stpm past year question

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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 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.

Contents

1 PAPER 1 QUESTIONS 2

STPM 2002 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3STPM 2003 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5STPM 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7STPM 2005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9STPM 2006 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11STPM 2007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13STPM 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15STPM 2009 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17STPM 2010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 PAPER 2 QUESTIONS 21

STPM 2002 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22STPM 2003 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25STPM 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27STPM 2005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30STPM 2006 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32STPM 2007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34STPM 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

STPM 2009 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40STPM 2010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

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

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PAPER 1 QUESTIONS Lee Kian Keong

1 PAPER 1 QUESTIONS

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PAPER 1 QUESTIONS Lee Kian Keong STPM 2002

STPM 2002

1. The function f is defined by

f : x 3x + 1, x R, x 13

.

Find f1 and state its domain and range. [4 marks]

2. Given that y = ex cos x, finddy

dxand

d2y

dx2when x = 0. [4 marks]

3. Determine the values ofa, b, and c so that the matrix 2b 1 a2 b22a 1 a bc

b b + c 2c 1

is a symmetric matrix. [5 marks]

4. By using suitable substitution, find

3x 1

x + 1dx. [5 marks]

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

6. Express

59 24

6 as p

2 + q

3 where p, q are integers. [7 marks]

7. Express1

4k2 1 as partial fraction. [4 marks]

Hence, find a simple expression for Sn =n

k=1

1

4k2 1 and find limnSn. [4 marks]

8. Given that PQRS is a parallelogram where P(0, 9), Q(2, 5), R(7, 0) and S(a, b) are points on theplane. Find a and b. [4 marks]

Find the shortest distance from P to QR and the area of the parallelogram PQRS. [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]

Calculate the area of the region bounded by the curve y = x2 + 3x and y = 2x3 x2 5x.[6 marks]

10. Matrices M and N are given as M =

10 4 915 4 14

5 1 6

, and N =

2 3 44 3 1

1 2 4

Find MN and deduce N1. [4 marks]

Products X, Y and Z are assembled from three components A, B and C according to differentproportions. Each product X consists of two components of A, four components of B, and onecomponent of C; each product of Y consists of three components of A, three components of B, andtwo components of C; each product of Z consists of four components of A, one component ofB, and

four components of C. A total of 750 components of A, 1000 components of B, and 500 components

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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. [2 marks]Hence,

(a) find all the real roots of 2x6 9x4 + 3x2 + 4 = 0. [5 marks](b) determine the set of values of x so that 2x3 9x2 + 3x + 4 < 12 12x. [6 marks]

12. Function f is defined by

f(x) =2x

(x + 1)(x 2) .

Show that f(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 ofy = |f(x)|. Explain how the number of the roots of the equation |f(x)| = k(x2)depends on k. [4 marks]

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STPM 2003

1. Show that 1 is the only one real root of the equation x3 + 3x2 + 5x + 3 = 0. [5 marks]

2. Ify = ln xy, find the value of dydx when y = 1. [5 marks]

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

6

0

cos

(3 + 2 sin )2d. [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 ofx such that 2 < x3 2x2 + x 2 < 0. [7 marks]


f(x) =

1 + ex, x < 1

3, x = 1

2 + e x, x > 1(a) Find lim

x1f(x) and lim

x1+f(x). Hence, determine whether f is continuous at x = 1. [4 marks]

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

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

5 : 3. [8 marks]

8. Express

1 + x

1 + 2x

12

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

By taking x =1

30, find

62 correct to four decimal places. [3 marks]

9. The matrix A is given by A =

1 2 33 1 1

0 1 2

(a) Find the matrix B such that B = A2 10I, where I is the 3 3 identity matrix. [3 marks](b) Find (A + I)B, and hence find (A + I)21B. [6 marks]

10. The curve y =a

2x(b x), where a = 0, has a turning point at point (2, 1). Determine the values of

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

1 + x. Show that the equation

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

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12. Express ur =2

r2 + 2rin partial fractions. [3 marks]

Using the result obtained,

(a) show that u2r

=

1

r

+1

r2

+1

r + 2

+1

(r + 2)2

, [2 marks]

(b) show thatn

r=1

ur =3

2 1

n + 1 1

n + 2and determine the values of

r=1

ur andr=1

ur+1 +

1

3r

.

[9 marks]

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STPM 2004

1. Show that

e

1

ln x dx = 1. [4 marks]

2. Expand (1 x) 12 in ascending powers of x up to the term in x3. Hence, find the value of7 correctto five decimal places. [5 marks]

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

(A B) (B A) = (A B) (A B)[6 marks]

4. Matrix A is given by A =

3 3 45 4 1

1 2 3

.

Find the adjoint of A. Hence, find A1. [6 marks]


f(x) =

x 1x + 2

, 0 x < 2ax2 1, x 2

where a R. Find the value of a if limx2

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

continuous at x = 2. [6 marks]

6. The sum of the distance of the point P from the point (4,0) and the distance of P from the origin is8 units. Show that the locus of P is the ellipse

(x 2)216

+y2

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

7. Sketch, on the same coordinate axes, the graphs of y = 2 x and y =2 + 1x

. [4 marks]Hence, solve the inequality 2 x >

2 + 1x. [4 marks]

8. Using the sketch graphs of y = x3 and x + y = 1, show that the equation x3 + x 1 = 0 has only onereal 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. [5 marks]

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

P =

2 2 00 0 2

a b c

and Q =

1 1 00 0 1

0 2 2

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

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

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

. Find the equation of the

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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]

11. Sketch on the same coordinates axes, the line y =1

2x and the curve y2 = x. Find the coordinates of

the points of intersection. [5 marks]

Find the area of region bounded by the line y =1

2x and the curve y2 = x. [4 marks]

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

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

1 r [3 marks]

(a) The sum of the first five terms of a geometric series is 33 and the sum of the first ten terms of

the geometric series is -1023. Find the common ratio and the first term of the geometric series.[5 marks]

(b) The sum of the first n terms and the sum to infinity of the geometric series 6 3 + 32

. . . areSn and S respectively. Determine the smallest value of n such that |Sn S| < 0.001.[7 marks]

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STPM 2005

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

(A B) (A B) = B

[4 marks]

2. Ify =cos x

x, where x = 0, show that x d

2y

dx2+ 2

dy

dx+ xy = 0. [4 marks]

3. The point R divides the line joining the points P(3, 2) and Q(5, 8) in the ratio 3 : 4. Find the equationof the line passing 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 differencebetween the sum of the first n terms and the sum to infinity is less than 0 .01. [6 marks]

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

where x = 0. [7 marks]

6. Find the perpendicular distance from the centre of the circle x2 + y2 8x + 2y + 8 = 0 to the straightline 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 + 3ex. [2 marks]

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

8. A, B and C are square matrices such that BA = B1

and ABC = (AB)1

. Show that A1

=B2 = C. [3 marks]

If B =

1 2 00 1 0

1 0 1

, find C and A. [7 marks]

9. The complex numbers z1 and z2 satisfy the equation z2 = 2 2

3i.

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

(b) Represent z1 and z2 in an Argand diagram. [1 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 exex + ex

and g(x) =2

ex + ex

(a) State the domains off and g, [1 marks]

(b) Without using differentiation, find the range off, [4 marks]

(c) Show that f(x)2 + g(x)2 = 1. Hence, find the range of g. [6 marks]

11. Express f(x) =x2 x 1

(x + 2)(x

3)

in partial fractions. [5 marks]

Hence, obtain an expansion of f(x) in ascending powers of1

xup to the term in

1

x3. [6 marks]

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

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

xwhere x > 0; give the x-

coordinate and y-coordinate correct to three decimal places. Determine whether the stationary pointis a minimum point or a maximum point. [5 marks]

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

xand y =

1

x2, where x > 0, is

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

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STPM 2006

1. IfA, B and C are arbitrary sets, show that [(A B) (B C)] (A C) = . [4 marks]

2. Ifx is so small that x2

and higher powers of x may be neglected, show that

(1 x)

2 +x

2

10 29(2 7x).

[4 marks]

3. Determine the values ofk such that the determinant of the matrix

k 1 32k + 1 3 2

0 k 2

is 0.[4 marks]

4. Using trapezium rule, with five ordinates, evaluate 104 x2 dx. [4 marks]

5. Ify = x ln(x + 1), find an approximation for the increase in y when x increases by x.

Hence, estimate the value of ln 2.01 given that ln 2 = 0.6931. [6 marks]

6. Express2x + 1

(x2 + 1)(2 x) in the formAx + B

x2 + 1+

C

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

Hence, evaluate

10

2x + 1

(x2 + 1)(2 x) dx. [4 marks]

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

(2)2( 10Tn17 ) is the nth term ofa geometric progression. [4 marks]

If Tn =1

2(17n 14), evaluate

n=1

Un. [4 marks]

8. Show that x2 + y2 2ax 2by + c = 0 is the equation of the circle with centre (a, b) and radiusa2 + b2 c. [3 marks]

C1

C2

C3

The above figure shows three circles C1, C2 and C3 touching one another, where their centres lie on astraight line. IfC1 and C2 have equations x

2 + y210x4y +28 = 0 and x2 + y216x + 4y + 52 = 0respectively. Find the equation of C3. [7 marks]

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9. Functions f, g and h are defined by

f : x xx + 1

; g : x x + 2x

; h : x 3 + 2x

(a) State the domains off 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 ofh. [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, hasremainder 60. Find the values of a and b and factorise p(x) completely. [9 marks]

Using the substitution y =1

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

11. If P =

5 2 31 4 33 1 2

, Q = a 1 18b 1 1213 1 c

and PQ = 2I, where I is the 3 3 identitymatrix, determine the values of a, b and c. Hence find P1. [8 marks]

Two groups of workers have their drinks at a stall. The first group comprising ten workers have fivecups of tea, two cups of coffee and three glasses of fruit juice at a total cost of RM11.80. The secondgroup of six workers have three cups of tea, a cup of coffee and two glasses of fruit juice at a totalcost of RM7.10. The cost of a cup of tea and three glasses of fruit juice is the same as the cost offour 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. Hencedetermine the cost of each drink. [6 marks]

12. The function f is defined by f(t) =4ekt 14ekt + 1

where k is a positive constant, t > 0,

(a) Find the value off(0) [1 marks]

(b) Show that f(t) > 0 [5 marks]

(c) Show that k[1 f(t)2] = 2f(t) and, hence, show that f(t) < 0. [6 marks](d) Find lim

tf(t). [2 marks]

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

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STPM 2007

1. Express the infinite recurring decimal 0.725 (= 0.7252525 . . . ) as a fraction in its lowest terms.[4 marks]

2. Ify = x1 + x2

, show that x2 dydx

= (1 x2)y2. [[ marks]4

3. If loga

xa2

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

4. Simplify

(a)(

7 3)22(

7 +

3)

, [3 marks]

(b)2(1 + 3i)

(1

3i)2

, where i =1. [3 marks]

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

x

x2 + y2,

y

x2 + y2

respectively, where x = 0

and y = 0. IfQ moves on a circle with centre (1, 1) and radius 3, show that the locus of P is also acircle. Find the coordinates of the centre and radius of the circle. [6 marks]

6. Find

(a)

x2 + x + 2

x2 + 2dx, [3 marks]

(b) xex+1

dx. [4 marks]

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

3x2 + 5x

(1 x2)(1 + x)2 =A

1 x +B

1 + x+

C

(1 + x)2+

D

(1 + x)3.

[8 marks]


f(x) =

x + 1, 1 x < 1,

|x

| 1, otherwise.

(a) Find limx1

f(x), limx1+

f(x), limx1

f(x) and limx1+

f(x). [4 marks]

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

9. The matrices A and B are given by

A =

1 2 13 1 4

0 1 2

, B =

35 19 1827 13 45

3 12 5

.

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

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Hence, solve the system of linear equations

x 2y z = 8,3x y 4z = 15,

y + 2z = 4.

[5 marks]

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

dx=

3x 52

x, where x > 0. If

the curve passes through the point (1, 4).(a) find the equation of the curve, [4 marks]

(b) sketch the curve, [2 marks]

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

11. Using the substitution y = x + 1x

, express f(x) = x3 4x 6 4x

+ 1x3

as a polynomial in y. [3 marks]

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

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

x2 1 and determine their nature.[10 marks]

Sketch the curve. [4 marks]

Determine the number of real roots of the equation x3 = k(x2 1), where k R, when k varies.[3 marks]

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STPM 2008

1. The function f and g are defined by

f : x

1

x

, x

R

\ {0

};

g : x 2x 1, x RFind f g and its domain. [4 marks]

2. Show that

32

(x 2)2x2

dx =5

3+ 4ln

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. Ifz is a complex number such that |z| = 1, find the real part of 11 z . [6 marks]

5. The polynomial p(x) = 2x3 + 4x2 +1

2x k has factor (x + 1).

(a) Find the value ofk. [2 marks]

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

6. Ify =sin x cos xsin x + cos x

, show thatd2y

dx2= 2y

dy

dx. [6 marks]

7. Matrix A is given by A =

1 0 01 1 0

1 2 1

.

(a) Show that A2 = I, where I is the 3 3 identity matrix, and deduce A1. [4 marks]

(b) Find matrix B which satisfies BA =

1 4 30 2 1

1 0 2

. [4 marks]

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

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

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

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

9. Find the solution set of the inequality

4

x 1

> 3 3x

. [10 marks]

10. Show that the gradient of the curve y = xx2 1 is always decreasing. [3 marks]

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Determine the coordinates of the point of inflexion of the curve, and state the intervals for which thecurve is concave upwards. [5 marks]

Sketch the curve. [3 marks]

11. Sketch, on the same coordinate axes, the curves y = 6 ex and y = 5ex, and find the coordinatesof 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 2 radians about thex-axis. [5 marks]

12. At the beginning of this year, Mr. Liu and Miss Dora deposited RM10 000 and RM2000 respectivelyin a bank. They receive an interest of 4% per annum. Mr Liu does not make any additional depositnor withdrawal, whereas, Miss Dora continues to deposit RM2000 at the beginning of each of thesubsequent 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]

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STPM 2009

1. Determine the set of values of x satisfying the inequalityx

x + 1 1

x + 1. [4 marks]

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

x, find limh0

f(x) f(x + h)h

. [4 marks]

3. For the geometric series 6 + 3 +3

2+ . . ., obtain the smallest value of n if the difference between the

sum of the first n + 4 terms and the sum of the first n terms is less than45

64. [6 marks]

4. The line y + x + 3 = 0 is a tangent to the curve y = px2 + qx, where p = 0 at the point x = 1. Findthe values of p and q. [6 marks]

5. Given that

loga

(3x 4a) + loga 3x =2

log2 a+ log

a(1 2a),

where 0 < a 0. Find f(x), and hence, determine the value of

2ee

ln x dx.

[6 marks]

5. Let A B denotes a set of elements which belongs to set A, but does not belong to set B. Withoutusing Venn diagram, show that A B = A B. [3 marks]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 off. [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 by2x2 + 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) ,

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(a) find the constants A, B, C and D such that f(x) = A +B

x 1 +Cx + D

x2 + 1, [5 marks]

(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]

9. Sketch, on the same coordinate axes, the graphs y = ex and y =4

2 x . Show that the equationx + 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 initialestimate x0 = 0.4. [5 marks]

10. A circle C1 passes 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 marks]

(c) If C2 is the circle (x 4)2

+ (y 11)2

= 25,i. find the distance between the centres of the two circles, [2 marks]

ii. find the coordinates of the point of intersection of C1 with C2. [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 ofy =1

h(x). [3 marks]

Hence, determine the set of values of k such that the equation1

h(x)= k has

i. one root, [1 marks]

ii. two roots, [1 marks]

iii. three roots. [1 marks]

12. Matrix P is given by P =

1 2 12 1 3

2 1 1

.

(a) Find the determinant and adjoint of P. Hence, find P1. [6 marks]

(b) A factory assembles three types of toys Q, R and S. The total time taken to assemble one unitof R and one unit of S exceeds the time taken to assemble two units of Q by 8 minutes. Oneunit of Q, two units of R and one unit of S take 31 minutes to be assembled. The time takento 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 Srespectively,

i. write a system of linear equations to represent the above information, [2 marks]

ii. using the results in (a), determine the time taken to assemble each type of toy. [5 marks]

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PAPER 2 QUESTIONS Lee Kian Keong

2 PAPER 2 QUESTIONS

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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 vectorOP 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 onthe sides Y Z and XZ respectively such that ABCD is a rhombus. Prove that XZ Y = 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. [3 marks]

(b) Show that the length of the chord RS is the same as the radius of the circle. [4 marks]

4. Express cos x +

3sin x in the form r cos(x ), with r > 0 and 0 < < 2

. [4 marks]

Hence, find the value of x with 0 x 2, which satisfies the inequality 0 < cos x +

3sin x < 1.[5 marks]

5. The rate of change of water temperature is described by the differential equation

d

dt= k( s)

where is the water temperature at time t, s is the surrounding temperature, and k is a positiveconstant.

A boiling water at 100C is left to cool in a kitchen that has a surrounding temperature of 25C.

The water takes 1 hour to decrease to the temperature of 75C. Show that k = ln3

2. [6 marks]

When the water reaches 50C, the water is placed in a freezer at 10C to be frozen to ice. Find thetime required, from the moment the water is put in the freezer until it becomes ice at 0C. [6 marks]

6. Wind is blowing with a speed ofw from the direction of N W. When a ship is cruising eastwardswith 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 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

W.

(a) Draw the triangles of velocity of both situations. [4 marks]

(b) Show that tan =tan 11 cot . [7 marks]

(c) Express v22 v21 in terms of u, w and . [2 marks]

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

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8. The table below shows the number of defective electronic components per lot for 500 lots that havebeen tested.

Numbers of defectivecomponents per lot

0 1 2 3 4 5 6 or more

Relative frequency 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 componentsper lot is 6.4. Find the mean number of defective electronic components per lot for the given500 lots. [2 marks]

9. Two percent of the bulb produced by a factory are not usable. Find the smallest number of bulbsthat 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 oftime is indicated in the following table.

Age in Years Number of teenagers12 - 413 - 1014 - 2715 - 11016 - 21217 - 23818 - 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 recre-ation 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 kgand a standard deviation of 800 g. The mass of red water melon produced by the farmer is normallydistributed 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 than4 kg. [5 marks]

(b) IfY = M 2K, where M represents the mass of a red water melon and K the mass of a yellowwater melon, determine the mean and variance of Y.

Assuming that Y is normally distributes, find the probability that the mass of a red water melonselected 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

P(X > x) =

1 ax, 0 x 3,b 1

2x, 3 < x 4,

with a and b as constants,

(a) Show that a =1

6and b = 2. [3 marks]

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(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]

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STPM 2003

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

that1

2|u v| = sin 1

2( ). [5 marks]

2. Vertices B and C of the triangle ABC lie on the circumference of a circle. AB and AC cut thecircumference of the circle at X and Y respectively. Show that CB X +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 andRAS 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 lineOB. The angle between OA and OB is 120. The resultant force has magnitude

3p N. Calculate

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

5. Starting from the formulae for sin(A

+B

) and cos(A

+B

), prove that tan(A

+B

) =

tan A + tan B

1 tan A tan B .[3 marks]

If 2x + y =

4, show that

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

By substituting x =

8, show that tan

8=

2 1. [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

dt

= (a

b)P,

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

(a) Assuming that P = P0 at time t = 0 and a > b, solve the above differential equation and sketchits solution curve. [7 marks]

(b) At a certain instant, there is an outbreak of an epidemic of a disease. The epidemic results inno more offspring of the fish being produced and the fish die at a rate directly proportional to

1

P. There are 900 fish before the outbreak of the epidemic and only 400 fish are alive after 6

weeks. Determine the length of time from the outbreak of the epidemic until all the fish of thatspecies die. [9 marks]

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7. The probability of a person allergic to a type of anaesthetic is 0.002. A total of 2000 persons areinjected with the anaesthetic. Using a suitable approximate distribution, calculate the probabilitythat 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 baghas mean 2.05 g and standard deviation 0.05 g. Assuming that the mass of tea powder of each bagis normally distributed, calculate the expected number of tea bags which contain 1.95 g to 2.10 g oftea 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 thanRM1000.00 a month and 17 female workers earning at least RM1000.00 a month. At the end ofthe year, workers earning less than RM1000.00 are given a bonus of RM1000.00 whereas the othersreceive a months salary.

(a) If two workers are randomly chosen, find the probability that exactly one worker receives abonus of one months salary. [3 marks]

(b) If a male worker and a female worker are randomly chosen, find the probability that exactly oneworker receives a bonus of one months salary. [3 marks]

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

x2 nx2.

[2 marks]

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

(x x)2 = 102.9.Calculate the values of p and q. [6 marks]

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

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

(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 densityfunction is given by

f(t) =

1

8e

18t, t > 0,

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 hasa lifespan of more than 20 days. [3 marks]

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

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STPM 2004

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

2. Find all values ofx, where 0

< x < 360

, which satisfy the equation tan x +4cot x = 4sec x.[5 marks]

3. The variables t and x are connected by

dx

dt= 2t(x 1),

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

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

Show thatBU =

BC +

CD and

BU = (1

)

BC +

1

2

CD also where and are constants.

Hence, show that BU = 13BD . [6 marks]

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 ADrespectively. Each triangle has base angles measuring 75, with BC and DE parallel and equal inlength. Show that

(a) DBC = BDE = 90, [4 marks]

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

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

6. A canal of width 2a has parallel straight banks and the water flows due north. The points A and Bare on opposite banks and B is due east of A, with the point O as the midpoint of AB. The x-axisand y-axis are taken in the east and north directions respectively with O as the origin. The speed ofthe current in the canal, vc, is given by

vc = v0

1 x

2

a2

,

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

thatdy

dx=

v0

vr

1 x

2

a2

. [3 marks]

If the width of the canal is 12 m, the speed of the current in the middle of the canal is 10 m s

1 andthe speed of the swimmer is 2 m s1 relative to the current in the canal,

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(a) find the distance of the swimmer from O when he is at the middle of the canal and his distancefrom 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 aseed per packet does not germinate. Find the probability that a packet chosen at random containsless than two seeds which do not germinate. [4 marks]

8. The continuous random variable X has the probability density function

f(x) =

4

27x2(3 x), 0 < x < 3,

0, otherwise.

(a) Calculate PX 0 and 0 < < 90. Hence, solve theequation 4 sin 3cos = 3 for 0 < < 360. [6 marks]

2. If the angle between the vectors a =

4

8

and b =

1

p

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

3. Find the general solution of the differential equation xdy

dx= y2 y 2. [6 marks]

4. The points P, Q, and R are the midpoints of the sides BC, CA and AB respectively of the triangleABC. The lines AP and BQ meet at the point G, where AG = mAP and BG = nBQ.

(a) Show thatAG =

1

2m

AB +

1

2m

AC and

AG = (1 n)AB + 1

2

AC. Deduce that AG =

2

3AP and

BG =2

3 BQ. [6 marks]

(b) Show that CR meets AP and BQ at G, where CG =2

3CR. [3 marks]

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 thepoint E and the lines AD and BC extended meet at the point F. Show that triangles ADE andCBE are similar. [3 marks]

If DA = DE, CF D = and BE C = 3, determine the value of . [4 marks]

6. A particle moves from rest along a horizontal straight line. At time t s, the displacement and velocityof the particle are x m and v ms1 respectively and its acceleration, in ms2, is given by

dv

dt= sin(t)

3cos(t).

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

Find the velocities of the particle when its acceleration is zero for the first and second times. Find

also the distance traveled by the particle between the first and second times its acceleration is zero.[7 marks]

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7. Two archers A and B take turns to shoot, with archer A taking the first shot. The probabilities

of archers A and B hitting the bulls-eye in each shot are1

6and

1

5respectively. Show that the

probability of archer A hitting the bull-eye first is1

2. [4 marks]

8. The probability that it rains in a certain area is1

5. The probability that an accident occurs at a

particular corner of a road in that area is1

20if it rains and

1

50if it does not rain. Find the probability

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. Therandom 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, foursurvive. [3 marks]

(b) Using a suitable approximate distribution, find the probability that more than 160 survive aftersurgery 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) 2 x < 5 5 x < 8 8 x < 11 11 x < 14 14 x < 17 17 x < 20Number of students 1 2 4 8 5 2(a) Draw a histogram of the grouped data. Comment on the shape of frequency distribution.[4 marks]

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

12. The continuous random variable X has probability density function

f(x) =

x 1

12, 1 x b,

0, otherwise,

where b is a constant.

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

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

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

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STPM 2007

1. Find, in terms of , all the values of x between 0 and which satisfies the equation

tan x + cot x = 8cos 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 andR are and respectively, where and are acute. The top A of a tower P A is at height h abovethe plane and the angle of elevation of A from R is . The height of a vertical pole QB is k and theangle of elevation of B from R is . Show that

h =k tan cos

tan sin( ) .

[5 marks]

3. The position vectors of the points A, B, C and D,relative to an origin, are i+3j, 5i3j, (x3)i6jand (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. IfOA = a and

OB = b,

O

R

A

P

BQ

(a) show that P Q = 115

a + 25

b, [5 marks]

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

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PB

A

C

Q

If the lines AQ and CB are parallel and T is the point of intersection of AB and CQ, show that thetriangles AT Q and BT C are isosceles triangles. Hence, show that the areas of the triangles AT Qand BT C are in the ratio AT2 : BT2. [7 marks]

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

x2dy

dx= y2 xy.

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

xdu

dx = u2

2u.

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

y =2x

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

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

7. There are eight parking bays in a row at a taxi stand. If one blue taxi, two red taxis and five yellowtaxis 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 maleand female students are 68.5 and 72.0 respectively. Find the ratio of the number of male to femalestudents. [4 marks]

9. The random variable X is normally distributed with mean and standard deviation 100. It is knownthat P(X > 1169) 0.117 and P(X > 879) 0.877. Determine the range of the values of.[7 marks]

10. Two events A and B are such that P(A) =3

8, P(B) =

1

4and P(A|B) = 1

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]

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(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 lemonsper packet so that the probability that a packet chosen at random does not contain rotten lemonsis more than 0.85. [5 marks]

(b) If the lemons in the fruit store are packed in boxed each containing 60 lemons, find, using asuitable approximation, the probability that a box chosen at random contains less than threerotten lemons. [5 marks]

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

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

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

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

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

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

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

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STPM 2008

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

(1

x)dy

dx

+ 2y + 2x = 0

into the differential equation

(1 x) dudx

= 2u.[3 marks]

2. In triangle ABC, the point X divides BC internally in the ratio m : n, where m + n = 1.

Express AX2 in terms of AB, BC, CA, m and n. [5 marks]

3. Ift = tan

2

, show that sin =2t

1 + t2

and cos =1 t2

1 + t2

. [4 marks]

Hence, find the values of between 0 and 360 that satisfy the equation 10 sin 5cos = 2.[3 marks]

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

T

B

R

A

CQP

The tangent to the circle at A meets the line BC extended to T. The angle bisector of the angleAT 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.

(a) Show that the position vector of any point on the line LM is1

2a +

1

2(b + c a) for some scalar

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

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

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

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

dQ

dt= Q

k

10 + (4 k)t .

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

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

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

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

8. The random variable X has a binomial distribution with parameters n = 500 and p =1

2. Using a

suitable approximate distribution, find P(|X

E(X)

| 25). [6 marks]

9. In a basket of mangoes and papayas, 70% of mangoes and 60% of papayas are ripe. If 40% of thefruits 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 passing increasing electric currentthrough them. The current at which they blow are recorded and the following cumulative frequencytable is obtained.

Currents (amperes) Cumulative frequency


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

i. no cars are requested, [2 marks]

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

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

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

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STPM 2009

1. Find the values ofx, where 0 x , which satisfy the equation sin3 x sec x = 2 tan x. [4 marks]

2. The circumscribed circle of the triangle J KL 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 angleJ T K cuts J L and J K at U and V respectively. Show that J V = J U. [4 marks]

3. Find the particular solution of the differential equation

exdy

dx y2(x + 1) = 0

for which y = 1 when x = 0. Hence, express y in terms 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. At1300 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 passes through D and is parallel to ACmeets 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]

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6. One of the rules at a training camp of 1000 occupants states that camp activities are to be suspendedif 10% of the occupants are infected with a virus. A trainee infected with a flu virus enrolls in thecamp causing an outbreak of flu. The rate of increase of the number of infected occupants x at tdays is given by differential equation

dxdt

= kx(1000 x),

where k is a constant.

Assume that the outbreak of flu begins at the time the infected trainee enrolls and no one leaves thecamp during the outbreak,

(a) Show that x =1000e1000kt

999 + e1000kt, [9 marks]

(b) Determine the value of k if it is found that, after one day, there are five infected occupants,[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 andthat of women engineers are 12 and 5 respectively. Four participants from this group are selectedrandomly to chair some sessions of panel 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 candidatesare shown in the table below.

School candidates Private candidates

Number of candidates 25 5Mean 55 40Standard deviation 4 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

respectively. Given that E(X) =4

3and Var(X) =

5

9, find the values of a, b and c. [6 marks]

10. The independent random variable Yi, where i = 1, 2, . . . , n, takes the values of 0 and 1 with theprobabilities 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 normallydistributed with mean 12 hours and standard deviation 5 hours for arts students, and mean 15 hoursand standard deviation 4 hours for science students.

A random sample of four arts students and six science students is chosen. Assuming that X is themean number of hours spent by these 10 students in a week,

(a) calculate E(X) and Var(X), [7 marks]

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(b) find the probability that in a given week, the mean number of hours spent by this sample ofstudents 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

f(x) =

0.01x p, 10 x < 20,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 ofRM100, express the total cost of repair in terms of X, and determine the expected total cost ofrepair. [3 marks]

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STPM 2010

1. Find the general solution of the differential equation

1

x

dy

dx

=2 ln x

cos y.

[5 marks]

2. Express 5 sin + 12 cos in the form r sin( + ), where r > 0 and 0 < < 90. Hence, find themaximum and minimum values of the expression

1

5sin + 12cos + 15.

[7 marks]

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

O

A

P

H K

B

Q

Prove that

(a) P A + P B = 2P O, [3 marks](b) P H2 = P A P B. [4 marks]

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

xydy

dx x2 y2 = 0

may be reduced to

vxdv

dx= 1.

[3 marks]

Hence, find the particular solution that satisfies y = 2 and x = 1. [6 marks]

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

(a) ABL = BC P, [2 marks]

(b) the trianglesABL

andBC K

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 thatQN : 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 lineP 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 umbrellais 0.6. The probability that it rains or the student does not carry an umbrella is 0.5. If it rains on aparticular 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 theleast integer k such that P(

|X

48

|> k) < 0.3. [5 marks]

9. A discrete random variable X has cumulative distribution function

F(x) =

0, x < 1,

0.6, 1 x < 3,0.9, 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 functionis given by

f(x) =

1

9xe

x

3 , x 0,0, 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 theprobability that all the eggs hatch is more than 0.75. [6 marks]

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(b) Using a suitable approximate distribution, find the probability that more than three out of 200eggs 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 Number of soil samples4.0 - 4.5 14.5 - 5.0 35.0 - 5.5 55.5 - 6.0 96.0 - 6.5 116.5 - 7.0 77.0 - 7.5 37.5 - 8.0 1

(a) Construct the cumulative frequency table for this distribution, and plot the cumulative frequencycurve. [3 marks]

(b) Using the cumulative frequency curve, estimate the median and semi-interquartile range of thedistribution. [4 marks]

(c) The pH value of a sample is wrongly recorded as 5.8, while its actual value is 5.0. State whetherthe 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 begreater than 8.0.

i. Out of the 44 samples, find the percentage of samples which have a pH value greater than7.0. [2 marks]

ii. Using the curve in (a), estimate the median of the 44 samples. State a reason why the samecurve is used. [2 marks]

stpm past year question

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