stpm mathematics t questions for revision

STPM Further Mathematics T Past Year Questions

Lee Kian Keong & [email protected]

http://www.facebook.com/akeong

Last Edited by December 24, 2011

Abstract

This is a document which shows all the STPM questions from year 2002 to year 2011 using LATEX.Students should use this document as reference and try all the questions if possible. Students areencourage to contact me via email1 or facebook2. Students also encourage to send me your collectionof papers or questions by email because i am collecting various type of papers. All papers are welcomed.Special thanks to Zhu Ming for helping me to check the questions.

Contents

1 PAPER 1 QUESTIONS 2STPM 2001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3STPM 2002 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5STPM 2003 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7STPM 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9STPM 2005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11STPM 2006 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13STPM 2007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15STPM 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17STPM 2009 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19STPM 2010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21STPM 2011 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 PAPER 2 QUESTIONS 25STPM 2002 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26STPM 2003 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28STPM 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30STPM 2005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32STPM 2006 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34STPM 2007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37STPM 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40STPM 2009 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43STPM 2010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46STPM 2011 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

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1

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FURTHER MATHEMATICS PAPER 1 QUESTIONS

1 PAPER 1 QUESTIONS

2

FURTHER MATHEMATICS PAPER 1 QUESTIONS STPM 2001

STPM 2001

1. Solve the equationcosh 2x+ 2 coshx = 5.

Give your answer correct to two decimal places. [4 marks]

2. Sketch the curve with polar equationr = 3 + 2 cos θ.

Find the area of the region bound by the curve. [6 marks]

3. Using Maclaurin expansion, find

limx→0

1− cos2 x

x(1− e−x).

[4 marks]

4. Use the Newton Raphson method with initial estimate, x0 = 0.5, find the root for equation 10x −2 sinx = 5 correct to three decimal places. [5 marks]

5. Using mathematical induction, prove that, for n ≥ 2, xn − nx+ (n− 1) divisible by (x− 1)2.[6 marks]

6. Shade the region in Argand diagram which satisfies the inequality |z − (2 + 2i)| ≤ 1. Hence, find thegreatest and least value of |z − (1 + i)| for z in this region. [4 marks]

7. If M =

(cos θ sin θ− sin θ cos θ

), show that (MT )3 =

(cos 3θ − sin 3θsin 3θ cos 3θ

). Hence, write down (MT )n in

terms of n and θ. Justify your answer. [5 marks]

8. The position vector of points A and B respect to O are a = 2i − 2j − 9k and b = −2i + 4j +15k respectively. Find the vector equation for the line passes through the midpoint of AB andperpendicular to the plane OAB. [3 marks]

Hence, determine the position vector, respect to O, of the points which are 7 units from the midpointof AB. [3 marks]

9. Given a curve with parametric equation

x = a(t− 3t3), y = 3at2,

with a > 0 and t ∈ R.

Determine the values of t when the curve cuts the y-axis and sketch the curve. [4 marks]

Show that

(dx

dt

)2

+

(dy

dt

)2

= a2(1 + 9t2)2. [3 marks]

Calculate the length of the loop in the curve. [4 marks].

Calculate the surface area generated when the loop is rotated through π radians about the y-axis.[4 marks]

10. Find the general solution for the differential equation

d2y

dx2+ 5

dy

dx+ 4y = e−2x + sinx.

3


[9 marks]

Find the particular solution which satisfies the condition y =6

17and

dy

dx=

3

34when x = 0. Show

that y =3

34sinx− 5

34cosx when x is positive and large. [6 marks]

11. (a) Find the expansion for ex cosx and ex sinx in ascending powers of x up to the term x3. Hence,

show that, if x4 and higher power of x can be ignored, sinhx cosx ≈ x− 1

3x3 and coshx sinx ≈

x+1

3x3. [6 marks]

(b) Using Maclaurin theorem, find the expansion for secx in ascending powers of x up to the termx4. [5 marks]

Deduce the first three non zero terms in the expansion sec2 x and tanx in ascending powers ofx. [4 marks]

12. (a) Describe the locus of the points represented by the complex numbers z if z satisfies

i.

∣∣∣∣z − 1− 2i

z + 1 + 4i

∣∣∣∣ = 1, [3 marks]

ii.1

z+

1

z∗= 3. [4 marks]

(b) Show that, for z 6= −1,

z − z2 + z3 − z4 + z5 =z3(z2 + 1

z2 + z3 + 1z3

)2 + z + 1

z

.

Using the substitution z = eiθ, show that

5∑k=1

(−1)k+1 cos kθ =cos 3θ cos 5

2θ

cos 12θ

,

where θ is odd multiple of π. [8 marks]

13. The transformation T on the x− y plane is given by

T :

(xy

)→ A2

(xy

),

where A =

(6 k1 2

), k ∈ R.

If the line y = mx+c is the invariant line under transformation T, show that c = 0 and km2+4m−1 =0. [9 marks]

Deduce the possible value of k such that there is only one invariant line, and determine equation ofthe line. [6 marks]

14. Given that origin, O and position vectors of P , Q, R, and R are 4i+3j+4k, 6i+ j−6k and −i+ j+krespectively. Find the equation of the plane OPQ. [2 marks]

Show that the point S lies on the plane OPQ. [4 marks]

Show that the line RS are perpendicular to the plane OPQ. [4 marks]

Find the acute angle between the line PR and the plane OPQ. [5 marks]

4


STPM 2002

1. Simplify the statement p ∨ (∼ p ∨ q) ∨ (∼ p∧ ∼ q). [4 marks]

2. Find the exact value of x that satisfies the equation 5 sechx− 12 tanhx = 13. [4 marks]

3. Solve the recurrence relation xn+1 = xn+1 + xn with x0 = 0, x1 = 1. [5 marks]

4. Point P is represented by the complex number z = cos θ + i sin θ, where 0 ≤ θ ≤ 2π, in Arganddiagram. Show that the locus of the point Q that is represented by ω = 3z2 is a circle, and find itscentre and radius. Find the minimum and maximum distance between the point P and Q, and statethe corresponding value of θ. [6 marks]

5. If In =

∫ π2

0

xn cosx dx, show that, for n ≥ 2,

In =(π

2

)n− n(n− 1)In−2.

[4 marks]

Hence, evaluate

∫ π2

0

x4 cosx dx. [2 marks]

6. The function f is defined by

f(x) =

a+ bx+ x2, x > 1,

1− x, x ≤ 1,

with a and b are constants. Determine the values of a and b such that f is differentiable at x = 1.[7 marks]

7. Find the particular solution for the differential equation

dy

dx+

x− 2

x(x− 1)y = − 1

x2(x− 1).

that satisfies the boundary condition y =3

4when x = 2. [8 marks]

8. Using mathematical induction, prove that

n∑r=1

cos 2rθ =sin(2n+ 1)θ

2 sin θ− 1

2.

[7 marks]

If sin θ =1

2

√2−√

3, find sin 5θ. [3 marks]

9. Show that cosnθ =1

2(einθ + e−inθ). [3 marks]

Hence, show that, for x ∈ R and |x| < 1,

∞∑n=0

xn cos(2n+ 1)θ =(1− x) cos θ

1− 2x cos 2θ + x2.

5


Deduce that∞∑n=0

cosn 2θ cos(2n+ 1)θ =1

2sec θ

for θ 6= 1

2kπ, where k is integer. [7 marks]

10. If xn = xn−1 + h and yn = y(xn), with h3 and higher powers of h can be neglected, show that therecurrence relation for differential equation y′′n − 2y′n + 2yn = 0 is

(1− h)yn+1 − 2(1− h2)yn + (1 + h)yn−1 = 0.

[7 marks]

If y0 = 0, y1 = 0.5, and h = 0.1, calculate y4 correct to three decimal places. [3 marks]

11. Matrix A is given by

A =

−2 1 3−3 2 3−1 1 2

.

Find the eigenvalue λ1, λ2, and λ3, where λ1 < λ2 < λ3 of matrix A. Find also the eigenvectors e1,e2, and e3 where ei corresponding to λi for i = 1, 2, 3. [8 marks]

Matrix P is a 3× 3 matrix where its columns are e1, e2 and e3 in sequence. Show that

P−1AP =

λ1 0 00 λ2 00 0 λ3

.

[5 marks]

Deduce the relationship between determinant of A and its eigen values. [2 marks]

12. If y = sin−1 x, show thatd2y

dx2= x

(dy

dx

)3

andd3y

dx3=

(dy

dx

)3

+ 3x2(dy

dx

)5

. [4 marks]

Using Maclaurin theorem, express sin−1 x as a series of ascending powers of x up to the term in x5.State the range of values of x for the expansion valid. [7 marks]

Hence,

(a) taking x = 0.5, find the approximation of π correct to two decimal places, [2 marks]

(b) find limx→0

x− sin−1 x

x− sinx. [2 marks]

6


STPM 2003

1. Determine whether each of the following propositions is true or false.

(a) If |2x− 3| > 9, where x is an integer, then |x| > 2. [2 marks]

(b) If A ∩ B = φ, where A and B are non-empty sets, then x /∈ B, ∀x ∈ A, or x /∈ A, ∀x ∈ B.[2 marks]

2. Find the integral ∫dx√

4x2 − 4x+ 3.

[4 marks]

3. Using Cramer’s rule, determine the set of values of k such that the following system of linear equationshas integer solutions.

2x − y + 3z = k,2x + y − z = 1,6x − 3y + z = 3k,

[6 marks]

4. Complex numbers z and w are such that |z|2 + 2Re wz = c, where c > −|w|2. Show that the locusof the point P which represents z in the Argand diagram is a circle, and state the centre and radiusof the circle in terms of w and c. [6 marks]

5. Use the expansion of1

1 + x2to express tan−1 x as a series of ascending powers of x up to the term

in x7. [4 marks]

Hence find, in terms of π, the sum of the infinite series

1

3× 3− 1

5× 32+

1

7× 33− . . .

[2 marks]

6. Let In =

∫ π

0

sinn x dx, where n ≥ 2. Show that

In =n− 1

nIn−2.

[4 marks] Express In in terms of I1, for odd integers n ≥ 3. Hence find the value of∫ π

0

sin7 x cos2 x dx.

[3 marks]

7. A loan of P ringgit is to be paid off over a period of N years. The loan carries a compound interestof 100k% a year. The yearly repayment is B ringgit. The balance after r years is ar ringgit, wherea0 = P . The interest is charged on the balance at the beginning of the year. Write down a recurrencerelation that involves ar+1 and ar and show that

ar =

(P − B

k

)(1 + k)r +

B

k.

7


[6 marks]

Calculate the yearly repayment if a loan of RM100 000 that carries a compound interest of 8% a yearis to be paid off over a period of 10 years. [3 marks]

8. Matrix M is given by

M =

(3 −14 −1

).

Use mathematical induction to prove that

Mn =

(2n+ 1 −n

4n 1− 2n

), n = 1, 2, . . .

[5 marks]

Show that M and Mn have the same eigenvalues. [4 marks]

9. If y = (sin−1 x)2, show that, for −1 < x < 1,

(1− x2)d2y

dx2− xdy

dx− 2 = 0.

[3 marks]

Find the Maclaurin series for (sin−1 x)2 up to the term in x6. [6 marks]

10. Find the general solution of the differential equation

d2y

dx2+ 2

dy

dx+ y = cos 2x− 7 sin 2x.

[7 marks]

Find the solution of the differential equation for which y =dy

dx= 2 when x = 0. Determine whether

this solution is finite as x→∞. [5 marks]

11. Let a coshx+ b sinhx = r sinh(x+ k), where a, b, and r are positive real numbers. Express k and rin terms of a and b, and determine the condition in order that k and r exist. [7 marks]

Find the coordinates of the point on the curve y = coshx + 3 sinhx which has a gradient of√

8 atthat point. [6 marks]

12. Obtain all the roots of the equation z5 = 1 in the form eiθ, where 0 < θ ≤ 2π. Hence show that theroots of the equation

(ω − 1)5 = ω5

are1

2+

1

2i cot

k

5π, where k = 1, 2, 3, 4. [8 marks]

Deduce the roots of the equation(ω − i)5 = ω5.

[2 marks]

Write down the roots of the equations (ω − 1)5 = ω5 and (ω − i)5 = ω5 in the form a + bi, with aand b in one decimal place. Describe and compare the positions of the roots of these two equationsin the Argand diagram. [5 marks]

8


STPM 2004

1. Show that the following pair of propositions are equivalent.

(p↔ q); [∼ (p∧ ∼ q)] ∧ [∼ (∼ p ∧ q)].

[3 marks]

2. Show that

x+ sin−1(cosx) =1

2π,

for 0 ≤ x ≤ π. [3 marks]

3. Using mathematical induction, prove that

n∑p=1

p(p!) = (n+ 1)!− 1.

[5 marks]

4. Solve the recurrence relation xn+1 = kxn + 4, where x0 = 7, x1 = 25 and k is a constant. [7 marks]

5. Show that tanh−1 x =1

2ln

(1 + x

1− x

)for −1 < x < 1 and find

d

dx(tanh−1 x). [7 marks]

6. Find the exact value of ∫ 2

1

x√x4 − 1 dx.

[7 marks]

7. Show that the general solution of the recurrence relation an+2 = 6an+1 − 25an is

an = 5n(α cosnθ + iβ sinnθ),

where α and β are arbitrary complex constants and θ = tan−14

3. [6 marks]

If a0 = a1 = 1, find the particular solution of the above recurrence relation. [2 marks]

8. If In =

∫xn√x2 + a2

dx, where n > 1 and a is a non-zero constant, show that

nIn + (n− 1)a2In−2 = xn−1√x2 + a2.

[4 marks]

Hence, find the exact value of

∫ 1

0

x3 + x2√x2 + 1

dx. [6 marks]

9. Using Taylor’s theorem, find the series expansion of sin

(1

6π + h

)in ascending powers of h up to

the term in h3. [3 marks]

If 0 < h <1

6π, show that the remainder term R is given by

1

48h4 < R <

√3

48h4.

9


[4 marks]

Hence, find the bounds for sin1

5π correct to seven decimal places. [3 marks]

10. For each of the following cases, find the equation of the locus of the complex number z and show thelocus in an Argand diagram.

(a)

∣∣∣∣z − (1 + i)

z − (1− i)

∣∣∣∣ = 2 [5 marks]

(b) arg

(z − (1 + i)

z − (1− i)

)=

1

2π [6 marks]

11. The matrix A is given by

A =

2 −3 2−3 3 32 3 2

.

Show that the eigenvalues of A are -3, 4 and 6. Find the corresponding eigenvectors. [8 marks]

Write down a matrix P such that PTAP is a diagonal matrix. Find this diagonal matrix and itsinverse. [6 marks]


d2y

dt2+ 4

dy

dt+ 12y = 8 cos 2t.

[8 marks]

(a) Find the approximate values of y when t = nπ and when t =

(n+

1

2

)π, where n is a large

positive integer. [3 marks]

(b) Show that, whatever the initial conditions, the limiting solution as t→∞ may be expressed inthe form

y = k sin(2t+ α),

where k is a positive integer and α an acute angle which are to be determined. [4 marks]

10


STPM 2005

1. Write down the negation, the converse and the contrapositive of the following implication

If a = 0 and b = 0, then ab = 0

Which of your answers is equivalent to the given implication? [4 marks]

2. The function f is defined by

f =

x2 − 1, x ≤ 1,

k(x+ 1), x > 1.

(a) If f is continuous, find the value of k. [2 marks]

(b) Determine, for this value of k, whether f is differentiable at x = 1. [3 marks]

3. Find the general solution of each of the following recurrence relations.

(a) ur+1 = αur + β2r, where α and β are arbitrary constants and α 6= 2. [3 marks]

(b) ur+1 = 2ur + γ2r, where γ is an arbitrary constant. [3 marks]

4. A differential equation has y = e3x and y = xe3x as solutions.

(a) Find the differential equation. [2 marks]

(b) Write down the general solution of the differential equation and determine the particular solution

satisfying the initial conditions y = 1 anddy

dx= 0 when x = 0. [4 marks]

5. A circular disc is partitioned into n equal sectors. Each sector is to be coloured with one of the fourdifferent colours provided and no two adjacent sectors are to be of the same colour. If an is thenumber of ways to colour the disc with n sectors, find a2, a3 and a4. [3 marks]

Given that an = 2an−1 + 3an−2, where n ≥ 4, find an explicit formula for an. [4 marks]

6. Find the Maclaurin expansion of x√

cosx up to the term in x3. State the range of values of x forwhich the expansion is valid. [7 marks]

7. If P is the point on an Argand diagram representing the complex number z and |z− 2|+ |z+ 2| = 5,find the cartesian equation of the locus of P and sketch this locus. [4 marks]

Find the points on this locus which satisfy the equation

|z| =∣∣∣∣z − 3

10+

3

10i

∣∣∣∣ .[4 marks]

8. Show that, for |x| > 1,

coth−1 x =1

2ln

(x+ 1

x− 1

).

[4 marks]

If coth 2y =5

4, find the value of coth y. [4 marks]

11



A =

k 1 52 k 88 −3 2

.

Determine all values of k for which the equation AX = B, where B is a 3× 1 matrix, does not havea unique solution. [3 marks]

For each of these values of k, find the solution, if any, of the equation

AX =

1−24

.

[7 marks]


xdy

dx− 3y = x3.

[4 marks]

Find the particular solution given that y has a minimum value when x = 1. [3 marks]

Sketch the graph of this particular solution. [3 marks]

11. A curve has equation y = 1− ln tanh1

2x, where x > 0.

(a) Show that, for this curve, y > 1. [2 marks]

(b) Show thatdy

dx= − cosechx. [3 marks]

(c) Sketch the curve. [2 marks]

(d) Show that the length of the curve between the points where x = 1 and x = 4 is ln(e6 + e4 + e2 +1)− 3. [6 marks]

12. Prove de Moivre’s theorem for positive integer exponents. [5 marks]

Using de Moivre theorem, show that

sin 5θ = a sin5 θ + b cos2 sin3 θ + c cos4 θ sin θ

where a, b and c are integers to be determined. Expresssin 5θ

sin θin terms of cos θ, where θ is not a

multiple of π. [5 marks]

Hence, find the roots of the equation 16x4 − 12x2 + 1 = 0 in trigonometric form. Deduce the value

of cos2(π

5

)+ cos2

(2π

5

). [6 marks]

12


STPM 2006

1. The matrix M is given by

M =

−3 1 2−6 2 30 2 1

.

By using Cayley Hamilton theorem, show that

M−1 =1

6(7I−M2),

where I is the 3× 3 identity matrix. [4 marks]

2. Prove that

tanh−1 x+ tanh−1 y = tanh−1(x+ y

1 + xy

).

[2 marks]

Hence, solve the equation

tanh−1 3x+ tanh−1 x = tanh−18

13.

[3 marks]

3. Let A = a, a + d, a + 2d and B = a, ar, ar2, with a 6= 0, d 6= 0 and r 6= 1. Show that A = B if

and only if r = −1

2and d = −3

4a. [7 marks]

4. Consider the system of equations

x + y + pz = q,3x − y − 2z = 1,6x + 2y + z = 4,

for the two cases: p = 2, q = 1 and p = 1, q = 2.

For each case, find the unique solution if it exists or determine the consistency of the system if thereis no unique solution. [7 marks]

5. Find the domain and the range of the function f defined by

f(x) = sin−12(x− 1)

x+ 1.

[4 marks]

Sketch the graph of f . [3 marks]

6. Show that ∫ 1

0

x− 1√x2 + 2x+ 3

dx =√

6−√

3− 2 ln

(2 +√

6

1 +√

3

).

[7 marks]

7. If y = ex cosx, prove by mathematical induction that

dny

dxn= 2

12nex

(x+

1

4nπ

),

for every positive integer n. [8 marks]

13


8. Determine the number of root(s) of the equation x− tanh ax = 0 for each of the following cases.

(a) a < 1

(b) a = 1

(c) a > 1

[8 marks]

9. Let an be the number of ways (where the order is significant) the natural number n can be writtenas a sum of l’s, 2’s or both. Find a1, a2, a3 and a4. [2 marks]

Explain why the recurrence relation for an, in terms of an−1 and an−2 is

an = an−1 + an−2, n > 2.

[2 marks]

Find an explicit formula for an. [6 marks]

10. Using the substitution x = z12 , transform the differential equation

d2y

dx2+

(4x− 1

x

)dy

dx+ 4x2y = 0

into one relating y and z. [5 marks]

Hence, find y in terms of x given that y = 2 anddy

dx= −2 when x = 1. [5 marks]

State the limiting value of y as x→∞. [1 marks]

11. Let zl = 1, z2 = x + iy, z3 = y + ix, where x, y ∈ R, x > 0 and i =√−1. If z1, z2, . . ., zn is a

geometric progression,

(a) find x and y, [3 marks]

(b) express z2 and z3 in the polar form, [2 marks]

(c) find the smallest positive integer n such that z1 + z2 + . . .+ zn = 0, [5 marks]

(d) find the product z1z2z3 . . . zn, for the value of n in (c). [3 marks]

12. Derive the Taylor series for ex expanded at x = 0 and show that 2 < e < 3. [6 marks]

Write the above series up to the term in xn together with the remainder term. Hence, determine thesmallest integer n to ensure that the estimated value of e is correct to four decimal places and findthe estimated value. [7 marks]

14


STPM 2007

1. If z = cos θ + i sin θ, show that1

1 + z2=

1

2(1− i tan θ) and express

1

1− z2in a similar form.[4 marks]

2. Show that

∫ π4

0

secx(secx+ tanx)2 dx = 1 +√

2. [4 marks]

3. Using the substitution x = e2, show that the differential equation

x2d2y

dx2+ px

dy

dx+ qy = 0,

where p and q are constants, can be transformed into the differential equation

d2y

dz2+ r

dy

dz+ sy = 0,

where r and s are constants to be determined in terms of p and q. [5 marks]

4. The matrices A and B are given by

A =

1 2 3 45 1 1 32 0 8 0−1 −1 3 9

,B =

4 2 3 45 4 1 32 0 11 0−1 −1 3 12

.

Given that

1111

is an eigenvector of the matrix A, find its corresponding eigenvalue. [3 marks]

Hence, find the eigenvalue of the matrix B corresponding to the eigenvector

1111

. [3 marks]

5. Show that

sec−1 x+ cosec−1 x =1

2π,

where 0 ≤ sec−1 x ≤ π, sec−1 x 6= 1

2π, −1

2π ≤ cosec−1 x ≤ 1

2π and cosec−1 x 6= 0. [3 marks]

Hence, find the value of x such that

sec−1 x cosec−1 x = −3

2.

[3 marks]

6. A sequence a0, a1, a2, . . . is defined by a0 = 1 and ar+1 = 2ar + b for r ≥ 0, where b ∈ R.

Express ar in terms of r and b, and verify your result by using mathematical induction. [7 marks]

7. Solve the recurrence relationar − ar−1 − 6ar−2 = (−2)r,

where a0 = 0 and a1 = 2. [8 marks]

15


8. Consider the system of equations

λx + y + z = 1,x + λy + z = λ,x + y + λz = λ2,

where λ is a constant.

(a) Determine the values of λ for which this system has a unique solution, infinitely many solutionsand no solution. [5 marks]

(b) Find the unique solution in terms of λ. [5 marks]

9. Solve the differential equationdy

dx+ y sinhx = sinhx,

given that y = 0 when x = 0. [5 marks]

Show that 0 ≤ y < 1 for all real values of x, and sketch the graph of the solution of the differentialequation. [5 marks]

10. Find the constants A, B, C and D such that

e2x + e−2x − 2ex − 2e−x − 1 = A cosh2 x+B coshx+ C sinh2 x+D sinhx.

[5 marks]

Hence, solve the equatione2x + e−2x − 2ex − 2e−x − 1 = 0.

[5 marks]

11. Show that the Maclaurin series for (1 + x)r is

1 +

∞∑k=1

r(r − 1) . . . (r − k + 1)

k!xk,

where r is a rational number. Write down the Maclaurin series for (1+x2)−1 and (1+x2)−12 .[8 marks]

Hence, find the Maclaurin series for tan−1 x and sinh−1 x. [5 marks]

Show that limx→0

x− tan−1 x

x2 sinh−1 x=

1

3. [2 marks]

12. Find the roots of the equation (z − iα)3 = i3, where α is a real constant. [3 marks]

(a) Show that the points representing the roots of the above equation form an equilateral triangle.[2 marks]

(b) Solve the equation [z − (1 + i)]3 = (2i)3. [5 marks]

(c) If ω is a root of the equation ax2 + bx + c = 0, where a, b, c ∈ R and a 6= 0, show that itsconjugate ω∗ is also a root of this equation. Hence, obtain a polynomial equation of degree sixwith three of its roots also the roots of the equation (z − i)3 = i3. [5 marks]

16


STPM 2008

1. If p⇒ q, r ⇒∼ q and ∼ s⇒ r, show that p⇒ s. [3 marks]

2. Using the result that tan−1 x =

∫ x

0

1

1 + t2dt, show that

tan−1 x = x− x3

3+x5

5− x7

7+ . . .

where |x| < 1. [2 marks]

Hence, find limx→0

tan−1 x

sinx. [2 marks]

3. If −π2< tan−1(sinhx) <

π

2, show that

cos(tan−1(sinhx)) = sechx.

[5 marks]

4. Use Taylor series to find the first four terms in the expansion of1

(2x+ 1)3at x = 1. [5 marks]

5. Let x1 = a and xr+1 = xr+d, where a and d are positive constants and r is a positive integer. Prove,by mathematical induction, that

k∑i=1

1

xixi+1=

k

x1xk+1,

for k ≥ 2. [6 marks]

6. Sketch, on an Argand diagram, the region in which 1 ≤ |z+2−2i| ≤ 3, where z is a complex number.[3 marks]

Determine the range of values of |z|. [4 marks]

7. The equation z4 − 2z3 + kz2 − 18z + 45 = 0 has an imaginary root. Obtain all the roots of theequation and the value of the real constant k. [8 marks]

8. Let Im,n =

∫ 0

−1xm(1 + x)n dx, where m,n ≥ 0. Show that, for m ≥ 0, n ≥ 1,

Im,n = − n

m+ 1Im+1,n−1.

[3 marks]

Hence, show that Im,n =(−1)mm!n!

(m+ n+ 1)!. [6 marks]

9. The cooling system of an engine has a capacity of 10 litres. It leaks at a rate of 100 ml per week. Attime 0. it is full and contains water only. Every week (at times 1, 2, 3. ...). the cooling system istopped up with 100 ml of a coolant mixture of concentration 20% (a mixture of 80 ml of water and20 ml of coolant).

Let vn, be the volume in millilitres and cn the concentration of the coolant in the cooling systemimmediately after the top up at time n.

17


(a) Express vn+1 in terms of vn, [3 marks]

(b) Show that cn+1 = 0.99cn + 0.002, and solve this recurrence relation. [8 marks]

(c) What is the concentration of the coolant as n becomes very large? [1 marks]

10. Find the solution of the differential equation

3d2y

dx2− 2

dy

dx− y = 4e−

13x − x2 + 24

for which y = 0 anddy

dx= 1 at x = 0. [12 marks]

11. A curve is defined by x = cos θ(1 + cos θ) , y = sin θ(1 + cos θ).

(a) Show that (dx

dθ

)2

+

(dy

dθ

)2

= 2(1 + cos θ).

[4 marks]

(b) Calculate the length of the arc of the curve between the points where θ = 0 and θ = π.[5 marks]

(c) Calculate the surface area generated when the arc is rotated completely about the x-axis.[4 marks]

12. Find the eigenvalues of the matrix A given by

A =

4 4 3−1 −1 −1−4 −4 −3

and an eigenvector corresponding to each eigenvalue. [8 marks]

Write down a matrix P and a diagonal matrix D such that A = PDP−1. [2 marks]

Hence, show that An = A when n is an odd positive integer, and find An when n is an even positiveinteger. [6 marks]

18


STPM 2009

1. Prove that, if x > k, then x3 − kx2 + ax− ak > 0 for every real number a > 0. [2 marks]

Write down the converse and contrapositive of the above statement. [2 marks]

2. The real matrix A is given by

A =

a 0 −a0 b 0−c 0 c

.

where 0 < a < b < c. Show that all the eigenvalues of matrix A are distinct. [4 marks]

3. Solve the recurrence relation un+1 = 3un + 1, where n ≥ 1 and u1 = 3. [5 marks]

4. Let a = a(2t + sinh 2t) and y = b tanh t, where a, b > 0 are constants and t ∈ R. Show that

0 <dy

dx≤ b

4a. [5 marks]

5. Show that tanh−1 x =1

2ln

(1 + x

1− x

)for |x| < 1. [4 marks]

Hence, solve the equation

2 tanh−1(

3

x

)= lnx.

where x > 0. [3 marks]

6. If tan y =x

1 + x, where −π

4< y <

π

4, expand y in ascending powers of (x − 1) up to the term in

(x− 1)2. [5 marks]

Hence, find the approximate value of tan−1(

1

2

)− tan−1

(1

3

). [2 marks]


A =

1 1 c1 2 31 c 1

and B is a 3× 1 matrix.

(a) Find the values of c for which the equation AX = B does not have a unique solution. [3 marks]

(b) For each value of c, find the solutions, if any, of the equation

AX =

1−3−11

.

[5 marks]

8. (a) Show that sin(x+ iy) = sinx cosh y + i cosx sinh y, nnd hence, find the values of x and y if the

imaginary part of sin(x+ iy) is zero, where x ≥ 0 and y ≤ π

2. [4 marks]

(b) Find the roots of ω4 = −16i, and sketch the roots on an Argand diagram. [5 marks]

19


9. A curve has parametric equations

x = t+ ln | sinh t| and y = cosech t,

where t 6= 0.

(a) Finddy

dx. [3 marks]

(b) Show thatd2y

dx2= −e−2t(cosh t+ cosech t). [2 marks]

(c) Show that the curve has a point of inflexion where t =1

2ln(√

5− 2). [5 marks]

10. Using the substitution z =1

y, show that the differential equation

dy

dx− 2y

x= y2

may be reduced todz

dx+

2z

x= −1.

[2 marks]

Hence, find the particular solution y in terms of x for the differential equation given that y = 3 whenx = 1. [6 marks]

Sketch the graph y. [3 marks]

11. Find the values of p and q so that −1, −1 and 2 are the characteristic roots of the recurrence relationan + pan−2 + qan−3 = 0. [3 marks]

Using the values of p and q, solve the recurrence relation

an + pan−2 + qan−3 = −1 +√bn+1,

where n ≥ 3, a0 =1

4, a1 =

3

2, a2 =

15

4, and bm satisfies the relation

bm+1 =

(m+ 1

m

)2

bm,

where m ≥ 1 and b1 = 9. [11 marks]

12. If x(t) and y(t) are variables satisfying the differential equations

dy

dt+ 2

dx

dt= 2x+ 5 and

dy

dt− dx

dt= 2y + t.

(a) show that 3d2y

dt2− 6

dy

dt+ 4y = 2− 2t. [4 marks]

(b) find the solution x in terms of t for the second order of differential equation given that y(0) =y′(0) = π. [12 marks]

20


STPM 2010

1. Find the contrapositive, converse, inverse and negation of the quantifier proposition

∀x ∈ R, if x(x+ 1) > 0, then x > 0 or x < −1.

[4 marks]

2. Using mathematical induction, show that 22n− 1 is a multiple of 3 for all positive integers n.[4 marks]

3. Solve the recurrence relations

(a) an = 22nan−1, where a0 = 1, [3 marks]

(b) an = an−1 + 3n−1 + 2, where a0 = 0. [3 marks]

4. Show that

tan−1(

1

x− 1

)− tan−1

(1

x

)= cot−1(x2 − x+ 1), x > 1.

[3 marks]

Hence, find the value of x such that

tan−1(

1

x− 1

)= tan−1

(1

x

)+ tan−1

(1

x+ 1

).

[3 marks]

[The principal values of each angle are to be considered.]

5. Given that y = x− cos−1 x, where −1 < x < 1.

(a) Show thatd2y

dx2= x

(dy

dx− 1

)3

. [2 marks]

(b) Find the Maclaurin series for y in ascending powers of x up to the term in x5. [5 marks]

6. The differential equation

RdQ

dt+Q

C= V

describes the charge Q on a capacitor of capacitance C during a charging process involving a resistanceR and electromotive force V .

(a) Given that Q = 0 when t = 0, express Q as a function of t. [5 marks]

(b) What happens to Q over a long period of time when R = 10Ω, V= 5 V and C= 2 F? [2 marks]

7. Find the eigenvalues and eigenvectors of the matrix

1 0 40 2 03 1 −3

. [8 marks]

8. A curve is defined parametrically by

x = 2 cosech3 t, y = 3 coth2 t.

(a) Show that

(dx

dt

)2

+

(dy

dt

)2

= 36 cosech4 t coth4 t. [5 marks]

21


(b) The points A and B in the curve are defined by t = ln 2 and t = ln 3 respectively. Show that

the length of the arc AB is4625

864. [3 marks]

9. Show that

coth−1 x =1

2ln

(x+ 1

x− 1

),

where |x| > 1. [3 marks]

Hence, solve the equation coth−1(

1 + y

y

)+ coth−1

(y

1− y

)=

1

2ln 2y. [6 marks]

10. A particle moves in a horizontal straight line. The displacement ar of the particle from a fixed pointat the r-th second (r ≥ 2) satisfies the recurrence relation

ar = 9ar−2 + br,

wherebr = br−1 + 6br−2,

with a0 = 0, a1 = 4, b0 = 0, and b1 = 10.

(a) Show that br = 2(3r) + (−2)r+1. [4 marks]

(b) Find ar in terms of r. [8 marks]

11. Find the general solution of the first order differential equation

dy

dx+ 5y = sinh 2x+ 2x.

[6 marks]

Hence, solve the second order differential equation

d2y

dx2+ 5

dy

dx= 4 cosh2 x,

given that x = 0, y = 1 anddy

dx= 3. [7 marks]

12. (a) Find the fifth roots of unity in the form cos θ + i sin θ, where −π < θ ≤ π. [4 marks]

(b) By writing the equation (z − 2i)5 = (z + 2i)5 in the form

(z − 2i

z + 2i

)5

= 1, show that its roots

are ±2 cot(π

5

)and ±2 cot

(2π

5

). [7 marks]

Hence, find the values of cot2(π

5

)+ cot2

(2π

5

)and cot2

(π5

)cot2

(2π

5

). [5 marks]

22


STPM 2011

1. Prove by contradiction that the sum of a rational number and an irrational number is an irrationalnumber. [3 marks]

2. Let pn be the pressure within an enclosed container on day n. The pressure on day n+ 1 is3

n+ 3pn

less than that on day n. Show that pn =6

n(n+ 1)(n+ 2)p1. [4 marks]

3. Show that cos(sin−1 x) =√

1− x2, for |x| ≤ 1, and hence, find the exact value of

tan

(sin−1

(2√

2

3

)).

[4 marks]

4. Given that In =

∫ e

1

(lnx)n dx. Show that In + nIn−1 = e for n ≥ 1, and hence, evaluate

∫ e

1

(lnx)4 dx.

[6 marks]

5. A complex number z satisfies the inequality |z −√

3− i| ≤ 1.

(a) Sketch, on an Argand diagram, to show the region defined by the inequality. [3 marks]

(b) Determine the range of the values of |z|. [2 marks]

(c) Determine the range of the values of arg z. [2 marks]

6. Find a second order Taylor polynomial for√

1 + x about x = 24. [4 marks]

Hence, find an approximate value of√

26, and estimate the size of the error. [3 marks]

7. The complementary function of the differential equation

d2y

dx2+ p

dy

dx+ 13y = ex cosx

is y = e3x(A cos rx+B sin rx).

(a) Determine the values of p and r. [3 marks]

(b) Find the general solution of the differential equation. [5 marks]

8. Construct a truth table to show that ∼ (p→ q) ≡ p ∧ (∼ q). [4 marks]

Write an equivalent form for ∼ (∀x, p(x)→ q(x)). Hence, show the logical step for a statement to beequivalent to

“It is not true that for every hot day, it rains.”

[5 marks]

23


9. The arc of a curve y = lnx from x = 1 to x = 8 is revolved about the y-axis through 2π radian.

(a) Show that the surface area of revolution is given by

S = 2π

∫ 8

1

√1 + p2 dp,

where p = ey. [3 marks]

(b) Using the substitution p = sinh θ, show that

S = π

[8√

65−√

2 + ln

(8 +√

65

1 +√

2

)].

[7 marks]

10. Show that1 + cos 2θ + i sin 2θ

1− cos 2θ − i sin 2θ= i cot θ. [2 marks]

Hence, show that the roots of the equation (z+ i)5 = (z− i)5 are ± cot(π

5

)and ± cot

(2π

5

).[6 marks]

Deduce the values of cot2(π

5

)+ cot2

(2π

5

)and cot2

(π5

)cot2

(2π

5

). [4 marks]

11. A square matrix A has an eigenvalue λ with corresponding eigenvector v. Show that λ + c is aneigenvalue of A + cI, where c ∈ R and I is identity matrix. Show that v is also an eigenvector ofA + cI corresponding to λ+ c. [3 marks]

Determine the values of k so that the eigenvalues of the matrix

P =

−1 0 20 k 0−5 0 6

,

where k ∈ R are all distinct. Find the eigenvalues and the eigenvectors of P for k = 1. [8 marks]

Hence, find the eigenvalues and eigenvectors of the matrix Q =

2 0 20 4 0−5 0 9

. [4 marks]

12. Using the definitions of sinhx and coshx,

(a) show that

i. sinh(x− y) = sinhx cosh y − coshx sinh y, [3 marks]

ii. tanh(n+ 1)x− tanhnx = sinhx sech(n+ 1)x sechnx. [3 marks]

(b) Given that Sr =r∑

n=1

sech(n+ 1)xsechnx. Show that

i. if x 6= 0, then Sr = 2 cosech 2x sinh rx sech(r + 1)x, [5 marks]

ii. if x > 0 and r is very large, then Sr ≈ 2e−x cosech 2x. [4 marks]

24

FURTHER MATHEMATICS PAPER 2 QUESTIONS

2 PAPER 2 QUESTIONS

25


STPM 2002

1. State the relationship between the sum of degree of vertices and the number of edges in a simplegraph. Deduce that a simple graph with five vertices of degree 1, 1, 2, 3, 4 is impossible. [4 marks]

2. Given that points O, P , and Q non-colinear, R lies on the line PQ. Position vector of P , Q, and Rrespect to O are p, q, and r respectively. Show that r = µp + (1 − µ)q, where µ is a real number.

[2 marks]

If R is the perpendicular foot form O to PQ, show that

r =|q|2 − p · q|p− q|2

p +|p|2 − p · q|p− q|2

q.

[4 marks]

3. Prove that the planes ax+ by+ cz = d and a′x+ b′y+ c′z = d′ are parallel if and only if a : b : c = a′ :b′ : c′. Find the equation of the plane π that is parallel to the plane 3x + 2y − 5z = 2 and containsthe point (-1, 1, 3). [5 marks]

Find the perpendicular distance between the plan π and the plane 3x+ 2y − 5z = 2. [2 marks]

4. Draw a connected graph with eight vertices, with no degree 1, if the graph is

(a) Eulerian and Hamiltonian, [2 marks]

(b) not Eulerian but Hamiltonian, [2 marks]

(c) Eulerian but not Hamiltonian, [2 marks]

(d) not Eulerian and not Hamiltonian. [2 marks]

5. Transformation P is defined by

P :

(xy

)→(

1 10 −1

)(xy

)(a) Find the invariant line that passes through origin under transformation P . [6 marks]

(b) Find the area of the image of triangle ABC with points A(0,−2), B(3, 0), and C(1, 4) underthe transformation P . [3 marks]

6. Let n ∈ N and n ≥ 2.

(a) If n = 6k + 1 or n = 6k + 5, with k integer, prove that 2n + n2 ≡ 0(mod 3). [9 marks]

(b) If 2n + n2 is a prime number, prove that n ≡ 3(mod 6). [4 marks]

7. Continuous random variable Z is standard normal random variable. State the exact name of thedistribution Z2, and state the mean and variance. [3 marks]

8. A survey carried out in an area to estimate the proportion of people who have more than one house.This proportion is estimated using 95% confidence interval. If the estimated proportion is 0.35,determine the smallest sample size required so that estimation error did not exceed 0.03 and deducethe smallest sample size required so that the estimation error did not exceed 0.01. [7 marks]

9. A random sample of 12 randomly selected apples from a box of apples. The mass of apples, in g, is

summarized by∑

x = 956.2 and∑

x2 = 81175.82. Find a 95% confidence interval for the mean

mass of apples in the box. State any assumptions you make. [7 marks]

26


10. Random variable X is normally distributed with mean µ and variance 36. The significance testsperformed on the null hypothesis H0 : µ = 70 versus the alternative hypothesis H1 : µ 6= 70 with aprobability of type I error equal to 0.01. A random sample of 30 observations of X are taken andsample mean barX taken as the test statistic. Find the range of the test statistic lies in the criticalregion. [8 marks]

11. State common conditions apply to the distribution of test statistics in the test of goodness of fit, χ2isapproximated to χ2 distribution. [2 marks]

A study was conducted in a factory on the number of accidents that happen on 100 factory workersin a period of time. The following data were obtained.

Number of accident 0 1 2 3 4 5Number of workers 25 31 23 13 5 3

Test, at 1% significant level, whether the data above is a sample from the Poisson distribution withparameter 1.5. [10 marks]

12. Explain the least squares method to obtain equation of regression line, using diagram. [3 marks]

Five pairs of value of variable x and y are given by

x 30 40 50 60 70y 90 84 82 77 68

(a) Obtain the equation of the least squares regression line of y on x in the form y = a + bx, bygiving the values of a and b correct to one decimal places. [7 marks]

(b) When the sixth pair (x6, y6) is combined with the five pairs of values above, the equation of the

least squares regression line of y on x from the six pairs of values is y = 105− 1

2x. Find x6 and

y6. [6 marks]

27


STPM 2003

1. Let T : R2 → R2 be a linear transformation such that

T :

(11

)→(

2−3

), T :

(01

)→(

12

).

Show that

T :

(ab

)→(

a+ b−5a+ 2b

).

[4 marks]

2. Solve the pair of congruences

x ≡ 1 (mod 6),x ≡ 5 (mod 11).

[5 marks]

3. The vector equations of two intersecting lines are given by r = 2i + j + λ(i + j + 2k) and r =2i + 2j− k + µ(i + 2j + k).

(a) Determine the coordinates of the point of intersection of the two lines. [3 marks]

(b) Find the acute angle between the two lines. [4 marks]

4. Find the greatest common divisor of 2501 and 2173 and express it in the form 2501m + 2173n, wherem and n are integers which are to be determined. [6 marks]

Find the smallest positive integer p such that 9977 + p = 2501x+ 2173y, where x and y are integers.[3 marks]

5. Let G be a simple graph with n vertices and m edges. Prove that m ≤ 1

2n(n− 1). [4 marks]

If n = 11 and m = 46, show that G is connected. [7 marks]

6. Describe the transformation in the xy-plane represented by the matrix√

5

3

2

32

3−√

5

3

.

[3 marks]

(a) Show that the equation of the invariant line which contains the invariant points is y =

(3−√

5

2

)x,

and find the equation of the other invariant line which passes through the origin. [7 marks]

(b) The points P and Q are (2,−1) and (3, 0) respectively. Find the coordinates of the point T

which lies on the invariant line y =

(3−√

5

2

)x such that the sum of the distance between P

and T and the distance between Q and T is minimum. [4 marks]

7. A random sample X1, X2, . . . , Xn is taken from a normal population with mean µ and variance 1.Determine the smallest sample size which is required so that the probability that X lies within 0.2of µ is at least 0.90. [5 marks]

28


8. In a chi-square test, the test statistic is given by χ2 =k∑i=1

(Oi − Ei)2

Ei, where Oi is the observed

frequency, Ei the expected frequency, and k > 2. Show that

k∑i=1

(Oi − Ei)2

Ei=

k∑i=1

O2i

Ei−N,

where N =k∑i=1

Oi =k∑i=1

Ei. [2 marks]

A car model has four colours: white, red, blue, and green. From a random sample of 60 cars of thatmodel in Kuala Lumpur, 13 are white, 14 are red, 16 are blue, and 17 are green. Test, at the 5%significance level, whether the number of cars of each colour of that model in Kuala Lumpur is thesame. [5 marks]

9. Explain what a least square regression line means. [3 marks]

The life (x thousand hours) and charge (y amperes) of a type of battery may be related by the leastsquare regression line y = 7.80− 1.72x.

(a) Determine, on the average, the reduction in the charge of the batteries after they have beenused for 1000 hours. [2 marks]

(b) Find the mean charge of the batteries after they have been used for 3000 hours. [2 marks]

10. The mean and standard deviation of the yield of a type of rice in Malaysia are 960 kg per hectare and192 kg per hectare respectively. From a random sample of 30 farmers in Kedah who plant this rice,the mean yield of rice is 996 kg per hectare. Test, at the 5% significance level, the hypothesis that themean yield of rice in Kedah is more than the mean yield of rice in Malaysia. Give any assumptionsthat need to be made in the test of this hypothesis. [8 marks]

11. Let Pearson’s correlation coefficient between variables x and y for a random sample be r.

(a) If all the points on the scatter diagram lie on the line y = c, where c is a constant, comment onthe values of r. [2 marks]

(b) If all the points on the scatter diagram lie on the line y = a + bx, where a and b are constants

and b 6= 0, show that r =b√b2

and deduce the possible values of r. [6 marks]

12. Out of 100 corn seeds of type A which are planted in a certain area, 24 seeds fail to germinate. Outof 50 corn seeds of type B which are planted in that area, 4 seeds fail to germinate.

(a) Find a 90% confidence interval for the proportion of corn seeds of type A which fail to germinatein that area. [5 marks]

(b) By considering an appropriate 2 × 2 contingency table, test, at the 5% significance level, thehypothesis that the proportions of corn seeds of type A and of type B which fail to germinatein that area arc the same. [10 marks]

29


STPM 2004

1. Write down an incidence matrix for the graph given below. What can be said about the sum of theentries in any row and the sum of the entries in any column of this incidence matrix? [4 marks]

v1 v2e1

v3v4

v5

e2

e3

e4

e5

e6

2. The straight line joining the points (0, 0, 1) and (1, 0, 0) intersects the plane x+ y+ 2z+ 2 = 0. Findthe coordinates of the point of intersection. [5 marks]

3. Define a simple graph. [1 marks]

State whether there exists a simple graph with

(a) eight vertices of degrees 7, 7, 7, 5, 4, 4, 4, 4;

(b) five vertices of degrees 2, 3, 4, 4, 5.

For each case, construct a simple graph if it exists, or give a reason if there does not exist a simplegraph. [5 marks]

4. The point P lies on the line r =

133

+ t

348

and the point Q lies on the plane whose equation is

r · (3i + 6j + 2k) = −22 such that PQ is perpendicular to the plane.

(a) Find the coordinates of P and Q in terms of t. [6 marks]

(b) Find the vector equation of the locus of the midpoint of PQ. [3 marks]

5. T : R2 → R2 is a linear transformation such that the images of the points (1, 2) and (−1, 0) are thepoints (3,−2) and (−1, 0) respectively.

(a) Find the matrix which represents T. [4 marks]

(b) Find the equation of the line which is the image of the line y = mx+ c under T. [6 marks]

6. Define the congruence a ≡ b(mod m). [1 marks]

Solve each of the congruences x3 ≡ 2(mod 3) and x3 ≡ 2(mod 5). Deduce the set of positive integerswhich satisfy both the congruences. [9 marks]

Hence, find the positive integers x and y which satisfy the equation x3 + 15xy = 12152. [5 marks]

7. The Pearson correlation coefficient between two variables X and Y for a random sample is 0.

30


(a) State whether this means that there is no linear relation between X and Y . Sketch a scatterdiagram for X and Y . [2 marks]

(b) State whether this means that X and Y are independent. Give a reason for your answer.[2 marks]

8. A normal population has mean µ and variance σ2.

(a) Explain briefly what a 95% confidence interval for µ means. [2 marks]

(b) From a random sample, it is found that the 95% confidence interval for µ is (−1.5, 3.8). Statewhether it is true that the probability that µ lies in the interval is 0.95. Give a reason. [2 marks]

(c) A total of 120 random samples of size 50 are taken from the population and for each sample a95% confidence interval for µ is calculated. Find the number of 95% confidence intervals whichare expected to contain µ. [1 marks]

9. The equation of the regression line of the variable X on variable Y is x = −2y + 11. The Pearson

correlation coefficient between X and Y is − 2√5

. The means of X and Y are 5 and 3 respectively.

Find the equation of the regression line of Y on X. [6 marks]

10. In order to investigate whether the level of education and the opinion on a social issue are independent,1300 adults are interviewed. The following table shows the results of the interviews.

Level of educationOpinion on the social issue

TotalAgree Disagree

University 450 18 468College 547 30 577High school 230 25 255Total 1227 73 1300

Determine, at the 1% significance level, whether the level of education and the opinion on the socialissue are independent. [9 marks]

11. A petroleum company claims that its petrol has a RON rating of at least 97. From a random sampleof 15 petrol stations selling the petrol, the mean RON rating is found to be 96.30.

Assuming that the RON rating of the petrol of the petroleum company has a normal distributionwith standard deviation 3.21,

(a) test the claim of the petroleum company at the 2.5% significance level, [6 marks]

(b) determine the smallest sample size required so that the null hypothesis in the test of thepetroleum company’s claim is rejected at the 5% significance level. [6 marks]

12. In a survey on the quality of service provided by a bank, 12 out of 150 customers think that theservice is unsatisfactory while the other 138 customers think otherwise.

Test the hypothesis that the proportion of customers who think that the service is unsatisfactory is0.10

(a) by using a 99% confidence interval, [7 marks]

(b) by carrying out a significance test at the 1% significance level. [6 marks]

Comment on methods (a) and (b) used in the test of the hypothesis. [2 marks]

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STPM 2005

1. What is a graph and what does the degree of a vertex of a graph mean? [3 marks]

2. Show that (x, 12) = 1 and (x, 15) = 1 if and only if (x, 30) = 1, where (a, b) denotes the greatestcommon divisor of a and b. Hence, find all integers x such that (x, 12) = 1 and (x, 15) = 1. [6 marks]

3. Four schools P , Q, R and S each has team A and team B in a tournament. The teams from thesame school do not play against each other. At a certain stage of the tournament, the numbers ofgames played by the teams, except team A of school P , are distinct. Determine the number of gamesplayed by team B of school P . [7 marks]

4. Using the definition of congruence, prove that if x, r and q are integers and x ≡ r(mod q), thenxn ≡ rn(mod q) where n is a positive integer. [4 marks]

Hence, show that 19n + 39n ≡ 2(mod 8) for every integer n ≥ 1. [6 marks]

5. T : R2 → R2 is a linear transformation such that

T :

(11

)→(

34

), and T :

(21

)→(

64

).

(a) By expressing

(10

)as a linear combination of

(11

)and

(21

), show that T

(10

)=

(30

). Find

also T

(01

). Hence, write down the matrix representing T. [6 marks]

(b) Find the image of the circle x2 + y2 = 1 under T. [4 marks]

6. The line l has equation r = 2i + j + λ(2i + k) and the plane π has equation r = i + 3j− k + µ(2i +k) + v(−i + 4j).

(a) The points P and Q lie on l and π respectively. The point R lies on the line PQ, wherePR = 2RQ. If P and Q move on l and π respectively, find the equation of the locus of R.

[6 marks]

(b) The points L and M have coordinates (0, 1,−1) and (1,−5,−2) respectively. Show that L lieson l and M lies on π. [3 marks]

Determine the sine of the acute angle between the line LM and the plane π and the shortest distancefrom L to π. [6 marks]

7. In a study on the petrol consumption of cars, it is found that the mean mileage per litre of petrol for24 cars of the same engine capacity is 15.2 km with a standard deviation of 4.2 km. Calculate thestandard error of the mean mileage and interpret this standard error. [3 marks]

8. The mass of a particular type of steel sheet produced by a factory has a normal distribution. Themean mass of a random sample of 14 sheets is 72.5 kg and the standard deviation is 3.2 kg. Calculatea 95% confidence interval for the mean mass of the steel sheets produced by the factory. [4 marks]

9. Explain what is meant by significance level in the context of hypothesis testing. [2 marks]

In a goodness-of-fit test for the null hypothesis that the binomial distribution is an adequate modelfor the data, the test statistic is found to have the value 19.38 with 7 degrees of freedom. Find thesmallest significance level at which the null hypothesis is rejected. [2 marks]

32


10. The lifespan of a type of bulb is known to be normally distributed with standard deviation 150 hours.The supplier of the bulbs claims that the mean lifespan is more than 5500 hours. The lifespans, inhours, of a random sample of 15 bulbs are as follows.

5260 5400 5820 5530 5380 5460 5510 55205600 5780 5520 5500 5360 5620 5430

(a) State the appropriate hypotheses to test the supplier’s claim and carry out the hypothesis testat the 5% significance level. [8 marks]

(b) If the true mean lifespan is 5550 hours, find the probability that the test gives a correct decision.[4 marks]

11. The following table shows the frequency distribution of passengers (excluding the driver) per car ina town for a particular period. The data could be a sample from a Poisson distribution.

Number of passengers 0 1 2 3 4 5 6Frequency 241 211 104 35 7 0 2

(a) Find all the expected frequencies for the distribution correct to two decimal places. [3 marks]

(b) Calculate the values of the χ2 goodness-of-fit statistic

i. without combining any frequency classes,

ii. with the last three frequency classes being combined.

Comment on the values obtained and explain any differences. [6 marks]

(c) Using the value of the statistic in (b)(ii), test for goodness of fit at the 5% significance level.[3 marks]

12. The following table shows the values of the variable y corresponding to seven accurately specifiedvalues of the variable x.

y 800 920 1280 1500 4020 6200 6800x 2 3 5 5 8 10 12

(a) Plot a scatter diagram of loge y against loge x. [3 marks]

(b) Find the equation of the least squares regression line of the form loge y = β0 + β1 loge x, withβ0 and β1 correct to two decimal places. [5 marks]

(c) Calculate the Pearson correlation coefficient r between loge y and loge x. Interpret your valueof r and comment on this value with respect to the scatter diagram in (a). [6 marks]

33


STPM 2006

1. The points A, B and C are three collinear points on a cartesian plane and T(A) = A1, T(B) = B1 andT(C) = C1, where T : R2 → R2 is a linear transformation. If AB : BC = m : n, find A1B1 : B1C1.

[3 marks]

2. Using congruence properties, prove that 2mn − 1 is divisible by 2m − 1 for all integers m, n ≥ l.[3 marks]

Deduce that, if 2p − 1 is a prime number, then p is a prime number. [3 marks]

3. The matrix M represents an anticlockwise rotation in the xy-plane about the origin through an angle

θ. The matrix N =

(cos θ 2 cos θ − sin θsin θ 2 sin θ + cos θ

)represents the combined effect of the transformation

represented by a matrix K followed by the transformation represented by M. Find K and describethe transformation represented by K. [7 marks]

4. Graphs G1, G2 and G3 are given as follows.

(a) Draw an eulerian circuit for the graph which is eulerian. [3 marks]

(b) Draw a hamiltonian cycle for each of the graphs which is hamiltonian. [3 marks]

(c) For the graph which is not hamiltonian, determine how it could be made into a hamiltoniangraph by adding an edge. [1 marks]

(d) For each of the graphs which is not eulerian, determine how it could be made into an euleriangraph by deleting or adding an edge. [2 marks]

5. Using Euclid’s algorithm, find g.c.d.(6893, 11 639). [3 marks]

If n is an integer between 20 000 and 700 000 such that the remainder is 6893 when n is divided by11 639, find g.c.d.(6893, n). [7 marks]

6. The planes π1 and π2 with equations x− y + 2z = 1 and 2x+ y − z = 0 respectively intersect in theline l. The point A has coordinates (1,0, 1).

(a) Calculate the acute angle between π1 and π2. [2 marks]

(b) Explain why the vector

1−12

× 2

1−1

is in the direction of l. Hence, show that the equation

of l is

r =

011

+ t

−153

.

where t is a parameter. [5 marks]

(c) Find the equation of the plane passing through A and containing l. [3 marks]

(d) Find the equation of the plane passing through A and perpendicular to l. [2 marks]

34


(e) Determine the distance from A to l. [3 marks]

7. The length of a species of fish is normally distributed with mean µ and standard deviation 10 cm.If the sample mean of 50 fish is greater than 35 cm, the null hypothesis µ = 32.5 cm is rejected infavour of the alternative hypothesis µ > 32.5 cm. Find the probability of making a type I error. Findalso the probability of making a type II error when µ = 34 cm. [5 marks]

8. A property agent believes that the price of a house in a certain district depends principally on itsbuilt-up area. The prices (p thousand ringgit) of eight houses in different parts of the district andtheir built-up areas (a square metres) are summarised as follows:∑

a = 2855,∑

p = 1689,∑

ap = 827550,∑

a2 = 1400925,∑

p2 = 489181.

(a) Find the equation of the regression line in the form p = β0 + β1a, with p as the dependentvariable and a the independent variable. [5 marks]

(b) Give a reason why it is not suitable to use your regression equation to make predictions whena = 0. [1 marks]

9. The average duration of an electronic device to retain information after the power is switched off isnormally distributed with a mean of µ0 and an unknown variance.

It is of interest to determine whether there is an improvement in the performance of this device whena component is added. A random sample of n such duration yields mean x and variance s2.

(a) State the appropriate null and alternative hypotheses. Explain briefly your choice of the alter-native hypothesis. [2 marks]

(b) Write down the test statistic and the critical region at the 5% significant level for each of thefollowing cases.

i. n = 100

ii. n = 16

[5 marks]

10. A survey is to be carried out to estimate the proportion p of households having personal computers.This estimate must be within 0.02 of the population proportion at a confidence level of 95%.

(a) If p is estimated to be 0.12, find the smallest sample size required. [4 marks]

(b) If the value of p is unknown, determine whether a sample size of 2500 is sufficient. [4 marks]

11. The following table shows the annual salary and experience of nine randomly selected engineers.

Salary (thousand of riggit) 88 48 60 70 62 78 100 52 110Experience (year) 12 4 6 7 5 10 18 5 19

(a) Plot a scatter diagram. What do you expect of the relationship between the salary and experi-ence based on the scatter diagram? [4 marks]

(b) Calculate the Pearson correlation coefficient between the salary and experience. Explain whetherthe value of the correlation coefficient is consistent with what you expect in part (a). [7 marks]

12. What is meant by a contingency table? [1 marks]

A survey is carried out on a random sample of 820 persons who are asked whether students whobreak school rules should be caned. The results of the survey are as follows.

35


hhhhhhhhhhhhhhhEducation levelOpinion

Yes No Not sure

High school 125 65 100College 98 68 64University 100 80 120

Carry out chi-squared tests to determine whether educational level is related to the opinion on caningat the significance levels of 1% and 2%. [10 marks]

Comment on the conclusions of these tests. [2 marks]

36


STPM 2007

1. Two graphs G1 and G2 are shown below.

(a) State, with reasons, which is an eulerian graph and which is not. [3 marks]

(b) Find an eulerian circuit for the eulerian graph. [2 marks]

2. If a simple graph has n vertices where n ≥ 2, show that at least two vertices are of the same degree.[4 marks]

Give a counter example to show that the result is not true for a graph which is not a simple graph.[2 marks]

3. The line which passes through the points A(−6,−1,−7) and B(6, 3, 1) cuts the plane 3x−y+ 2z = 5at the point P . Show that the points A and B are on the opposite sides of the plane, and find theratio AP : BP . [6 marks]

4. (a) Find all integers x that satisfy the congruence

2x ≡ 0 (mod 6).

[3 marks]

(b) Find all integers x and y that satisfy the pair of congruences

2x+ y ≡ 1 (mod 6),x+ 3y ≡ 3 (mod 6).

[6 marks]

5. A linear transformation f : R2 → R2 is defined by f(u) = v, where u =

(xy

)and v =

(x+ yx− y

).

(a) If f(u) = Au, where A is a 2× 2 matrix, find A. [2 marks]

(b) If f2(u) = Bu, where f2(u) = (f f)(u) and B is a 2× 2 matrix, find B and verify that B = A2.[3 marks]

(c) By using mathematical induction, show that fn(u) = Anu, where fn(u) = (f fn−1)(u), for allintegers n ≥ 1. [4 marks]

(d) The points P ′, Q′ and R′ are the images of the vertices P , Q and R respectively of a triangleunder the transformation fn. If the area of the triangle PQR is 8 units2, find the area of thetriangle P ′Q′R′. [3 marks]

6. Two straight lines l1 and l2 have equations −2x + 4 = 2y − 4 = z − 4 and 2x = y + 1 = −z + 3respectively. Determine whether l1 and l2 intersect. [7 marks]

The points P and Q lie on l1 and l2 respectively, and the point R divides PQ in the ratio 2:3. Findthe equation of the locus of R. [6 marks]

37


7. (a) Explain briefly why it is advisable to plot a scatter diagram before interpreting a Pearsoncorrelation coefficient for a sample of bivariate distribution. [2 marks]

(b) Sketch a scatter diagram with five data-points for each of the following cases.

i. The Pearson correlation coefficient of two variables is close to 0 but there is an obviousrelation between them. [1 marks]

ii. The Pearson correlation coefficient of two variables is close to 1 but there is no linear relationbetween them. [1 marks]

8. The lengths of petals taken from a particular species of flowers have mean 80 cm and variance 30cm2. Determine the sampling distribution of the sample mean if 100 petals are chosen at random.

[3 marks]

Hence, find the probability that the sample mean is at least two standard deviations from the mean.[3 marks]

9. It is found that 5% of doctors in a particular country play golf. Find, to three decimal places, theprobability that, in a random sample of 50 doctors, two play golf. [2 marks]

Hence, state the sampling distribution of the proportion of the doctors who play golf, and constructa 98% confidence interval for the proportion. [5 marks]

10. A farmer wishes to find out the effect of a new feed on his calves. The average weight gain of a calfon the original feed in a month is normally distributed with mean 10 kg. In a particular month, thefarmer gives the new feed to 16 calves. It is found that the average weight gain of a calf on the newfeed is 11.5 kg with a standard deviation of 2.5 kg.

(a) State appropriate hypotheses for a significant test. [2 marks]

(b) Carry out the test at the 5% significance level. [7 marks]

11. In a psychological study, 50 persons are asked to answer four multiple-choice questions. The an-swers obtained are compared with the predetermined answers. The data is recorded as a frequencydistribution as follows:

Number of matched answers 0 1 2 3 4Frequency 10 15 8 5 12

A psychologist suggests that the data fits the following probability distribution.

Number of matched answers 0 1 2 3 4Probability 0.08 0.10 0.15 0.25 0.42

(a) Calculate the expected frequencies based on the probability distribution. [2 marks]

(b) Determine whether there is significant evidence, at the 5% significant level, to reject the sugges-tion. [8 marks]

12. A study is carried out to determine the effect of exercise frequency on lung capacity. The exercisefrequency in weeks is denoted by X and the percentage increase in lung capacity by Y . The dataobtained from 10 volunteers are summarised as follows:∑

x = 319,∑

x2 = 11053,∑

y = 530,∑

y2 = 30600,∑

xy = 18055.

(a) Calculate the Pearson correlation coefficient and describe the relationship between X and Y .[5 marks]

38


(b) Calculate the coefficient of determination and interpret its value. [2 marks]

(c) Find the equation of the regression line in the form y = a+ bx, where a and b are correct to twodecimal places. [4 marks]

(d) Predict the mean value of the percentage increase in lung capacity if the exercise frequency is25 weeks. State your assumption in obtaining the predicted value. [2 marks]

39


STPM 2008

1. If n is an odd integer, prove that x2 − y2 = 2n has no solution in the set of integers. [4 marks]

2. The adjacency matrix of a graph G is 0 1 0 1 11 0 1 0 00 1 0 1 11 0 1 0 01 0 1 0 0

.

(a) Draw the graph G. [2 marks]

(b) Distinguish the vertices of the graph G using minimum number ofc olours such that vertices ofthe same colour are not adjacent. [2 marks]

(c) State the type of the graph G. [1 marks]

3. Find the equation of the plane which is parallel to the plane 3x+2y−6z−24 = 0 and passes throughthe point (1, 0, 0). Hence, determine the distance between these two planes. [6 marks]

4. A graph is given as follows:

a b

(a) Find the total number of paths with the end vertices a and b. [3 marks]

(b) Write down the degree sequence of the graph, (d1, d2, . . . , d9), in ascending order. [2 marks]

(c) Show that the total number of paths consisting of three vertices is

9∑i=2

(di2

)and determine its

value. [4 marks]

5. The points A and B lie on the line r =

036

+λ

1−1−4

, and the distance of each point is three units

from the origin O.

(a) Determine the coordinates of A and B. [6 marks]

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

6. A linear transformation T : R2 → R2 maps

(0−1

)into

(22

),

(11

)into

(36

)and

(10

)into

(35

).

(a) Find the matrices A and C such that

T

(xy

)= A

(xy

)+ C.

[6 marks]

40


(b) The transformation T is equivalent to a transformation U followed by a transformation V.Determine the images of the points P1(0, 0), P2(0, 2), P3(1, 2) and P4(1, 0) under T, and hencedescribe U and V. [6 marks]

(c) Determine the image of the straight line y = 2x under T. [3 marks]

7. A set of sample data for two random variables X and Y gives the results∑(x− x)2 = 72,

∑(y − y)2 = 57,

∑(x− x)(x− y) = −36.

(a) Calculate the Pearson correlation coefficient r between X and Y , and interpret the value of robtained. [3 marks]

(b) What is the value of r if each value of x is increased by 0.3 whereas the value of y remainunchanged? Give a reason. [2 marks]

8. A machine is regulated to dispense a chocolate drink into cups. From a random sample of 100 cupsof the chocolate drink dispensed, it is found that the cocoa content in one cup of the chocolate drinkhas mean 5 g and standard deviation 0.5 g. The owner of the machine uses the confidence interval(4.900 g, 5.100 g) to estimate the mean cocoa content in one cup of the chocolate drink.

(a) Identify the population parameter under study. [1 marks]

(b) Determine the confidence level for the confidence interval used. [5 marks]

9. The masses of watermelon produced by a farmer are normally distributed with mean 3 kg. Thefarmer decides to use a new organic fertiliser for his crop if the mean mass of a random sample of 10watermelons using the new organic fertiliser exceeds k kg. For this random sample of 10 watermelons,the standard deviation is 0.65 kg. The farmer uses a probability of Type I error equal to 0.01 inmaking his decision.

(a) State what is meant by a Type I error. [1 marks]

(b) Determine the value of k. [5 marks]

10. For a set of data, the least-squares regression line of y on x is

y = 100.15 + 0.25x.

(a) Explain the method used to estimate the coefficient of x and the constant in the equation of theregression line. State an appropriate assumption. [4 marks]

(b) What is the percentage error in the estimation of value of y when x = 2500 using the regressionline, given that the actual value of y is 850? [4 marks]

11. Two hundred patients with a certain skin disorder are treated for five days either with cream A orcream B. The following table shows the number of patients recorded as no recovery, partial recoveryor complete recovery.

Type of creamNumber of patients

No recovery Partial recovery Complete recoveryA 25 30 45B 10 25 65

(a) Calculate the percentage of patients with improvement for each type of cream used and commenton your answers. [2 marks]

(b) Determine, at the 5% significance level, whether the condition of a patient is independent of thetype of cream used. [9 marks]

41


(c) What conclusion can be made based on the results in (a) and (b)? [1 marks]

12. A manufacturer produces a new type of paint. He claims that the paint has 0.04% lead content byweight. A random sample of 25 tins (1 kg per tin) of the paint is analysed to determine the leadcontent. Sample mean and standard deviation of the lead content in a kilogramme of paint are 0.38g and 0.1 g respectively.

(a) Construct a 99% confidence interval for the mean lead content in a kilogramme of paint. Stateyour assumption. [5 marks]

(b) What is the effect on the confidence interval obtained in (a) if the sample size is increased to100? [2 marks]

(c) Carry out a test, at the 1% significance level, to test the manufacturers claim. [6 marks]

(d) Relate the confidence interval obtained in (a) with the result of the test in (c). [2 marks]

42


STPM 2009

1. Let Krs be a complete bipartite graph, where r ≤ s. Each of the r vertices in a partite set is connectedto each of the s vertices in another partite set. If Krs has 36 edges and 15 vertices. find r and s.

[4 marks]

2. T : R2 → R2 is a linear transformation represented by a matrix

(a b−3 a

). If the images of the points

(a,−1) and (−2, a) are (a+ 2, 2b) and (2b, 7) respectively under T, find the values of a and b.[6 marks]

3. For any integers a and b, prove that if a divides b then a3 divides b3. [3 marks]

Hence, determine whether 64 divides [(5n+ 1)2 − (3n+ 5)2]3, where n is an integer. [4 marks]

4. (a) Define a walk of a graph. [2 marks]

(b) Graph G is given as follows:

V1 V2 V3

V4V5

i. Find all walks of length two from vertex V2 to vertex Vi where i = 1, 2, 3, 4, 5, and representthem by a row matrix. [2 marks]

ii. Find all walks of length two from vertex Vr. where r = 1, 2, 3, 4, 5, to vertex V4, andrepresent them by a column matrix. [1 marks]

iii. Find the total number of possible walks of length four from vertex V2 to vertex V4.[3 marks]

5. (a) If g.c.d.(a, m)=1, show that there exists a solution for ax ≡ b (mod m). [3 marks]

(b) Deduce that, if g.c.d.(ad− bc, n)=1, then a solution exists for the system of linear congruences

ax+ by ≡ k (mod n),cx+ dy ≡ l (mod n).

Hence, solve the system of linear congruences

7x+ 3y ≡ 10 (mod 16),2x+ 5y ≡ 9 (mod 16).

[7 marks]

6. (a) Find the equation of line l1, passing through points A and B, where the position vectors ofpoints A and B are a and b respectively. [1 marks]

(b) R is a point on the line l1 in (a). If point C has position vector c,

i. find−→CR in terms of vectors a, b and c. [1 marks]

ii. prove that−→CR×

−−→AB = a× b + b× c + c× a, [3 marks]

iii. deduce the shortest distance of the point C from the line l1. [3 marks]

(c) i. Find the distance of point T (1+3k,−2+k, 5−2k) to line l2 passing through points P (4, 4, 4)and Q(3, 2, 1) in terms of k. [3 marks]

43


ii. Deduce the distance between skew lines l2 and l3, where the equation of line l3, is given by

t =

1−25

+ k

31−2

.

[4 marks]

7. A local automobile manufacturing company studies the relationship between the age (x years) andthe price (y thousand RM) of one of its car models. The data obtained from a random sample of tencars of this model are as follows:

(1, 70), (2, 65), (3, 60), (4, 55), (5, 50), (6, 45), (7, 40), (8, 35), (9, 30) and (10, 20)

Calculate the Pearson correlation coefficient between the age and the price of this car model, andinterpret the value obtained. [5 marks]

8. A medical researcher wishes to study whether the severity of a certain lung ailment is related to thesmoking habit of a patient. A patient who smokes more than 20 cigarettes a day on the average isclassified as a heavy smoker. A random sample of 96 patients was taken. It is found that from 43patients with mild lung ailment, 23 are light smokers while the rest are heavy smokers. The remaining53 patients with severe lung ailment, 39 patients are heavy smokers.

(a) Tabulate the data in an appropriate contingency table. [1 marks]

(b) Determine, at 5% significance level, whether the severity of the lung ailment and the smokinghabit are independent. [5 marks]

9. Tte monthly salary of engineers working in a town has a normal distribution. The mean monthlysalary of a random sample of 16 engineers is RM5280 and the standard deviation is RM480.

(a) Construct a 95% confidence interval for the mean monthly salary of an engineer in the town.Interpret your answer. [4 marks]

(b) A statistician finds that the confidence interval obtained in (a) is too wide. Suggest, with reasonsa method for reducing the width of the confidence interval but maintaining the confidence level.

[3 marks]

10. An electronic component produced by a factory is found to have at most five defects. A supervisorat the factory conducts a study on the number of defects found in the electronic components. Onethousand electronic components have been selected at random and inspected. The number of defectsfound and their respective frequency arc given as follows:

Number of defects 0 1 2 3 4 5Frequency 33 145 337 286 174 25

(a) Test, at 5% and 20% significance levels, whether the binomial distribution with probability ofdefect p = 0.5 fits the data. [7 marks]

(b) Recommend, with a reason, an appropriate conclusion for part (a). [2 marks]

11. The lifespan (in kilometres) of a tyre is defined as the distance travelled before wearing out. A tyremanufacturer claims that the mean lifespan µ of its tyre is at least 50 000 km. In order to test thisclaim, a consumer association takes a random sample of 121 tyres. The mean and standard deviationof the lifespan for the sample are 49 200 km and 2500 km respectively.

(a) Determine the distribution of the sample mean lifespan of the tyres if the lifespan of the tyresis assumed to have

44


i. a normal distribution, [1 marks]

ii. an unknown distribution. [1 marks]

(b) Assuming that the lifespan of the tyres has a normal distribution, state the appropriate hypothe-ses to test the manufacturer’s claim, and carry out the hypothesis test at the 1% significancelevel. [5 marks]

(c) If the true mean lifespan of a tyre is 49 400 km. determine the probability of type II error in(b). [3 marks]

12. A study on the relationship between amount of time spent on revision X (in hours) and performancein a final examination Y (scores of 0 to 100) for 10 female undergraduates are summarised as follows:∑

x = 725,∑

x2 = 54625,∑

y = 696,∑

y2 = 49376,∑

xy = 51705

(a) Find the equation of the regression line in the form y = β0 + β1x, where y is the dependentvariable and x the independent variable. Interpret the value of β1 obtained. [7 marks]

(b) Using the regression line obtained, predict the mean value of y when x = 70, and state yourassumption. [2 marks]

(c) Find the coefficient of determination D between the amount of time spent on revision and theperformance in a final examination of the undergraduates. Hence, interpret the value of D.

[4 marks]

45


STPM 2010

1. Show that 9n − 1 is divisible by 8 for every positive integer n. [3 marks]

2. Find the coordinates of the point P on the linex

5=y − 1

1=z − 3

−2which is closest to the point

Q(9, 4,−3). [6 marks]

3. T : R2 → R2 is a linear transformation such that the matrix which represents T is

(4 2−1 2

). Find

and describe the image of the circle x2 + y2 = 1 under the transformation T. [5 marks]

Hence, determine the area of the image. [2 marks]

4. The transformation M is a rotation 45 anticlockwise about the origin. The transformation N is anenlargement with the origin as the centre of enlargement and scale factor k. The transformation NMmaps the points (0, 1) and (3,−1) into the points (−1, 1) and (4, 2) respectively.

(a) Find the matrix representing the transformation M. [2 marks]

(b) Find the matrix representing the transformation NM. [4 marks]

(c) Determine the value of k. [2 marks]

5. The line l has the equationx+ 7

1=y − 4

−3=z − 5

2and the plane π has the equation 4x−2y−5z = 8.

(a) Determine whether the line l is parallel to the plane π. [5 marks]

(b) Find the equation of the plane that is perpendicular to the plane π and contains the pointsQ(−2, 0, 3) and R(2, 1, 7). [6 marks]

6. The simple graph G with its vertex set V (G) = a, b, c, d, e, f, g, h, i, j is shown below.

(a) State, with a reason, whether G is an Eulerian graph. [2 marks]

(b) Find the largest cycle in G, and determine whether G is a Hamiltonian graph. [4 marks]

(c) The subgraph S of G is obtained by removing the vertex g together with all the edges adjacentto it.

i. Draw the subgraph S. [2 marks]

ii. Determine whether S is a connected graph. [2 marks]

iii. Determine whether S is a bipartite graph. [3 marks]

7. The mean mark of an English test for a random sample of 50 form five students in a particular stateis 47.7. A hypothesis test is to be carried out to determine whether the mean mark for all the form

46


five students in the state is greater than 45.0. Using a population standard deviation of 13 marks,carry out the hypothesis test at the 10% significance level. [6 marks]

8. The relationships between two variables x and y are shown in the graphs below.

Suggest a value for Pearson correlation coefficient r if the outlier is not taken into account, andcomment on the effect of the outlier on the value of r in each of the above graphs. [6 marks]

9. A doctor claims that doctors in government hospitals work at least 12 extra hours in a week. Arandom sample of 25 doctors is taken and it is found that the mean and standard deviation are 10.9hours and 2.3 hours respectively.

State the appropriate hypotheses to test the doctor’s claim, and carry out the test at the 5% signifi-cance level. [7 marks]

10. A food company carries out a market survey in a state on its new flavoured yoghurt. Three hundredrandomly chosen consumers taste the yoghurt. Their responses are shown in the table below.

Response Like Dislike NeutralNumber of consumers 195 70 35

(a) Estimate the proportion of consumers in the state who like the yoghurt. Hence, calculate theprobability that the proportion of consumers who like the yoghurt is at least 0.70. [5 marks]

(b) Construct a 95% confidence interval for the proportion of consumers in the state who like theyoghurt. [4 marks]

11. In developing a new drug for an allergy, an experiment is carried out to study how different dosages ofthe drug affect the duration of relief from the allergic symptoms. A random sample of eight patientsis taken and each patient is given a specified dosage of the drug. The duration of relief for the patientsis shown in the table below.

47


Dosage (mg) Duration of relief (hours)3 94 105 126 146 167 188 229 24

(a) State the independent and dependent variables. [1 marks]

(b) Find the equation of the least-squares regression line in the form y = a+ bx, where x and y arethe independent and dependent variables respectively. Write down your answers correct to twodecimal places. [6 marks]

(c) Calculate the coefficient of determination for the regression line, and comment on the adequacyof the straight line fit. [4 marks]

12. A thread always breaks during the weaving of cloth in a factory. The number of breaks per threadwhich occur for 100 threads of equal length are tabulated as follows:

Number of breaks per thread 0 1 2 3 4 5Number of threads 15 22 31 18 8 6

(a) Calculate the expected number of threads with respect to the number of breaks based on aPoisson distribution having the same mean as the observed distribution. [6 marks]

(b) Carry out a χ2 test, at the 5% significance level, to determine whether the data fits the proposedmodel in (a). [7 marks]

48


STPM 2011

1. When integers a and b are divided by 8, the reminaders are 3 and 7 respectively. Find the remainderwhen 5a+ b divided by 8. [4 marks]

2. If n is an integer such that 7 divides n+ 1 and 19 divides n+ 3, show that 12n ≡ 2(mod 133). Hence,determine the smallest value of n. [5 marks]

3. The adjacency matrix of a simple graph G is0 0 0 1 10 0 1 1 00 1 0 1 m2

1 1 1 0 01 0 4m− 3 0 0

.

(a) Determine the value of m, and draw the graph G. [4 marks]

(b) State the length of the longest cycle in the graph G. [1 marks]

4. The line r =

123

+λ

3p1

is perpendicular to the plane r =

501

+ s

1−1q

+ t

012

, where p and

q are constants.

(a) Determine the values of p and q. [5 marks]

(b) Using the values of p and q in (a), find the position vector of the point of intersection of the lineand the plane [5 marks]

5. The transformation T: R2 → R2 is defined by T:

(xy

)7→ A

(xy

), where A =

(−2 33 6

).

(a) Determine the invariant line under T. [7 marks]

(b) Find the image of the line y = x− 1 under T. [5 marks]

6. The transformation T: R2 → R2 is defined by F

(xy

)=

(x− yx+ y

).

(a) Show that F is a linear transformation. [4 marks]

(b) If the image of the point (x, y) under F is the point (u, v), show that 2x = u+ v and 2y = v−u.Hence find and sketch the image of the region (x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1. [6 marks]

(c) Given that F is a composition of transformations M and N,

i. Describe M and N, [4 marks]

ii. state the matrices representing M and N. [2 marks]

7. X1, and X2 are two independent observations of a random variable X. Two estimators for the meanvalue of x are defined by

θ1 =kX1 + 2kX2

3and θ2 = kX1 + (1− k)X2,

where k is a constant and k 6= 1. Determine whether the estimators are unbiased. [4 marks]

49


8. Five pairs of values of variables x and y are taken from a bivariate sample. The scatter diagrams ofy against x and ln y against x are shown below.

The Pearson correlation coefficient between x and y is 0.901, and the Pearson correlation coefficientbetween x and ln y is 0.998. Comment on these two values with respect to the scatter diagrams.

[4 marks]

9. A random sample of size n is taken to estimate the mean length of a particular aluminum rodproduced by a factory. Assuming that the length of the rod is normally distributed with a standarddeviation of 2 mm, determine the smallest value of n so that the width of the confidence interval forthe mean length of the rod is 1 mm with a confidence level of at least 90%. [5 marks]

10. A nutritionist wishes to determine whether the mean intake of fat by women in a city exceeds therecommended 25 g of fat intake per day. A random sample of 20 women from the city gives a meanof 27.5 g and a standard deviation of 12 g of fat intake per day.

(a) State the appropriate hypotheses for a significance test. [1 marks]

(b) Carry out the test at the 5% significance level. [6 marks]

(c) State any assumption required. [1 marks]

11. For several pairs of observations of variables x and y, where 5 ≤ x ≤ 28 and x = 16.8, the Pearsoncorrelation coefficient is 0.92. The equation of the regression line of y on x is y = −1.172 + 0.115x.

(a) Estimate the values of y when x = 10 and x = 30. Comment on the reliability of the twoestimates. [5 marks]

(b) Find the equation of the regression line of x on y. [5 marks]

12. In a physics experiment, the number of emissions per hour from a radioactive substance is recordedfor 100 hours. The results of the experiment are shown in the following table.

Number of emissions per hour 0 1 2 3 4 5 6 7 8Frequency 2 15 21 16 19 12 9 3 3

(a) Calculate the sample mean and an unbiased estimate for the variance of the number of emissionsper hour from the radioactive substance. Give a reason why the number of emissions per hourmay have a Poisson distribution. [6 marks]

(b) Carry out a χ2 goodness-of-fit test, at the 5% significance level, to determine whether the numberof emissions per hour has a Poisson distribution. [11 marks]

50

stpm mathematics t questions for revision

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