Stpm Past Year Question j1v5i

STPM Mathematics T Past Year Questions Lee Kian Keong & LATEX [email protected] http://www.facebook.com/akeong Last Edited by June 15, 2011 Abstract This is a document which shows all the STPM questions from year 2002 to year 2010 using LATEX. Students should use this document as reference and try all the questions if possible. Students are encourage to 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.

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

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2 3 5 7 9 11 13 15 17 19

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

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21 22 25 27 30 32 34 37 40 43

1 2

[email protected] http://www.facebook.com/akeong

1

PAPER 1 QUESTIONS

1

Lee Kian Keong

PAPER 1 QUESTIONS

2

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2002

STPM 2002 1. The function f is defined by f :x→

√

1 3x + 1, x ∈ R, x ≥ − . 3

Find f −1 and state its domain and range.

2. Given that y = e−x cos x, find

[4 marks]

d2 y dy when x = 0. and dx dx2

[4 marks]

3. Determine the values of a, b, and c so that the matrix   2b − 1 a2 b2  2a − 1 a bc  b b + c 2c − 1 is a symmetric matrix.

[5 marks]

Z 4. By using suitable substitution, find

3x − 1 √ dx. x+1

[5 marks]

5. Determine the set of x such that the geometric series 1 + ex + e2x + . . . converges. Find the exact value of x so that the series converges to 2. [6 marks] q 6. Express

7. Express

√ √ √ 59 − 24 6 as p 2 + q 3 where p, q are integers.

[7 marks]

1 as partial fraction. 4k 2 − 1

[4 marks]

Hence, find a simple expression for Sn =

n X k=1

1 and find lim Sn . n→∞ 4k 2 − 1

[4 marks]

8. Given that P QRS is a parallelogram where P (0, 9), Q(2, −5), R(7, 0) and S(a, b) are points on the plane. Find a and b. [4 marks] Find the shortest distance from P to QR and the area of the parallelogram P QRS. [6 marks] 9. Find the point of intersection of the curves y = −x2 + 3x and y = 2x3 − x2 − 5x. Sketch on the same coordinate system these two curves. [5 marks] 2 3 2 Calculate the area of the region bounded by the curve y = −x + 3x and y = 2x − x − 5x.[6 marks] 

−10 4 10. Matrices M and N are given as M =  15 −4 −5 1

  9 2 −14 , and N =  4 6 1

3 3 2

 4 1  4

Find MN and deduce N−1 . [4 marks] Products X, Y and Z are assembled from three components A, B and C according to different proportions. Each product X consists of two components of A, four components of B, and one component of C; each product of Y consists of three components of A, three components of B, and two components of C; each product of Z consists of four components of A, one component of B, and four components of C. A total of 750 components of A, 1000 components of B, and 500 components

3

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2002

of C are used. With x, y and z representing the number of products of X, Y , and Z assembled, obtain a matrix equation representing the information given. [4 marks] Hence, find the number of products of X, Y , and Z assembled. [4 marks] 11. Show that polynomial 2x3 − 9x2 + 3x + 4 has x − 1 as factor. Hence, (a) find all the real roots of 2x6 − 9x4 + 3x2 + 4 = 0. 3

2

(b) determine the set of values of x so that 2x − 9x + 3x + 4 < 12 − 12x.

[2 marks]

[5 marks] [6 marks]

12. Function f is defined by f (x) =

2x . (x + 1)(x − 2)

Show that f 0 (x) < 0 for all values of x in the domain of f . [5 marks] Sketch the graph of y = f (x). Determine if f is a one to one function. Give reasons to your answer. [6 marks]

Sketch the graph of y = |f (x)|. Explain how the number of the roots of the equation |f (x)| = k(x−2) depends on k. [4 marks]

4

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2003

STPM 2003 1. Show that −1 is the only one real root of the equation x3 + 3x2 + 5x + 3 = 0.

[5 marks]

dy √ 2. If y = ln xy, find the value of when y = 1. dx

[5 marks]

Z

π 6

3. Using the substitution u = 3 + 2 sin θ, evaluate 0

cos θ dθ. (3 + 2 sin θ)2

[5 marks]

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

[6 marks]

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

[7 marks]

6. The function f is defined by  x  x<1 1 + e , f (x) = 3, x=1   2 + e − x, x > 1 (a) Find lim f (x) and lim f (x). Hence, determine whether f is continuous at x = 1.

[4 marks]

(b) Sketch the graph of f .

[3 marks]

x→1−

x→1+

7. The straight line l1 which es through the points A(4, 0) and B(2, 4) intersects the y-axis at the point P . The straight line l2 is perpendicular to l1 and es through√B. If l2 intersects the x-axis and y-axis at the points Q and R respectively, show that P R : QR = 5 : 3. [8 marks] 1 1+x 2 8. Express as a series of ascending powers of x up to the term in x3 . 1 + 2x √ 1 , find 62 correct to four decimal places. By taking x = 30



1 9. The matrix A is given by A =  3 0

2 1 1

[6 marks] [3 marks]

 −3 1  −2

(a) Find the matrix B such that B = A2 − 10I, where I is the 3 × 3 identity matrix. 21

(b) Find (A + I)B, and hence find (A + I) B.

[3 marks] [6 marks]

a 10. The curve y = x(b − x), where a 6= 0, has a turning point at point (2, 1). Determine the values of 2 a and b. [4 marks] Calculate the area of the region bounded by the x-axis and the curve. [4 marks] Calculate the volume of the solid formed by revolving the region about the x-axis. [4 marks]

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

2 . Show that the equation 1+x

(1 + x)ex − 2 = 0 has a root in the interval [0, 1]. [7 marks] Use the Newton-Raphson method with the initial estimate x0 = 0.5 to estimate the root correct to three decimal places. [6 marks] 5

PAPER 1 QUESTIONS

Lee Kian Keong

2 in partial fractions. + 2r Using the result obtained,

12. Express ur =

STPM 2003

[3 marks]

r2

1 1 1 1 , [2 marks] (a) show that u2r = − + 2 + + r r r + 2 (r + 2)2 n ∞ ∞ X X X 3 1 1 1 (b) show that ur = − − and determine the values of ur and ur+1 + r . 2 n+1 n+2 3 r=1 r=1 r=1 [9 marks]

6

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2004

STPM 2004 Z

e

ln x dx = 1.

1. Show that

[4 marks]

1

1

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

√

7 correct [5 marks]

3. Using the laws of the algebra of sets, show that, for any sets A and B, (A − B) ∪ (B − A) = (A ∪ B) − (A ∩ B) [6 marks]



3 4. Matrix A is given by A =  5 1

3 4 2

 4 1 . 3

Find the adt of A. Hence, find A−1 .

[6 marks]

5. The function f is defined by  x − 1, 0≤x<2 f (x) = x + 2 ax2 − 1, x ≥ 2 where a ∈ R. Find the value of a if lim f (x) exists. With this value of a, determine whether f is x→2 continuous at x = 2. [6 marks] 6. The sum of the distance of the point P from the point (4,0) and the distance of P from the origin is (x − 2)2 y2 8 units. Show that the locus of P is the ellipse + = 1 and sketch the ellipse. [7 marks] 16 12 1 7. Sketch, on the same coordinate axes, the graphs of y = 2 − x and y = 2 + . x 1 Hence, solve the inequality 2 − x > 2 + . x

[4 marks]

[4 marks]

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

Use the Newton-Raphson method to find the real root correct to three decimal places. 9. The matrices P and Q, where PQ = QP, are given by    2 −2 0 −1 P =  0 0 2  and Q =  0 a b c 0

1 0 −2

[5 marks]

 0 −1  2

Determine the values of a, b and c. [5 marks] Find the real numbers m and n for which P = mQ + nI, where I is the 3 × 3 identity matrix.[5 marks] 2 10. A curve is defined by the parametric equations x = 1 − 2t, y = −2 + . Find the equation of the t 7

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2004

normal to the curve at the point A(3, −4). [7 marks] The normal to the curve at the point A cuts the curve again at point B. Find the coordinates of B. [4 marks]

1 11. Sketch on the same coordinates axes, the line y = x and the curve y 2 = x. Find the coordinates of 2 the points of intersection. [5 marks] 1 2 [4 marks] Find the area of region bounded by the line y = x and the curve y = x. 2 Find the volume of the solid formed when the region is rotated through 2π radians about the y-axis. [4 marks]

12. Prove that the sum of the first n of a geometric series a + ar + ar2 + . . . is

a(1 − rn ) [3 marks] 1−r

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

3 (b) The sum of the first n and the sum to infinity of the geometric series 6 − 3 + − . . . are 2 Sn and S∞ respectively. Determine the smallest value of n such that |Sn − S∞ | < 0.001.[7 marks]

8

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2005

STPM 2005 1. Using the laws of the algebra of sets, show that (A ∩ B)0 − (A0 ∩ B) = B 0 [4 marks]

2. If y =

cos x dy d2 y , where x 6= 0, show that x 2 + 2 + xy = 0. x dx dx

[4 marks]

3. The point R divides the line ing the points P (3, 2) and Q(5, 8) in the ratio 3 : 4. Find the equation of the line ing through R and perpendicular to P Q. [5 marks] 4. For the geometric series 7 + 3.5 + 1.75 + 0.875 + ..., find the smallest value of n for which the difference between the sum of the first n and the sum to infinity is less than 0.01. [6 marks]

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

1 where x 6= 0. x

[7 marks]

6. Find the perpendicular distance from the centre of the circle x2 + y 2 − 8x + 2y + 8 = 0 to the straight line 3x + 4y = 28. Hence, find the shortest distance between the circle and the straight line. [7 marks] 7. Sketch, on the same coordinate axes, the curves y = ex and y = 2 + 3e−x . Calculate the area of the region bounded by the y-axis and the curves. 8. A, B and C B2 = C.  1 If B =  0 1

[2 marks] [6 marks]

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

2 −1 0

 0 0 , find C and A. 1

[7 marks]

√ 9. The complex numbers z1 and z2 satisfy the equation z 2 = 2 − 2 3i. (a) Express z1 and z2 in the form a + bi, where a and b are real numbers.

[6 marks]

(b) Represent z1 and z2 in an Argand diagram.

[1 marks]

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

[4 marks]

10. The functions f and g are given by f (x) =

ex − e−x ex + e−x

and g(x) =

ex

2 + e−x

(a) State the domains of f and g,

[1 marks]

(b) Without using differentiation, find the range of f ,

[4 marks]

2

2

(c) Show that f (x) + g(x) = 1. Hence, find the range of g.

11. Express f (x) =

x2 − x − 1 in partial fractions. (x + 2)(x − 3)

Hence, obtain an expansion of f (x) in ascending powers of 9

[6 marks]

[5 marks]

1 1 up to the term in 3 . x x

[6 marks]

PAPER 1 QUESTIONS

Lee Kian Keong

Determine the set of values of x for which this expansion is valid.

STPM 2005 [2 marks]

1 12. Find the coordinates of the stationary point on the curve y = x2 + where x > 0; give the xx coordinate and y-coordinate correct to three decimal places. Determine whether the stationary point is a minimum point or a maximum point. [5 marks] 1 1 The x-coordinate of the point of intersection of the curves y = x2 + and y = 2 , where x > 0, is x x p. Show that 0.5 < p < 1. Use the Newton-Raphson method to determine the value of p correct to three decimal places and, hence, find the point of intersection. [9 marks]

10

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2006

STPM 2006 1. If A, B and C are arbitrary sets, show that [(A ∪ B) − (B ∪ C)] ∩ (A ∪ C)0 = ∅.

[4 marks]

2. If x is so small that x2 and higher powers of x may be neglected, show that x 10 (1 − x) 2 + ≈ 29 (2 − 7x). 2 [4 marks]



k 3. Determine the values of k such that the determinant of the matrix  2k + 1 0 Z 4. Using trapezium rule, with five ordinates, evaluate

1

p

1 −3 k

 3 2  is 0.[4 marks] 2

4 − x2 dx.

[4 marks]

0

5. If y = x ln(x + 1), find an approximation for the increase in y when x increases by δx. Hence, estimate the value of ln 2.01 given that ln 2 = 0.6931. Ax + B C 2x + 1 in the form 2 + where A, B and C are constants. (x2 + 1)(2 − x) x +1 2−x Z 1 2x + 1 dx. Hence, evaluate 2 + 1)(2 − x) (x 0

6. Express

[6 marks]

[3 marks]

[4 marks]

10−Tn 5 7. The nth term of an arithmetic progression is Tn , show that Un = (−2)2( 17 ) is the nth term of 2 a geometric progression. [4 marks] ∞ X 1 If Tn = (17n − 14), evaluate Un . [4 marks] 2 n=1

8. p Show that x2 + y 2 − 2ax − 2by + c = 0 is the equation of the circle with centre (a, b) and radius a2 + b2 − c. [3 marks] C3 C1

C2

The above figure shows three circles C1 , C2 and C3 touching one another, where their centres lie on a straight line. If C1 and C2 have equations x2 + y 2 − 10x − 4y + 28 = 0 and x2 + y 2 − 16x + 4y + 52 = 0 respectively. Find the equation of C3 . [7 marks]

11

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2006

9. Functions f , g and h are defined by f :x→

x ; x+1

g:x→

x+2 ; x

h:x→3+

2 x

(a) State the domains of f and g.

[2 marks]

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

[5 marks]

(c) State the domain and range of h.

[2 marks]

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

[2 marks]

10. The polynomial p(x) = x4 + ax3 − 7x2 − 4ax + b has a factor x + 3 and when divided by x − 3, has remainder 60. Find the values of a and b and factorise p(x) completely. [9 marks] 1 [3 marks] Using the substitution y = , solve the equation 12y 4 − 8y 3 − 7y 2 + 2y + 1 = 0. x 

   5 2 3 a 1 −18 −1 12  and PQ = 2I, where I is the 3 × 3 identity 11. If P =  1 −4 3 , Q =  b 3 1 2 −13 −1 c matrix, determine the values of a, b and c. Hence find P−1 . [8 marks] Two groups of workers have their drinks at a stall. The first group comprising ten workers have five cups of tea, two cups of coffee and three glasses of fruit juice at a total cost of RM11.80. The second group of six workers have three cups of tea, a cup of coffee and two glasses of fruit juice at a total cost of RM7.10. The cost of a cup of tea and three glasses of fruit juice is the same as the cost of four cups of coffee. If the costs of a cup of tea, a cup of coffee and a glass of fruit juice are RM x, RM y and RM z respectively, obtain a matrix equation to represent the above information. Hence determine the cost of each drink. [6 marks]

12. The function f is defined by f (t) =

4ekt − 1 where k is a positive constant, t > 0, 4ekt + 1

(a) Find the value of f (0)

[1 marks]

0

(b) Show that f (t) > 0

[5 marks] 2

0

00

(c) Show that k[1 − f (t) ] = 2f (t) and, hence, show that f (t) < 0.

[6 marks]

(d) Find lim f (t).

[2 marks]

(e) Sketch the graph of f .

[2 marks]

t→∞

12

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2007

STPM 2007 1. Express the infinite recurring decimal 0.72˙ 5˙ (= 0.7252525 . . . ) as a fraction in its lowest .[4 marks]

2. If y =

x dy , show that x2 = (1 − x2 )y 2 . 2 1+x dx

3. If loga

x = 3 loga 2 − loga (x − 2a), express x in of a. a2

[[ marks]4

[6 marks]

4. Simplify √ √ ( 7 − 3)2 √ √ , (a) 2( 7 + 3) (b)

[3 marks]

√ 2(1 + 3i) −1. , where i = (1 − 3i)2

[3 marks]



x y 5. The coordinates of the points P and Q are (x, y) and , respectively, where x 6= 0 x2 + y 2 x2 + y 2 and y 6= 0. If Q moves on a circle with centre (1, 1) and radius 3, show that the locus of P is also a circle. Find the coordinates of the centre and radius of the circle. [6 marks] 6. Find x2 + x + 2 dx, x2 + 2 Z x (b) dx. ex+1 Z

(a)

[3 marks]

[4 marks]

7. Find the constants A, B, C and D such that 3x2 + 5x A B C D = + + + . (1 − x2 )(1 + x)2 1 − x 1 + x (1 + x)2 (1 + x)3 [8 marks]

8. The function f is defined by

(√ f (x) =

(a) Find

lim f (x),

x→−1−

x + 1, −1 ≤ x < 1, |x| − 1, otherwise.

lim f (x), lim− f (x) and lim+ f (x).

x→−1+

x→1

[4 marks]

x→1

(b) Determine whether f is continuous at x = −1 and x = 1. 9. The matrices A and B are given by  −1 A =  −3 0

  2 1 −35 19 1 4  , B =  −27 −13 1 2 −3 12

Find the matrix A2 B and deduce the inverse of A.

13

[4 marks]

 18 45  . 5 [5 marks]

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2007

Hence, solve the system of linear equations x 3x

− 2y − y y

− z − 4z + 2z

= −8, = −15, = 4. [5 marks]

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

dy 3x − 5 = √ , where x > 0. If dx 2 x

the curve es through the point (1, −4). (a) find the equation of the curve,

[4 marks]

(b) sketch the curve,

[2 marks]

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

[5 marks]

1 1 4 11. Using the substitution y = x + , express f (x) = x3 − 4x − 6 − + 3 as a polynomial in y. [3 marks] x x x Hence, find all the real roots of the equation f (x) = 0. [10 marks]

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

x3 and determine their nature. −1

x2

[10 marks]

Sketch the curve. [4 marks] 3 2 Determine the number of real roots of the equation x = k(x − 1), where k ∈ R, when k varies. [3 marks]

14

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2008

STPM 2008 1. The function f and g are defined by f :x→

1 , x ∈ R \ {0}; x

g : x → 2x − 1, x ∈ R Find f ◦ g and its domain. Z 2. Show that 2

3

[4 marks]

(x − 2)2 5 dx = + 4 ln x2 3

2 . 3

[4 marks]

3. Using definitions, show that, for any sets A, B and C, A ∩ (B ∪ C) ⊂ (A ∩ B) ∪ (A ∩ C) [5 marks]

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

1 . 1−z

[6 marks]

1 5. The polynomial p(x) = 2x3 + 4x2 + x − k has factor (x + 1). 2 (a) Find the value of k.

[2 marks]

(b) Factorise p(x) completely.

[4 marks]

6. If y =

sin x − cos x d2 y dy , show that = 2y . sin x + cos x dx2 dx 

1 7. Matrix A is given by A =  1 1

[6 marks]

 0 0 −1 0 . −2 1

(a) Show that A2 = I, where I is the 3 × 3 identity  1 4 (b) Find matrix B which satisfies BA =  0 2 −1 0

matrix, and deduce A−1 .  3 1 . 2

[4 marks]

[4 marks]

8. The lines y = 2x and y = x intersect the curve y 2 + 7xy = 18 at points A and B respectively, where A and B lie in the first quadrant. (a) Find the coordinates of A and B.

[4 marks]

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

[2 marks]

(c) Find the area of the OAB triangle.

[3 marks]

4 > 3 − 3. 9. Find the solution set of the inequality x − 1 x

10. Show that the gradient of the curve y =

x is always decreasing. x2 − 1 15

[10 marks]

[3 marks]

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2008

Determine the coordinates of the point of inflexion of the curve, and state the intervals for which the curve is concave upwards. [5 marks] Sketch the curve. [3 marks] 11. Sketch, on the same coordinate axes, the curves y = 6 − ex and y = 5e−x , and find the coordinates of the points of intersection. [7 marks] Calculate the area of the region bounded by the curves. [4 marks] Calculate the volume of the solid formed when the region is rotated through 2π radians about the x-axis. [5 marks] 12. At the beginning of this year, Mr. Liu and Miss Dora deposited RM10 000 and RM2000 respectively in a bank. They receive an interest of 4% per annum. Mr Liu does not make any additional deposit nor withdrawal, whereas, Miss Dora continues to deposit RM2000 at the beginning of each of the subsequent years without any withdrawal. (a) Calculate the total savings of Mr. Liu at the end of n-th year.

[3 marks]

(b) Calculate the total savings of Miss Dora at the end of n-th year.

[7 marks]

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

16

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2009

STPM 2009 1. Determine the set of values of x satisfying the inequality

2. Given x > 0 and f (x) =

√

x 1 ≥ . x+1 x+1

[4 marks]

f (x) − f (x + h) . h→0 h

x, find lim

[4 marks]

3 + . . ., obtain the smallest value of n if the difference between the 2 45 sum of the first n + 4 and the sum of the first n is less than . [6 marks] 64

3. For the geometric series 6 + 3 +

4. The line y + x + 3 = 0 is a tangent to the curve y = px2 + qx, where p 6= 0 at the point x = −1. Find the values of p and q. [6 marks] 5. Given that loga (3x − 4a) + loga 3x = where 0 < a <

2 + loga (1 − 2a), log2 a

1 , find x. 2

[7 marks]

1

Z

1

x2 (1 − x) 3 dx.

6. Using an appropriate substitution, evaluate

[7 marks]

0

7. The parametric equations of a straight line l are given by x = 4t − 2 and y = 3 − 3t. 3 [2 marks] (a) Show that the point A(1, ) lies on line l, 4 (b) Find the Cartesian equation of line l, [2 marks] (c) Given that line l cuts the x and y-axes at P and Q respectively, find the ratio P A : AQ.[4 marks] 8. Find the values of x if y = |3 − x| and 4y − (x2 − 9) = −24. 9. (a) The matrices P, Q and  1 5  2 −2 P= 1 −3

[9 marks]

R are given by     6 −13 −50 −33 4 4  , Q =  −1 −6 −5  , R =  1 2 7 20 15 −2

Find the matrices PQ and PQR and hence, deduce (PQ)−1 . (b) Using the result in (a), solve the system of linear equations 6x + x − x +

10y 2y 2y

+ 8z + z + 3z

7 −5 1

 −13 −1  . 11 [5 marks]

= 4500 = 0 = 1080

.

[5 marks]

10. A curve is defined by the parametric equations x=t−

2 t

and y = 2t +

where t 6= 0. 17

1 t

PAPER 1 QUESTIONS

Lee Kian Keong

dy 5 1 dy =2− 2 , and hence, deduce that − < < 2. dx t +2 2 dx dy 1 (b) Find the coordinates of points when = . dx 3 (a) Show that

STPM 2009

[8 marks] [3 marks]

11. Given a curve y = x2 − 4 and straight line y = x − 2, (a) sketch, on the same coordinates axes, the curve and the straight line,

[2 marks]

(b) determine the coordinate of their points of intersection,

[2 marks]

(c) calculate the area of the region R bounded by the curve and the straight line,

[4 marks]

(d) find the volume of the solid formed when R is rotated through 360◦ about the x-axis. [5 marks] 12. The polynomial p(x) = 6x4 − ax3 − bx2 + 28x + 12, where a and b are real constants, has factors (x + 2) and (x − 2). (a) Find the values of a and b, and hence, factorise p(x) completely.

[7 marks]

3

(b) Give that p(x) = (2x − 3)[q(x) − 41 + 3x ], find q(x), and determine its range when x ∈ [−2, 10]. [8 marks]

18

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2010

STPM 2010 1. Solve the following simultaneous equations: log3 (xy) = 5

and

log9

x2 y

= 2. [4 marks]

2. Given that u = show that

1 x −x dy 1 , (e +e ), where x > 0 and y = f (u) is a differentiable function f . If =√ 2 2 du u −1

dy = 1. dx

[5 marks]

3. Determine the set of values of x such that the geometric series e−x + e−2x + e−3x + . . . converges. Find the exact value of x if the sum to infinity of the series is 3. [6 marks]

4. Given that f (x) = x ln x, where x > 0. Find f 0 (x), and hence, determine the value of

Z

2e

ln x dx. e

[6 marks]

5. Let A − B denotes a set of elements which belongs to set A, but does not belong to set B. Without using Venn diagram, show that A − B = A ∩ B 0 . [3 marks] 0 0 Hence, prove that (A ∪ B ) − (B ∩ C) = B ∪ (A − C). [4 marks] 6. The graph of a function f is as follows:

(a) State the domain and range of f .

[2 marks]

(b) State whether f is a one-to-one function or not. Give a reason for your answer.

[2 marks]

(c) Determine whether f is continuous or not at x = −1. Give a reason for your answer. [3 marks] 7. The polynomial p(x) = 2x4 − 7x3 + 5x2 + ax + b, where a and b are real constants, is divisible by 2x2 + x − 1. (a) Find a and b.

[4 marks]

(b) For these values of a and b, determine the set of values of x such that p(x) ≤ 0.

[4 marks]

8. Given f (x) =

x3 − 3x − 4 , (x − 1)(x2 + 1) 19

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2010

B Cx + D + 2 , [5 marks] x−1 x +1 (b) when x is sufficiently small such that x4 and higher powers can be neglected, show that f (x) ≈ 4 + 7x + 3x2 − x3 . [4 marks] (a) find the constants A, B, C and D such that f (x) = A +

9. Sketch, on the same coordinate axes, the graphs y = e−x and y =

4 . Show that the equation 2−x

x + 4ex = 2 has a root in the interval [-1,0]. [6 marks] Estimate the root correct to three decimal places by using Newton-Raphson method with initial estimate x0 = −0.4. [5 marks] 10. A circle C1 es through the points (-6, 0), (2, 0) and (-2, 8). (a) Find the equation of C1 .

[4 marks]

(b) Determine the coordinates of the centre and the radius of C1 . 2

[2 marks]

2

(c) If C2 is the circle (x − 4) + (y − 11) = 25, i. find the distance between the centres of the two circles, ii. find the coordinates of the point of intersection of C1 with C2 .

[2 marks] [3 marks]

11. The functions f and g are defined by f : x → x3 − 3x + 2, x ∈ R. g : x → x − 1, x ∈ R. (a) Find h(x) = (f ◦ g)(x), and determine the coordinates of the stationary points of h.

[5 marks]

(b) Sketch the graph of y = h(x).

[2 marks]

(c) On a separate diagram, sketch the graph of y =

1 . h(x)

Hence, determine the set of values of k such that the equation

[3 marks]

1 = k has h(x)

i. one root, ii. two roots, iii. three roots.  1 12. Matrix P is given by P = 2 2

[1 marks] [1 marks] [1 marks]

2 1 −1

 1 3 . −1

(a) Find the determinant and adt of P. Hence, find P−1 .

[6 marks]

(b) A factory assembles three types of toys Q, R and S. The total time taken to assemble one unit of R and one unit of S exceeds the time taken to assemble two units of Q by 8 minutes. One unit of Q, two units of R and one unit of S take 31 minutes to be assembled. The time taken to assemble two units of Q, one unit of R and three units of S is 48 minutes. If x, y and z represent the time, in minutes, taken to assemble each unit of toys Q, R and S respectively, i. write a system of linear equations to represent the above information, ii. using the results in (a), determine the time taken to assemble each type of toy.

20

[2 marks] [5 marks]

PAPER 2 QUESTIONS

2

Lee Kian Keong

PAPER 2 QUESTIONS

21

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2002

STPM 2002 1. Position vectors of the points P and Q relative to the origin O are 2i and 3i + 4j respectively. Find −−→ −−→ the angle between vector OP and vector OQ. [4 marks] 2. Points A and B are on the side XY of triangle XY Z with XA = AB = BY . Points C and D are on the sides Y Z and XZ respectively such that ABCD is a rhombus. Prove that ∠XZY = 90◦ .[6 marks] 3. The points P , Q, R, S are on the circumference of a circle, such that ∠P QR = 80◦ and ∠RP S = 30◦ as shown in the diagram below. The tangent to the circle at P and the chord RS which is produced, meet at T .

(a) Show that P R = P T . (b) Show that the length of the chord RS is the same as the radius of the circle. √

[3 marks] [4 marks]

π . [4 marks] 2 √ Hence, find the value of x with 0 ≤ x ≤ 2π, which satisfies the inequality 0 < cos x + 3 sin x < 1.

4. Express cos x +

3 sin x in the form r cos(x − α), with r > 0 and 0 < α <

[5 marks]

5. The rate of change of water temperature is described by the differential equation dθ = −k(θ − θs ) dt where θ is the water temperature at time t, θs is the surrounding temperature, and k is a positive constant. A boiling water at 100◦ C is left to cool in a kitchen that has a surrounding temperature of 25◦ C. 3 The water takes 1 hour to decrease to the temperature of 75◦ C. Show that k = ln . [6 marks] 2 When the water reaches 50◦ C, the water is placed in a freezer at −10◦ C to be frozen to ice. Find the time required, from the moment the water is put in the freezer until it becomes ice at 0◦ C. [6 marks] 6. Wind is blowing with a speed of w from the direction of N θ◦ W. When a ship is cruising eastwards with a speed of u, the captain of ship found that the wind seemed to be blowing with a speed of v1 , from the direction N α◦ W. When the ship is cruising north with a speed of u, the captain of the ship, however, found that the wind seemed to be blowing with a speed of v2 from the direction N β ◦ W. (a) Draw the triangles of velocity of both situations. tan α − 1 (b) Show that tan θ = . 1 − cot β (c) Express v22 − v12 in of u, w and θ.

[4 marks] [7 marks] [2 marks]

7. Three balls are selected at random from one blue ball, three red balls and six white balls. Find the probability that all the three balls selected are of the same color. [3 marks] 22

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2002

8. The table below shows the number of defective electronic components per lot for 500 lots that have been tested. Numbers of defective components per lot Relative frequency

0

1

2

3

4

5

6 or more

0.042

0.054

0.392

0.318

0.148

0.014

0.032

(a) State the mode and the median number of defective electronic components per lot.

[2 marks]

(b) For the lots with defective components of more than 5, the mean number of defective components per lot is 6.4. Find the mean number of defective electronic components per lot for the given 500 lots. [2 marks] 9. Two percent of the bulb produced by a factory are not usable. Find the smallest number of bulbs that must be examined so that the probability of obtaining at least one non-usable bulb exceeds 0.5. [6 marks]

10. The number of teenagers, according to age, that patronize a recreation centre for a certain period of time is indicated in the following table. Age in Years 12 13 14 15 16 17 18 -

Number of teenagers 4 10 27 110 212 238 149

[ Age 12 - means age 12 and more but less than 13 years ] (a) Display the above data using histogram.

[3 marks]

(b) Find the median and semi-interquartile range for the age of teenagers who patronize the recreation centre. Give your answer to the nearest months. [7 marks] 11. The mass of yellow water melon produced by a farmer is normally distributed with a mean of 4 kg and a standard deviation of 800 g. The mass of red water melon produced by the farmer is normally distributed with a mean of 6 kg and a standard deviation of 1 kg. (a) Find the probability that the mass of a red water melon, selected at random, is less than 5 kg. Hence, find the probability that a red water melon with mass less than 5 kg has mass less than 4 kg. [5 marks] (b) If Y = M − 2K, where M represents the mass of a red water melon and K the mass of a yellow water melon, determine the mean and variance of Y . Assuming that Y is normally distributes, find the probability that the mass of a red water melon selected at random is more than twice the mass of yellow water melon selected at random.[6 marks] 12. Continuous random variable X is defined in the interval 0 to 4, with  1 − ax, 0 ≤ x ≤ 3, P (X > x) = 1 b − x, 3 < x ≤ 4, 2 with a and b as constants, (a) Show that a =

1 and b = 2. 6

[3 marks]

23

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2002

(b) Find the cumulative distribution function of X and sketch its graph.

[4 marks]

(c) Find the probability density function of X.

[2 marks]

(d) Calculate the mean and standard deviation of X.

[6 marks]

24

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2003

STPM 2003 1. Let u = cos φ i + sin φ j and v = cos θ i + sin θ j, where i and j are perpendicular unit vectors. Show 1 1 that |u − v| = sin (φ − θ). [5 marks] 2 2 2. Vertices B and C of the triangle ABC lie on the circumference of a circle. AB and AC cut the circumference of the circle at X and Y respectively. Show that ∠CBX + ∠CY X = 180◦ . [3 marks] If AB = AC, show that BC is parallel to XY . [3 marks] 3. The diagram below shows two circles ABRP and ABQS which intersect at A and B. P AQ and RAS are straight lines. Prove that the triangles RP B and SQB are similar. [7 marks]

4. A force of magnitude 2p N acts along the line OA and a force of magnitude 10 N acts √ along the line OB. The angle between OA and OB is 120◦ . The resultant force has magnitude 3p N. Calculate the value of p and determine the angle between the resultant force and OA. [8 marks]

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

tan A + tan B . 1 − tan A tan B [3 marks]

π If 2x + y = , show that 4

1 − 2 tan x − tan2 x . 1 + 2 tan x − tan2 x π π √ By substituting x = , show that tan = 2 − 1. 8 8 tan y =

[6 marks]

6. The rate of increase in the number of a species of fish in a lake is described by the differential equation dP = (a − b)P, dt where P is the number of fish at time t weeks, a is the rate of reproduction, and b is the mortality rate, with a and b as constants. (a) Assuming that P = P0 at time t = 0 and a > b, solve the above differential equation and sketch its solution curve. [7 marks] (b) At a certain instant, there is an outbreak of an epidemic of a disease. The epidemic results in no r more offspring of the fish being produced and the fish die at a rate directly proportional to 1 . There are 900 fish before the outbreak of the epidemic and only 400 fish are alive after 6 P weeks. Determine the length of time from the outbreak of the epidemic until all the fish of that species die. [9 marks]

25

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2003

7. The probability of a person allergic to a type of anaesthetic is 0.002. A total of 2000 persons are injected with the anaesthetic. Using a suitable approximate distribution, calculate the probability that more than two persons are allergic to the anaesthetic. [5 marks] 8. Tea bags are labelled as containing 2 g of tea powder. In actual face, the mass of tea powder per bag has mean 2.05 g and standard deviation 0.05 g. Assuming that the mass of tea powder of each bag is normally distributed, calculate the expected number of tea bags which contain 1.95 g to 2.10 g of tea powder in a box of 100 tea bags. [5 marks] 9. A factory has 36 male workers and 64 female workers, with 10 male workers earning less than RM1000.00 a month and 17 female workers earning at least RM1000.00 a month. At the end of the year, workers earning less than RM1000.00 are given a bonus of RM1000.00 whereas the others receive a month’s salary. (a) If two workers are randomly chosen, find the probability that exactly one worker receives a bonus of one month’s salary. [3 marks] (b) If a male worker and a female worker are randomly chosen, find the probability that exactly one worker receives a bonus of one month’s salary. [3 marks] 10. Show that, for the numbers x1 , x2 , x3 , . . . , xn with mean x ¯, X X (x − x ¯ )2 = x2 − n¯ x2 . [2 marks]

The numbers 4, 6, 12, 5, 7, 9, 5, 11, p, q, where p < q, have mean x ¯ = 6.9 and Calculate the values of p and q.

X

2

(x − x ¯) = 102.9. [6 marks]

11. The number of ships which anchor at a port every week for 26 particular weeks are as follows. 32 26

28 27

43 38

21 42

35 18

19 37

25 50

45 46

35 23

32 40

18 20

26 29

30 46

(a) Display the data in a stemplot.

[2 marks]

(b) Find the median and interquartile range.

[4 marks]

(c) Draw a boxplot to represent the data.

[3 marks]

(d) State the shape of the frequency distribution. Give a reason for your answer.

[2 marks]

12. The lifespan of a species of plant is a random variable T (tens of days). The probability density function is given by   1 e− 18 t , t > 0, f (t) = 8 0, otherwise. (a) Find the cumulative distribution function of T and sketch its graph.

[6 marks]

(b) Find the probability, to three decimal places, that a plant of that species randomly chosen has a lifespan of more than 20 days. [3 marks] (c) Calculate the expected lifespan of that species of plant.

26

[5 marks]

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2004

STPM 2004 1. Express cos θ + 3 sin θ in the form r cos(θ − α), where r > 0 and 0◦ < α < 90◦ .

[4 marks]

2. Find all values of x, where 0◦ < x < 360◦ , which satisfy the equation tan x + 4 cot x = 4 sec x.[5 marks] 3. The variables t and x are connected by dx = 2t(x − 1), dt where x 6= 1. Find x in of t if x = 2 when t = 1.

[5 marks]

4. The points S and T are midpoints of the sides AB and AD respectively of a parallelogram ABCD. The lines CS and CT cut the diagonal BD at the points U and V respectively. −−→ −−→ −−→ −−→ −−→ 1 −−→ Show that BU = λBC + λCD and BU = (1 − µ)BC + µCD also where λ and µ are constants. 2 −−→ 1 −−→ [6 marks] Hence, show that BU = BD. 3 Deduce that the lines CS and CT trisect the diagonal BD. [3 marks] 5. The diagram below shows two isosceles triangles ABC and ADE which have bases AB and AD respectively. Each triangle has base angles measuring 75◦ , with BC and DE parallel and equal in length. Show that

(a) ∠DBC = ∠BDE = 90◦ , (b) the triangle ACE is an equilateral triangle, (c) the quadrilateral BCED is a square.

[4 marks] [4 marks] [4 marks]

6. A canal of width 2a has parallel straight banks and the water flows due north. The points A and B are on opposite banks and B is due east of A, with the point O as the midpoint of AB. The x-axis and y-axis are taken in the east and north directions respectively with O as the origin. The speed of the current in the canal, vc , is given by x2 vc = v0 1 − 2 , a where v0 is the speed of the current in the middle of the canal and x is the distance eastwards from the middle of the canal. A swimmer swims from A towards the east at speed vr relative to the current in thecanal. Taking y to denote the distance northwards travelled by the swimmer, show v0 x2 dy = 1− 2 . [3 marks] that dx vr a If the width of the canal is 12 m, the speed of the current in the middle of the canal is 10 m s−1 and the speed of the swimmer is 2 m s−1 relative to the current in the canal, 27

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2004

(a) find the distance of the swimmer from O when he is at the middle of the canal and his distance from B when he reaches the east bank of the canal, [7 marks] (b) sketch the actual path taken by the swimmer.

[3 marks]

7. A type of seed is sold in packets which contain ten seeds each. On the average, it is found that a seed per packet does not germinate. Find the probability that a packet chosen at random contains less than two seeds which do not germinate. [4 marks] 8. The continuous random variable X has the probability density function   4 x2 (3 − x), 0 < x < 3, f (x) = 27 0, otherwise. 3 . X< 2 (b) Find the cumulative distribution function of X. (a) Calculate P

[3 marks] [3 marks]

9. Two transistors are chosen at random from a batch of transistors containing ninety good and ten defective ones. (a) Find the probability that at least one out of the two transistors chosen is defective.

[3 marks]

(b) If at least one out of the two transistors chosen is defective, find the probability that both transistors are defective. [4 marks] 10. The lifespan of an electrical instrument produced by a manufacturer is normally distributed with a mean of 72 months and a standard deviation of 15 months. (a) If the manufacturer guarantees that the lifespan of an electrical instrument is at least 36 months, calculate the percentage of the electrical instruments which have to be replaced free of charge. [4 marks]

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

11. The discrete random variable X has the probability function ( k(4 − x)2 , x = 1, 2, 3, P (X = x) = 0, otherwise where k is a constant. (a) Determine the value of k and tabulate the probability distribution of X.

[3 marks]

(b) Find E(7X − 1) and Var(7X − 1).

[7 marks]

12. The following data show the masses, in kg, of fish caught by 22 fishermen on a particular day. 23 69

48 22

51 42

25 46

39 23

37 52

41 41

38 40

37 59

20 68

88 59

(a) Display the above data in an ordered stemplot.

[2 marks]

(b) Find the mean and standard deviation.

[5 marks]

28

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2004

(c) Find the median and interquartile range.

[4 marks]

(d) Draw a boxplot to represent the above data.

[3 marks]

(e) State whether the mean or the median is more suitable as a representative value of the above data. Justify your answer. [2 marks]

29

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2005

STPM 2005 1. The diagram below shows two intersecting circles AXY B and CBOX, where O is the centre of the circle AXY B. AXC and BY C are straight lines. Show that ∠ABC = ∠BAC. [5 marks]

2. In the triangle ABC, the point P lies on the side AC such that ∠BP C = ∠ABC. Show that the triangles BP C and ABC are similar. [3 marks] If AB = 4 cm, AC = 8 cm and BP = 3 cm, find the area of the triangle BP C. [4 marks]

3. Using the substitution y =

dy 2y v , show that the differential equation + y2 = − may be reduced 2 x dx x

dv v2 = − 2. dx x Hence, find the general solution of the original differential equation.

to

[3 marks] [4 marks]

√ 4. In the tetrahedron ABCD, AB = BC = 10 cm, AC = 8 2 cm, AD = CD = 8 cm and BD = 6 cm. Show that the line from C perpendicular to AB and the line from D perpendicular to AB meet at a point on AB. Hence, calculate the angle between the face ABC and the face ABD. [8 marks] d 2 (ln tan x) = . dx sin 2x Hence, find the solution of the differential equation

5. Show that

(sin 2x) for which y =

[2 marks]

dy = 2y(1 − y) dx

1 1 when x = π. Express y explicitly in of x in your answer. 3 4

[8 marks]

6. The points P and Q lie on the diagonals BD and DF respectively of a regular hexagon ABCDEF such that BP DQ = = k. BD DF −−→ −−→ −−→ −−→ Express and CQ in of k, a and b, where AB = a and BC = b. [7 marks] If the points C, P and Q lie on a straight line, determine the value of k. Hence, find : P Q. [7 marks]

7. The mass of a small loaf of bread produced in a bakery may be modelled by a normal random variable with mean 303 g and standard deviation 4 g. Find the probability that a randomly chosen loaf has a mass between 295 g and 305 g. [3 marks] 8. A four-digit number, in the range 0000 to 9999 inclusive, is formed. Find the probability that (a) the number begins or ends with 0,

[3 marks]

30

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2005

(b) the number contains exactly two non-zero digits.

[3 marks]

9. A computer accessories distributor obtains its supply of diskettes from manufacturers A and B, with 60% of the diskettes from manufacturer A. The diskettes are packed by the manufacturers in packets of tens. The probability that a diskette produced by manufacturer A is defective is 0.05 whereas the probability that a diskette produced by manufacturer B is defective is 0.02. Find the probability that a randomly chosen packet contains exactly one defective diskette. [7 marks] 10. The continuous random variable X has probability density function  1 1   (1 − 2x), −2 ≤ x ≤   25 2 3 1 f (x) = (2x − 1), ≤x≤3   25 2  0, otherwise. (a) Sketch the graph of y = f (x). 13 (b) Given that P (0 ≤ X ≤ k) = , determine the value of k. 100

[2 marks] [6 marks]

11. The probability distribution function of the discrete random variable Y is P (Y = y) =

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

(a) Show that E(Y ) = 67 and find Var(Y ).

[5 marks]

(b) Find P (|Y − E(Y )| ≤ 30).

[4 marks]

12. Overexposure to a certain metal dust at the workplace of a factory is detrimental to the health of its workers. The workplace is considered safe if the level of the metal dust is less than 198 µg m−3 . The level of the metal dust at the workplace is recorded at a particular time of day for a period of 90 consecutive working days. The results are summarised in the table below. Metal dust level (µg m−3 ) 170 - 174 175 - 179 180 - 184 185 - 189 190 - 194 195 - 199 200 - 204

Number of days 8 11 25 22 15 7 2

(a) State what the number 11 in the table means.

[1 marks]

(b) Calculate estimates of the mean and standard deviation of the levels of the metal dust. [5 marks] (c) Plot a cumulative frequency curve of the above data. Hence, estimate the median and the interquartile range. [7 marks] (d) Find the percentage of days for which the workplace is considered unsafe.

31

[3 marks]

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2006

STPM 2006 1. Express 4 sin θ − 3 cos θ in the form R sin(θ − α), where R > 0 and 0◦ < α < 90◦ . Hence, solve the equation 4 sin θ − 3 cos θ = 3 for 0◦ < α < 360◦ . [6 marks]

2. If the angle between the vectors a =

4 1 and b = is 135◦ , find the value of p. 8 p

3. Find the general solution of the differential equation x

dy = y 2 − y − 2. dx

[6 marks]

[6 marks]

4. The points P , Q, and R are the midpoints of the sides BC, CA and AB respectively of the triangle ABC. The lines AP and BQ meet at the point G, where AG = mAP and BG = nBQ. −→ −−→ 1 −→ −→ 1 −−→ 1 −→ 2 (a) Show that AG = mAB + mAC and AG = (1 − n)AB + AC. Deduce that AG = AP and 2 2 2 3 2 BG = BQ. [6 marks] 3 2 [3 marks] (b) Show that CR meets AP and BQ at G, where CG = CR. 3 5. Prove that an exterior angle of a cyclic quadrilateral is equal to the opposite interior angle. [3 marks]

In the above diagram, ABCD is a cyclic quadrilateral. The lines AB and DC extended meet at the point E and the lines AD and BC extended meet at the point F . Show that triangles ADE and CBE are similar. [3 marks] If DA = DE, ∠CF D = α and ∠BEC = 3α, determine the value of α. [4 marks] 6. A particle moves from rest along a horizontal straight line. At time t s, the displacement and velocity of the particle are x m and v ms−1 respectively and its acceleration, in ms−2 , is given by √ dv = sin(πt) − 3 cos(πt). dt Express v and x in of t. [7 marks] Find the velocities of the particle when its acceleration is zero for the first and second times. Find also the distance traveled by the particle between the first and second times its acceleration is zero. [7 marks]

32

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2006

7. Two archers A and B take turns to shoot, with archer A taking the first shot. The probabilities 1 1 of archers A and B hitting the bull’s-eye in each shot are and respectively. Show that the 6 5 1 probability of archer A hitting the bull-eye first is . [4 marks] 2 1 8. The probability that it rains in a certain area is . The probability that an accident occurs at a 5 1 1 particular corner of a road in that area is if it rains and if it does not rain. Find the probability 20 50 that it rains if an accident occurs at the corner. [4 marks] 9. The independent Poisson random variables X and Y have parameters 0.5 and 3.5 respectively. The random variable W is defined by W = X − Y . (a) Find E(W ) and Var(W ).

[4 marks]

(b) Give one reason why W is not a Poisson random variable.

[1 marks]

10. The probability that a heart patient survives after surgery in a country is 0.85. (a) Find the probability that, out of five randomly chosen heard patients undergoing surgery, four survive. [3 marks] (b) Using a suitable approximate distribution, find the probability that more than 160 survive after surgery in a random sample of 200 heart patients. [6 marks] 11. The times taken by 22 students to breakfast are shown in the following table. Time (x minutes) Number of students

2≤x<5 1

5≤x<8 2

8 ≤ x < 11 4

11 ≤ x < 14 8

14 ≤ x < 17 5

17 ≤ x < 20 2

(a) Draw a histogram of the grouped data. Comment on the shape of frequency distribution.[4 marks] (b) Calculate estimates of the mean, median, and mode of the breakfast times. Use your calculations to justify your statement about the shape of the frequency distribution. [7 marks] 12. The continuous random variable X has probability density function r  x−1 , 1 ≤ x ≤ b, f (x) = 12  0, otherwise, where b is a constant. (a) Determine the value of b.

[4 marks]

(b) Find the cumulative distribution function of X and sketch its graph.

[5 marks]

(c) Calculate E(X).

[6 marks]

33

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2007

STPM 2007 1. Find, in of π, all the values of x between 0 and π which satisfies the equation tan x + cot x = 8 cos 2x. [4 marks]

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

k tan α cos θ . tan β sin(θ − φ) [5 marks]

3. The position vectors of the points A, B, C and D,relative to an origin, are i+3j, −5i−3j, (x−3)i−6j and (x + 3)i respectively. (a) Show that, for any value of x, ABCD is a parallelogram.

[3 marks]

(b) Determine the value of x for which ABCD is a rectangle.

[4 marks]

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

Q

B

A

P O −−→ 1 2 (a) show that P Q = − a + b, 15 5 (b) find the position vector of R, relative to O, in of b.

[5 marks] [5 marks]

5. The diagram below shows two intersecting circles AP Q and BP Q, where AP B is a straight line. The tangents at the points A and B meet at a point C. SHow that ACBQ is a cyclic quadrilateral. [4 marks]

34

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2007

C

B

P

A

Q

If the lines AQ and CB are parallel and T is the point of intersection of AB and CQ, show that the triangles AT Q and BT C are isosceles triangles. Hence, show that the areas of the triangles AT Q and BT C are in the ratio AT 2 : BT 2 . [7 marks] 6. The variables x and y, where x > 0, satisfy the differential equation x2

dy = y 2 − xy. dx

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

du = u2 − 2u. dx

Hence, show that the general solution of the given differential equation may be expressed in the form 2x y= , where A is an arbitrary constant. [10 marks] 1 + Ax2 Find the equation of the solution curve which es through the point (1,4) and sketch this solution curve. [4 marks] 7. There are eight parking bays in a row at a taxi stand. If one blue taxi, two red taxis and five yellow taxis are parked there, find the probability that two red taxis are parked next to each other.[3 marks] [Assume that a taxi may be parked at any of the parking bays.] 8. The mean mark for a group of students taking a statistics test is 70.6. The mean marks for male and female students are 68.5 and 72.0 respectively. Find the ratio of the number of male to female students. [4 marks] 9. The random variable X is normally distributed with mean µ and standard deviation 100. It is known that P (X > 1169) ≤ 0.117 and P (X > 879) ≥ 0.877. Determine the range of the values of µ.[7 marks]

10. Two events A and B are such that P (A) =

3 1 1 , P (B) = and P (A|B) = . 8 4 6

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

[2 marks]

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

[3 marks]

35

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2007

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

[4 marks]

11. The probability that a lemon sold in a fruit store is rotten is 0.02. (a) If the lemons in the fruit store are packed in packets, determine the maximum number of lemons per packet so that the probability that a packet chosen at random does not contain rotten lemons is more than 0.85. [5 marks] (b) If the lemons in the fruit store are packed in boxed each containing 60 lemons, find, using a suitable approximation, the probability that a box chosen at random contains less than three rotten lemons. [5 marks] 12. The masses (in thousands of kg) of solid waste collected from a town for 25 consecutive days are as follows: 41 48

53 33

44 46

55 55

48 49

57 50

50 52

38 47

53 39

50 51

43 49

56 52

51

(a) Construct a stemplot to represent the data.

[2 marks]

(b) Find the median and interquartile range.

[4 marks]

(c) Calculate the mean and standard deviation.

[5 marks]

(d) Draw a boxplot to represent the data.

[3 marks]

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

[2 marks]

36

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2008

STPM 2008 1. Show that the substitution u = x2 + y transforms the differential equation (1 − x)

dy + 2y + 2x = 0 dx

into the differential equation (1 − x)

du = −2u. dx [3 marks]

2. In triangle ABC, the point X divides BC internally in the ratio m : n, where m + n = 1. Express AX 2 in of AB, BC, CA, m and n.

[5 marks]

1 − t2 2t θ and cos θ = . [4 marks] 3. If t = tan , show that sin θ = 2 1 + t2 1 + t2 Hence, find the values of θ between 0◦ and 360◦ that satisfy the equation 10 sin θ −5 cos θ = 2.[3 marks] 4. The diagram below shows the circumscribed circle of the triangle ABC. B

R

C

Q P

T A The tangent to the circle at A meets the line BC extended to T . The angle bisector of the angle AT B cuts AC at P , AB at Q and the circle at R. Show that (a) triangles AP T and BQT are similar,

[4 marks]

(b) P T · BT = QT · AT ,

[2 marks]

(c) AP = AQ.

[4 marks]

5. The position vectors of the points A, B and C, with respect to the origin O, are a, b and c respectively. The points L, M , P and Q are the midpoints of OA, BC, OB, and AC respectively. 1 1 (a) Show that the position vector of any point on the line LM is a + λ(b + c − a) for some scalar 2 2 λ, and express the position vector of any point on the line P Q in of a, b and c. [6 marks] (b) Find the position vector of the point of intersection of the line LM and the line P Q. [4 marks] 6. A 50 litre tank is initially filled with 10 litres of brine solution containing 20 kg of salt. Starting from time t = 0, distilled water is poured into the tank at a constant √ rate of 4 litres per minute. At the same time, the mixture leaves the tank at a constant rate of k litres per minute, where k > 0. The time taken for overflow to occur is 20 minutes.

37

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2008

(a) Let Q be the amount of salt in the tank at time t minutes. Show that the rate of change of Q is given by √ dQ Q k √ . =− dt 10 + (4 − k)t Hence, express Q in of t.

[7 marks]

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

(c) Sketch the graph of Q against t for 0 ≤ t ≤ 20.

[2 marks]

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

8. The random variable X has a binomial distribution with parameters n = 500 and p = suitable approximate distribution, find P (|X − E(X)| ≤ 25).

1 . Using a 2 [6 marks]

9. In a basket of mangoes and papayas, 70% of mangoes and 60% of papayas are ripe. If 40% of the fruits in the basket are mangoes, (a) find the percentage of the fruits which are ripe,

[3 marks]

(b) find the percentage of the ripe fruits which are mangoes.

[4 marks]

10. A sample of 100 fuses, nominally rated at 13 amperes, are tested by ing increasing electric current through them. The current at which they blow are recorded and the following cumulative frequency table is obtained. Currents (amperes) <10 <11 <12 <13 <14 <15 <16 <17

Cumulative frequency 0 8 30 63 88 97 99 100

Calculate the estimates of the mean, median and mode. Comment on the distribution.

[8 marks]

11. The continuous random variable X has probability density function   0, x < 0,   5 f (x) = 4 − x, 0 ≤ x < 1,     1 , x ≥ 1. 4x2 (a) Find the cumulative distribution function of X.

[7 marks]

(b) Calculate the probability that at least one of two independent observed values of X is greater than three. [4 marks] 12. A car rental shop has four cars to be rented out on a daily basis at RM50.00 per car. The average daily demand for cars is four. 38

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2008

(a) Find the probability that, on a particular day, i. no cars are requested, ii. at least four requests for cars are received. (b) Calculate the expected daily income received from the rentals.

[2 marks] [2 marks] [5 marks]

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

39

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2009

STPM 2009 1. Find the values of x, where 0 ≤ x ≤ π, which satisfy the equation sin3 x sec x = 2 tan x.

[4 marks]

2. The circumscribed circle of the triangle JKL is shown in the diagram below.

The tangent to the circle at J meets the line KL extended to T . The angle bisector of the angle JT K cuts JL and JK at U and V respectively. Show that JV = JU . [4 marks] 3. Find the particular solution of the differential equation ex

dy − y 2 (x + 1) = 0 dx

for which y = 1 when x = 0. Hence, express y in of x.

[7 marks]

4. A boat is travelling at a speed of 30 knots. A yacht is sailing northwards at a speed of 10 knots. At 1300 hours, the boat is 14 nautical miles to the north-east of the yacht. (a) Determine the direction in which the boat should be travelling in order to intercept the yacht. [3 marks]

(b) At what time does the interception occur?

[4 marks]

5. A parallelogram ABCD with its diagonals meeting at the point O is shown in the diagram below.

AB is extended to P such that BP = AB. The line that es through D and is parallel to AC meets P C produced at point R and ∠CRD = 90◦ . (a) Show that the triangles ABD and BP C are congruent.

[3 marks]

(b) Show that ABCD is a rhombus.

[6 marks]

(c) Find the ratio CR : P C.

[3 marks]

40

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2009

6. One of the rules at a training camp of 1000 occupants states that camp activities are to be suspended if 10% of the occupants are infected with a virus. A trainee infected with a flu virus enrolls in the camp causing an outbreak of flu. The rate of increase of the number of infected occupants x at t days is given by differential equation dx = kx(1000 − x), dt where k is a constant. Assume that the outbreak of flu begins at the time the infected trainee enrolls and no one leaves the camp during the outbreak, 1000e1000kt , [9 marks] 999 + e1000kt (b) Determine the value of k if it is found that, after one day, there are five infected occupants, (a) Show that x =

[3 marks]

(c) Determine the number of days before the camp activities will be suspended.

[4 marks]

7. There are 20 doctors and 15 engineers attending a conference. The number of women doctors and that of women engineers are 12 and 5 respectively. Four participants from this group are selected randomly to chair some sessions of discussion. (a) Find the probability that three doctors are selected.

[2 marks]

(b) Given that two women are selected, find the probability that both of them are doctors.[2 marks] [4 marks] 8. The mean and standard deviation of Physics marks for 25 school candidates and 5 private candidates are shown in the table below. Number of candidates Mean Standard deviation

School candidates 25 55 4

Private candidates 5 40 5

Calculate the overall mean and standard deviation of the Physics marks.

[5 marks]

9. A discrete random variable X takes the values of 0, 1 and 2 with the probabilities of a, b and c 5 4 [6 marks] respectively. Given that E(X) = and Var(X) = , find the values of a, b and c. 3 9 10. The independent random variable Yi , where i = 1, 2, . . . , n, takes the values of 0 and 1 with the probabilities of q and p respectively, where q = 1 − p. (a) Show that E(Yi ) = p and Var(Yi ) = pq.

[3 marks]

(b) If X = Y1 + Y2 + . . . + Yn , determine E(X) and Var(X). Comment on the distribution of X. [5 marks]

11. The number of hours spent in a library per week by arts and science students in a college is normally distributed with mean 12 hours and standard deviation 5 hours for arts students, and mean 15 hours and standard deviation 4 hours for science students. A random sample of four arts students and six science students is chosen. Assuming that X is the mean number of hours spent by these 10 students in a week, (a) calculate E(X) and Var(X),

[7 marks]

41

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2009

(b) find the probability that in a given week, the mean number of hours spent by this sample of students is between 11 hours and 15 hours. [3 marks] 12. The time to repair a certain type of machine is a random variable X (in hours). The probability density function is given by   0.01x − p, 10 ≤ x < 20, f (x) = q − 0.01x, 20 ≤ x ≤ 30,   0, otherwise, where p and q are constants. (a) Show that p = 0.1 and q = 0.3.

[6 marks]

(b) Find the probability that the repair work takes at least 15 hours.

[4 marks]

(c) Determine the expected value of X.

[4 marks]

(d) If the total cost of repair of the machine comprises a surcharge of RM500 and an hourly rate of RM100, express the total cost of repair in of X, and determine the expected total cost of repair. [3 marks]

42

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2010

STPM 2010 1. Find the general solution of the differential equation 1 dy 2 ln x = . x dx cos y [5 marks]

2. Express 5 sin θ + 12 cos θ in the form r sin(θ + α), where r > 0 and 0◦ < α < 90◦ . Hence, find the maximum and minimum values of the expression 1 . 5 sin θ + 12 cos θ + 15 [7 marks]

3. The diagram below shows a circle with centre O and tangents at points H and K meeting at the point P . The diameter AB intersects the chord HK at the point Q. The points P , B, Q, O and A lie on a straight line. A

O

H

Q

K

B

P

Prove that (a) P A + P B = 2P O,

[3 marks]

2

(b) P H = P A ◦ P B.

[4 marks]

4. Using the substitution y = vx, show that the differential equation xy

dy − x2 − y 2 = 0 dx

may be reduced to vx

dv = 1. dx [3 marks]

Hence, find the particular solution that satisfies y = 2 and x = 1.

43

[6 marks]

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2010

5. The point K lies on the side AB of a square ABCD such that AK : KB = 3 : 2. The point L lies on the AD such that BL and CK intersect perpendicularly at the point P . Show that (a) ∠ABL = ∠B ,

[2 marks]

(b) the triangles ABL and BCK are congruent,

[3 marks]

(c) the triangles KP B and BP C are similar,

[2 marks]

(d) KP : P C = 4 : 25.

[4 marks]

6. The position vectors of the points Q and R, relative to the origin O, are q and r respectively. (a) Find an expression for the position vector of the point N which lies on the line QR such that QN : N R = 3 : 2. [2 marks] (b) The point P with position vector p, relative to O, is such that O is the midpoint of the line P N . Prove that 5p + 2q + 3r = 0. [2 marks]

(c) The point L lies on the line RP such that RL : LP = 5 : 3. Show that the points Q, O and L are collinear, and find QO : OL. [4 marks] (d) The point M lies on the line P Q such that M OR is a straight line. Find P M : M Q. [5 marks] 7. The probability that it rains in a day is 0.25, and the probability that a student carries an umbrella is 0.6. The probability that it rains or the student does not carry an umbrella is 0.5. If it rains on a particular day, find the probability that the student does not carry an umbrella. [4 marks] 8. The random variable X is normally distributed with mean 48 and standard deviation 10. Find the least integer k such that P(|X − 48| > k) < 0.3. [5 marks] 9. A discrete random variable X has cumulative distribution function  0, x < 1,    0.6, 1 ≤ x < 3, F (x) =  0.9, 3 ≤ x < 5,    1, x ≥ 5. (a) Construct the probability distribution table for X.

[2 marks]

(b) Find the mean and variance of X.

[4 marks]

10. The lifespan, in months, of a type of bulb is a random variable X. The probability density function is given by   1 xe− x3 , x ≥ 0, f (x) = 9 0, x < 0. (a) Find the cumulative distribution function of X, and hence, sketch the graph.

[6 marks]

(b) Determine the probability that a randomly chosen bulb has a lifespan of more than 9 months. [3 marks]

11. The probability that a chicken egg placed in an incubator fails to hatch is 0.01. (a) Determine the maximum number of eggs that may be placed in the incubator so that the probability that all the eggs hatch is more than 0.75. [6 marks] 44

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2010

(b) Using a suitable approximate distribution, find the probability that more than three out of 200 eggs fail to hatch. [5 marks] 12. Forty soil samples are collected from a certain area and tested for their pH values. The pH values and the number of soil samples tested are given in the table below. pH value 4.0 - 4.5 4.5 - 5.0 5.0 - 5.5 5.5 - 6.0 6.0 - 6.5 6.5 - 7.0 7.0 - 7.5 7.5 - 8.0

Number of soil samples 1 3 5 9 11 7 3 1

(a) Construct the cumulative frequency table for this distribution, and plot the cumulative frequency curve. [3 marks] (b) Using the cumulative frequency curve, estimate the median and semi-interquartile range of the distribution. [4 marks] (c) The pH value of a sample is wrongly recorded as 5.8, while its actual value is 5.0. State whether the wrong pH value affects the median and semi-interquartile range. Justify your answers. [2 marks]

(d) Another four soil samples are collected from the same area, and their pH values are found to be greater than 8.0. i. Out of the 44 samples, find the percentage of samples which have a pH value greater than 7.0. [2 marks] ii. Using the curve in (a), estimate the median of the 44 samples. State a reason why the same curve is used. [2 marks]

45