mathematics t

A Collection of Mathematics T (954) Papersfor Sijil Tinggi Persekolahan Malaysia (STPM) 2003-2008

Compiled byYau Ching Koon

Last update on December 30, 2009

Note

All questions were carefully compiled in this little collection. Nevertheless, the author does not warrant the infor-mation contained therein to be free of errors. Readers are advised to keep in mind that the questions, illustrations,procedural details or other items may inadvertently be inaccurate. Readers may kindly inform me the errors foundin this collection to: [email protected]. Your information are greatly appreciated. The collection was pre-pared using LATEX. More materials can be found in the editor’s website: http://www.freewebs.com/yauchingkoon.

Contents

1 2008 2

2 2007 5

3 2006 9

4 2005 13

5 2004 16

6 2003 20

1

Collection of Mathematics T (956) Papers 2003–2008 1 2008

1 2008

Adapted from Koleksi Kertas Soalan STPM 2008 Jurusan Sains, Oxford Fajar Sdn. Bhd.

Instructions:

1. DO NOT OPEN THIS QUESTIONS PAPER UNTIL YOU ARE TOLD TO DO SO.

2. Answer all questions. Answers may be written in either English or Malay.

3. All necessary working should be shown clearly.

4. Non-exact numerical answers may be given correct to three significant figures, or one decimal place in thecase of angles in degrees, unless a different level of accuracy is specified in the questions.

5. Mathematical tables, a list of mathematical formulae and graph paper are provided.

Paper 1

1 The functions f and g are defined by

f : x 7→1x, x ∈ R\{0};

g : x 7→ 2x − 1, x ∈ R.

Find f ◦ g and its domain. [4 marks]

2 Show that∫ 3

2

(x − 2)2

x2 dx = 53 + 4 ln

(23

). [4 marks]

3 Using definitions, show that, for any sets A, B and C,

A ∩ (B ∪C) ⊂ (A ∩ B) ∪ (A ∩C) .

[5 marks]

4 If z is a complex number such that |z| = 1, find the real part of1

1 − z. [6 marks]

5 The polynomial p (x) = 2x3 + 4x2 + 12 x − k has factor (x + 1).

(a) Find the value of k. [2 marks]

(b) Factorise p (x) completely. [4 marks]

6 If y =sin x − cos xsin x + cos x

, show thatd2ydx2 = 2y

dydx

. [6 marks]

7 The matrix A is given by A =

1 0 01 −1 01 −2 1

.(a) Show that A2 = I, where I is the 3 × 3 identity matrix, and deduce A−1. [4 marks]

(b) Find matrix B which satisfies BA =

1 4 30 2 1−1 0 2

. [4 marks]

2


8 The lines y = 2x and y = x intersect the curve y2 + 7xy = 18 at points A and B respectively, where A and B liein the first quadrant.

(a) Find the coordinates of A and B. [4 marks]

(b) Calculate the perpendicular distance of A to OB, where O is the origin. [2 marks]

(c) Find the area of the OAB triangle. [3 marks]

9 Find the solution set of the inequality ∣∣∣∣∣ 4x − 1

∣∣∣∣∣ > 3 −3x.

[10 marks]

10 Show that the gradient of the curve y =x

x2 − 1is always decreasing. [3 marks]

Determine the coordinates of the point of inflexion of the curve, and state the intervals for which the curve isconcave upward. [5 marks]

Sketch the curve. [3 marks]

11 Sketch, on the same coordinate axes, the curves y = 6−ex and y = 5e−x, and find the coordinates of the pointsof intersection. [7 marks]

Calculate the area of the region bounded by the curves. [4 marks]

Calculate the volume of the solid formed when the region is rotated through 2π radians about the x-axis.[5 marks]

12 At the beginning of this year, Mr. Liu and Miss Dora deposited RM10 000 and RM2000 respectively in abank. The receive an interest of 4% per annum. Mr. Liu does not make any additional deposit nor withdrawal,whereas, Miss Dora continues to deposit RM2000 at the beginning of each of the subsequent years without anywithdrawal.

(a) Calculate the total savings of Mr. Liu at the end of nth year. [3 marks]

(b) Calculate the total savings of Miss Dora at the end of nth year. [7 marks]

(c) Determine in which year the total savings of Miss Dora exceeds the total savings of Mr. Liu. [5 marks]

Paper 2

1 Show that the substitution u = x2 + y transforms the differential equation

(1 − x)dydx

+ 2y + 2x = 0

into the differential equation

(1 − x)dudx

= −2u.

[3 marks]

2 In the triangle ABC, the point X divides BC internally in the ratio m : n, where m + n = 1. Express AX2 interms of AB, BC, CA, m and n. [5 marks]

3 If t = tanθ

2, show that sin θ =

2t1 + t2 and cos θ =

1 − t2

1 + t2 . [4 marks]

Hence, find the values of θ between 0◦ and 360◦ that satisfy the equation

10 sin θ − 5 cos θ = 2.

[3 marks]

3


4 The diagram below shows the circumscribed circle of the triangle ABC.

A

B

C

T

PQ

R

The tangent to the circle at A meets the line BC extended to T . The angle bisector of the angle AT B cuts ACat P, AB at Q and the circle at R. Show that

(a) triangles APT and BQT are similar, [4 marks]

(b) PT · BT = QT · AT , [2 marks]

(c) AP = AQ. [4 marks]

5 The position vectors of the points A, B and C, with respect to the origin O, are a, b and c respectively. Thepoints L, M, P and Q are the midpoints of OA, BC, OB, and AC respectively.

(a) Show that the position vector of any point on the line LM is12

a +12λ (b + c − a) for some scalar λ, and

express the position vector of any point on the line PQ in terms of a, b and c. [6 marks]

(b) Find the position vector of the point of intersection of the line LM and the line PQ. [4 marks]

6 A 50 litre tank is initially filled with 10 litres of brine solution containing 20 kg of salt. Starting from timet = 0, distilled water is poured into the tank at a constant rate of 4 litres per minute. At the same time, the mixtureleaves the tank at a constant rate of

√k litres per minute, where k > 0. The time taken for overflow to occur is 20

minutes.

(a) Let Q be the amount of salt in the tank at time t minutes. Show that the rate of change of Q is given by

dQdt

= −Q√

k

10 +(4 −√

k)

t.

Hence, express Q in terms of t. [7 marks]

(b) Show that k = 4, and calculate the amount of salt in the tank at the instant overflow occurs. [6 marks]

(c) Sketch the graph of Q against t for 0 ≤ t ≤ 20. [2 marks]

7 There are 12 towels, two of which are red. If five towels are chosen at random, find the probability that at leastone is red. [4 marks]

8 The random variable X has a binomial distribution with parameters n = 500 and p = 12 . Using a suitable

approximate distribution, find P (|X − E (X)| ≤ 25). [6 marks]

9 In a basket of mangoes and papayas, 70% of mangoes and 60% of papayas are ripe. If 40% of the fruits in thebasket are mangoes,

(a) find the percentage of the fruits which are ripe, [3 marks]

(b) find the percentage of the ripe fruits which are mangoes. [4 marks]

4


10 A sample of 100 fuses, nominally rated at 13 amperes, are tested by passing increasing electric currentthrough them. The current at which they blow are recorded and the following cumulative frequency table isobtained.

Current (amperes) Cumulative frequency< 10 0< 11 8< 12 30< 13 63< 14 88< 15 97< 16 99< 17 100

Calculate the estimates of the mean, median and mode. Comment on the distribution. [8 marks]

11 The continuous random variable X has probability density

f (x) =

0, x < 0,54− x, 0 ≤ x < 1,

14x2 , x ≥ 1.

(a) Find the cumulative distribution function of X. [7 marks]

(b) Calculate the probability that at least one of two independent observed values of X is greater than three.[4 marks]

12 A car rental shop has four cars to be rented out on a daily basis at RM50.00 per car. The average daily demandfor cars is four.

(a) Find the probability that, on a particular day,

(i) no cars are requested, [2 marks]

(ii) at least four requests for cars are received. [2 marks]

(b) Calculate the expected daily income received from the rentals. [5 marks]

(c) If the shop wishes to have one more car, the additional cost incurred is RM20.00 per day. Determine whetherthe shop should buy another car for rental. [5 marks]

2 2007


Instructions:






5


Paper 1

1 Express the infinite recurring decimal 0.72̇5̇ (= 0.7252525 . . . ) as a fraction in its lowest terms. [4 marks]

2 If y =x

1 + x2 , show that x2 dydx

=(1 − x2

)y2. [4 marks]

3 If loga

( xa2

)= 3 loga 2 − loga (x − 2a), express x in terms of a. [6 marks]

4 Simplify

(a)

(√7 −√

3)2

2(√

7 +√

3) , [3 marks]

(b)2 (1 + 3i)(1 − 3i)2 , where i =

√−1. [3 marks]

5 The coordinates of the points P and Q are (x, y) and(

xx2 + y2 ,

yx2 + y2

)respectively, where x , 0 and y , 0. If

Q moves on a circle with centre (1, 1) and radius 3, show that the locus of P is also a circle. Find the coordinatesof the centre and radius of that circle. [6 marks]

6 Find

(a)∫

x2 + x + 2x + 2

dx, [3 marks]

(b)∫

xex+1 dx. [4 marks]

7 Find the constants A, B, C and D such that

3x2 + 5x(1 − x2) (1 + x)2 =

A1 − x

+B

1 + x+

C(1 + x)2 +

D(1 + x)3

[8 marks]

8 The function f is defined by

f (x) =

√

x + 1, −1 ≤ x < 1,|x| − 1, otherwise.

(a) Find limx→−1−

f (x), limx→−1+

f (x), limx→1−

f (x) and limx→1+

f (x). [4 marks]

(b) Determine whether f is continuous at x = −1 and x = 1. [4 marks]

9 The matrices A and B are given by

A =

−1 2 1−3 1 4

0 1 2

, B =

−35 19 18−27 −13 45−3 12 5

.Find the matrix A2B and deduce the inverse of A. [5 marks]

6


Hence, solve the system of linear equations

x − 2y − z = −8,3x − y − 4z = −15,

y + 2z = 4.

[5 marks]

10 The gradient of the tangent to a curve at any point (x, y) is given bydydx

=3x − 52√

x, where x > 0. If the curve

passes through the point (1,−4),

(a) find the equation of the curve, [4 marks]

(b) sketch the curve, [2 marks]

(c) calculate the area of the region bounded by the curve and the x-axis. [5 marks]

11 Using the substitution y = x +1x

, express f (x) = x3 − 4x − 6 −4x

+1x3 as a polynomial in y. [3 marks]

Hence, find all the real roots of the equation f (x) = 0. [10 marks]

12 Find the coordinates of the stationary points on the curve y =x3

x2 − 1and determine their nature. [10 marks]

Sketch the curve. [4 marks]

Determine the number of real roots of the equation x3 = k(x2 − 1

), where k ∈ R, when k varies. [3 marks]

Paper 2

1 Find, in terms of π, all the values of x between 0 and π which satisfy the equation

tan x + cot x = 8 cos 2x.

[4 marks]

2 The triangle PQR lies in a horizontal plane, with Q due west of R. The bearings of P from Q and R are θ andφ respectively. where θ and φ are acute. The top A of a tower PA is at a height h above the plane and the angle ofelevation of A from R is α. The height of a vertical pole QB is k and the angle of elevation of B from R is β. Showthat

h =k tanα cos θ

tan β sin (θ − φ).

[5 marks]

3 The position vectors of the points A, B, C and D, relative to an origin, are i + 3j, −5i − 3j, (x − 3) i − 6j and(x + 3) i respectively.

(a) Show that, for any value of x, ABCD is a parallelogram. [3 marks]

(b) Determine the value of x for which ABCD is a rectangle. [4 marks]

4 The diagram above shows non-collinear points O, A and B, with P on the line OA such that OP : PA = 2 : 1and Q on the line AB such that AQ : QB = 2 : 3. The lines PQ and OB produced meet at the point R. If

−−→OA = a

and−−→OB = b,

(a) show that−−→PQ = −

115

a +25

b, [5 marks]

(b) find the position vector of R, relative to O, in terms of b. [5 marks]

7


R

P

QAB

O

A

B

C

P

Q

5 The diagram above shows two intersecting circles APQ and BPQ, where APB is a straight line. The tangentsat the points A and B meet at a point C. Show that ACBQ is a cyclic quadrilateral. [4 marks]

If the lines AQ and CB are parallel and T is the point of intersection of AB and CQ, show that the trianglesAT Q and BTC are isosceles triangles. Hence, show that the areas of the triangles AT Q and BTC are in the ratioAT 2 : BT 2. [7 marks]

6 The variables X and y, where x > 0, satisfy the differential equation

x2 dydx

= y2 − xy.

Using the substitution y = ux, show that the given differential equation may be reduced to

xdudx

= u2 − 2u.

Hence, show that the general solution of the given differential equation may be expressed in the form y =2x

1 + Ax2 ,where A is an arbitrary constant. [10 marks]

Find the equation of the solution curve which passes through the point (1, 4) and sketch this solution curve.[4 marks]

7 There are eight parking bays in a row at a taxi stand. If one blue taxi, two red taxis and five yellow taxis areparked there, find the probability that two red taxis are parked next to each other.

8


[Assume that a taxi may be parked at any of the parking bays.] [3 marks]

8 The mean mark for a group of students taking a statistics test is 70.6. The mean marks for male and femalestudents are 68.5 and 72.0 respectively. Find the ratio of the number of male to female students. [4 marks]

9 The random variable X is normally distributed with mean µ and standard deviation 100. It is known thatP (X > 1169) ≤ 0.117 and P (X > 879) ≥ 0.877. Determine the range of values of µ.

10 Two events A and B are such that P (A) = 38 , P (B) = 1

4 and P (A|B) = 16 .

(a) Show that the events A and B are neither independent nor mutually exclusive. [2 marks]

(b) Find the probability that at least one of events A and B occurs. [3 marks]

(c) Find the probability that either one of the events A and B occurs. [4 marks]

11 The probability that a lemon sold in a fruit store is rotten is 0.02.

(a) If the lemons in the fruit store are packed in packets, determine the maximum number of lemons per packetso that the probability that a packet chosen at random does not contain rotten lemons is more than 0.85.

[5 marks]

(b) If the lemons in the fruit store are packed in boxes each containing 60 lemons, find, using a suitable approx-imation, the probability that a box chosen at random contains less than three rotten lemons. [5 marks]

12 The masses (in thousands of kg) of solid waste collected from a town for 25 consecutive days are as follows:

41 53 44 55 48 57 50 38 53 50 43 56 5148 33 46 55 49 50 52 47 39 51 49 52

(a) Construct a stemplot to represent the data. [2 marks]

(b) Find the median and interquartile range. [4 marks]

(c) Calculate the mean and standard deviation. [5 marks]

(d) Draw a boxplot to represent the data. [3 marks]

(e) Comment on the shape of the distribution and give a reason for your answer. [2 marks]

3 2006


Instructions:






9


Paper 1

1 If A, B and C are arbitrary sets, show that

[(A ∪ B) − (B ∪C)] ∩ (A ∪C)′ = ∅.

[4 marks]

2 If x is so small that x2 and higher powers of x may be neglected, show that

(1 − x)6(2 +

2x

)10

≈ 29 (2 − 7x) .

[4 marks]

3 Determine the values of k such that the determinant of the matrix

k 1 32k + 1 −3 2

0 k 2

is 0. [4 marks]

4 Using the trapezium rule, with five ordinates, evaluate∫ 1

0

√4 − x2 dx. [4 marks]

5 If y = x ln (x + 1), find an approximation for the increase in y when x increases by δx. Hence, estimate thevalue of ln 2.01 given that ln 2 = 0.6931. [6 marks]

6 Express2x + 1(

x2 + 1)

(2 − x)in the form

Ax + Bx2 + 1

+C

2 − x, where A, B and C are constants. [3 marks]

Hence, evaluate∫ 1

0

2x + 1(x2 + 1

)(2 − x)

dx. [4 marks]

7 The nth term of an arithmetic progression is Tn. Show that Un =52

(−2)2(

10−Tn17

)is the nth term of a geometric

progression. [4 marks]

If Tn =12

(17n − 14), evaluate∞∑

n=1

Un. [4 marks]

8 Show that x2 + y2 − 2ax − 2by + c = 0 is the equation of the circle with centre (a, b) and radius√

a2 + b2 − c.[3 marks]

C1

C2C3

The above figure shows three circles C1, C2 and C3 touching one another, where their centres lie on a straightline. If C1 and C2 have equations x2 + y2 − 10x − 4y + 28 = 0 and x2 + y2 − 16x + 4y + 52 = 0 respectively, findthe equation of C3. [7 marks]

10


9 Functions f, g and h are defined by

f : x 7→x

x + 1, g : x 7→

x + 2x

, h : x 7→ 3 +2x.

(a) State the domains of f and g. [2 marks]

(b) Find the composite function g ◦ f and state its domain and range. [5 marks]

(c) State the domain and range of h. [2 marks]

(d) State whether h = g ◦ f. Give a reason for your answer. [2 marks]

10 The polynomial p (x) = x4 + ax3 − 7x2 − 4ax + b has a factor x + 3 and, when divided by x− 3, has remainder60. Find the values of a and b, and factorise p (x) completely. [9 marks]

Using the substitution y =1x

, solve the equation 12y4 − 8y3 − 7y2 + 2y + 1 = 0. [3 marks]

11 If P =

5 2 31 −4 33 1 2

, Q =

a 1 −18b −1 12

−13 −1 c

and PQ = 2I, where I is the 3×3 identity matrix, determine

the values of a, b and c. Hence find P−1. [8 marks]

Two groups of workers have their drinks at a stall. The first group comprising ten workers have five cups oftea, two cups of coffee and three glasses of fruit juice at a total cost of RM11.80. The second group of six workershave three cups of tea, a cup of coffee and two glasses of fruit juice at a total cost of RM7.10. The cost of a cup oftea and three glasses of fruit juice is the same as the cost of four cups of coffee. If the costs of a cup of tea, a cupof coffee and a glass of fruit juice are RMx, RMy and RMz respectively, obtain a matrix equation to represent theabove information. Hence, determine the cost of each drink. [6 marks]


f (t) =4ekt − 14ekt + 1

,

where k is a positive constant.

(a) Find the value of f (0). [1 mark]

(b) Show that f′ (t) > 0. [5 marks]

(c) Show that k{1 − [f (t)]2

}= 2 f′ (t) and, hence, show that f′′ (t) < 0. [6 marks]

(d) Find limt→∞

f (t). [2 marks]

(e) Sketch the graph of f. [2 marks]

Paper 2

1 Express 4 sin θ − 3 cos θ in the form r sin (θ − α), where r > 0 and 0◦ < α < 90◦. Hence, solve the equation

4 sin θ − 3 cos θ = 3

for 0◦ < θ < 360◦. [6 marks]

2 If the angle between the vectors a =

(48

)and b =

(1p

)is 135◦, find the value of p. [6 marks]

3 Find the general solution of the differential equation

xdydx

= y2 − y − 2.

[6 marks]

11


4 The points P, Q and R are the midpoints of the sides BC, CA and AB respectively of the triangle ABC. Thelines AP and BQ meet at the point G, where AG = m · AP and BG = n · BQ.

(a) Show that−−→AG =

12

m−−→AB +

12

m−−→AC and

−−→AG = (1 − n)

−−→AB +

12

n−−→AC. Deduce that AG =

23

AP and BG =23

BQ.[6 marks]

(b) Show that CR meets AP and BQ at G, where CG =23

CR. [3 marks]

5 Prove that an exterior angle of a cyclic quadrilateral is equal to the opposite interior angle. [3 marks]

A

B

C

D

E

F

In the above diagram, ABCD is a cyclic quadrilateral. The lines AB and DC extended meet at the point E andthe lines AD and BC extended meet at the point F. Show that the triangles ADE and CBE are similar. [3 marks]

If DA = DE, ∠CFD = α and ∠BEC = 3α, determine the value of α. [4 marks]

6 A particle moves from rest along a horizontal straight line. At time t s, the displacement and velocity of theparticle are x m and v m s−1 respectively and its concentration, in m s−2, is given by

dvdt

= sin πt −√

3 cos πt.

Express v and x in terms of t. [7 marks]

Find the velocities of the particle when its acceleration is zero for the first and second times. Find also thedistance travelled by the particle between the first and second times its acceleration is zero. [7 marks]

7 Two archers A and B take turns to shoot, with archer A taking the first shot. The probabilities of archers Aand B hitting the bull’s-eye in each shot are 1

6 and 15 respectively. Show that the probability of archer A hitting the

bull’s-eye first is 12 . [4 marks]

8 The probability that it rains in a certain area is 15 . The probability that an accident occurs at a particular corner

of a road in that area is 120 if it rains and 1

50 if it does no rain. Find the probability that it rains if an accident occursat the corner. [5 marks]

9 The independent Poisson random variables X and Y have parameters 0.5 and 3.5 respectively. The randomvariable W is defined by W = X − Y .

(a) Find E (W) and Var (W). [4 marks]

(b) Give one reason why W is not a Poisson random variable. [1 mark]

10 The probability that a heart patient survives after surgery in a country is 0.85.

(a) Find the probability that, out of five randomly chosen heart patients undergoing surgery, four survive.[3 marks]

(b) Using a suitable approximate distribution, find the probability that more than 160 survive after surgery in arandom sample of 200 heart patients. [6 marks]

12


11 The times taken by 22 students to breakfast are shown in the following table.

Time (x minutes) 2 ≤ x < 5 5 ≤ x < 8 8 ≤ x < 11 11 ≤ x < 14 14 ≤ x < 17 17 ≤ x < 20Number of students 1 2 4 8 5 2

(a) Draw a histogram of the grouped data. Comment on the shape of the frequency distribution. [4 marks]

(b) Calculate estimates of the mean, median and mode of the breakfast times. Use your calculations to justifyyour statement about the shape of the frequency distribution. [7 marks]

12 The continuous random variable X has probability density function

f (x) =

√

x − 112

, 1 ≤ x ≤ b,

0, otherwise,

where b is a constant.

(a) Determine the value of b. [4 marks]

(b) Find the cumulative distribution function of X and sketch its graph. [5 marks]

(c) Calculate E (X). [6 marks]

4 2005

Adapted from Koleksi Kertas Soalan STPM 2005 Jurusan Sains, Penerbit Fajar Bakti Sdn. Bhd.

Instructions:






Paper 1

1 Using the laws of the algebra of sets, show that

(A ∩ B)′ −(A′ ∩ B

)= B′.

[4 marks]

2 If y =cos x

x, where x , 0, show that x

d2ydx2 + 2

dydx

+ xy = 0. [4 marks]

3 The point R divides the line joining the points P(3, 2) and Q(5, 8) in the ratio 3 : 4. Find the equation of theline passing through R and perpendicular to PQ. [5 marks]

4 For the geometric series 7 + 3.5 + 1.75 + 0.875 + . . . , find the smallest value of n for which the differencebetween the sum of the first n terms and the sum to infinity is less than 0.01. [6 marks]

13


5 Find the solution set of the inequality |x − 2| <1x

, where x , 0. [7 marks]

6 Find the perpendicular distance from the centre of the circle x2 + y2 − 8x + 2y + 8 = 0 to the straight line3x + 4y = 28. Hence, find the shortest distance between the circle and the straight line. [7 marks]

7 Sketch, on the same coordinate axes, the curves y = ex and y = 2 + 3e−x. [2 marks]

Calculate the area of the region bounded by the y-axis and the curves. [6 marks]

8 A, B and C are square matrices such that BA = B−1 and ABC = (AB)−1. Show that A−1 = B2 = C. [3 marks]

If B =

1 2 00 −1 01 0 1

, find C and A. [7 marks]

9 The complex number z1 and z2 satisfy the equation z2 = 2 − 2√

3i.

(a) Express z1 and z2 in the form a + bi, where a and b are real numbers. [6 marks]

(b) Represent z1 and z2 in an Argand diagram. [1 mark]

(c) For each of z1 and z2, find the modulus, and the argument in radians. [4 marks]

10 The functions f and g are given by

f : x 7→ex − e−x

ex + e−x , g : x 7→2

ex + e−x .

(a) State the domains of f and g. [1 mark]

(b) Without using differentiation, find the range of f. [4 marks]

(c) Show that (f (x))2 +(g (x)

)2= 1. Hence, find the range of g. [6 marks]

11 Express f (x) =x2 − x − 1

(x + 2) (x − 3)in partial fractions. [5 marks]

Hence, obtain an expression of f (x) in ascending powers of1x

up to the term in1x3 . [6 marks]

Determine the set of values of x for which this expansion is valid. [2 marks]

12 Find the coordinates of the stationary point on the curve y = x2 +1x

, where x > 0; give the x-coordinate andy-coordinate correct to three decimal places. Determine whether the stationary point is a minimum or a maximumpoint. [5 marks]

The x-coordinate of the point of intersection of the curves y = x2 +1x

and y =1x2 , where x > 0, is p. Show

that 0.5 < p < 1. Use the Newton-Raphson method to determine the value of p correct to three decimal placesand, hence, find the point of intersection. [9 marks]

Paper 2

1 The diagram above shows two intersecting circles AXYB and CBOX, where O is the centre of the circle AXYB.AXC and BYC are straight lines. Show that ∠ABC = ∠BAC. [5 marks]

14


A

B

C

X

YO

2 In the triangle ABC, the point P lies on the side AC such that ∠BPC = ∠ABC. Show that the triangles BPCand ABC are similar. [3 marks]

If AB = 4 cm, AC = 8 cm and BP = 3 cm, find the area of the triangle BPC. [4 marks]

3 Using the substitution y =vx2 , show that the differential equation

dydx

+ y2 = −2yx

may be reduced todvdx

= −v2

x2 .

[3 marks]

Hence, find the general solution of the original differential equation. [4 marks]

4 In the tetrahedron ABCD, AB = BC = 10 cm, AC = 8√

2 cm, AD = CD = 8 cm and BD = 6 cm. Showthat the line from C perpendicular to AB and the line from D perpendicular to AB meet at a point on AB. Hence,calculate the angle between the face ABC and the face ABD. [8 marks]

5 Show thatddx

(ln tan x) =2

sin 2x.

[2 marks]

Hence, find the solution of the differential equation

(sin 2x)dydx

= 2y (1 − y)

for which y = 13 when x = 1

4π. Express y explicitly in terms of x in your answer. [8 marks]

6 The points P and Q lie on the diagonals BD and DF respectively of a regular hexagon ABCDEF such that

BPBD

=DQDF

= k.

Express−−→CP and

−−→CQ in terms of k, a and b, where

−−→AB = a and

−−→BC = b. [7 marks]

If the points C, P and Q lie on a straight line, determine the value of k. Hence, find CP : PQ. [7 marks]

7 The mass of a small loaf of bread produced in a bakery may be modelled by a normal random variable withmean 303 g and standard deviation 4 g. Find the probability that a randomly chosen loaf has a mass between 295g and 305 g. [3 marks]

15


8 A four-digit number, in the range 0000 to 9999 inclusive, is formed. Find the probability that

(a) the number begins or ends with 0, [3 marks]

(b) the number contains exactly two non-zero digits. [3 marks]

9 A computer accessories distributor obtains its supply of diskettes from manufacturers A and B, with 60% of thediskettes from manufacturer A. The diskettes are packed by the manufacturers in packets of tens. The probabilitythat a diskette produced by manufacturer A is defective is 0.05 whereas the probability that a diskette producedby manufacturer B is defective is 0.02. Find the probability that a randomly chosen packet contains exactly onedefective diskette. [7 marks]

10 The continuous random variable X has probability density function

f (x) =

125 (1 − 2x) , −2 ≤ x ≤ 1

2 ,325 (2x − 1) , 1

2 ≤ x ≤ 3,0, otherwise.

(a) Sketch the graph of y = f (x). [2 marks]

(b) Given that P (0 ≤ X ≤ k) =13100

, determine the value of k. [6 marks]

11 The probability distribution function of the discrete random variable Y is

P (Y = y) =y

5050, y = 1, 2, 3, . . . , 100.

(a) Show that E (Y) = 67 and find Var (Y). [5 marks]

(b) Find P (|Y − E (Y)| ≤ 30). [4 marks]

12 Overexposure to a certain metal dust at the workplace of a factory is detrimental to the health of its workers.The workplace is considered safe if the level of the metal dust is less than 198 µg m−3. The level of the metaldust at the workplace is recorded at a particular time of day for a period of 90 consecutive days. The results aresummarised in the table below.

Metal dust level (µg m−3) Number of days170 – 174 8175 – 179 11180 – 184 25185 – 189 22190 – 194 15195 – 199 7200 – 204 2

(a) State what the number 11 in the table means. [1 mark]

(b) Calculate estimates of the mean and standard deviation of the levels of the metal dust. [5 marks]

(c) Plot a cumulative frequency curve of the above data. Hence, estimate the median and the interquartile range.[7 marks]

(d) Find the percentage of days for which the workplace is considered unsafe. [3 marks]

5 2004

Adapted from Koleksi Kertas Soalan STPM 2004 Jurusan Sains, Pearson Malaysia Sdn. Bhd.

16


Instructions:






Paper 1

1 Show that ∫ e

1ln x dx = 1.

[4 marks]

2 Expand (1 − x)12 in ascending powers of x up to the term in x3. Hence, find the value of

√7 correct to five

decimal places. [5 marks]

3 Using the laws of the algebra of sets, show that, for any sets A and B,

(A − B) ∪ (B − A) = (A ∪ B) − (A ∩ B) .

[6 marks]

4 Matrix A is given by

A =

3 3 45 4 11 2 3

.Find the adjoint of A. Hence, find A−1. [6 marks]

5 The functionf is defined by

f (x) =

x − 1x + 2

, 0 ≤ x < 2,

ax2 − 1, x ≥ 2,

where a ∈ R. Find the value of a if limx→2

f (x) exists. With this value of a, determine whether f is continuous at

x = 2. [6 marks]

6 The sum of the distance of the point P from the point (4, 0) and the distance of P from the origin is 8 units.

Show that the locus of P is the ellipse(x − 2)2

16+

y2

12= 1 and sketch the ellipse. [7 marks]

7 Sketch, on the same coordinate axes, the graphs of y = 2 − x and y =

∣∣∣∣∣2 +1x

∣∣∣∣∣. [4 marks]

Hence, solve the inequality

2 − x >∣∣∣∣∣2 +

1x

∣∣∣∣∣ .[4 marks]

17


8 Using the sketch graphs of y = x3 and x + y = 1, show that the equation x3 + x − 1 = 0 has only one real rootand state the successive integers a and b such that the real root lies in the interval (a, b). [4 marks]

Use the Newton-Raphson method to find the real root correct to three decimal places. [5 marks]

9 The matrices P and Q, where PQ = QP, are given by

P =

2 −2 00 0 2a b c

, Q =

−1 1 00 0 −10 −2 2

.Determine the values of a, b and c. [5 marks]

Find the real numbers m and n for which P = mQ + nI, where I is the 3 × 3 identity matrix. [5 marks]

10 A curve is defined by the parametric equations x = 1− 2t, y = −2 +2t

. Find the equation of the normal to thecurve at the point A(3,−4). [7 marks]

The normal to the curve at the point A cuts the curve again at the point B. Find the coordinates of B. [4 marks]

11 Sketch, on the same coordinate axes, the line y = 12 x and the curve y2 = x. Find the coordinates of the points

of intersection. [5 marks]

Find the area of the region bounded by the line y = 12 x and the curve y2 = x. [4 marks]

Find the volume of the solid formed when the region is rotated through 2π radians about the y-axis. [4 marks]

12 Prove that the sum of the first n terms of a geometric series a + ar + ar2 + . . . isa (1 − rn)

1 − r. [3 marks]

(a) The sum of the first five terms of a geometric series is 33 and the sum of the first ten terms of the geometricseries is −1023. Find the common ratio and the first term of the geometric series. [5 marks]

(b) The sum of the first n terms and the sum to infinity of the geometric series 6 − 3 + 32 − . . . are S n and S∞

respectively. Determine the smallest value of n such that |S n − S∞| < 0.001. [7 marks]

Paper 2

1 Express cos θ + 3 sin θ in the form r cos (θ − α), where r > 0 and 0◦ < α < 90◦. [4 marks]

2 Find all values of x, where 0◦ < x < 360◦, which satisfy the equation

tan x + 4 cot x = 4 sec x.

[5 marks]

3 The variables t and x are connected bydxdt

= 2t (x − 1) ,

where x , 1. Find x in terms of t if x = 2 when t = 1. [5 marks]

4 The points S and T are midpoints of the sides AB and AD respectively of a parallelogram ABCD. The linesCS and CT cut the diagonal BD at the points U and V respectively.

Show that−−→BU = λ

−−→BC + λ

−−→CD and also

−−→BU = (1 − µ)

−−→BC +

12µ−−→CD, where λ and µ are constants. Hence, show

that−−→BU =

13−−→BD. [6 marks]

Deduce that the lines CS and CT trisect the diagonal BD. [3 marks]

18


A

B C

D E

75◦

75◦

75◦

75◦

5 The above diagram shows two isosceles triangles ABC and ADE which have bases AB and AD respectively.Each triangle has base angles measuring 75◦, with BC and DE parallel and equal in length. Show that

(i) ∠DBC = ∠BDE = 90◦, [4 marks]

(ii) the triangle ACE is an equilateral triangle, [4 marks]

(iii) the quadrilateral BCED is a square. [4 marks]

6 A canal of width 2a has parallel straight banks and the water flows due north. The points A and B are onopposite banks and B is due east of A, with the point O as the midpoint of AB. The x-axis and y-axis are taken inthe east and north directions respectively with O as the origin. The speed of the current in the canal, ve, is givenby

ve = v0

(1 −

x2

a2

),

where v0 is the speed of the current in the middle of the canal and x the distance eastwards from the middle ofthe canal. A swimmer swims from A towards the east at speed vr relative to the current in the canal. Taking y todenote the distance northwards travelled by the swimmer, show that

dydx

=v0

vr

(1 −

x2

a2

).

[3 marks]

If the width of the canal is 12 m, the speed of the current in the middle of the canal is 10 m s−1 and the speedof the swimmer is 2 m s−1 relative to the current in the canal,

(i) find the distance of the swimmer from O when he is at the middle of the canal and his distance from B whenhe reaches the east bank of the canal, [7 marks]

(ii) sketch the actual path taken by the swimmer. [3 marks]

7 A type of seed is sold in packets which contain ten seeds each. On the average, it is found that a seed perpacket does not germinate. Find the probability that a packet chosen at random contains less than two seeds whichdo not germinate. [4 marks]

8 The continuous random variable X has the probability density function

f (x) =

427

x2 (3 − x) , 0 < x < 3,

0, otherwise.

(i) Calculate P(X <

32

). [3 marks]

(ii) Find the cumulative distribution function of X. [3 marks]

19


9 Two transistors are chosen at random from a batch of transistors containing ninety good and ten defective ones.

(i) Find the probability that at least one out of the two transistors chosen is defective. [3 marks]

(ii) If at least one out of the two transistors chosen is defective, find the probability that both transistors aredefective. [4 marks]

10 The lifespan of an electrical instrument produced by a manufacturer is normally distributed with a mean of72 months and a standard deviation of 15 months.

(i) If the manufacturer guarantees that the lifespan of an electrical instrument is at least 36 months, calculatethe percentage of the electrical instruments which have to be replaced free of charge. [4 marks]

(ii) If the manufacturer specifies that less than 0.1% of the electrical instruments have to be replaced free ofcharge, determine the greatest lenght of guarantee period correct to the nearest month. [5 marks]

11 The discrete random variable X has the probability function

P (X = x) =

k (4 − x)2 , x = 1, 2, 3,0, otherwise,

where k is a constant.

(i) Determine the value of k and tabulate the probability distribution of X. [3 marks]

(ii) Find E (7X − 1) and Var (7X − 1). [7 marks]

12 The following data show the masses, in kg, of fish caught by 22 fishermen on a particular day.

23 48 51 25 39 37 41 38 37 20 8869 22 42 46 23 52 41 40 59 68 59

(i) Display the above data in an ordered stemplot. [2 marks]

(ii) Find the mean and standard deviation. [5 marks]

(iii) Find the median and interquartile range. [4 marks]

(iv) Draw a boxplot to represent the above data. [3 marks]

(v) State whether the mean of the median is more suitable as a representative value of the above data. Justifyyour answer. [2 marks]

6 2003

Adapted from Koleksi Kertas Soalan STPM 2003 Jurusan Sains, Pearson Malaysia Sdn. Bhd.

Instructions:






20


Paper 1

1 Show that −1 is the only real root of the equation

x3 + 3x2 + 5x + 3 = 0.

[5 marks]

2 If y = ln√

xy, find the value ofdydx

when y = 1. [5 marks]

3 Using the substitution u = 3 + 2 sin θ, evaluate∫ π6

0

cos θ(3 + 2 sin θ)2 dθ.

[5 marks]

4 If (x + iy)2 = i, find all the real values of x and y. [6 marks]

5 Find the set of values of x such that −2 < x3 − 2x2 + x − 2 < 0. [7 marks]


f (x) =

1 + ex, x < 1,3, x = 1,2 + e − x, x > 1.

(i) Find limx→1−

f (x) and limx→1+

f (x). Hence determine whether f is continuous at x = 1. [4 marks]

(ii) Sketch the graph of f. [3 marks]

7 The straight line l1 which passes through the points A(4, 0) and B(2, 4) intersects the y-axis at the point P. Thestraight line l2 is perpendicular l1 and passes through B. If l2 intersects the x-axis and y-axis at the points Q and Rrespectively, show that PR : QR =

√5 : 3. [8 marks]

8 Express(

1 + x1 + 2x

) 12

as a series of ascending powers of x up to the term in x3. [6 marks]

By taking x = 130 , find

√62 correct to four decimal places. [3 marks]

9 The matrix A is given by

A =

1 2 −33 1 10 1 2

.(i) Fin the matrix B such that B = A2 − 10I, where I is the 3 × 3 identity matrix. [3 marks]

(ii) Find (A + I) B, and hence find (A + I)21 B. [6 marks]

10 The curve y =a2

x (b − x), where a , 0, has a turning point at the point (2, 1). Determine the values of a andb. [4 marks]

Calculate the area of the region bounded by the x-axis and the curve. [4 marks]

Calculate the volume of the solid formed by revolving the region about the x-axis. [4 marks]

21


11 Sketch, on the same coordinate axes, the graphs y = ex and y =2

1 + x. Show that the equation (1 + x) ex−2 =

0 has a root in the interval [0, 1]. [7 marks]

Use the Newton-Raphson method with the initial estimate x0 = 0.5 to estimate the root correct to three decimalplaces. [6 marks]

12 Express ur =2

r2 + 2rin partial fractions. [3 marks]

Using the result obtained,

(i) show that u2r = −

1r

+1r2 +

1r + 2

+1

(r + 2)2 , [2 marks]

(ii) show thatn∑

r=1

ur =32−

1n + 1

−1

n + 2, and determine the values of

∞∑r=1

ur and∞∑

r=1

(ur+1 +

13r

). [9 marks]

Paper 2

1 Let u = cos φ i + sin φ j and v = cos θ i + sin θ j, where i and j are perpendicular unit vectors. Show that

12|u − v| = sin

12

(φ − θ) .

[5 marks]

2 Vertices B and C of the triangle ABC lie on the circumference of a circle. AB and AC cut the circumference ofthe circle at X and Y respectively. Show that ∠CBX + ∠CYX = 180◦. [3 marks]

If AB = AC, show that BC is parallel to XY . [3 marks]

R

PS

QA

B

3 The above diagram shows two circles ABRP and ABQS which intersect at A and B. PAQ and RAS are straightlines. Prove that the triangles RPB and S QB are similar. [7 marks]

4 A force of magnitude 2p N acts along the line OA and a force of magnitude 10 N acts along the line OB.The angle between OA and OB is 120◦. The resultant force has magnitude

√3p N. Calculate the value of p and

determine the angle between the resultant force and OA. [8 marks]

5 Starting from the formulae for sin (A + B) and cos (A + B), prove that

tan (A + B) =tan A + tan B

1 − tan A tan B.

[3 marks]

If 2x + y =π

4, show that

tan y =1 − 2 tan x − tan2 x1 + 2 tan x − tan2 x

.

By substituting x =π

8, show that tan

π

8=√

2 − 1. [6 marks]

22


6 The rate of increase in the number of a species of fish in a lake is described by the differential equation

dPdt

= (a − b) P,

where P is the number of fish at the time t weeks, a the rate of reproduction, and b the mortality rate, with a and bas constants.

(i) Assuming that P = P0 at time t = t0 and a > b, solve the above differential equation and sketch its solutioncurve.

(ii) At a certain instant, there is an outbreak of an epidemic of a disease. The epidemic results in no more

offspring of the fish being produced and the fish die at a rate directly proportional to

√1P

. There are 900

fish before the outbreak of the epidemic and only 400 fish are alive after 6 weeks. Determine the length oftime from the outbreak of the epidemic until all the fish of that species die. [9 marks]

7 The probability that a person allergic to a type of anaesthetic is 0.002. A total of 2000 persons are injected withthe anaesthetic. Using a suitable approximate distribution, calculate the probability that more than two personsare allergic to the anaesthetic. [5 marks]

8 Tea bags are labelled as containing 2 g of tea powder. In actual fact, the mass of tea powder per bag has mean2.05 g and standard deviation 0.05 g . Assuming that the mass of tea powder of each bag is normally distributed,calculate the expected number of tea bags which contain 1.95 g to 2.10 g of tea powder in a box of 100 tea bags.

[5 marks]

9 A factor has 36 male workers and 64 female workers, with 10 male workers earning less than RM1000.00 amonth and 17 female workers earning at least RM1000.00 a month. At the end of the year, worker earning lessthan RM1000.00 are given a bonus of RM1000.00 whereas the others receive a month’s salary.

(i) If two workers are randomly chosen, find the probability that exactly one worker receives a bonus of onemonth’s salary. [3 marks]

(ii) If a male worker and a female worker are randomly chosen, find the probability that exactly one workerreceives a bonus of one month’s salary. [3 marks]

10 Show that, for numbers x1, x2, x3, . . . , xn with mean x̄,∑(x − x̄)2 =

∑x2 − nx̄2.

[2 marks]

The numbers 4, 6, 12, 5, 7, 9, 5, 11, p, q, where p < q, have mean x̄ = 6.9 and∑

(x − x̄)2 = 102.9. Calculatethe values of p and q. [6 marks]

11 The number of ships which anchor at a port every week for 26 particular weeks are as follows.

32 28 43 21 35 19 25 45 35 32 18 26 3026 27 38 42 18 37 50 46 23 40 20 29 46

(i) Display the data in a stemplot. [2 marks]

(ii) Find the median and interquartile range. [4 marks]

(iii) Draw a boxplot to represent the data. [3 marks]

(iv) State the shape of the frequency distribution. Give a reason for your answer. [2 marks]

23


12 The lifespan of a species of plant is a random variable T (tens of days). The probability density function isgiven by

f (t) =

18

e−18 t, t > 0,

0, otherwise.

(i) Find the cumulative distribution function of T and sketch its graph. [6 marks]

(ii) Find the probability, to three decimal places, that a plant of that species randomly chosen has a lifespan ofmore than 20 days. [3 marks]

(iii) Calculate the expected lifespan of that species of plant. [5 marks]

24

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