h2 math (topical set1)

© 2009 Mr Teo | www.teachmejcmath-sg.webs.com

Topic: APGP

Question

a) A geometric series has first term 48 and common ratio 5

3. The sum of the first n terms of the

series is denoted by Sn and the sum to infinity is denoted by S. calculate the least value of n

for which the difference between the sum to infinity and the sum of the first n terms is less

than 5.

b) The sum to n terms of a series is given by nnSn 32 −= . Find an expression for the nth term.

Show that the series is in arithmetic progression.

Hence, find the sum of the first 150 even numbered terms (i.e. second, fourth, sixth term etc).

Guide to solving part (a):

Since the question states that the series is geometric, students must recall the following formulas:

r

raS

n

n−

−=

1

)1( , where Sn denotes the sum of the first n terms, and

r

aS

−=∞

1.

Guide to solving part (b):

To prove an AP, students must recall that 1−− nn TT = constant (i.e. independent of n).

The following formulas are useful in solving AP problems:

dnaTn )1( −+= and [ ]dnan

Sn )1(22

−+=


Topic: Binomial Expansion and Partial Fractions

Question 1

a) Write down and simplify the expression of (1 + x) 2

1−

in ascending powers

of x up to and including the term in x3.

b) Hence, or otherwise, show that

+≈

−+

+2

2

4

32

111

a

b

ababaif a and b are

positive and a

b is small enough such that powers of

a

b higher than the third can

be neglected.

Use your result to deduce an approximate value for 2

1+

6

1

Question 2

Express )41)(21(

2)(

2xx

xf++

= in partial fractions.

Guide to Q1(a):

Use the binomial series expansion to solve. Refer to MF15 for the exact expansion.

Guide to Q1(b):

For every ‘Hence’ question, students are supposed to use the previous set of working or answers to

solve. Here, the result of 1(a) will be useful to determine the expansion of baba −

++

11.

Compare and make the necessary substitution. Also, use the hint that a

bis small, appropriately.

To solve for 2

1+

6

1, students are advised to determine the correct values of a and b by

simultaneous equation.

Guide to Q2:

Consider the factors at the denominator. Are they linear, quadratic and/or repeat factors? Check

MF15 if necessary.


Topic: Complex Numbers

Question 1

a) Find the modulus and argument of the complex number i

i

−

−

1

23.

b) The complex number q is given by q =θ

θ

i

i

e

e

−1, where 0 < θ < 2π. In either order,

i) find the real part of q.

ii) show that the imaginary part of q is

θ

2

1tan

2

1.

c) The complex numbers z and w are such that |z| = 2, arg(z) = π3

2− , |w| = 5, arg(w) =

π4

3. Find the exact values of

i) the real and imaginary part of z.

ii) the modulus and argument of 2

z

w.

Question 2

a) Two complex numbers p and q are given respectively by p = 3 – 2i and q = 1 + 3i.

Show on separate diagrams

i) |q| < |z| < |p|

ii) arg(z) = arg(pq).

b) Solve the equation 013 36 =+− zz , giving your answers in the form θire .

Guide to Q1:

(a) Use conjugate to express the complex number in a + ib form.

(b) Express q in terms of polar (or trigonometric) form. Apply trigonometric formulae to

determine real and imaginary part.

(c) Use |zw| = |z||w|, ||

||

w

z

w

z= , arg(zw) = arg(z) + arg(w) and arg(

w

z) = arg(z) - arg(w)

appropriately.

Guide to Q2:

(a) Identify the correct loci i.e. circle, half-line or perpendicular bisector.

(b) Make z3 the subject. Hence, determine the modulus and argument to solve for all possible

values of z.


Topic: Differential Equation

Question 1

Salt is dissolved in a tank filled with 120 litres of water. Salt water containing 20 g of salt per

litre is poured into the tank at a constant rate of 3 litres per minute and the mixture flows out

at a constant rate of 3 litres per minute. The contents of the tank are kept well mixed at all

times. Let the amount of salt in the tank (in grams) be denoted by S and the time (in minutes)

be denoted by t.

i) Show that ( )Sdt

dS−= 2400

40

1

ii) Given that 400 g of salt was dissolved in the tank initially, find the amount of salt

in the tank after 1 hour, giving your answer to the nearest gram.

Question 2

By means of a substitution, 2yu = , solve the differential equation

dx

dyy2 - x=xy2

Guide to Q1:

dt

dS can be obtained by determining the difference between the rate of content coming in and the

rate of content going out. This applies to most of the rate of change problems.

Guide to Q2:

The original DE is not variable separable (VS). The use of substitution will change the DE to a

VS form. Implicit differentiation is required.


Topic: Differentiation Techniques

Question 1

Differentiate with respect to x:

a) x1sin −

b) 22ln xk + , where k is a constant

c) 2

5lnxe

x

d)

+

−

2

1

1

1tan

x

e) xln2

Question 2

i) For the curve yxyx +=− 2)( , show that 122

122

+−

−−=

yx

yx

dx

dy

ii) Find the gradient of the curve at the points where it cuts the x-axis.

Guide to Q1:

Recall all differentiation techniques. Apply product or quotient rule where necessary.

Guide to Q2:

Apply implicit differentiation.


Topic: Applications of Differentiation

Question 1

The vertical cross-section of a water trough is in the shape of an equilateral triangle with one

vertex pointing down. The trough is 15m long. When the water in the trough is 1m deep, its depth

is increasing at a rate of 1

5ms-1. At what rate is water flowing into the trough at that instant?

Question 2

The parametric equations of a curve are )ln(sinθ=x , 3ln(cos )y = θ , 2

0π

θ << .

Without the use of graphic calculator, find the equation of the tangent to the curve at the point

where 6

πθ = .

Question 3

A beam is to be cut from a cylindrical log so that its cross-section is a rectangle. The log has

diameter d and the beam is to have breadth x and depth y . Given that the stiffness of such a

beam is proportional to3

xy , find, in terms of d , the values of x and y for the stiffest beam that

can be cut from the log.

Question 4

The diagram shows a rectangular block ABCDEFGH with AB = BC, AG = 32 and

AE = 2( x−3 ) where x < 3 .

i) Show that AB = )32(2 xx − .

Hence write down the volume of the block in terms of x.

ii) Show that the volume of the block is a maximum when it is a

cube.

Guide to Q1:

Draw a simple sketch of the diagram to visualize the problem. Find the volume of the diagram

and determine its rate of change. Chain rule is necessary to solve the problem.

C

G

D

A B

F

H

E


Guide to Q2:

Determine dx

dyusing chain rule. Use

6

πθ = to define the gradient and determine the equation of the

tangent using y = mx + c.

Guide to Q3:

Draw a simple sketch of the diagram to visualize the problem. Express x and/or y in terms of d.

This question is a maximum/minimum problem, hence the use dx

dy and

2

2

dx

yd in relation to the

problem is necessary.

Guide to Q4:

(i) Use Pythagoras’ Theorem to solve for AB.

(ii) In order to show that the volume is a maximum, students must determine dx

dV and

2

2

dx

Vd.


Topic: Functions

Question

a) The functions f and g are defined by

f(x) = lnx, x +ℜ∈ and g(x) = 12 −x , x +ℜ∈

i) Define 1−g in similar form.

ii) On the same axes, sketch the graph of g and 1−g , indicating clearly the

relationship between them.

iii) Explain briefly why fg1− does not exist. If the domain of f is restricted to the

subset A of R defined by A = { }kxx >ℜ∈ : , find the least value of k for which

fg1− exists. With A as the domain of f, define fg

1− .

b) Find the maximal domain for which f is a function where f(x) = )2-( 2xx + .

Hence, state the range of f for which x α≥ , where α is positive.

Find 1-f and state its domain and range.

Guide to part (a):

i) To find inverse function, students must let y be the function and rearrange to make x the

subject.

ii) The graph of g and g-1 can be found using GC.

iii) For composite function (e.g. fg) to exists, Rg ⊆ Df.

Also, Rg can be adjusted to satisfy Rg ⊆ Df which correspondingly changes the domain of g.

Guide to part (b):

For f(x) to be a function, it must be defined for all real values of x. The following theorems are

useful in solving inverse function problems:

1−=ff RD and 1−=

ff DR


Topic: Graphing Techniques

Question 1

The curve C has equation 1

52

+

+=

x

xy

λ where λ is a non-zero constant.

i) Obtain the equations of asymptotes of C.

ii) Show that C has exactly two points at which 0=dx

dy

iii) Draw a sketch of C to illustrate the case λ > 0, and a separate sketch to illustrate

the case λ < 0.

iv) Find the set of values of λ for which the line y = 4x and C have at least one point

in common. Adapted from FM P1 Nov 99

Question 2

The graph y = f(x) is shown below:

On separate diagrams, sketch the following graphs, showing clearly all relevant

coordinates:

a) f (1 2 )y x= +

b) 2 f ( )y x=

c) 1

f ( )y

x=

d) ( )f 1y x= +

2 0 x

y

(1, -1)

(3, 1)

2.5


Guide to Q1:

i) Apply long division.

ii) Find dx

dy. Use discriminant i.e. b2-4ac to determine prove.

iii) Use GC to obtain an approximate sketch of 1

52

+

+=

x

xy

λ. Students may assume any values

that satisfy λ > 0 and λ < 0 to use in GC.

iv) Use discriminant to obtain the values of λ.

Guide to Q2:

Apply the correct transformation(s). Students are advised to refer to lecture notes.


Topic: Inequalities

Question 1

Solve the following inequality 1)3)(1(

41

2

<+−

+<−

xx

x.

Hence, or otherwise, solve 0)3)(1-(

4ln

2

<+

+

xx

x

Question 2

i) By completing the square or otherwise, show that 3+x2+x2 is always positive

for all real values of x.

ii) Without using GC, solve the inequality x

3< x - 2.

Guide to Q1:

Since exact answer is not required, students can use GC to obtain the answer.

Guide to Q2:

i) Use complete the square as recommended in the question.

ii) Solve the inequality algebraically.


Topic: Integration

Question 1

a) Find ( )( )

dxxex

∫+ tancos

12

b) Show that ( )28

12cos

4/

0

−=∫ ππ

dxxx . Hence, evaluate dxxx∫4/

0

2cosπ

.

c) By means of substitutionu

x12 = , evaluate the integral dx

xx∫

−

1

2/122 15

1.

Question 2

A curve has equation y = ( ) 2/124−

− x for -1≤ x ≤1. The region R is enclosed by C, the x-axis

and the lines x = -1 and x = 1.

a) Find the exact area of R.

b) Show that the volume generated when R is rotated through two right angles about

the y-axis is )324( −π .

Guide to Q1:

a) Consider integration by using standard form.

b) Apply integration by parts. Use trigonometric formula to determine the second integral.

c) Apply substitution correctly.

Guide to Q2:

a) Use GC to provide a sketch and locate the correct region R.

b) Apply dxy2

∫π or dyx2

∫π to determine volume of revolution.


Topic: Maclaurin’s Series

Question 1

Given that y =x2cos1

2

+, show that 02sin)2cos1( =−+ xy

dx

dyx .

Hence, by further differentiation of the result and using Maclaurin’s theorem, expand y in

ascending powers of x, up to the term in x².

Hence, by putting x = π/6, find an approximate value of3

2, to three decimal places.

Question 2

Using standard series expansion or otherwise, find the series expansion of exsinx, up to

and including the term in x5.

Hence, find the series expansion of excosx.

Guide to Q1:

Differentiate y and replace expression in the result with y accordingly. Use the expansion to

determine an approximation to 3

2 by replacing all x with

6

π.

Guide to Q2:

Refer to MF15 for series expansion of xe and sinx. Consider the derivative of the result to obtain

expansion of excosx.


Topic: Mathematical Induction

Question 1

The sequence of numbers u1, u2, u3, … is given by u1 = 1 and 3un+1 = 2un – 1 for all

integral values of n. Prove by induction that 13

23 −

=

n

nu .

Question 2

Prove by mathematical induction that ( ) ( )

1

1

2 21

2 ! 2 !

r nn

r

r

r n

+

=

= −+ +

∑ for all positive integers n.

Guide to Q1 and Q2:

Follow the steps recommended in the lecture notes. Recall the properties of summation series

and factorial, use them where appropriate.


Topic: Summation and Method of Difference

Question 1

Show that )54)(1(6

1)12(

1

++=+∑=

nnnrrn

r

.

Hence, evaluate∑

30

10

)12(=

+r

rr .

Question 2

Let )(,)2(3

)1(4)(

1

Njj

jj

j

∈+

−=

+

φ

i) Show that, for j ≥ 1, )2)(1(

4)1()(

2

++=−−

jj

jjj

j

φφ .

ii) Hence find, in terms of n, the sum of the series ∑= ++

n

j

j

jj

j

1

2

)2)(1(

4.

Guide to Q1:

Use )1(2

1

1

+=∑=

nnrn

r

, )12)(1(6

1

1

2 ++=∑=

nnnrn

r

and apply ∑∑∑−

===

−=1

11

)()()(m

r

n

r

n

mr

rfrfrf

where necessary.

Guide to Q2:

Use )2(3

)1(4)(

1

+

−=

+

j

jj

j

φ to determine )1( −jφ i.e. replace all j on RHS with (j-1). Determine the

cancellation pattern of the MOD.


Topic: Vectors

Question 1

Two straight lines l1 and l2 have equations given by

+

−=

5

1

3

14

8

20

:1 λrl and

+

−

−=

0

3

4

1

2

23

:2 µrl

a) Show that l1 and l2 intersect and find the coordinates of the point of intersection, C.

b) Given that A is the point on l1 with parameter λ=2. Obtain the position vector of the

foot F of the perpendicular from A to l2.

c) Show that an equation for the common perpendicular, p, to l1 and l2 through C is

given by

−

+

−

−=

1

4

3

1

11

11

: trp

d) D is the point with parameter 3 on p. Obtain the length of projection of vector AD on

l2.

Question 2

Consider the two lines

=+

=+

0

0:1

zy

yxl and

+=

=

+=

tz

y

tx

l

1

1

21

:2

a) Find the equation of the plane 1Π which contains l1 and is parallel to l2.

b) Find the equation of the plane 2Π which contains l2 and is perpendicular to 1Π .

c) Find the coordinates of the foot of perpendicular, Q, from the point P (1, 1, 1) to the

plane 1Π . Hence determine the shortest distance between l1 and l2.

Guide to Q1:

a) Solve for λ and µ by forming three set of equations. For both lines to intersect, values of λ

and µ must satisfy the three equations.

b) Use scalar dot product to obtain the answer.

c) Use vector cross product to determine the common perpendicular. A common

perpendicular is perpendicular to both lines.

d) Apply the length of projection formula.


Guide to Q2:

a) Use direction vectors of l1 and l2 to determine the equation of plane.

b) Use direction vectors of l2 and normal vector of 1Π to determine the equation of plane.

c) Determine the equation of the line that contains point P and is perpendicular to the plane

1Π . Find the intersection between this line and the plane 1Π .


Topic: Probability

Question

There are 36 people at a gathering of two families. There are 25 people with the name

Lee and 11 people with the name Chan. Of the 25 people named Lee, 4 are single men, 5

are single women and there are 8 married couples. Of the 11 people named Chan, 2 are

single men, 3 are single women and there are 3 married couples. Two people are chosen

at random from the gathering.

a) Show that the probability that they both have the name Lee is 21

10.

b) Find the probability that they are married to each other.

c) Find the probability that they both have the name Lee, given that they are married to

each other.

d) Find the probability that they are a man and a woman with the same name.

e) Find the probability that they are married to each other, given that they are a man and

woman with the same name.

{Leave your answers as an exact fraction in its lowest terms or as a decimal correct to 3 places}

Adapted from TYS

Guide to (a), (b) & (d):

Use R

nC approach to determine the probability. Venn and tree diagrams are not appropriate for

these questions.

Guide to (c):

Apply conditional probability i.e. )(

)()|(

BP

BAPBAP

∩= .

Guide to (e):

Apply conditional probability i.e. )(

)()|(

BP

BAPBAP

∩= . You may use answers to part (b) and (d)

where necessary.


Topic: Binomial Distribution

Question 1

a) On your way to school, you have to pass through 12 sets of traffic lights that operate

independently. The chance that any set is green when you reach is 0.5. If you are stopped fewer

than 3 sets, you have time to take your breakfast at MacTucky. You will be late if you are

stopped by more than 8 sets of traffic lights. If you are late more than twice in a 5-day week, you

would have to see your discipline teacher for counseling.

What is the probability that

i) you have time for breakfast for two consecutive mornings,

ii) you have to be counseled by your discipline teacher.

b) In an experiment, 3 numbers are selected at random from the sequence 1, 2, 3, 4, 5, 6, the

numbers being sampled without replacement. The greatest and least of the three numbers are

then rejected. X denotes the number retained.

i) Prove that P(X=3) = 0.3.

If the experiment is carried out n times, find

ii) The least value of n such that the probability that X = 3 occurs at least once

exceeds 0.99.

Guide to 1(a):

i) Use GC to determine the probability that you have time to have breakfast, which is

defined by a binomial distribution. Apply probability theorem where necessary.

ii) Identify the type of distribution i.e. Binomial. Determine n and p of the distribution.

Solve the probability using GC.

Guide to 1(b):

i) There are altogether 6 ways to obtain P(X=3).

ii) Use the definition of binomial distribution to work out n i.e. P(X=x) = xnx

x

nppC

−− )1(


Topic: Poisson Distribution

Question 1

a) A car salesman receives $300 commission for each new cars and $140 for each

used cars he sells. He sells, on average, three new cars in two weeks and two used cars in two

weeks. The number of new cars he sells is independent of the number of old car he sells.

Show that the probability that he sells two new cars in a week is 0.251.

iii) Determine the probability that he sells at most five cars in a four-week period, leaving

your answers in 3 significant figures,

iv) Determine the probability, to 3 significant figures, that his commission for a four-

week period is $1460.

b) Your mathematics teacher decides to bring all mathematics students to the

Botanic Gardens to breathe in some fresh air and relax. There are 100 students and the

probability that a student is absent is 0.01.

ii) Find the probability that less than 3 students are absent,

iii) Using a suitable approximation, find the probability that less than 3 students are

absent.

Guide to 1(a):

(i) Students must determine the average number of cars sold in a four week period. Do note

that the sales of the new and old cars are independent of each other.

(ii) Work out the possibilities of obtaining $1460 commission in a four week period.

Guide to 1(b):

(i) Determine the appropriate distribution and find the probability using GC.

(ii) Check n, np and nq where necessary. Determine the most appropriate approximation

based on the values of n, np and nq.


Topic: Normal Distribution

Question 1

a) The manager of a company producing a large number of bags of flour with a nominal

weight of 2 kg has two possible production processes, A and B. In both cases, the weights of flour

are normally distributed about a mean weight of 2.05 kg.

In A, the standard deviation is 0.025 kg and the profit per bag of flour is 50 cents. In B, the

standard deviation is σ kg. Under process B, the probability that the bag of flour will weigh less

than 2 kg is 0.01 and that the profit per bag of flour is only 40 cents.

i) Find the value of σ correct to 3 significant figures.

For each bag of flour sold which weighs less than 2 kg, there is a probability of 0.01 that the

purchaser will initiate legal proceedings against the manager, which under the unfair trading laws,

will lead the company having to pay a fine of $50.

ii) Find, correct to four decimal places, the probability that in the process A, the weight of

the flour is less than 2 kg.

iii) Determine which process will, in the long run, produce the greater profit.

(Assume that all bags if flour produced are sold and that the weight of an empty bag is

negligible.)

b) There are 10 questions in a certain examination. Each question has 5 suggested answers,

and the candidate has to choose the right one in each question. Suppose that candidate X chooses

answers totally at random, so that he is equally likely to choose any one of the five answers in

each question.

i) Calculate the probability that he will score at least 3 correct answers out of 10.

In a similar examination there are 100 questions, each with 5 suggested answers. X again chooses

entirely at random.

By using a suitable approximation,

ii) estimate the probability of X scoring at least 30 questions out of 100.


Guide to Q1(a):

(i) Use invNorm to determine σ.

(ii) Solve the probability using normalcdf.

(iii) A tree diagram is necessary to determine the process with the better profit.

Guide to Q1(b):

(i) Identify the appropriate distribution to determine the probability.

(ii) Identify the appropriate conditions i.e. n, np and nq. Choose the correct approximation

based on these conditions. Use continuity correction where appropriate.


Topic: Sampling

Question 1

The exhaustion time of alkaline batteries manufactured by Durazer and Energie-cell have

the following population parameters:

Mean

(in hours)

Standard deviation

(in hours)

Durazer 19.87 0.24

Energie-cell 20.16 0.19

a)

i) 50 Durazer and 80 Energie-cell batteries are tested. Determine the distribution of

the sample mean exhaustion time for the two brands.

ii) Hence find the probability that the difference in the mean exhaustion time of the

two brands of batteries is less than 0.2 hours.

b) A survey was carried out to determine the popularity of both brands. The survey is

conducted by 10 surveyors, each assigned to interview 50 people. Each surveyor is to

interview 30 males and 20 females in three different age ranges as follows: below 25, 26

to 35, and above 36

i) State the sampling method used.

ii) Give one advantage and disadvantage of the method used.

MJC 2007 Prelim (Modified)

Guide to Q1:

(a) Apply Central Limit Theorem where possible. Use GC to obtain the probability.

(b) Refer to lecture notes on characteristics/advantages/disadvantages of each sampling

method.


Topic: Hypothesis Testing

Question 1

The ‘reading age’ of children about to start secondary school is a measure of how good

they are at reading and understanding printed text. A child’s reading age, measured in

years, is denoted by the random variable X. The distribution of X is assumed to be

N(µ,σ2) . The reading ages of a random sample of 80 children were measured, and the

data obtained is summarised by ∑ x = 892.7, ∑ 2x = 10 266.82.

(i) Calculate unbiased estimates of µ and σ2, giving your answers correct to 2

decimal places.

(ii) Previous research has suggested that the value of µ was 10.75. Determine whether

the evidence of this sample indicates that the value of µ is now different from

10.75. Use a 10% significance level for your test.

(iii) State, giving a brief reason, whether your conclusion in part (ii) would remain

valid if

(a) the distribution of X could not be assumed to be normal,

(b) the 80 children were all chosen from those starting at one particular

secondary school.

Guide to Q1:

i) Use the formulas in MF15.

ii) Apply the steps as described in the lecture notes. Use GC to determine the value of p.


Topic: Correlation and Linear Regression

Question

A study comparing the amount of advertising time on TV per week for a product and the

number of sales per week for the same product was conducted. The results over eight

weeks are given below:

Advertising

time, x in

mins

10 12 15 14 17 16 22 20

Sales, y in

thousands 2.3 2.8 k 3.1 3.2 2.9 5.0 4.0

i) Find the coordinates of the point through which the regression line y on x and that of

x on y both pass. Give your answer in terms of k.

ii) Given that the regression y on x is y = 0.197x + 0.184, find k. Hence, find the

product moment correlation coefficient r between advertising time and sales per

week.

iii) Plot a scatter plot of y against x and the regression line y on x on the same diagram.

iv) State with a reason, the effect on r if the advertising time was in hours instead of

minutes.

v) By using an appropriate least square regression line, determine the advertising time

if the sales for a particular week was 3400. Comment on the accuracy of the value

obtained. Adapted from NYJC Prelim 2007

Guide to parts:

i) Both regression lines pass through yandx .

ii) Use regression equation to determine k and obtain r using GC.

ii) Use GC to obtain the scatter plot.

v) Using either regression y on x or x on y to determine the answer.

h2 math (topical set1)

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