STPM Further Mathematics T Past Year Questions

STPM Further Mathematics T Past Year Questions Lee Kian Keong & LATEX [email protected] http://www.facebook.com/akeong

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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 are encourage to contact me via email1 or facebook2 . Students also encourage to send me your collection of 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.

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2 3 5 7 9 11 13 15 17 19 21 23

2 PAPER 2 QUESTIONS STPM 2002 . . . . . . . . . STPM 2003 . . . . . . . . . STPM 2004 . . . . . . . . . STPM 2005 . . . . . . . . . STPM 2006 . . . . . . . . . STPM 2007 . . . . . . . . . STPM 2008 . . . . . . . . . STPM 2009 . . . . . . . . . STPM 2010 . . . . . . . . . STPM 2011 . . . . . . . . .

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25 26 28 30 32 34 37 40 43 46 49

1 2

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1 PAPER 1 QUESTIONS STPM 2001 . . . . . . . . . STPM 2002 . . . . . . . . . STPM 2003 . . . . . . . . . STPM 2004 . . . . . . . . . STPM 2005 . . . . . . . . . STPM 2006 . . . . . . . . . STPM 2007 . . . . . . . . . STPM 2008 . . . . . . . . . STPM 2009 . . . . . . . . . STPM 2010 . . . . . . . . . STPM 2011 . . . . . . . . .

FURTHER MATHEMATICS

1

PAPER 1 QUESTIONS

PAPER 1 QUESTIONS

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FURTHER MATHEMATICS

PAPER 1 QUESTIONS

STPM 2001

STPM 2001 1. Solve the equation cosh 2x + 2 cosh x = 5. Give your answer correct to two decimal places.

[4 marks]

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

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Find the area of the region bound by the curve.

[6 marks]

3. Using Maclaurin expansion, find

1 − cos2 x . x→0 x(1 − e−x ) lim

[4 marks]

4. Use the Newton Raphson method with initial estimate, x0 = 0.5, find the root for equation 10x − 2 sin x = 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]

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

  cos θ sin θ cos 3θ T 3 7. If M = , show that (M ) = − sin θ cos θ sin 3θ terms of n and θ. Justify your answer.

 − sin 3θ . Hence, write down (MT )n in cos 3θ [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 and perpendicular to the plane OAB. [3 marks] Hence, determine the position vector, respect to O, of the points which are 7 units from the midpoint of AB. [3 marks] 9. Given a curve with parametric equation

x = a(t − 3t3 ), y = 3at2 ,

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with a > 0 and t ∈ R. Determine the values of t when the curve cuts the y-axis and sketch the curve. [4 marks]  2  2 dx dy Show that + = a2 (1 + 9t2 )2 . [3 marks] dt dt 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 d2 y dy +5 + 4y = e−2x + sin x. dx2 dx 3

FURTHER MATHEMATICS

PAPER 1 QUESTIONS

STPM 2001 [9 marks]

6 dy 3 Find the particular solution which satisfies the condition y = and = when x = 0. Show 17 dx 34 5 3 that y = sin x − cos x when x is positive and large. [6 marks] 34 34

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11. (a) Find the expansion for ex cos x and ex sin x in ascending powers of x up to the term x3 . Hence, 1 show that, if x4 and higher power of x can be ignored, sinh x cos x ≈ x − x3 and cosh x sin x ≈ 3 1 x + x3 . [6 marks] 3 (b) Using Maclaurin theorem, find the expansion for sec x in ascending powers of x up to the term x4 . [5 marks] Deduce the first three non zero terms in the expansion sec2 x and tan x in ascending powers of x. [4 marks]

12. (a) Describe the locus of the points represented by the complex numbers z if z satisfies z − 1 − 2i = 1, i. z + 1 + 4i 1 1 + ∗ = 3. z z (b) Show that, for z 6= −1, ii.

[3 marks] [4 marks]

1 z3




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z 3 z 2 + z12 + z 3 + z−z +z −z +z = 2 + z + z1 2

3

4

5

.

Using the substitution z = eiθ , show that 5 X

(−1)k+1 cos kθ =

k=1

cos 3θ cos 52 θ , cos 12 θ

where θ is odd multiple of π.

[8 marks]

13. The transformation T on the x − y plane is given by     x 2 x T: →A , y y 

where A =

6 1

 k , k ∈ R. 2

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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 of the 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 + k respectively. Find the equation of the plane OP Q. [2 marks] Show that the point S lies on the plane OP Q. [4 marks] Show that the line RS are perpendicular to the plane OP Q. [4 marks] Find the acute angle between the line P R and the plane OP Q. [5 marks]

4

FURTHER MATHEMATICS

PAPER 1 QUESTIONS

STPM 2002

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 sech x − 12 tanh x = 13.

[4 marks]

3. Solve the recurrence relation xn+1 = xn+1 + xn with x0 = 0, x1 = 1.

[5 marks]

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

Z

5. If In =

xn cos x dx, show that, for n ≥ 2,

0

In =

Z

π 2

Hence, evaluate

 π n 2

− n(n − 1)In−2 . [4 marks]

x4 cos x dx.

[2 marks]

0

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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

x−2 1 dy + y=− 2 . dx x(x − 1) x (x − 1)

that satisfies the boundary condition y =

3 when x = 2. 4

[8 marks]

8. Using mathematical induction, prove that n X

cos 2rθ =

r=1

sin(2n + 1)θ 1 − . 2 sin θ 2

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[7 marks]

q √ 1 If sin θ = 2 − 3, find sin 5θ. 2

[3 marks]

1 inθ (e + e−inθ ). 2 Hence, show that, for x ∈ R and |x| < 1,

9. Show that cos nθ =

∞ X

[3 marks]

xn cos(2n + 1)θ =

n=0

5

(1 − x) cos θ . 1 − 2x cos 2θ + x2

FURTHER MATHEMATICS

PAPER 1 QUESTIONS

Deduce that

∞ X

cosn 2θ cos(2n + 1)θ =

n=0

STPM 2002

1 sec θ 2

1 for θ 6= kπ, where k is integer. 2

[7 marks]

10. If xn = xn−1 + h and yn = y(xn ), with h3 and higher powers of h can be neglected, show that the recurrence relation for differential equation yn00 − 2yn0 + 2yn = 0 is

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(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

 −2 A = −3 −1

1 2 1

 3 3 . 2

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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   λ1 0 0 P−1 AP =  0 λ2 0  . 0 0 λ3 [5 marks]

Deduce the relationship between determinant of A and its eigen values.

12. If y = sin−1 x, show that

d2 y =x dx2



dy dx

3

and

d3 y = dx3



dy dx

3

+ 3x2



dy dx

[2 marks]

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]

−1

x − sin x . x→0 x − sin x

(b) find lim

[2 marks]

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FURTHER MATHEMATICS

PAPER 1 QUESTIONS

STPM 2003

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

AK

Z

√

4x2

dx . − 4x + 3 [4 marks]

3. Using Cramer’s rule, determine the set of values of k such that the following system of linear equations has integer solutions. 2x − y + 3z = k, 2x + y − z = 1, 6x − 3y + z = 3k, [6 marks]

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

5. Use the expansion of

1 to express tan−1 x as a series of ascending powers of x up to the term 1 + x2

in x7 . Hence find, in terms of π, the sum of the infinite series

[4 marks]

1 1 1 − + − ... 2 3×3 5×3 7 × 33

[2 marks]

Z 6. Let In =

π

sinn x dx, where n ≥ 2. Show that

0

In =

[4 marks]

n−1 In−2 . n

Express In in terms of I1 , for odd integers n ≥ 3. Hence find the value of Z π sin7 x cos2 x dx.

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0

[3 marks]

7. A loan of P ringgit is to be paid off over a period of N years. The loan carries a compound interest of 100k% a year. The yearly repayment is B ringgit. The balance after r years is ar ringgit, where a0 = P . The interest is charged on the balance at the beginning of the year. Write down a recurrence relation that involves ar+1 and ar and show that   B B ar = P − (1 + k)r + . k k 7

FURTHER MATHEMATICS

PAPER 1 QUESTIONS

STPM 2003 [6 marks]

Calculate the yearly repayment if a loan of RM100 000 that carries a compound interest of 8% a year is to be paid off over a period of 10 years. [3 marks] 8. Matrix M is given by M=

 3 4

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Use mathematical induction to prove that  2n + 1 n M = 4n

 −1 . −1

 −n , n = 1, 2, . . . 1 − 2n [5 marks]

n

Show that M and M have the same eigenvalues.

[4 marks]

9. If y = (sin−1 x)2 , show that, for −1 < x < 1, (1 − x2 )

−1

Find the Maclaurin series for (sin

d2 y dy − 2 = 0. −x 2 dx dx [3 marks]

2

6

x) up to the term in x .

[6 marks]

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10. Find the general solution of the differential equation

d2 y dy + y = cos 2x − 7 sin 2x. +2 dx2 dx

[7 marks]

dy Find the solution of the differential equation for which y = = 2 when x = 0. Determine whether dx this solution is finite as x → ∞. [5 marks] 11. Let a cosh x + b sinh x = r sinh(x + k), where a, b, and r are positive real numbers. Express k and r in 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 = cosh x + 3 sinh x which has a gradient of 8 at that point. [6 marks] 12. Obtain all the roots of the equation z 5 = 1 in the form eiθ , where 0 < θ ≤ 2π. Hence show that the roots of the equation (ω − 1)5 = ω 5 1 1 k + i cot π, where k = 1, 2, 3, 4. 2 2 5 Deduce the roots of the equation

are

[8 marks]

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(ω − i)5 = ω 5 .

[2 marks]

5

5

5

5

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

8

FURTHER MATHEMATICS

PAPER 1 QUESTIONS

STPM 2004

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 (cos x) =

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1 π, 2

for 0 ≤ x ≤ π.

[3 marks]

3. Using mathematical induction, prove that n X

p(p!) = (n + 1)! − 1.

p=1

[5 marks]

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

5. Show that tanh−1 x =

1 ln 2



1+x 1−x



for −1 < x < 1 and find

d (tanh−1 x). dx

[7 marks]

[7 marks]

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6. Find the exact value of

Z

2

x

p

x4 − 1 dx.

1

[7 marks]

7. Show that the general solution of the recurrence relation an+2 = 6an+1 − 25an is an = 5n (α cos nθ + iβ sin nθ),

4 where α and β are arbitrary complex constants and θ = tan−1 . 3 If a0 = a1 = 1, find the particular solution of the above recurrence relation. Z 8. If In =

√

[6 marks] [2 marks]

xn dx, where n > 1 and a is a non-zero constant, show that x2 + a2 p nIn + (n − 1)a2 In−2 = xn−1 x2 + a2 .

[4 marks]

Z

0

3

2

x +x √ dx. x2 + 1

[6 marks]

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

 1 π + h in ascending powers of h up to 6

the term in h3 . 1 If 0 < h < π, show that the remainder term R is given by 6 √ 1 4 3 4 h 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. r

(b) ur+1 = 2ur + γ2 , where γ is an arbitrary constant.

[3 marks] [3 marks]

4. A differential equation has y = e3x and y = xe3x as solutions. (a) Find the differential equation.

[2 marks]

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(b) Write down the general solution of the differential equation and determine the particular solution dy = 0 when x = 0. [4 marks] satisfying the initial conditions y = 1 and dx 5. A circular disc is partitioned into n equal sectors. Each sector is to be coloured with one of the four different colours provided and no two adjacent sectors are to be of the same colour. If an is the number 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 cos x up to the term in x3 . State the range of values of x for which 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 3 3 |z| = z − + i . 10 10

G [4 marks]

8. Show that, for |x| > 1, coth−1 x =

1 ln 2



x+1 x−1

 .

[4 marks]

5 If coth 2y = , find the value of coth y. 4

[4 marks]

11

FURTHER MATHEMATICS

PAPER 1 QUESTIONS

STPM 2005

9. The matrix A is given by  k A = 2 8

1 k −3

 5 8 . 2

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Determine all values of k for which the equation AX = B, where B is a 3 × 1 matrix, does not have a unique solution. [3 marks] For each of these values of k, find the solution, if any, of the equation   1 AX = −2 . 4 [7 marks]

10. Find the general solution of the differential equation x

dy − 3y = x3 . dx [4 marks]

Find the particular solution given that y has a minimum value when x = 1. Sketch the graph of this particular solution.

[3 marks] [3 marks]

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1 11. A curve has equation y = 1 − ln tanh x, where x > 0. 2 (a) Show that, for this curve, y > 1. dy (b) Show that = − cosech x. dx (c) Sketch the curve.

[2 marks] [3 marks]

[2 marks]

6

(d) Show that the length of the curve between the points where x = 1 and x = 4 is ln(e + e4 + e2 + 1) − 3. [6 marks] 12. Prove de Moivre’s theorem for positive integer exponents. Using de Moivre theorem, show that

[5 marks]

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

sin 5θ where a, b and c are integers to be determined. Express in terms of cos θ, where θ is not a sin θ multiple of π. [5 marks] 4 2 Hence, find the roots of the equation 16x − 12x + 1 = 0 in trigonometric form. Deduce the value   π 2π of cos2 + cos2 . [6 marks] 5 5

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FURTHER MATHEMATICS

PAPER 1 QUESTIONS

STPM 2006

STPM 2006 1. The matrix M is given by  −3 M = −6 0

 1 2 2 3 . 2 1

By using Cayley Hamilton theorem, show that M−1 =

1 (7I − M2 ), 6

AK 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−1

8 . 13 [3 marks]

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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 3 1 [7 marks] and only if r = − and d = − a. 2 4 4. Consider the system of equations

x 3x 6x

+ y − y + 2y

+ − +

pz 2z z

= q, = 1, = 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 there is no unique solution. [7 marks] 5. Find the domain and the range of the function f defined by f (x) = sin−1

2(x − 1) . x+1

[4 marks]

Sketch the graph of f .

[3 marks]

6. Show that

1

0

√ √ x−1 √ dx = 6 − 3 − 2 ln x2 + 2x + 3

√ ! 2+ 6 √ . 1+ 3

G

Z

[7 marks]

7. If y = ex cos x, prove by mathematical induction that   1 dn y 1 n x 2 = 2 e x + nπ , dxn 4 for every positive integer n.

[8 marks]

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FURTHER MATHEMATICS

PAPER 1 QUESTIONS

STPM 2006

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]

AK

9. Let an be the number of ways (where the order is significant) the natural number n can be written as 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]

1

10. Using the substitution x = z 2 , transform the differential equation   d2 y 1 dy + 4x2 y = 0 + 4x − dx2 x dx into one relating y and z.

[5 marks]

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dy Hence, find y in terms of x given that y = 2 and = −2 when x = 1. dx State the limiting value of y as x → ∞.

11. Let zl = 1, z2 = x + iy, z3 = y + ix, where x, y ∈ R, x > 0 and i = geometric progression,

[5 marks] [1 marks]

√

−1. If z1 , z2 , . . ., zn is a

(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 z1 z2 z3 . . . 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] n Write the above series up to the term in x together with the remainder term. Hence, determine the smallest integer n to ensure that the estimated value of e is correct to four decimal places and find the estimated value. [7 marks]

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FURTHER MATHEMATICS

PAPER 1 QUESTIONS

STPM 2007

STPM 2007 1. If z = cos θ + i sin θ, show that

Z 2. Show that

π 4

1 1 1 = (1 − i tan θ) and express in a similar form.[4 marks] 1 + z2 2 1 − z2

sec x(sec x + tan x)2 dx = 1 +

√

2.

[4 marks]

0

AK

3. Using the substitution x = e2 , show that the differential equation x2

d2 y dy + px + qy = 0, 2 dx dx

where p and q are constants, can be transformed into the differential equation d2 y dy +r + sy = 0, 2 dz dz

where r and s are constants to be determined in terms of p and q.

  3 4 4 5 1 3 ,B =  2 8 0 3 9 −1

2 4 0 −1

 3 4 1 3 . 11 0  3 12

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4. The matrices A and B are given by  1 2 5 1 A= 2 0 −1 −1

[5 marks]

  1 1  Given that  1 is an eigenvector of the matrix A, find its corresponding eigenvalue. 1   1 1  Hence, find the eigenvalue of the matrix B corresponding to the eigenvector  1. 1 5. Show that

sec−1 x + cosec−1 x =

[3 marks]

[3 marks]

1 π, 2

1 1 1 π, − π ≤ cosec−1 x ≤ π and cosec−1 x 6= 0. 2 2 2 Hence, find the value of x such that

where 0 ≤ sec−1 x ≤ π, sec−1 x 6=

[3 marks]

3 sec−1 x cosec−1 x = − . 2

G [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. Solve the recurrence relation

[7 marks]

ar − ar−1 − 6ar−2 = (−2)r , where a0 = 0 and a1 = 2.

[8 marks]

15

FURTHER MATHEMATICS

PAPER 1 QUESTIONS

STPM 2007

8. Consider the system of equations λx x x

+ y + λy + y

+ + +

z z λz

= 1, = λ, = λ2 ,

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

AK

(b) Find the unique solution in terms of λ.

[5 marks]

9. Solve the differential equation

dy + y sinh x = sinh x, dx

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 differential equation. [5 marks]

10. Find the constants A, B, C and D such that e2x + e−2x − 2ex − 2e−x − 1 = A cosh2 x + B cosh x + C sinh2 x + D sinh x.

N EO

[5 marks]

Hence, solve the equation

e2x + e−2x − 2ex − 2e−x − 1 = 0.

[5 marks]

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

∞ X r(r − 1) . . . (r − k + 1)

k!

k=1

xk ,

1

where r is a rational number. Write down the Maclaurin series for (1 + x2 )−1 and (1 + x2 )− 2 .[8 marks] Hence, find the Maclaurin series for tan−1 x and sinh−1 x. [5 marks] x − tan−1 x 1 [2 marks] = . Show that lim 2 x→0 x sinh−1 x 3 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]

G

3

3

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

[5 marks]

2

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

16

FURTHER MATHEMATICS

PAPER 1 QUESTIONS

STPM 2008

STPM 2008 1. If p ⇒ q, r ⇒∼ q and ∼ s ⇒ r, show that p ⇒ s.

2. Using the result that tan−1 x =

Z 0

x

[3 marks]

1 dt, show that 1 + t2

tan−1 x = x −

x3 x5 x7 + − + ... 3 5 7

AK

where |x| < 1.

[2 marks]

−1

Hence, find lim

x→0

3. If −

tan x . sin x

[2 marks]

π π < tan−1 (sinh x) < , show that 2 2 cos(tan−1 (sinh x)) = sech x. [5 marks]

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

1 at x = 1. (2x + 1)3

[5 marks]

N EO

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 X k 1 = , x x x x 1 k+1 i=1 i i+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 z 4 − 2z 3 + kz 2 − 18z + 45 = 0 has an imaginary root. Obtain all the roots of the equation and the value of the real constant k. [8 marks] Z

0

8. Let Im,n =

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

−1

Im,n = −

n Im+1,n−1 . m+1

G

[3 marks]

m

Hence, show that Im,n =

(−1) m!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. At time 0. it is full and contains water only. Every week (at times 1, 2, 3. ...). the cooling system is topped up with 100 ml of a coolant mixture of concentration 20% (a mixture of 80 ml of water and 20 ml of coolant). Let vn , be the volume in millilitres and cn the concentration of the coolant in the cooling system immediately after the top up at time n. 17

FURTHER MATHEMATICS

PAPER 1 QUESTIONS

STPM 2008

(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 3

1 dy d2 y −2 − y = 4e− 3 x − x2 + 24 2 dx dx

AK for which y = 0 and

dy = 1 at x = 0. dx

[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]

N EO

12. Find the eigenvalues of the matrix A given by 

4 A = −1 −4

4 −1 −4

 3 −1 −3

and an eigenvector corresponding to each eigenvalue. [8 marks] −1 Write down a matrix P and a diagonal matrix D such that A = PDP . [2 marks] n n Hence, show that A = A when n is an odd positive integer, and find A when n is an even positive integer. [6 marks]

G 18

FURTHER MATHEMATICS

PAPER 1 QUESTIONS

STPM 2009

STPM 2009 1. Prove that, if x > k, then x3 − kx2 + ax − ak > 0 for every real number a > 0. Write down the converse and contrapositive of the above statement.

[2 marks] [2 marks]

2. The real matrix A is given by 

a 0 A=0 b −c 0

 −a 0 . c

AK

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

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

[4 marks]

[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 dy b 0< ≤ . [5 marks] dx 4a 1 ln 2 Hence, solve the equation

5. Show that tanh−1 x =



1+x 1−x



for |x| < 1.

−1

2 tanh

[4 marks]

  3 = ln x. x

N EO

where x > 0.

6. If tan y = (x − 1)2 .

[3 marks]

π π x , where − < y < , expand y in ascending powers of (x − 1) up to the term in 1+x 4 4 [5 marks]

    1 1 −1 −1 Hence, find the approximate value of tan − tan . 2 3

[2 marks]

7. The matrix A is given by



1 A = 1 1

1 2 c

 c 3 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]

G

(b) For each value of c, find the solutions, if any, of the equation   1 AX =  −3  . −11

[5 marks]

8. (a) Show that sin(x + iy) = sin x cosh y + i cos x sinh y, nnd hence, find the values of x and y if the π [4 marks] imaginary part of sin(x + iy) is zero, where x ≥ 0 and y ≤ . 2 (b) Find the roots of ω 4 = −16i, and sketch the roots on an Argand diagram. [5 marks]

19

FURTHER MATHEMATICS

PAPER 1 QUESTIONS

STPM 2009

9. A curve has parametric equations x = t + ln | sinh t| and y = cosech t, where t 6= 0. (a) Find

dy . dx

(b) Show that

[3 marks]

AK

d2 y = −e−2t (cosh t + cosech t). dx2

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

10. Using the substitution z =

[2 marks]

√ 1 ln( 5 − 2). 2

[5 marks]

1 , show that the differential equation y 2y dy − = y2 dx x

may be reduced to

2z dz + = −1. dx x [2 marks]

N EO

Hence, find the particular solution y in terms of x for the differential equation given that y = 3 when x = 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 relation an + pan−2 + qan−3 = 0. [3 marks] Using the values of p and q, solve the recurrence relation p an + pan−2 + qan−3 = −1 + bn+1 , where n ≥ 3, a0 =

3 15 1 , a1 = , a2 = , and bm satisfies the relation 4 2 4  2 m+1 bm+1 = bm , m

where m ≥ 1 and b1 = 9.

[11 marks]

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

dx dy dx dy +2 = 2x + 5 and − = 2y + t. dt dt dt dt

G

d2 y dy + 4y = 2 − 2t. [4 marks] −6 2 dt dt (b) find the solution x in terms of t for the second order of differential equation given that y(0) = y 0 (0) = π. [12 marks] (a) show that 3

20

FURTHER MATHEMATICS

PAPER 1 QUESTIONS

STPM 2010

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]

AK 3. Solve the recurrence relations

(a) an = 22n an−1 , where a0 = 1,

(b) an = an−1 + 3

n−1

[3 marks]

+ 2, where a0 = 0.

4. Show that

tan

−1



1 x−1



[3 marks]

−1

− tan

  1 = cot−1 (x2 − x + 1), x > 1. x [3 marks]

Hence, find the value of x such that       1 1 1 −1 −1 −1 tan = tan + tan . x−1 x x+1

N EO

[3 marks]

[The principal values of each angle are to be considered.] 5. Given that y = x − cos−1 x, where −1 < x < 1. 3  d2 y dy − 1 . (a) Show that = x dx2 dx

[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

dQ Q + =V dt C describes the charge Q on a capacitor of capacitance C during a charging process involving a resistance R and electromotive force V . R

(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] 0 2 1

 4 0 . −3

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.

21

G

 1 7. Find the eigenvalues and eigenvectors of the matrix 0 3

[8 marks]

[5 marks]

FURTHER MATHEMATICS

PAPER 1 QUESTIONS

STPM 2010

(b) The points A and B in the curve are defined by t = ln 2 and t = ln 3 respectively. Show that 4625 the length of the arc AB is . [3 marks] 864 9. Show that −1

coth

1 x = ln 2



x+1 x−1

 ,

where |x| > 1.

[3 marks]

1+y y

AK

Hence, solve the equation coth−1





+ coth−1



y 1−y



1 = ln 2y. 2

[6 marks]

10. A particle moves in a horizontal straight line. The displacement ar of the particle from a fixed point at the r-th second (r ≥ 2) satisfies the recurrence relation ar = 9ar−2 + br ,

where

br = 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]

N EO

11. Find the general solution of the first order differential equation dy + 5y = sinh 2x + 2x. dx

[6 marks]

Hence, solve the second order differential equation

d2 y dy = 4 cosh2 x, +5 2 dx dx

given that x = 0, y = 1 and

dy = 3. dx

[7 marks]

G

12. (a) Find the fifth roots of unity in the form cos θ + i sin θ, where −π < θ ≤ π. [4 marks]  5 z − 2i (b) By writing the equation (z − 2i)5 = (z + 2i)5 in the form = 1, show that its roots z + 2i   π 2π and ±2 cot . [7 marks] are ±2 cot 5 5     π π 2π 2π Hence, find the values of cot2 + cot2 and cot2 cot2 . [5 marks] 5 5 5 5

22

FURTHER MATHEMATICS

PAPER 1 QUESTIONS

STPM 2011

STPM 2011 1. Prove by contradiction that the sum of a rational number and an irrational number is an irrational number. [3 marks]

2. Let pn be the pressure within an enclosed container on day n. The pressure on day n + 1 is less than that on day n. Show that pn =

AK 3. Show that cos(sin−1 x) =

Z

4. Given that In =

6 p1 . n(n + 1)(n + 2)

3 pn n+3 [4 marks]

p 1 − x2 , for |x| ≤ 1, and hence, find the exact value of √ !! 2 2 −1 tan sin . 3 [4 marks]

e

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

1

Z

e

(ln x)4 dx.

1

[6 marks]

N EO

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. √ Hence, find an approximate value of 26, and estimate the size of the error.

[4 marks]

[3 marks]

7. The complementary function of the differential equation

d2 y dy + 13y = ex cos x +p dx2 dx

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

[3 marks]

(b) Find the general solution of the differential equation.

[5 marks]

G

(a) Determine the values of p and r.

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

[5 marks]

23

FURTHER MATHEMATICS

PAPER 1 QUESTIONS

STPM 2011

9. The arc of a curve y = ln x 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 Z S = 2π

8

p

1 + p2 dp,

1

where p = ey .

[3 marks]

AK

(b) Using the substitution p = sinh θ, show that " √ √ S = π 8 65 − 2 + ln

10. Show that

√ !# 8 + 65 √ . 1+ 2 [7 marks]

1 + cos 2θ + i sin 2θ = i cot θ. 1 − cos 2θ − i sin 2θ

[2 marks]

π

Hence, show that the roots of the equation (z + i)5 = (z − i)5 are ± cot 5     π π 2π 2π + cot2 and cot2 cot2 . Deduce the values of cot2 5 5 5 5

 and ± cot



2π .[6 marks] 5 [4 marks]

N EO

11. A square matrix A has an eigenvalue λ with corresponding eigenvector v. Show that λ + c is an eigenvalue of A + cI, where c ∈ R and I is identity matrix. Show that v is also an eigenvector of A + cI corresponding to λ + c. [3 marks] Determine the values of k so that the eigenvalues of the matrix   −1 0 2 P =  0 k 0 , −5 0 6 where k ∈ R are all distinct. Find the eigenvalues and the eigenvectors of  2 0 Hence, find the eigenvalues and eigenvectors of the matrix Q =  0 4 −5 0

P for k = 1.  2 0. 9

[8 marks] [4 marks]

12. Using the definitions of sinh x and cosh x, (a) show that

i. sinh(x − y) = sinh x cosh y − cosh x sinh y, ii. tanh(n + 1)x − tanh nx = sinh x sech(n + 1)x sech nx. (b) Given that Sr =

r X

[3 marks] [3 marks]

sech(n + 1)xsechnx. Show that

i. if x 6= 0, then Sr = 2 cosech 2x sinh rx sech(r + 1)x, ii. if x > 0 and r is very large, then Sr ≈ 2e−x cosech 2x.

24

G

n=1

[5 marks]

[4 marks]

FURTHER MATHEMATICS

2

PAPER 2 QUESTIONS

PAPER 2 QUESTIONS

AK G

N EO 25

FURTHER MATHEMATICS

PAPER 2 QUESTIONS

STPM 2002

STPM 2002 1. State the relationship between the sum of degree of vertices and the number of edges in a simple graph. 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 P Q. Position vector of P , Q, and R respect 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 P Q, show that

AK r=

|q|2 − p · q |p|2 − p · q p + q. |p − q|2 |p − q|2 [4 marks]

3. Prove that the planes ax + by + cz = d and a0 x + b0 y + c0 z = d0 are parallel if and only if a : b : c = a0 : b0 : c0 . Find the equation of the plane π that is parallel to the plane 3x + 2y − 5z = 2 and contains the 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] [2 marks]

(c) Eulerian but not Hamiltonian,

[2 marks]

(d) not Eulerian and not Hamiltonian.

[2 marks]

N EO

(b) not Eulerian but Hamiltonian,

5. Transformation P is defined by

   x 1 P : → y 0

  1 x −1 y

(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) under the 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). n

2

(b) If 2 + n is a prime number, prove that n ≡ 3(mod 6).

[9 marks] [4 marks]

7. Continuous random variable Z is standard normal random variable. State the exact name of the distribution Z 2 , and state the mean and variance. [3 marks]

G

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 deduce the smallest sample size required so that the estimation error did not exceed 0.01. [7 marks] 9. A random sample apples from a box of apples. The mass of apples, in g, is Xof 12 randomly selected X 2 summarized by x = 956.2 and x = 81175.82. Find a 95% confidence interval for the mean mass of apples in the box. State any assumptions you make. [7 marks] 26

FURTHER MATHEMATICS

PAPER 2 QUESTIONS

STPM 2002

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

AK

11. State common conditions apply to the distribution of test statistics in the test of goodness of fit, χ2 is approximated to χ2 distribution. [2 marks] A study was conducted in a factory on the number of accidents that happen on 100 factory workers in a period of time. The following data were obtained. Number of accident Number of workers

0 25

1 31

2 23

3 13

4 5

5 3

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

12. Explain the least squares method to obtain equation of regression line, using diagram. Five pairs of value of variable x and y are given by x y

30 90

40 84

50 82

60 77

[3 marks]

70 68

N EO

(a) Obtain the equation of the least squares regression line of y on x in the form y = a + bx, by giving 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 1 least squares regression line of y on x from the six pairs of values is y = 105 − x. Find x6 and 2 y6 . [6 marks]

G 27

FURTHER MATHEMATICS

PAPER 2 QUESTIONS

STPM 2003

STPM 2003 1. Let T : R2 → R2 be a linear transformation such that         1 0 2 1 . → , T: → T: 2 1 −3 1 Show that

    a a+b T: → . b −5a + 2b

AK

[4 marks]

2. Solve the pair of congruences

x x

≡ 1 (mod 6), ≡ 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]

N EO

4. Find the greatest common divisor of 2501 and 2173 and express it in the form 2501m + 2173n, where m 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 n(n − 1). 2

If n = 11 and m = 46, show that G is connected.

[4 marks] [7 marks]

6. Describe the transformation in the xy-plane represented by the matrix √  5 2  3 3  √  . 2 5 − 3 3

[3 marks]

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

√ ! 3− 5 x, 2

G

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 √ ! 3− 5 which lies on the invariant line y = x such that the sum of the distance between P 2 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. ¯ lies within 0.2 Determine the smallest sample size which is required so that the probability that X of µ is at least 0.90. [5 marks] 28

FURTHER MATHEMATICS

PAPER 2 QUESTIONS

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

STPM 2003

k X (Oi − Ei )2 i=1

Ei

, where Oi is the observed

frequency, Ei the expected frequency, and k > 2. Show that k X (Oi − Ei )2 i=1

where N =

k X

k X

k X O2 i

i=1

Ei

− N,

Ei .

[2 marks]

AK

Oi =

Ei

=

i=1

i=1

A car model has four colours: white, red, blue, and green. From a random sample of 60 cars of that model 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 the same. [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 least square regression line y = 7.80 − 1.72x. (a) Determine, on the average, the reduction in the charge of the batteries after they have been used for 1000 hours. [2 marks]

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

[2 marks]

N EO

10. The mean and standard deviation of the yield of a type of rice in Malaysia are 960 kg per hectare and 192 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 the mean yield of rice in Kedah is more than the mean yield of rice in Malaysia. Give any assumptions that 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 on the 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 b [6 marks] and b 6= 0, show that r = √ and deduce the possible values of r. b2 12. Out of 100 corn seeds of type A which are planted in a certain area, 24 seeds fail to germinate. Out of 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 germinate in that area. [5 marks]

G

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

29

FURTHER MATHEMATICS

PAPER 2 QUESTIONS

STPM 2004

STPM 2004 1. Write down an incidence matrix for the graph given below. What can be said about the sum of the entries in any row and the sum of the entries in any column of this incidence matrix? [4 marks]

v1

v2

e1

AK

e2

e6

e5

e3

v3

v4 e4

v5

2. The straight line joining the points (0, 0, 1) and (1, 0, 0) intersects the plane x + y + 2z + 2 = 0. Find the coordinates of the point of intersection. [5 marks] 3. Define a simple graph. State whether there exists a simple graph with

[1 marks]

N EO

(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 simple graph. [5 marks]     1 3 4. The point P lies on the line r = 3 + t 4 and the point Q lies on the plane whose equation is 3 8 r · (3i + 6j + 2k) = −22 such that P Q 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 P Q.

[3 marks]

5. T : R2 → R2 is a linear transformation such that the images of the points (1, 2) and (−1, 0) are the points (3, −2) and (−1, 0) respectively. [4 marks]

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

[6 marks]

G

(a) Find the matrix which represents T.

6. Define the congruence a ≡ b(mod m). [1 marks] 3 3 Solve each of the congruences x ≡ 2(mod 3) and x ≡ 2(mod 5). Deduce the set of positive integers which 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

FURTHER MATHEMATICS

PAPER 2 QUESTIONS

STPM 2004

(a) State whether this means that there is no linear relation between X and Y . Sketch a scatter diagram 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 σ 2 . (a) Explain briefly what a 95% confidence interval for µ means.

[2 marks]

AK

(b) From a random sample, it is found that the 95% confidence interval for µ is (−1.5, 3.8). State whether 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 a 95% confidence interval for µ is calculated. Find the number of 95% confidence intervals which are 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 2 correlation coefficient between X and Y is − √ . The means of X and Y are 5 and 3 respectively. 5 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. Opinion on the social issue Agree Disagree 450 18 547 30 230 25 1227 73

Total

N EO

Level of education University College High school Total

468 577 255 1300

Determine, at the 1% significance level, whether the level of education and the opinion on the social issue are independent. [9 marks] 11. A petroleum company claims that its petrol has a RON rating of at least 97. From a random sample of 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 distribution with 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 the petroleum company’s claim is rejected at the 5% significance level. [6 marks]

G

12. In a survey on the quality of service provided by a bank, 12 out of 150 customers think that the service is unsatisfactory while the other 138 customers think otherwise. Test the hypothesis that the proportion of customers who think that the service is unsatisfactory is 0.10 (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]

31

FURTHER MATHEMATICS

PAPER 2 QUESTIONS

STPM 2005

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 greatest common divisor of a and b. Hence, find all integers x such that (x, 12) = 1 and (x, 15) = 1. [6 marks]

AK

3. Four schools P , Q, R and S each has team A and team B in a tournament. The teams from the same school do not play against each other. At a certain stage of the tournament, the numbers of games played by the teams, except team A of school P , are distinct. Determine the number of games played 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), then xn ≡ 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         1 3 2 6 T: → , and T : → . 1 4 1 4

N EO

          1 1 2 1 3 (a) By expressing as a linear combination of and , show that T = . Find 0 1 1 0 0   0 also T . Hence, write down the matrix representing T. [6 marks] 1

(b) Find the image of the circle x2 + y 2 = 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 P Q, where P R = 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 lies on 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 distance from 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 for 24 cars of the same engine capacity is 15.2 km with a standard deviation of 4.2 km. Calculate the standard error of the mean mileage and interpret this standard error. [3 marks]

G

8. The mass of a particular type of steel sheet produced by a factory has a normal distribution. The mean mass of a random sample of 14 sheets is 72.5 kg and the standard deviation is 3.2 kg. Calculate a 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 model for the data, the test statistic is found to have the value 19.38 with 7 degrees of freedom. Find the smallest significance level at which the null hypothesis is rejected. [2 marks] 32

FURTHER MATHEMATICS

PAPER 2 QUESTIONS

STPM 2005

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, in hours, of a random sample of 15 bulbs are as follows. 5260 5600

5400 5780

5820 5520

5530 5500

5380 5360

5460 5620

5510 5430

5520

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

AK

(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 in a town for a particular period. The data could be a sample from a Poisson distribution. Number of passengers Frequency

0 241

1 211

2 104

3 35

4 7

5 0

6 2

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

[3 marks]

2

(b) Calculate the values of the χ 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.

N EO

[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 specified values of the variable x. y x

800 2

920 3

1280 5

1500 5

(a) Plot a scatter diagram of loge y against loge x.

4020 8

6200 10

6800 12

[3 marks]

(b) Find the equation of the least squares regression line of the form loge y = β0 + β1 loge x, with β0 and β1 correct to two decimal places. [5 marks] (c) Calculate the Pearson correlation coefficient r between loge y and loge x. Interpret your value of r and comment on this value with respect to the scatter diagram in (a). [6 marks]

G 33

FURTHER MATHEMATICS

PAPER 2 QUESTIONS

STPM 2006

STPM 2006 1. The points A, B and C are three collinear points on a cartesian plane and T(A) = A1 , T(B) = B1 and T(C) = C1 , where T : R2 → R2 is a linear transformation. If AB : BC = m : n, find A1 B1 : B1 C1 . [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]

AK

3. The matrix M represents in the xy-plane about the origin through an angle   an anticlockwise rotation cos θ 2 cos θ − sin θ represents the combined effect of the transformation θ. The matrix N = sin θ 2 sin θ + cos θ represented by a matrix K followed by the transformation represented by M. Find K and describe the transformation represented by K. [7 marks] 4. Graphs G1 , G2 and G3 are given as follows.

N EO

(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 hamiltonian graph by adding an edge. [1 marks] (d) For each of the graphs which is not eulerian, determine how it could be made into an eulerian graph 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 by 11 639, 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 the line l. The point A has coordinates (1,0, 1).

G

(a) Calculate the acute angle between π1 and π2 . [2 marks]     1 2 (b) Explain why the vector −1 ×  1  is in the direction of l. Hence, show that the equation 2 −1 of l is     0 −1 r = 1 + t  5  . 1 3 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

FURTHER MATHEMATICS

PAPER 2 QUESTIONS

STPM 2006

(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 in favour of the alternative hypothesis µ > 32.5 cm. Find the probability of making a type I error. Find also the probability of making a type II error when µ = 34 cm. [5 marks]

AK

8. A property agent believes that the price of a house in a certain district depends principally on its built-up area. The prices (p thousand ringgit) of eight houses in different parts of the district and their built-up areas (a square metres) are summarised as follows: X X X X X a = 2855, p = 1689, ap = 827550, a2 = 1400925, p2 = 489181. (a) Find the equation of the regression line in the form p = β0 + β1 a, with p as the dependent variable and a the independent variable. [5 marks]

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

9. The average duration of an electronic device to retain information after the power is switched off is normally 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 when a component is added. A random sample of n such duration yields mean x ¯ and variance s2 .

N EO

(a) State the appropriate null and alternative hypotheses. Explain briefly your choice of the alternative hypothesis. [2 marks] (b) Write down the test statistic and the critical region at the 5% significant level for each of the following 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) Experience (year)

88 12

48 4

60 6

70 7

62 5

78 10

100 18

52 5

110 19

G

(a) Plot a scatter diagram. What do you expect of the relationship between the salary and experience based on the scatter diagram? [4 marks] (b) Calculate the Pearson correlation coefficient between the salary and experience. Explain whether the 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 who break school rules should be caned. The results of the survey are as follows. 35

FURTHER MATHEMATICS

PAPER 2 QUESTIONS

hhhh

hhh Opinion hhhh hhh Education level h High school College University

STPM 2006

Yes

No

Not sure

125 98 100

65 68 80

100 64 120

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

AK G

N EO 36

FURTHER MATHEMATICS

PAPER 2 QUESTIONS

STPM 2007

STPM 2007 1. Two graphs G1 and G2 are shown below.

AK

(a) State, with reasons, which is an eulerian graph and which is not. (b) Find an eulerian circuit for the eulerian graph.

[3 marks] [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]

N EO

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

≡ ≡

1 (mod 6), 3 (mod 6).

[6 marks]

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

    x x+y and v = . y x−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]

n

n

n

n−1

G

(c) By using mathematical induction, show that f (u) = A u, where f (u) = (f ◦ f )(u), for all integers n ≥ 1. [4 marks] 0 0 0 (d) The points P , Q and R are the images of the vertices P , Q and R respectively of a triangle under the transformation fn . If the area of the triangle P QR is 8 units2 , find the area of the triangle P 0 Q0 R0 . [3 marks]

6. Two straight lines l1 and l2 have equations −2x + 4 = 2y − 4 = z − 4 and 2x = y + 1 = −z + 3 respectively. Determine whether l1 and l2 intersect. [7 marks] The points P and Q lie on l1 and l2 respectively, and the point R divides P Q in the ratio 2:3. Find the equation of the locus of R. [6 marks] 37

FURTHER MATHEMATICS

PAPER 2 QUESTIONS

STPM 2007

7. (a) Explain briefly why it is advisable to plot a scatter diagram before interpreting a Pearson correlation 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 obvious relation between them. [1 marks] ii. The Pearson correlation coefficient of two variables is close to 1 but there is no linear relation between them. [1 marks]

AK

8. The lengths of petals taken from a particular species of flowers have mean 80 cm and variance 30 cm2 . 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, the probability 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 construct a 98% confidence interval for the proportion. [5 marks]

N EO

10. A farmer wishes to find out the effect of a new feed on his calves. The average weight gain of a calf on the original feed in a month is normally distributed with mean 10 kg. In a particular month, the farmer gives the new feed to 16 calves. It is found that the average weight gain of a calf on the new feed 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 answers obtained are compared with the predetermined answers. The data is recorded as a frequency distribution as follows: Number of matched answers Frequency

0 10

1 15

2 8

3 5

4 12

A psychologist suggests that the data fits the following probability distribution. Number of matched answers Probability

0 0.08

1 0.10

2 0.15

3 0.25

4 0.42

(a) Calculate the expected frequencies based on the probability distribution.

[2 marks]

G

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

12. A study is carried out to determine the effect of exercise frequency on lung capacity. The exercise frequency in weeks is denoted by X and the percentage increase in lung capacity by Y . The data obtained from 10 volunteers are summarised as follows: X X X X X x = 319, x2 = 11053, y = 530, y 2 = 30600, xy = 18055. (a) Calculate the Pearson correlation coefficient and describe the relationship between X and Y . [5 marks]

38

FURTHER MATHEMATICS

PAPER 2 QUESTIONS

STPM 2007

(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 two decimal places. [4 marks] (d) Predict the mean value of the percentage increase in lung capacity if the exercise frequency is 25 weeks. State your assumption in obtaining the predicted value. [2 marks]

AK G

N EO 39

FURTHER MATHEMATICS

PAPER 2 QUESTIONS

STPM 2008

STPM 2008 1. If n is an odd integer, prove that x2 − y 2 = 2n has no solution in the set of integers.

[4 marks]

2. The adjacency matrix of a graph G is

AK

 0 1  0  1 1

1 0 1 0 0

0 1 0 1 1

1 0 1 0 0

 1 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 of the 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 through the point (1, 0, 0). Hence, determine the distance between these two planes. [6 marks] 4. A graph is given as follows:

N EO 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]   9 X di (c) Show that the total number of paths consisting of three vertices is and determine its 2 i=2 value. [4 marks]     1 0 5. The points A and B lie on the line r = 3 + λ −1, and the distance of each point is three units −4 6 from the origin O. [6 marks]

(b) Find the area of the triangle OAB.

[3 marks]

6. A linear transformation T : R2 → R2 maps



G

(a) Determine the coordinates of A and B.

           0 2 1 3 1 3 into , into and into . −1 2 1 6 0 5

(a) Find the matrices A and C such that     x x T =A + C. y y

[6 marks]

40

FURTHER MATHEMATICS

PAPER 2 QUESTIONS

STPM 2008

(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 hence describe 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 X (x − x ¯)2 = 72, (y − y¯)2 = 57, (x − x ¯)(x − y¯) = −36.

AK

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

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

8. A machine is regulated to dispense a chocolate drink into cups. From a random sample of 100 cups of the chocolate drink dispensed, it is found that the cocoa content in one cup of the chocolate drink has 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]

N EO

9. The masses of watermelon produced by a farmer are normally distributed with mean 3 kg. The farmer decides to use a new organic fertiliser for his crop if the mean mass of a random sample of 10 watermelons 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 in making 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 the regression 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 regression line, given that the actual value of y is 850? [4 marks]

Type of cream A B

No recovery 25 10

G

11. Two hundred patients with a certain skin disorder are treated for five days either with cream A or cream B. The following table shows the number of patients recorded as no recovery, partial recovery or complete recovery. Number of patients Partial recovery Complete recovery 30 45 25 65

(a) Calculate the percentage of patients with improvement for each type of cream used and comment on your answers. [2 marks] (b) Determine, at the 5% significance level, whether the condition of a patient is independent of the type of cream used. [9 marks] 41

FURTHER MATHEMATICS

PAPER 2 QUESTIONS

STPM 2008

(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 by weight. A random sample of 25 tins (1 kg per tin) of the paint is analysed to determine the lead content. Sample mean and standard deviation of the lead content in a kilogramme of paint are 0.38 g and 0.1 g respectively. (a) Construct a 99% confidence interval for the mean lead content in a kilogramme of paint. State your assumption. [5 marks]

AK

(b) What is the effect on the confidence interval obtained in (a) if the sample size is increased to 100? [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]

G

N EO 42

FURTHER MATHEMATICS

PAPER 2 QUESTIONS

STPM 2009

STPM 2009 1. Let Krs be a complete bipartite graph, where r ≤ s. Each of the r vertices in a partite set is connected to each of the s vertices in another partite set. If Krs has 36 edges and 15 vertices. find r and s. [4 marks]

 a b . If the images of the points −3 a (a, −1) and (−2, a) are (a + 2, 2b) and (2b, 7) respectively under T, find the values of a and b.[6 marks]

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



AK

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

[3 marks]

(b) Graph G is given as follows:

V4

V5

V2

V3

N EO

V1

i. Find all walks of length two from vertex V2 to vertex Vi where i = 1, 2, 3, 4, 5, and represent them 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 , and represent 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 ≡ k ≡ l

ax + by cx + dy

(mod n), (mod n).

Hence, solve the system of linear congruences 7x + 3y 2x + 5y

≡ 10 ≡ 9

(mod 16), (mod 16).

[7 marks]

G

6. (a) Find the equation of line l1 , passing through points A and B, where the position vectors of points 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. −→ −−→ ii. prove that CR × AB = a × b + b × c + c × a, iii. deduce the shortest distance of the point C from the line l1 . (c)

[1 marks]

[3 marks]

[3 marks]

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

FURTHER MATHEMATICS

PAPER 2 QUESTIONS

STPM 2009

ii. Deduce the distance between skew lines l2 and l3 , where the equation of line l3 , is given by     3 1 t = −2 + k  1  . −2 5 [4 marks]

AK

7. A local automobile manufacturing company studies the relationship between the age (x years) and the price (y thousand RM) of one of its car models. The data obtained from a random sample of ten cars 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, and interpret the value obtained. [5 marks]

8. A medical researcher wishes to study whether the severity of a certain lung ailment is related to the smoking habit of a patient. A patient who smokes more than 20 cigarettes a day on the average is classified as a heavy smoker. A random sample of 96 patients was taken. It is found that from 43 patients with mild lung ailment, 23 are light smokers while the rest are heavy smokers. The remaining 53 patients with severe lung ailment, 39 patients are heavy smokers. (a) Tabulate the data in an appropriate contingency table.

[1 marks]

N EO

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

9. Tte monthly salary of engineers working in a town has a normal distribution. The mean monthly salary 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 reasons a 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 supervisor at the factory conducts a study on the number of defects found in the electronic components. One thousand electronic components have been selected at random and inspected. The number of defects found and their respective frequency arc given as follows: Number of defects Frequency

0 33

1 145

2 337

3 286

4 174

5 25

G

(a) Test, at 5% and 20% significance levels, whether the binomial distribution with probability of defect 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 tyre manufacturer claims that the mean lifespan µ of its tyre is at least 50 000 km. In order to test this claim, a consumer association takes a random sample of 121 tyres. The mean and standard deviation of 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 tyres is assumed to have 44

FURTHER MATHEMATICS

PAPER 2 QUESTIONS

i. a normal distribution, ii. an unknown distribution.

STPM 2009 [1 marks] [1 marks]

(b) Assuming that the lifespan of the tyres has a normal distribution, state the appropriate hypotheses to test the manufacturer’s claim, and carry out the hypothesis test at the 1% significance level. [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]

AK

12. A study on the relationship between amount of time spent on revision X (in hours) and performance in a final examination Y (scores of 0 to 100) for 10 female undergraduates are summarised as follows: X X X X X x = 725, x2 = 54625, y = 696, y 2 = 49376, xy = 51705 (a) Find the equation of the regression line in the form y = β0 + β1 x, where y is the dependent variable 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 your assumption. [2 marks] (c) Find the coefficient of determination D between the amount of time spent on revision and the performance in a final examination of the undergraduates. Hence, interpret the value of D. [4 marks]

G

N EO 45

FURTHER MATHEMATICS

PAPER 2 QUESTIONS

STPM 2010

STPM 2010 1. Show that 9n − 1 is divisible by 8 for every positive integer n.

2. Find the coordinates of the point P on the line Q(9, 4, −3).

[3 marks]

x y−1 z−3 = = which is closest to the point 5 1 −2 [6 marks]

AK

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



4 −1

and describe the image of the circle x2 + y 2 = 1 under the transformation T. Hence, determine the area of the image.

 2 . Find 2 [5 marks] [2 marks]

4. The transformation M is a rotation 45◦ anticlockwise about the origin. The transformation N is an enlargement with the origin as the centre of enlargement and scale factor k. The transformation NM maps 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]

y−4 z−5 x+7 = = and the plane π has the equation 4x−2y −5z = 8. 1 −3 2

N EO

5. The line l has the equation

(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 points Q(−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.

[2 marks]

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

[4 marks]

G

(a) State, with a reason, whether G is an Eulerian graph.

(c) The subgraph S of G is obtained by removing the vertex g together with all the edges adjacent to it. i. Draw the subgraph S. ii. Determine whether S is a connected graph. iii. Determine whether S is a bipartite graph.

[2 marks] [2 marks]

[3 marks]

7. The mean mark of an English test for a random sample of 50 form five students in a particular state is 47.7. A hypothesis test is to be carried out to determine whether the mean mark for all the form 46

FURTHER MATHEMATICS

PAPER 2 QUESTIONS

STPM 2010

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.

AK N EO

Suggest a value for Pearson correlation coefficient r if the outlier is not taken into account, and comment 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. A random sample of 25 doctors is taken and it is found that the mean and standard deviation are 10.9 hours and 2.3 hours respectively. State the appropriate hypotheses to test the doctor’s claim, and carry out the test at the 5% significance level. [7 marks] 10. A food company carries out a market survey in a state on its new flavoured yoghurt. Three hundred randomly chosen consumers taste the yoghurt. Their responses are shown in the table below. Response Number of consumers

Like 195

Dislike 70

Neutral 35

G

(a) Estimate the proportion of consumers in the state who like the yoghurt. Hence, calculate the probability 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 the yoghurt. [4 marks]

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

47

FURTHER MATHEMATICS

PAPER 2 QUESTIONS Dosage (mg) 3 4 5 6 6 7 8 9

STPM 2010

Duration of relief (hours) 9 10 12 14 16 18 22 24

AK

(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 are the independent and dependent variables respectively. Write down your answers correct to two decimal places. [6 marks] (c) Calculate the coefficient of determination for the regression line, and comment on the adequacy of the straight line fit. [4 marks]

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

0 15

1 22

2 31

3 18

4 8

5 6

N EO

(a) Calculate the expected number of threads with respect to the number of breaks based on a Poisson 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 proposed model in (a). [7 marks]

G 48

FURTHER MATHEMATICS

PAPER 2 QUESTIONS

STPM 2011

STPM 2011 1. When integers a and b are divided by 8, the reminaders are 3 and 7 respectively. Find the remainder when 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]

AK

3. The adjacency matrix of a simple graph  0 0  0  1 1

G is 0 0 1 1 0

0 1 0 1 4m − 3

1 1 1 0 0

 1 0   m2  . 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]

          0 1 5 3 1          −1 0 p 2 + t 1, where p and +s is perpendicular to the plane r = +λ 4. The line r = 2 q 1 1 3 q are constants.

N EO

(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 line and the plane [5 marks]      x x −2 5. The transformation T: R → R is defined by T: 7→ A , where A = y y 3 2

2

 3 . 6

(a) Determine the invariant line under T.

[7 marks]

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

[5 marks]

    x x−y 6. The transformation T: R → R is defined by F = . y x+y 2

2

(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,

[4 marks]

G

i. Describe M and N, 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 mean value of x are defined by θ1 =

kX1 + 2kX2 3

and θ2 = kX1 + (1 − k)X2 ,

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

49

[4 marks]

FURTHER MATHEMATICS

PAPER 2 QUESTIONS

STPM 2011

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

AK

The Pearson correlation coefficient between x and y is 0.901, and the Pearson correlation coefficient between x and ln y is 0.998. Comment on these two values with respect to the scatter diagrams. [4 marks]

N EO

9. A random sample of size n is taken to estimate the mean length of a particular aluminum rod produced by a factory. Assuming that the length of the rod is normally distributed with a standard deviation of 2 mm, determine the smallest value of n so that the width of the confidence interval for the 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 the recommended 25 g of fat intake per day. A random sample of 20 women from the city gives a mean of 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 Pearson correlation 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 two estimates. [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 recorded for 100 hours. The results of the experiment are shown in the following table. 0 2

1 15

2 21

3 16

G

Number of emissions per hour Frequency

4 19

5 12

6 9

7 3

8 3

(a) Calculate the sample mean and an unbiased estimate for the variance of the number of emissions per hour from the radioactive substance. Give a reason why the number of emissions per hour may have a Poisson distribution. [6 marks] (b) Carry out a χ2 goodness-of-fit test, at the 5% significance level, to determine whether the number of emissions per hour has a Poisson distribution. [11 marks]

50