t + 3 = 0 for.

EDEXC 96 (9 marks)

Exponentials and Logarithms

A function of the form y = ax is an

exponential function. The graph of y = ax is positive for all

values of x and it passes through

the point (0, 1)

Laws of indices: a0 = 1

( =

=

( =

=

=

A logarithm is the inverse of an exponential function.

y = ax x =

= x

= x

Laws of logs: + =

= 1

( =

= 0

=

eg = = = = ( 1 =

eg + = = = 1

An equation of the form = b can be solved by taking logs of both sides.1.(a)Given that 8 = , write down the value of k.

(b) Given that = , find the value of x.


2.(a)Write + ( as a single term.

(b)Hence obtain an expression for y in terms of x if

+ ( = 0

SMP 96 (4 marks)

3.Express in the form p + where p and q are integers to be

determined.

AQA (3 marks)

4.(a)Show that

(b)Find the value of:(i)

(ii)

AQA 2002 (4 marks)

5.Given that = and =

(a)find the exact value of x and the exact value of y,

(b) calculate the exact value of .


6.The sequence , , ... is defined by the recurrence relation

= + 5, = 2, where p is a constant.

Given that = 8,

(a)show that one possible value of p is and find the other value of p.

(b) Using p = , write down the value of .

(c)Given also that = t, express in terms of t.


7.Solve the equations

(a)

= 64

(b)

= 12

8.The number of bacteria in a colony is initially 400. After t hours there are 400 ( .

(a)How many will there be after 12 hours? Give your answer to 3 s.f.

(b)How long does it take for the number of bacteria to double?

SMP (5 marks)

9.Solve the equations

(a)

=

(b) ( + 5 = 0

(c) + = 2

Differentiation and integration

You need to know all the ideas from Core 1

If y = then = for all values of n, including fractions and

negative numbers.

dx = + c for all values of n except n = (1

The trapezium rule can be used to estimate the area under a graph.

dx

EMBED Equation.DSMT4 +

EMBED Equation.DSMT4 + ... +

EMBED Equation.DSMT4

= h

The vertical heights are called 'ordinates'. For n trapezia there are

n + 1 ordinates. The width of each trapezium is h = .

To improve the estimate, make h smaller, ie increase the number of

trapezia.

1. (a)Write + 8x + 5 in completed square form.

(b) Find the coordinates of the minimum point of the graph of

(c)Hence write down the coordinates of the minimum point of

+ 8.

SMP 95 (6 marks)2.Given that ,

(a)find the value of at the point where ,

(b)find dx

AQA 2001 (5 marks)

3.(a)Express in the form .

(b)Given that y = , find the value of at the point where .

AQA 2001 (4 marks)4.A manufacturer produces cartons for fruit juice.

Each carton is in the shape of a closed cuboid with

base dimensions 2x cm by x cm and height h cm,

as shown.

Given that the capacity of a carton has to be 1030 cm3,

(a)express h in terms of x,

(b)show that the surface area, A cm2, of a carton

is given by A = + .

The manufacturer needs to minimise the surface area of a carton.

(c) Use calculus to find the value of x for which A is a minimum.

(d) Calculate the minimum value of A.

(e) Prove that this value of A is a minimum.


5.It is given that .

(a)Find .

(b) (i) Find dx

(ii) Hence evaluate dxAQA 2002 (6 marks)6.The diagram shows the curves and which intersect at the origin

and at the point for which .

(a)(i) Find dx and dx.

(ii)Hence calculate the area of the region enclosed by the two graphs.

(b)(i)Find for each of the two curves.

(ii)Hence calculate the value of x for which the two curves have the same

gradient.

AQA (11 marks)

7.The function f, defined for x > 0, is such that f '(x) = ( 2 + .

(a)Find the value of f ''(x) at x = 4.

(b)Given that f(3) = 0, find f(x).

(c)Prove that f is an increasing function.


8.The following is a table of values for , where x is in radians.

x00.511.52

y11.216p1.413q

(a)Find the value of p and the value of q.

(b)Use the trapezium rule and all the values of y in the completed table to obtain

an estimate of I , where

I = dx

EDEXC 2002 (6 marks)9.An engineer estimated the area of the vertical cross-section of water flowing under a

bridge. For her model she measured the depth of water at 4m intervals from one end

of the bridge to the other end. Her results are given in the table.

distance from one end (m)04812162024

depth (m)1.22.33.84.93.21.90.6

She used the trapezium rule to estimate the area of the cross-section. Calculate the

estimate she obtained.

EDEXC (5 marks)10. Given that y = , find and .

11. (a)Find dx.

(b)Use your answer to part (a) to evaluate dx,

giving your answer as an exact fraction.


Proof

A B

Statement A implies statement B

If A is true then B is true

is correct

BUT

does not necessarily imply

as is a possible solution.

In this case:

is sufficient for

But

is not necessary for

A B

Statement A implies and is implied by statement B

If A is true then B is true and vice versa

x = 2 2x ( 1 = 3 and 2x ( 1 = 3 x = 2

so

x = 2 2x ( 1 = 3

implies and is implied by 2x ( 1 = 3 In this case is necessary and sufficient for 2x ( 1 = 3 1.The function f is defined for all real numbers by f(x) =

(a)Show that f(x) can be written in the form where and

are constants to be determined.

(b)Find all the solutions in the interval of the equation f(x) = 0.

(c)State, with reasons, whether is

(i)a necessary condition for f(x) to be zero,

(ii)a sufficient condition for f(x) to be zero.

AQA (8 marks)2.A student attempted to solve the equation

=

for .

The students solution appeared as follows:

=

= 1

= 1

=

(a)State which one of the three symbols is used incorrectly.

(b)State whether the reasoning would be correct if all the symbols were replaced

by symbols.

(c)

Write out a correct solution of the equation from the point where the students

mistake occurred.AQA 2001 (5 marks)

Solutions

Sequences and Series 1. (a) 36 (b) n = 16, d = 6 2. (a) 1000 (b)(ii)

3. (a) 9 cm (b) 150 cm 4. (a) 75150 (b) ( 1

5. (a) (b)(i) 0, , , oscillatory (ii) 7, 11, 15, 19 divergent

(iii) 12, 11, , , converges to 10 6. (a) 12 + 3n (c) 16 miles

7. (a)(i) 196 (ii) 30 8. (a) 0.8 (b) 10 (c) 50 (d) 0.189

9. (a) 151.20 (b) 54600

The Binomial theorem 1. (a) 1024 ( 15360x + 103680x2 ( 414720x3 (b) 880.35

2. n = 9, p = (2 3. (a) 1 + 3nx + x2 + x3

(b) 12 (c) 40095 4. (a) 2 + 65x + 986x2 + 9248x3

Transformations of graphs 1. (a)

(b)

(c)

2. f(x + 3) =

3. (a)

(b)

(c)

4. (a)

(b)(i) y = (ii) y =

Trigonometry1. 84.3(, 19.9 cm 2 2. (a) 22( (b) 2.52 km (c) 098(

(d)(i) 4.4 km (ii) 11.1 km 3. (b) 6.768 cm2 (c) 15.682 cm2

(d) 22.512 cm 4. (a) 24 cm2 (b) 104 cm2 5. (a) 4.5 cm (b) cm2

6. area OBC = , angle AOB = ( ( 1, area AOB ( 2 = sin(( ( 1)

total 1.84 7. (b) 12 + 3( (b) 18 + ( (c) 12.9 mmTrig functions1. (a) stretch factor in the x-direction (b) x = 90

(c)(i) cos(x ( 30) + 1 (ii) ( cos x

Solving trig equations 1. (a) 225(, 345( (b) 22.2(, 67.8(, 202.2(, 247.8(

2. (b) 13.3, 103.3, 193.3, 283.3 3. (b) , . 4. (a)

(b) stretch factor 2 in the y-direction, stretch factor in the x-direction

(c) x = or 5. (a) x-intecepts at 0(, 60(, 120(,180( (b) (30(, 5)

(150(, 5) (90(, -5) (c) x = 10(, 50(, 130(, 170( 6. (a)(i) ( (ii) 3

(b) x = 0.365 C or 1.206 C 8. (a) , 3 (b) , (, Exponentials and Logarithms 1. (a) 3 (b) 1.2 2. (a) (b)

3. 3 ( 2 4. (b)(i) 12 (ii) 5. (a) x = , y = 2 (b) 8

6. (a) (3 (b) (1 (c) (3 (t 7. (a) 1 (b) 3.54 8. (a) 233000

(b) 1 hr 18 mins 9. (a) (3.32 (b) 0 or 1.46 (c) 0Differentiation and integration 1. (a) (b) ((4, (11) (c) ((5.2, (16)

2. (a) 4 (b)

EMBED Equation.DSMT4 + + c 3. (a) (b) 67.5 4. (a) h =

(c) x = 7.28 (d) A = 636 cm2 (e) = 16, > 0, so minimum

5. (a)

EMBED Equation.DSMT4 (b)(i) + c (ii) 45 6. (a)(i)

EMBED Equation.DSMT4 + c,

EMBED Equation.DSMT4 + c

(ii)

EMBED Equation.DSMT4 (b)(i) ,

EMBED Equation.DSMT4 (ii) x = 7. (a) 7

(b) f(x) =

EMBED Equation.DSMT4 ( 2x ( ( (c) f '(x) = (x ( )2, always positive

8. (a) p = 1.357, q = 1.382 (b) 2.59 9. 68 m2

10. =

EMBED Equation.DSMT4 (

EMBED Equation.DSMT4 + 3 =

EMBED Equation.DSMT4 +

EMBED Equation.DSMT4 (

EMBED Equation.DSMT4 11. (a) 5x (8 (

EMBED Equation.DSMT4 + c (b)

Proof1. (a) a = 1, b = (1 (b) x = 0, , , (c)(i) no, cosx = 1 is a possible

solution (ii) yes, cosx = 0 ( f(x) = 0 2. (a) = 1 = 1 is

wrong, since = (1 is also possible (b) yes

(c) = 1 = 1 or = (1 ( = or

c

y = ax

A

r

(

a is the side opposite angle A, b is the side opposite angle B etc

B

C

a

b

C



2

x

y

9 cm

10 cm

x

y

check you are working in the correct mode, degrees or radians.

(

1

cos(

sin(

C

B

O

A

B

O

1.5c

A

P

EMBED Equation.DSMT4 is true implies

EMBED Equation.DSMT4 is true

x

degrees

y

L

5 cm

Q

A

x

y

r

1

B

A

2x

x

h

4.The diagram shown is of a prism whose cross-section is a sector of a circle, of radius 5cm, with angle at the centre equal to 0.8 radians.

The prism has length 6 cm.

1

0

2

y

x

1

f(x)

2

x


x

5

2

0

(1

y

-3

0

0

3

1

(3

9 cm

-3

d

3

1

8 cm

-1

R

1

P

5 cm

-3

Q

3

B

2d

2d

C

A

P

O

D

Q

(

Shape X

Shape Y

l

PAGE 19

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core 2 revision booklet

Documents