Lesson 1: (a) Statistics Revisited >> Redo Good Tutorial, Lecture notes and Assignment Questions below:  Binomial and Poisson: 1. State the distribution of X. State an assumption you need to make for the distribution to

be valid. (a) X denotes the number of people with blood group O among 100 randomly selected

people. (b) X denotes the number of printing errors on the front page of a newspaper. (c) X denotes the number of cups of black coffee sold by a drink stall in a day. (d) X denotes the number of rainy days in a week. (e) A pupil answers 10 True/False questions. He gets 1 mark for a correct answer and 0

mark for a wrong answer. X denotes the score of the pupil. (f) X denotes the number of goals scored in a World Cup finals.

2. In a college with a large population, 203 of the population watched the World Cup Finals on

television. Suppose each class in this college has exactly 26 students. What is the distribution of the number of students who watched the Finals in a randomly

chosen class? Give one assumption for your distribution to be valid. [2] Using the distribution stated above, find the probability that (i) more than 2 students watched the Finals in a randomly chosen class. [1]

(ii) there are at least 2 classes with more than 2 students who watched the Finals among 5 classes chosen. [2]

(iii) not more than 2 students watched the Finals given that not more than 5 students watched the Finals in a randomly chosen class. [3]

A random sample of n students is chosen from the college. How large should the sample be so that the probability of at least one student watching the Finals exceeds 99%. [2]

3. A car rental company has n cars for hire on a daily basis. The number of cars on demand

in a day follows a Poisson distribution with variance 2.5. (a) Determine the least value of n such that all demands are met on at least 90% of

days. (b) If find the probability that ,4=n

(i) less than 2 cars are hired out on any one day, (ii) all the cars are in use on any one day, (iii) in a five-day week, there are exactly two days on which all the cars are in use. Find the least value of r, given that the probability that there is no demand in r consecutive days is less than 0.001.

1

4. On the average, there are 1.2 accidents occurring along Adam Road in a day. State a possible reason why the number of accidents occurring along Adam Road in a day, X may not follow a Poisson distribution. [1]

Taking the Poisson distribution to be an appropriate model, find (i) the probability that among 5 days, there are 2 accidents occurring along Adam Road on one of the days and none on the other days. [3] (ii) the least integer k such that P(X > k) < 0.01. [3] (iii) the least number of consecutive days for the probability of at least one accident occurring along Adam Road exceeds 0.99. [3] The number of accidents along Eve Road on a day is also assumed to follow Poisson

distribution with variance 2.3. (iv) Find the probability that on a day, there are more accidents occurring along Eve Road than

along Adam Road given that there are altogether 4 accidents occurring along both the roads. [4] State an assumption for your working to be valid. [1] (v) Show that the probability of at least 3 accidents occurring along Eve Road on three

consecutive days, p1 is higher than the probability of at least 1 accident occurring along Eve Road on each of the three consecutive days, p2 by calculation. [3] Explain why p1 > p2 by reasoning. [1]

5. The number of flaws per roll of manufactured material follows the Poisson distribution with

mean 0.5. Find the (i) probability that 5 randomly chosen rolls will have no flaws, (ii) probability that the fourth roll chosen is the only one which contains 1 or more flaws if

five rolls are chosen at random, one after another, (iii) value of m such that the probability of having m or more flaws in a randomly chosen roll,

is less than 0.1, (iv) probability that in 5 randomly chosen rolls, exactly 2 of the rolls contain 1 or more flaws, (v) largest number of rolls that can be chosen at random such that there is a probability of at

least 0.005 that none of the rolls contain a flaw. Answers: 1. Look out for my solutions on SMB or wikispaces. 2. Assume that the probability of a student in the class watching Finals is also 3/20. Assume that whether one student watch Finals is independent of another student. (i) 0.770 (ii) 0.989 (iii) 0.282 sample size must be at least 29. 3. (a) 5 (bi) 0.287 (ii) 0.242 (iii) 0.256; 3

2

4. The accidents may not be independent of one another eg chain car crash OR the rate at which the accidents occur may not be constant throughout the day as there are more cars during peak hours which may result in more accidents. Give a reason in the context of question!

(i) 0.00892 (ii) 4 (iii) 4 (iv) 0.576 The number of accidents occurring along both the road must be independent of each other.

(v) p2= 0.728 p1=0.968 Explain that one event is the subevent of the other. Therefore the former includes the case of the latter, so the probability, p1 is higher.

5. (i) 0.0821 (ii) 0.0533 (iii) 2 (iv) 0.345 (v) largest 10n = Normal and its approximations: Look up the answers in TYS or tutorial.  1. Lecture note Eg 8:

If X ~ N(70 , 25), find the value of a such that P(|X - 70| < a) = 0.8.

2. (TYS N2002/II/28)

Melons are sold by weight at a price of $1.50 per kilogram. The masses of melons are normally distributed with a mean of 0.8 kg and a standard deviation of 0.1 kg. Pumpkins are sold by weight at a price of $0.50 per kilogram. The masses of pumpkins are normally distributed with a mean of 1.2 kg and a standard deviation of 0.2 kg . Find the probability that the total price of 5 randomly chosen melons and 3 randomly chosen pumpkins exceeds $8.

3. (RJC03/2/29) Aluminium foils are packaged in two sizes, Jumbo Rolls and Regular Rolls,

which has lengths (in metres) that are normally distributed with means and variance as given:

Mean (m) Variance (m2)

Jumbo 19 0.25

Regular 6 0.09

Find the probability that

(i) the length of a randomly chosen Jumbo Roll is more than 3 times that of a randomly chosen Regular Roll,

(ii) the total length of 2 randomly chosen Jumbo Rolls and 3 randomly chosen Regular Rolls is more than 57 m,

(iii) at most 2 out of 10 randomly chosen Regular Rolls have lengths that are less

than 5.5 m.

3

4. pg 101 Qn 9 (TYS N01/II/8)

The random variable X has a normal distribution and P(X > 7.460) = 0.01, P(X < −3.120) = 0.25. Find the standard deviation of X. 200 independent observations of X are taken. (i) Using a Poisson approximation, find the probability that at least 197 of these observations are less than 7.460. (ii) Using a suitable approximation, find the probability that at least 40 of these observations are less than −3.120.

5. pg 101 Qn 10 (TYS N03/II/26)

The random variable X has the binomial distribution B(20, 0.4), and the independent random variable Y has the binomial distribution B(30, 0.6). State the approximate distribution of Y − X, and hence find an approximate value for P(Y − X > 13). 6. pg 95 Qn 4 (TYS N01/II/9)

A company sends a leaflet to 8000 customers. The leaflet describes two offers and a special prize. The company estimates that the probability that a randomly chosen customer will claim the ‘free offer’ is 0.4 and that, independently, the probability that the customer will claim the ‘cheap offer’ is 0.2. Each free offer costs the company $5 and cheap offer costs the company $3. Using a suitable approximation, find the probability that the total cost to the company of the offers exceeds $20 700. 7. (8863 N08/Q7)

An examination is marked out of 100. It is taken by a large number of candidates. The mean mark, for all candidates, is 72.1 and the standard deviation is 15.2. Give a reason why a normal distribution, with this mean and standard deviation, would not give a good approximation to the distribution of marks.

8. A hospital sees a number of babies born. For each gender, it may be assumed that the

weights of newborn babies are normally distributed, with average weights and standard deviations as given in the following table.

Gender Average weight Standard deviation

Boy 2.9 kg 0.6 kg Girl 2.6 kg 0.4 kg

(a) State the distribution of the average weight of a randomly chosen baby boy and two randomly chosen baby girls born in the hospital. [3]

4

(b) One baby boy and two baby girls are chosen at random from the hospital. Find the probability that the average weight of the two baby girls is at least 500 grams less than the weight of the baby boy. [4]

Answers: 8(a) N(2.7, 17 / 225) (b) 0.382

(b) Mathematical Induction

1. A sequence is defined by 1 2

1 and 1,r rru u ur+ 1+

= = prove by induction that

for 1,2,3,...( 1)!n

nu nn

= =−

State the value of as . nu n → ∞ 2. (ACJC06/H2 Promo/9) A sequence is defined by 0 1 2, , , ...u u u 0 3u = − and

for . Prove by mathematical induction that for all , 1 2 3 5nn nu u+ = + + n 0n ≥ 0≥n

2 3 5 5n nnu n= + − − .

3. (N2003/I/11) Prove by induction that ∑=

+−=+−n

rnnnrr

1).52)(1(

61)1)(1(

Use this result to prove that ∑=

++=n

r

nnnr1

2 ).12)(1(61

4. (N2001/I/13(b)) Use induction to prove that

.1)!()1(!)1(...)!3(13)!2(7)!1(3 22 −+=++++++ nnnnn

5

Lesson 2: Sequences and Series >> Important Questions for conceptual understanding‐ (a) APGP and Sigma Notation 1. A geometric series has first 2 terms and common ratio 0.95. 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 S−Sn < 1. [4]

2. A convergent geometric series has first term a and common ratio r. Given that twice the

sum of the first and fourth term is equal to three times the sum of the second and third terms, find the value of r. Hence state the sum to infinity of the series in terms of a. [5]

3. The first, second and fourth terms of a convergent geometric progression are consecutive

terms of an arithmetic progression. Prove that the common ratio of the geometric

progression is 1 52

− + . [4]

4. A bank has an account for investors. Interest is added to the account at the end of each

year at a fixed rate of 5% of the amount in the account at the beginning of that year. A man decides to invest $x at the beginning of one year and then a further $x at the beginning of the second and each subsequent year. He also decides that he will not draw any money out of the account, but just leave it, and any interest, to build up. (i) How much will there be in the account at the end of 1 year, including the interest? (ii) Show that, at the end of n years, when the interest for the last year has been added,

he will have a total of ( )xn 105.121$ − in his accounts. (iii) After how many complete years will he have, for the first time, at least x12$ in his

accounts?

5. The sum of the first n terms of a series is 623

1

1−+

−

n

n . Obtain an expression for the nth term

of the series. Prove that the series is geometric and state its first term and common ratio. [5]

6. Find , simplifying your answer. [4] ( 1

03

Nr

rN r−

=

+ −∑ )2

7. The nth term of a series is 2 12 n− + 3n + ln n. Find the sum of the first N terms. [4]  >> Term 2 Revision Package:  APGP Q6, 7  Answers: 1. 72 2. r = ½ , 2; 2a 4. (i) 1.05x (iii) 10 years

5. 2first term 2,3

r= = 6. 136

N + 1− 7. ( )2 3 ( 1)4 1 ln3 2

N N N N !+− + +

6

(b) Methods Of Difference 1 HCI 2007/P1/7 [Methods of Difference with 3 terms, to see cancellation, write out first 3

rows, do not oversimplify so that you can spot which 3 to cancel out]

Express )2)(1(

107++

+rrr

r in partial fractions.

Hence, find ∑= ++

+n

r rrrr

1 )2)(1(107 , giving your answer in the form f( )M n− , where M is a

constant. Deduce the exact value of 22

21 30( 1)( 2r

rr r r

∞

=

+)+ +∑ .

2. VJC 2007/P1/9 [Involving 2 terms for cancellation]

(i) Given that 12

1)(+

=r

rf , show that )12)(12(

2)()1(+−

=−−rr

rfrf .

(ii) Hence find the sum to n terms of the series ...75

153

131

1+

×+

×+

×.

(iii) Determine the sum to infinity of the series.

(iv) Show that ...161

91

411 ++++ is less than 2.

3. [2007/PJC/P1/Q3] Find 2

1 12 21

n

r

r nr r=

⎡ ⎤⎛− − −⎜⎢ ⎥−⎝ ⎠⎣ ⎦∑ ⎞

⎟ in terms of n. [4]

Hence state the value(s) of n for which the sum is zero. [1]

4. Prove that .1

11

Nnn

N

n

=−+

∑=

Deduce, or prove otherwise, that .211

Nn

N

n

<∑=

Answers:

1. 2

21

35+

−+

−rrr

, ⎥⎦⎤

⎢⎣⎡

++

+−

22

156

nn,

506477 2 (ii) ⎥⎦

⎤⎢⎣⎡

+−

1211

21

n (iii)

21

3. 2

)6)(1( −− nn , n = 6

7

(c) Sequence in general 1. [RJC 2007/P1/3]

A sequence of negative numbers is defined by 12 3

4n

nn

xxx+

−=

−, where 1

17

x = − .

(i) Write down the values of 7x , giving your answers correct to 4 significant figures. [Learn how to use GC from lecture notes] (ii) Given that as n → , , find, without the use of a graphic calculator, the value

of . ∞ nx →

2. [Specimen Paper Q8(b) part] The positive numbers nx satisfy the relation

( )12

1 5n nx x+ = + for 1, 2, 3,n = … . As , n → ∞ nx l→ . Find the the exact value of l.

(i) Prove that ( )2 21n nx l x+ − = − l .

(ii) Hence show that, if nx l> , then 1nx l+ > . Answers:

1 (i) 0.9947 (ii) 2. l =1= −1 2

2+ 1

(d) Complex Loci:

1. Sketch the locus of the points in an Argand diagram representing the complex number z where 1 2 1 2z i− + = − i . [2]

2. [Assignment] Sketch the locus of the points in an Argand diagram representing the complex

number z where 2

arg 01iz

i⎛ ⎞

=⎜ ⎟+⎝ ⎠. [3]

3. [Assignment] Shade, on a single Argand diagram, the region representing the complex number w for which 1w w i≥ + − and |w + 1 − i| ≤ 2 . [3]

Hence find (a) the maximum value of 3w + in exact form, [2] (b) the maximum value of arg( 3)w + , giving your answer to 3 significant figures. [2] 4. Special Loci

(a) 4 . (b) * 4z z i( 2)( * 2)w w− − = − = (c) Re( ) Im( )z z≤

8

Answers:

1. Sketch circle centre (1,−2) radius 5 pass thru origin 2. Sketch arg8

z π= − (half line)

3. (a) 2 5+ (b) 1.15 radians 4a) Circle centred at 2 and radius 2 b) Sketch y = 2 c) Sketch x y≤ .

Lesson 3: Complex Numbers >> Redo Assignments and good questions:  1. Given that ( ) , find the real numbers 22 3 ( 2 3 ) 0i iλ− + + − + + =μ and .λ μ [4] 2. The complex number x iy+ is such that ( )2x iy i+ = . Find the possible values of x and y

without using graphing calculator. Hence find the possible values of complex number w such that . [2] 2w = −i 3. Solve the simultaneous equations

3)1(23

=−+=−

wizwiz

giving z and w in the form a+ib, where a and b are real. [5] 4. A graphic calculator is not to be used in answering this question.

(i) It is given that . Find the value of , showing clearly how you obtain your answer. [3]

1 1 iz = − 31z

(ii) Given that 1 is a root of the equation i− 3 2 4 0z az bz+ + − = , show that a = – 4 and b = 6. [4]

(iii) For these values of a and b, solve the equation in part (ii). [4] 5. Given that w 3 i= − − , find the modulus and argument of w, giving your answers in

exact form. [2] Find the smallest positive integer, such that is real. [2] n nw

6 (a) Given that w = −1 + i, find the smallest positive integer, n so that wn is purely imaginary. [3]

(b) Determine the value of k such that 1 i3 i

kz −=

+ is purely imaginary. [3]

9

Answers: 1. 4, 13λ μ= = 2. 1 1,

2 2x y= ± = ± ; 1 1 1 1,

2 2 2 2w i= − − + i

3. 14 23 19 42,25 25 25 25

z i w= − = + i 4. (i) 2 2i− − (ii) a = −4, b = 6 (iii) 1 ,1 , 2i i− +

5. 52,6π

− , n = 6 6. (a) i, 1 2iz w= = − (b) 3k =

7. The complex number z is defined by cos sinz iθ θ= + where 02

< <πθ .

(i) Write down the modulus and argument of z. [1] Let the point A, Z, P and Q represent the complex numbers 1, z, z+1 and z −1 respectively. (ii) Show the points A, Z, P and Q on a single Argand diagram, [3] (iii) Hence deduce the arg(z +1) and arg(z −1), [2] (iv) Describe the transformation when z is multiplied by i, [1]

(v) Given that w = z + 2z

, find the real and imaginary parts of w in terms of θ. [4]

8. Find cube roots of 1 i− + 3 , giving your answers in exact exponential form. Hence find the roots to the equation 3 1z i= − − 3 . [4]

9. The complex number z is given by 3

2

(1 ) ,2( )

iza i+

=+

where a < 0.

Given that 12

z = , find the value of a. [3]

10. Find 2

2eRe

1 e

i

i

⎛ ⎞⎜ ⎟⎜ ⎟−⎝ ⎠

θ

θ . [3]

11. (a) O, A, B and C represents the complex numbers 0, 1+ 3i, −2 + 5i and c respectively. Given that OABC (labeled anticlockwise) is a parallogram, find the complex number c.

[2] (b) O, P, Q and R represents the complex numbers 0, p, qand r respectively. Given that

OPQR (labeled anticlockwise) is a square, explain why (1 )q i p= + . [2]

10

Answers:

7. (i) 1, θ (iii) 2θ ,

2 2π θ

+

(iv) Point P representing complex number z is rotated 2π radians anticlockwise about the origin.

(v) R e( ) 3cos Im( ) sinw w= θ = − θ

8. 2 219 332 , 1,

kie k

π π⎛ ⎞+⎜ ⎟⎝ ⎠ = − 0,1;

2 219 332 ,

kie k

π π⎛ ⎞− +⎜ ⎟⎝ ⎠ = −1,0,1

9. a 3= − 10. −1/2 11.(a) −3+2i (b)Refer to assignment >> Extra Exercise on Loci:  12. On a single diagram, shade the region R satisfied by 1 2 2z − ≤ and 1 i .z − − ≥ z Given

that z represents the complex number of a point in R, α is the minimum value arg( 3i)z + and β is the maximum value of arg( 3i)z + , find, correct to 3 significant figures, the value of β α− .

13. Find the complex number w such that arg( )3

w π= and |w −2 −2 i| = 5. [2]

Answers: 12. 2.21 radians 13. ( )3 2 3 i+ +

>> Term 2 Block Test 2 Revision Package:    Complex numbers Q1, 4, 8 and 9 (do it on the spot if you have no time to prepare for Q9). 

11

Lesson 4: (a)Graphing (b)Differentiation (a) Graphing >> Redo Assignments  1 Find the equations of the asymptotes of

(a) 2

45

xyx

+=

− (b)

2 21

x xyx−

=+

(c) 2 2 4y x− = [6]

2. Sketch and state the axes of symmetry of [3] 2 2 2 2 212 13 13x y+ = Label all intersections with axes.

3. Sketch 2

2( 2)4x y− + =1.

Label all intersections with axes and asymptotes if any. [3]

4. Prove, using an algebraic method, that the curve 3 6( 6)xy

x x−

=+

cannot lie between two

certain values of y (which is to be determined). [4] 5. The graph, 2 3 2xy y x+ = undergoes transformations in the following sequence:

(A) Reflect graph in the y-axis, (B) Scaling of graph by a factor of 3 parallel to the y-axis.

State the equation of the resulting curve. [2] 6. Describe a sequence of transformations geometrically to obtain y = f(2x −1) from

y = f(x). [2] Hence describe a sequence of transformation geometrically to obtain y = f(x) from y = f(2x −1). [2]

7. The graph of y = h(x) is as shown below.

Sketch the graph of

y = 2

0

y

1

A(2,4)

x

(a) y2 = h(x) (b) y2 = − h(x) (c) 1f ( )

yx

= . [9]

12

8. The graph of y = f(x) is shown below.

(a) Sketch the graph of 1f ( )

yx

= . [3]

(b) Sketch the graph of f ( | |)y x= − . [3] 9(a) Sketch the graph given by the parametric equations 3cosx t= and 3siny t= . You should label the points of intersection with the axes. [2] Find the exact y-ordinate of the point where x = 1 / 8. [2]

(b) Find the Cartesian equation of the curve given by 13tan and cos2

x t y= = t . [2]

>> Term 2 Revision Package:    Graphing  Q3, 6, 12  Answers: 1(a) 0, 5y x= = ± (b) 3, 1y x x= − = (c) y x= ±

2 Ellipse centre (0,0) with horizontal distance from centre = 13 and vertical distance from

centre = 1. x- and y- axes are the axes of symmetry. 12

3 Hyperbola “centre” (0, −2). Asymptotes: 22xy = − ±

4 The two values are 1 3and 6 2

.

5 3

223 3y yx x⎛ ⎞ ⎛ ⎞− + =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

6 Translate 1 unit in the negative x-direction and then scale by a factor of ½ parallel to the

x-axis.

x (−4, ½)

y = − ¼

x = −2

13

Construct your sequence backwards.

7a) 8.

9 (a) , 3 38

(b) 2

2

419x

y+ =

14

(b) Differentiation >> Redo Assignment 7A and good questions:  1 Differentiate the following with respect to x, simplifying your answers.

(a) 3 2

1ln( 1x

e x−

+ ) , [3]

(b) ( )1 2sin 1 x− − for . [3] 0x <

2 Given that 2tanx t= ° and secy t= ° where t is a parameter, find dydx

in terms of t°.

[4]

3 The equation of a curve is 1 2tan ( ) 2 1xy y− + = . Find dydx

in terms of x and y.

Find the gradient of the tangent to the curve where 12

y = [5]

Answers:

1a) 2

1 12 1 1

xx x

⎛ ⎞−⎜ ⎟− +⎝ ⎠ b)

2

11 x−

2. 1 cos2

t°

3. 2 24 (1 )y

x y x y−

+ +; 1

4−

>> Redo Assignment 7B and good questions:  1. A spherical balloon is being inflated and, at the instant when its radius is 3m, its surface

area is increasing at the rate of 2 m2 s−1. Find the rate of increase, at the same instant, of (a) the radius, [2]

(b) the volume. [2]

[The volume and surface area of a sphere with radius r is 343

rπ and 24 rπ respectively.]

2. A quadratic equation y = Ax2 + Bx + C passes through the point (2, 8) and the tangent to the curve at the point (1, 3) is y = 4x – 1. By setting up a system of linear equations, find the values of A, B and C. [4]

15

3. A curve is defined by the equation 2 33 20y x y x 0+ − = . Find dydx

in terms of x and y. Find

the equations of the tangents to the curve that are parallel to the y-axis. [5] 4. (RJC03/1/5) The parametric equations of a curve are x = t2, y = t3. Show that the equation

of the tangent to the curve at the point with parameter t is 32 3y tx t 0− + = .

The tangent to the curve at (9, 27) meets the curve again at P. Find the coordinates of P. [5] 5. (TJC06/H2 Promo/10) A curve C is defined by the parametric equations

ttytx 1, −== , where t > 0.

(i) Find the coordinates of A, where C intersects the x-axis. [2] (ii) The tangent and normal to the curve at the point A meet the y-axis at T and N respectively. Find the area of the triangle ATN. [6]

Answers:

1. (a) 112π

ms−1 (b) 3 m3 s−1

2. A = 2, B=1, C=0 3. x = 0 and x =−2.99

4. 9 27( , )4 8

−

5. (i) A(1, 0) (ii) 53

Lesson 5: (a) Differentiation (b) Binomial and Maclaurin’s Series >> Redo Assignment 7C and good questions:  1. A student wrote the following working in his Promotional Exam script:

( ) ( )2

3 22

2

2

6 1 18 6 1

1 10 . When , 0 It is a point of inflexion.6 6

dy d yx xdx d xdy d yx xdx d x

= − = −

= ⇒ = = = ⇒

The furious Mrs Yap put a big cross beside his working. Identify his mistake and explain briefly why he is wrong. [1]

Replace his mistake with a correct solution. [2]

16

2. A carpenter, Chua Chu Keng wanted to extract a cylinder, radius r cm and height h cm

from a wooden sphere of radius 20 cm so that wood wastage is minimized (according to figure below).

Show that . [2] 2 24 1600r h+ =

Using an analytical method, find the exact value of r so that the volume of the cylinder extracted is the largest. [5]

3. TPJC/07/1/Q9a

The figure below shows a rectangle which is inscribed within a semi-circle with radius r = 5cm. The top corners of the rectangle are to be in contact with the circumference of the semi- circle at all times. Let 2x cm and y cm be the length of the base and the height of the rectangle respectively. Find the maximum area of the rectangle that can be formed. [5]

r = 5

2x

y

4.

45°

A hollow vertical cone of semi-vertical angle of 45° is held with its axis vertical and vertex downwards (see diagram). At the beginning of an experiment, it is filled with 390 cm3 of liquid. The liquid runs out through a small hole at the vertex at a constant rate of 2 cm3s−1. Find the rate at which the depth of the liquid is decreasing 3 minutes after the start of the experiment. [6]

17

5. The diagram shows the graph of y = f(x) which has a turning point at A.

y

x 0

x=6

y =−2 A (3, −2)

State the set of values of x such that (i) f , (ii) . [2] '( ) 0x > f "( ) 0x >

Sketch the graph of f '( )y x= . [3] Answers: 1. Second derivative test in this case is inconclusive of the nature of the stationary point.

Must use first derivative test.

2. 403

h = .

3. 225A cm=

4. 0.0680 cms−1 5. (i) (ii) x < 6. 3, 6x x> ≠ >> Term 2 Revision Package:  Differentiation  Q4, 5, 9 

18

(b) Binomial and Maclaurin’s Series

Formula list Maclaurin’s expansion:

…… +++′′+′+= )0(f!

)0(f!2

)0(f)0f()f( )(2

nn

nxxxx

……… ++−−

++−

++=+ rn xr

rnnnxnnnxx!

)1()1(!2

)1(1)1( 2 ( )1 <x

…… ++++++=!!3!2

1e32

rxxxx

rx (all x)

…… ++

−+−+−=

+

)!12()1(

!5!3sin

1253

rxxxxx

rr (all x)

…… +−

+−+−=)!2(

)1(!4!2

1cos242

rxxxx

rr (all x)

…… +−

+−+−=++

rxxxxx

rr 132 )1(32

)1ln( ( 11 ≤<− x )

>> Redo Maclaurin Assignment:  1. The Maclaurin’s series for a particular function, f(x) is given by 31 3 7 ...x x+ + + .

(i) Find the values of and . [2] f ''(0) f ''' (0)(ii) Write down the equation of the tangent to the curve f ( )y x= at x = 0. [1]

2. Express sin 2

22 tan

π θ

θ

⎛ ⎞+⎜ ⎟⎝−

⎠ in the form where 2θθ cba ++ θ is small such that θ 3 and higher

powers of θ can be neglected. [4]

3. Given that = 1+ sin x, show that ye22

2d d e 1

dd−⎛ ⎞+ = −⎜ ⎟

⎝ ⎠yy y

xx. [3]

By further differentiation of this result, find the Maclaurin series, g(x) for y in ascending powers of x, up to and including the term in x3. [3] *Sketch the graph of y = ln(1+ sin x) = f(x) and the graph of the expansion found above in one diagram. Find values of x so that |f(x) −g(x)| < 0.5. [4] Verify that the same result is obtained if the standard series for ln(1+x) and sin x are used. [3]

>> Term 2 Revision Package:  Maclaurin’s Series  Q1, Q5. 

19

Answers 1. (i) 0, 42 (ii) y = 1+3x

2. 21 72 4 8

θ+ − θ

3. 2 31 12 6

x x x− +

>> Redo Binomial Assignment  1. Obtain the binomial expansion of ( )4 x− up to and including the terms in 2x . [3]

(i) By putting 116

x = in your above expansion, find an approximate value for 7 , giving

your answer correct to 2 decimal places. [3]

(ii) Student A found the approximate value for 7 by putting 94

x = in the above

expansion. State, with a reason, whether this value of x is suitable without performing any computation. [2]

(iii) Deduce the expansion of ( )

14 x−

from the above expansion of ( )4 x− . [2]

2. Express 3

2

2( 2)( 1)

xx x

+− +

in partial fractions. (Caution! Improper!) [5]

(i) Find the expansion of )1)(2(

842

3

+−+xx

x in ascending powers of x up to including

term containing x2. [4] (ii) Find the range of the values of x for which the expansion in (i) is valid. [2] 3. Find the coefficient of , in the simplest form, in the expansion of rx

(a) 11 3x−

(b) 2

1(1 )x−

in the ascending powers of x. [4]

Answers:

1. 2

24 64x x

− − (i) 2.65 (ii) Not suitable (iii) 12 16

x+

2. 2

2 112 1x x

+ −− +

, (i) 231

2 4x x

− + + , (ii) |x| < 1

3 (a) 3 (b) r r

20

Lesson 6: Probability and Statistics (a) Probability  >> Redo tutorial and assignment:  1. The medical test for Swine flu is not completely reliable: Given that an individual has

Swine flu, there is a probability of 0.95 that the test will prove positive. If the individual does not have the flu, there is a probability of 0.1 that the test will prove positive. In a certain country M, the probability that an individual chosen at random will have Swine flu is p. Draw a tree diagram to model the above scenario. It is known that the probability of the test being positive is 0.27. (i) Find the value of p. [2] (ii) Find the probability that a randomly chosen individual who is tested negative has

Swine flu. [3] 2. Two fair dice, one red and the other green are thrown.

A is the event: the score on the red die is divisible by 3 B is the event: the sum of two scores is 9

Justify your conclusion, determine whether A and B are independent. [3] Find P( ). [2] A B∪

>> Good Questions  3. [N04/II/23]

A and B are events such that P(A ∪ B) = 0.9, P( ) 0.2A B∩ = and Find P( | ) 0.8.A B =P( )A and P(B′). Are the events A and B mutually exclusive? Independent? Justify your answers.

4. [N03/II/25]

In the first stage of a computer game, the player chooses, at random, one of 5 icons, only one of which is correct. If the correct icon is chosen then, in the second stage, the player chooses, at random, one of 8 icons, only one of which is correct. If an incorrect icon is chosen in the first stage then, in the second stage, the player chooses, at random, one of 10 icons, only one of which is correct. The events A and B are defined as follows:

A: the first icon chosen is correct.

B: the second icon chosen is correct. Find (i) P , (ii) P(( )A B∩ ),B (iii) P(A ∪ B), (iv) P( | )A B

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5. An unbiased die is thrown thrice. Find the probability that (i) they all show different numbers, (ii) at least two show the same number. Suppose now the die is thrown until a six is shown. (iii) Find the probability that 3 throws are needed. (iv) Find the probability that more than 3 throws are needed. (v) Find the least number of n so that the P(at most n throws needed to show a six) > 0.9. 6. A box contains 5 white balls, 3 red balls and 2 green balls.

(i) Three balls are taken from the box, at random and with replacement. Find the probability that they all have different colours.

(ii) Five balls are taken, at random and without replacement, from the box. Find the probability that [Hint: try combination methods – less tedious]

(a) exactly 2 are white; (b) at least 2 are white; (c) exactly 2 are white given that at least 2 are white.

7. [N99/II/6] A set of 30 cards is made up of cards chosen from a number of packs of ordinary playing cards. The numbers of cards of each type are given in the following table.

Spades Hearts Diamonds Clubs King 2 3 1 3 Queen 3 3 5 2 Jack 1 2 3 2

Thus, for example, there are 2 Kings of Spades and 3 Jacks of Diamonds. (a) One card is taken at random from the set. Events H, K, J are defined as follows:

H: The card taken is a Heart K: The card taken is a King J : The card taken is a Jack.

(i) Describe in words what the event K ∪ H represents and state the probability of this

event. (ii) Describe in words what the event J’ ∩ H’ represents and state the probability of this

event. (iii) Find the conditional probability that the card is a Diamond, given that it is a King.

(b) Two cards are taken from the set, at random and without replacement. Find the probability that both cards are Jacks. Give your answer correct to 3 places of decimals. (c) Three cards are taken from the set, at random and without replacement. Find the probability that they are three Kings or three Queens or three Jacks. Give your answer correct to 3 places of decimals.

22

8. [N88 P2 Q6] In a sales campaign, a petrol company gives each motorist who buys their petrol a card with a picture of a film star on it. There are 10 different pictures, one each of 10 different film stars, and any motorist who collects a complete set of all 10 pictures get a free gift. On any occasion when a motorist buys petrol, the card received is equally likely to carry any one of the 10 pictures in the set. At a certain stage, the motorist has collected 9 out of the 10 pictures.

(a) Find the probability that the first 4 cards the motorist receives all carry different pictures. (b) Find the probability the first 4 cards received result in the motorist having exactly three different pictures. (c) Two of the ten film stars in the set are Tom Cruise and Will Smith. Find the probability that the first 4 cards received result in the motorist having a picture of Tom Cruise or Will Smith or both. (d) Find the least value of n such that P(at most n more cards are needed to complete the set) > 0.99. Answers: 1(i) 0.2 (ii) 1/73

2. Not indept , 7/18

3. 0.85; 0.75, no for both.

4. (i) 140

(ii) 21200

(iii) 725

(iv) 521

5. (i) 59

(ii) 49

(iii) 25216

(iv) 125216

(v) 13

6. (i) 950

(ii) (a) 2563

(b) 113126

(c) 50113

7. (a) (i) 715

(ii) 815

(iii) 19

(b) 0.064 (c) 0.105

8(a) 63/125 (b) 54/125 (c) 369/625 (d) 44

23

(b) Binomial Poisson >> Good questions  1. [N05/II/22]

In the UK, the failure rate for treatment by IVF is 80%. Find the probability that there are exactly 6 failures in 10 randomly chosen patients receiving the treatment. It is given that there are fewer than 8 failures in the 10 treatments. Find the conditional probability that there are exactly 6 failures.

2. In a large consignment of mangoes, 10% of the mangoes are damaged. (a) Find the most likely number of damaged mangoes in a bag of 15 mangoes. [2] (b) What is the largest number of mangoes that can be packed in a bag so that the probability

that there are no damaged mangoes in a bag exceeds 0.5? [3] (c) In a very large consignment of mangoes, 10% of the mangoes are damaged. Mrs Yap went to pick 10 mangoes. Find the probability that the 3rd damaged mango is

picked before picking the 10th mango. [2] (d) Suppose Mrs Yap went to pick 10 mangoes, find the probability that the 5th mango she

picked is not damaged and the 10th mango picked is her 3rd damaged mango. [3] 3. [N84 P2 Q11] An administrative centre has three independent telephone lines A, B and C.

During the period 1000 to 1015 hors on a working day, the number of telephone calls coming in on lines , B and C are X1, X2 and X3 respectively. Each of these independent random variables has a Poisson distribution. Also it is known that E(X1) = 1.2, E(X2) = 1.5 and that on 1 in 200 working days there are no telephone calls made to the centre between 1000 and 1015 hrs. Find

(a) the value of E(X3), (b) the probability that, on any given day, there will be exactly one incoming call to the centre between 1000 and 1015 hrs.

(c) the probability that not more than 2 out of 100 working days, there will be exactly one incoming call to the centre between 1000 and 1015 hrs.

Answers: 1. 0.0881; 0.273 2. (a) Mode = 1 (b) 6 (c) 0.0134 (d) 0.0702 3. (a) 2.60 (b) 0.0265 (c) 0.506  >> Term 2 Revision Package:   Binomial Poisson Q4, 5, 6, 7 

24

(c) Normal and its Approximations 1. [2002 P2 Q10] A machine grades apples according to their mass. Apples with a mass

exceeding 125g are rejected as too large and apples with a mass less than 75g are rejected as too small. A large batch of apples is graded and it is found that 10% are rejected as too large and 13% are rejected as too small. Assuming a normal distribution, find the mean mass of a randomly chosen apple from the batch.

2. [N95 P2 Q9] The length of time which an ordinary light-bulb will last may be taken to

have a normal distribution with mean 600 hours and standard deviation 100 hours. The length for which a new “long-life” light bulb will last may be taken to have a normal distribution with mean 2000 hours and standard deviation 200 hours.

(a) Two ordinary bulbs chosen at random. Find the probability that (i) sum of the times for which they last will be less than 1100 hours. (ii) one last longer than its mean and the other last less than its mean. (b) One hundred ordinary light bulbs are chosen at random. Find the probability that the

mean of the times for which they last will be more than 595 hours. (c) One ordinary light bulb and one long-life light bulb are chosen at random. Find the

probability that the long-life bulb lasts more than three times as long as the ordinary bulb. 3. [N98 P2 Q9] The random variable X has a normal distribution with mean 3 and variance

4. The random variable S us the sum of 100 independent observations of X, and the random variable T is the sum of another 300 independent observations of X. Find

(a) State the distribution of S and T (b) P(S > 310)

(c) P(3S > 50 +T)

25

)The random variable N is the sum of n independent observations of X. State the value of

as n becomes very large, justifying your answer. ( 3.5P N n> Answers: 1. 98.4 2. (a) (i) 0.240 (ii) 0.5 (b) 0.692 (c) 0.710 3. (a) (b) 0.309 (c) 0.235 ; 0 >> Term 2 Revision Package:    Normal distribution and its approx Q 1, 4, 6