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Contents 1 Numbers and Sets

2

2 Polynomials

3

3 Sequences and series

4

4 Matrices

6

5 Coordinate geometry

8

6 Functions

10

7 Differentiation

11

8 Integration

13

9 Differential Equations

15

10 Trigonometry

18

11 Geometry Deduction

19

12 Vectors

23

13 Data Description

25

14 Probability

28

15 Discrete Probability Distributions

30

16 Continuous Probability Distributions

32

1 2

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

1

Numbers and Sets

1

Lee Kian Keong

Numbers and Sets 1. If loga

x = 3 loga 2 − loga (x − 2a), express x in terms of a. a2 [Answer :

2. Given that loga (3x − 4a) + loga 3x =

x = 4a ]

2 1 + loga (1 − 2a), where 0 < a < , find x. log2 a 2 [Answer :

2 ] 3

3. Find the values of x if y = |3 − x| and 4y − (x2 − 9) = −24. [Answer :

x = 7, x = −9 ]

4. Using the laws of the algebra of sets, show that (A ∩ B)0 − (A0 ∩ B) = B 0 5. Using definitions, show that, for any sets A, B and C, A ∩ (B ∪ C) ⊂ (A ∩ B) ∪ (A ∩ C) 6. Using the laws of the algebra of sets, show that, for any sets A and B, (A − B) ∪ (B − A) = (A ∪ B) − (A ∩ B) 7. If A, B and C are arbitrary sets, show that [(A ∪ B) − (B ∪ C)] ∩ (A ∪ C)0 = ∅ 8. If z is a complex number such that |z| = 1, find the real part of

1 . 1−z [Answer :

9. Express

1 ] 2

q √ √ √ 59 − 24 6 as p 2 + q 3 where p and q are integers. [Answer :

√ √ 4 2−3 3 ]

10. Simplify √ √ ( 7 − 3)2 √ √ , (a) 2( 7 + 3) √ 2(1 + 3i) (b) −1. , where i = (1 − 3i)2 [Answer :

√ √ 13 9 (a) 2 7 − 3 3 ; (b) − − i] 25 25

√ 11. 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. (b) Represent z1 and z2 in an Argand diagram. (c) For each of z1 and z2 , find the modulus, and the argument in radians. [Answer :

(a) z1 =

√

√ π 5π 3 − i, z2 = − 3 + i ; (c) |z1 | = 2, |z2 | = 2 , arg(z1 )=− , arg(z2 )= ] 6 6

12. If (x + iy)2 = i, find all the real values of x and y. [Answer :

2

1 1 x = ±√ ,y = ±√ ] 2 2

Polynomials

2

Lee Kian Keong

Polynomials 1 1. The polynomial p(x) = 2x3 + 4x2 + x − k has factor (x + 1). 2 (a) Find the value of k. (b) Factorise p(x) completely. [Answer :

(a) k =

1 3 ; (b) (x + 1)(2x − 1)(2x + 3) ] 2 2

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

3. 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. 1 Using the substitution y = , solve the equation 12y 4 − 8y 3 − 7y 2 + 2y + 1 = 0. x [Answer : 4. Using the substitution y = x +

1 1 1 a = 2, b = 12; y = − , − , 1, ] 3 2 2

1 4 1 , express f (x) = x3 − 4x − 6 − + 3 as a polynomial in y. x x x [Answer :

3

y − 7y − 6 ]

Hence, find all the real roots of the equation f (x) = 0. [Answer :

−1, −1,

√ √ 3+ 5 3− 5 , ] 2 2

5. 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. (b) Give that p(x) = (2x − 3)[q(x) − 41 + 3x3 ], find q(x), and determine its range when x ∈ [−2, 10]. [Answer :

2

(a) a = 7, b = 27, (x − 2)(2x − 3)(3x + 1)(x + 2); (b) q(x) = x − 12x + 37, {y : 1 ≤ y ≤ 65} ]

6. 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. (b) determine the set of values of x so that 2x3 − 9x2 + 3x + 4 < 12 − 12x. [Answer :

x = 1, x = −1, x = 2, x = −2 ; x < 1 ]

7. Find the set of values of x such that −1 < x3 − 2x2 + x − 2 < 0. [Answer : 8. Find the solution set of inequality |x − 2| <

1 where x 6= 0. x [Answer :

3

{x : 0 < x < 1, 1 < x < 2} ]

{x : 0 < x < 1, 1 < x < 1 +

√

2} ]

Sequences and series

Lee Kian Keong

9. Determine the set of values of x satisfying the inequality x 1 ≥ x+1 x+1 [Answer :

{x : x < −1, x ≥ 1} ]

10. Find the solution set of the inequality 4 3 x − 1 > 3 − x. [Answer :

{x : 0 < x < 1, 1 < x < 3} ]

1 11. Sketch, on the same coordinate axes, the graph of y = 2 − x and y = |2 + |. x 1 Hence, solve the inequality 2 − x > |2 + | x [Answer :

{x : x < 2 −

√

5} ]

12. Find the constants A, B, C and D such that A B C D 3x2 + 5x = + + + 2 2 2 (1 − x )(1 + x) 1 − x 1 + x (1 + x) (1 + x)3 . [Answer :

3

A = 1, B = 1, C = −1, D = −1 ]

Sequences and series 1. For the geometric series 7 + 3.5 + 1.75 + 0.875 + ..., find the smallest value of n for which the different between the sum of the first n terms and the sum to infinity is less than 0.01. [Answer :

11 ]

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

2. For geometric series 6 + 3 +

[Answer :

5]

3. Express the infinite recurring decimal 0.72˙ 5˙ (= 0.7252525 . . . ) as a fraction in its lowest terms. [Answer :

359 ] 495

4. 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. [Answer :

{x : x < 0} ; x = − ln 2 ]

5. Prove that the sum of the first n terms of a geometric series a + ar + ar2 + . . . is a(1 − rn ) 1−r 4

Sequences and series

Lee Kian Keong

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

(a) r = −2, a = 3 ; (b) n = 12 ]

10−Tn 5 6. The nth term of an arithmetic progression is Tn , show that Un = (−2)2( 17 ) is the nth 2 term of a geometric progression. ∞ X 1 If Tn = (17n − 14), evaluate Un . 2 n=1

[Answer :

−

10 ] 3

7. 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. (b) Calculate the total savings of Miss Dora at the end of n-th year. (c) Determine in which year the total savings of Miss Dora exceeds the total savings of Mr. Liu. [Answer : 8. Express

1 as partial fraction. 4k 2 − 1

Hence, find a simple expression for Sn =

n X k=1

n

n

(a) 10000(1.04) ; (b) 52000[1.04 − 1]; (c) 6 ]

1 and find lim Sn n→∞ −1

4k 2

[Answer :

1 1 1 − ; Sn = 2(2k − 1) 2(2k + 1) 2

1−

1 2n + 1

;

1 ] 2

9. 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

1 1+x 2 10. Express as a series of ascending powers of x up to the term in x3 . 1 + 2x √ 1 By taking x = , find 62 correct to four decimal places. 30

[Answer : 11. Express ur =

1−

√ 1 7 2 25 3 x+ x − x + . . . ; 62 = 7.8740 ] 2 8 16

2 in partial fractions. r2 + 2r [Answer : 5

1 1 − ] r r+2

Matrices

Lee Kian Keong

Using the result obtained, 1 1 1 1 , (a) show that u2r = − + 2 + + r r r + 2 (r + 2)2 ∞ ∞ ∞ 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 [Answer :

(b)

3 4 , ] 2 3

1

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

4

1−

√

7

1 1 2 1 3 √ x− x − x ; 7 = 2.64609 ] 2 8 16

Matrices

−10 4 9 2 3 4 1. Matrix M and N is given by M = 15 −4 −14 , and N = 4 3 1 −5 1 6 1 2 4 Find M N and deduce N −1 . Product X, Y , Z are assembled from three components A, B, 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 of C are used. With X, Y , Z representing the number of products of X, Y , and Z assembled, obtain a matrix equation representing the information given. Hence, find the number of products of X, Y , and Z assembled. [Answer : 2. A, B, C are B2 = C. 1 If B = 0 1

x=200, y=50, z=50. ]

square matrices such that BA = B−1 and ABC = (AB)−1 . Show that A−1 = 2 0 −1 0 , find C and A. 0 1 [Answer :

3. (a) The matrix P, Q and R are given by 1 5 6 −13 P = 2 −2 4 , Q = −1 1 −3 2 7

1 A= 0 −2

−50 −33 4 −6 −5 , R = 1 20 15 −2

Find 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

. 6

0 1 −2

+ 8z + z + 3z

= 4500 = 0 = 1080

0 1 0 , C = 0 1 2

7 −5 1

0 1 2

−13 −1 11

0 0 ] 1

Matrices

Lee Kian Keong

[Answer :

24 (a) 4 4

40 −8 8

32 72 4,0 0 12

0 1/18 0 , 1/72 72 −1/36

0 72 0

−13/72 −1/72 ; (b) x = 220, y = 190, z = 160 ] 11/72

7/72 −5/72 1/72

2b − 1 4. Determine the values of a, b, c so that the matrix 2a − 1 b matrix.

a2 b2 a bc is a symmetric b + c 2c − 1 [Answer :

5. The matrices A and B are given by −1 2 1 −35 19 A = −3 1 4 , B = −27 −13 0 1 2 −3 12

a = 1, b = 0, c = 0 ]

18 45 . 5

Find the matrix A2 B and deduce the inverse of A. Hence, solve the system of linear equations x − 2y − z 3x − y − 4z y + 2z

[Answer :

121 0 0

1 6. The matrix A is given by A = 3 0

2 1 1

0 121 0

= −8, = −15, = 4.

0 −2/11 0 , 6/11 121 −3/11

−3/11 −2/11 1/11

7/11 1/11 ; x = −3, y = 2, z = 1 ] 5/11

−3 1 −2

(a) Find the matrix B such that B = A2 − 10I, where I is the 3 × 3 identity matrix. (b) Find (A + I)B, and hence find (A + I)21 B. [Answer :

−3 (a) 6 3

1 −2 −1

5 −3 −10 ; (b) 6 −5 3

7. The matrices P and Q, where P Q = QP , are given by 2 −2 0 −1 P = 0 0 2 and Q = 0 a b c 0

1 0 −2

1 −2 −1

5 −3 −10 , 6 −5 3

1 −2 −1

5 −10 ] −5

0 −1 2

Determine the values of a, b and c. Find the real numbers m and n for which P = mQ + nI, where I is the 3 × 3 identity matrix. [Answer :

3 3 8. Matrix A is given by A = 5 4 1 2 Find the adjoint of A. Hence, find

a = 0, b = 4, c = −4 ; m = −2, n = 0 ]

4 1 . 3 A−1 .

[Answer :

7

10 −14 6

−1 5 −3

−13 5/6 17 ; −7/6 −3 1/2

−1/12 5/12 −1/4

−13/12 17/12 ] −1/4

Coordinate geometry

Lee Kian Keong

5 2 3 a 1 −18 −1 12 and P Q = 2I, where I is the 3 × 3 identity 9. 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 . 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. [Answer :

5/2 a = 11, b = −7, c = 22 ; 1/2 3/2

1 −2 1/2

3/2 3/2 ; x=RM 1, y=RM 1.30, z=RM 1.40 ] 1

k 10. Determine the values of k such that the determinant of the matrix 2k + 1 0

1 −3 k

[Answer :

1 11. Matrix A is given by A = 1 1

1 k = − ,k = 2 ] 4

0 0 −1 0 . −2 1

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

matrix, and deduce A−1 . 4 3 2 1 . 0 2

[Answer :

5

3 2 is 0. 2

1 (a) 1 1

0 −1 −2

0 8 0 ; (b) 3 1 1

−10 −4 −4

3 1 ] 2

Coordinate geometry 1. 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. Find the shortest distance from P to QR and the area of the parallelogram. [Answer :

√ 2 a = 5, b = 14; shortest distance=8 2 units; Area=80 unit ]

2. The straight line l1 which passes through the points A(4, 0) and B(2, 4) intersects the y-axis at point P . The straight line l2 is perpendicular to l1 and passes through B. √ If l2 intersects the x-axis and y-axis at points Q and R respectively, show that P R : QR = 5 : 3.

3. The sum of distance of the point P from the point (4,0) and the distance of P from the origin (x − 2)2 y2 is 8 units. Show that the locus of P is the ellipse + = 1 and sketch the ellipse. 16 12

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

Coordinate geometry

Lee Kian Keong [Answer :

7x + 21y − 123 = 0 ]

5. 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. [Answer :

4;1]

6. 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. (b) Calculate the perpendicular distance of A to OB, where O is the origin. (c) Find the area of the triangle OAB. [Answer :

(a) A(1, 2) and B

3 3 , 2 2

√

; (b)

3 2 ; (c) ] 2 4

7. The parametric equations of a straight line l are given by x = 4t − 2 and y = 3 − 3t. 3 (a) Show that the point A(1, ) lies on line l, 4 (b) Find the Cartesian equation of line l, (c) Given that line l cuts the x and y-axes at P and Q respectively, find the ratio P A : AQ. [Answer :

(b) 3x + 4y − 6 = 0 ; (c) 1:1 ]

x y , respectively, where x2 + y 2 x2 + y 2 x 6= 0 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.

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

[Answer :

1 1 3 centre = (− , − ) ; radius = ] 7 7 7

9. 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.

The above figure shows three circles C1 , C2 and C3 touching one another, where their centres lies 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 . [Answer : 9

2

2

5x + 5y − 74x + 12y + 156 = 0 ]

Functions

6

Lee Kian Keong

Functions 1. The function f is defined by f : x →

√

3x + 1, x ∈ R, x ≥ −

Find f −1 and state its domain and range. [Answer :

f

−1

:x→

1 3

1 x2 − 1 , Df −1 = {x : x ≥ 0}, Rf −1 = {x : x ≥ − } ] 3 3

2. 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. x→1

x→1

(b) Sketch the graph of f . [Answer :

(a) 1 + e , 1 + e ; not continuous ]

3. 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 x→2

f is continuous at x = 2. [Answer :

a=

5 ; continuous at x = 2 ] 16

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

2 ex − e−x andg(x) = x ex + e−x e + e−x

(a) State the domains of f and g, (b) Without using differentiation, find the range of f , (c) Show that f (x)2 + g(x)2 = 1. Hence, find the range of g. [Answer :

(a) Df = {x : x ∈ R} , Dg = {x : x ∈ R} ; (b) {y : −1 < y < 1} ; (c) {y : 0 < y ≤ 1} ]

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

√

x, find lim

h→0

f (x) − f (x + h) . h [Answer :

6. The function f is defined by (√ f (x) = (a) Find

lim f (x),

x→−1−

x + 1, |x| − 1,

−1 ≤ x < 1, otherwise.

lim f (x), lim− f (x) and lim+ f (x).

x→−1+

x→1

x→1

(b) Determine whether f is continuous at x = −1 and x = 1. 10

1 − √ ] 2 x

Differentiation

Lee Kian Keong [Answer :

(a) 0 , 0 ,

√

2 , 0 ; (b) continuous at x = −1 , discontinuous at x = 1 ]

7. 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. (b) Find the composite functions g ◦ f and state its domain and range. (c) State the domain and range of h. (d) State whether h = g ◦ f . Give a reason for your answer [Answer :

(a) Df = {x : x ∈ R, x 6= −1} , Dg = {x : x ∈ R, x 6= 0} ; 2 (b) 3 + , D = {x : x ∈ R, x 6= 0, x 6= −1} , R = {y : y ∈ R, y 6= 3, y 6= 1} ; x (c) D = {x : x ∈ R, x 6= 0} , R = {y : y ∈ R, y 6= 3} ; (d) No. Different domain ]

8. 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. [Answer :

7

1 1 , D = {x : x ∈ R, x 6= } ] 2x − 1 2

Differentiation 1. Given that y = e−x cos x, find

dy d2 y and when x = 0. dx dx2 [Answer :

dy d2 y = 1, =0] dx dx2

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

1 ] e2

2x . (x + 1)(x − 2) Show that f 0 (x) < 0 for all values of x in the domain of f . Sketch the graph of y = f (x). Determine if f is a one to one function. Give reasons to your answer. 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.

3. Function f if defined by f (x) =

[Answer :

f is not one to one function. If k ≥ 0, 1 root. If k < 0, 3 roots. ]

4. If y =

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

5. If y =

x dy , show that x2 = (1 − x2 )y 2 . 1 + x2 dx 11

Differentiation

6. If y =

Lee Kian Keong

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

2 7. A curve is defined by the parametric equations x = 1 − 2t, y = −2 + . t Find the equation of the normal to the curve at the point A(3, −4). The normal of the curve at the point A cuts the curve again at point B. Find the coordinates of B. [Answer :

x + y + 1 = 0 ; B(−1, 0) ]

8. Using the sketch 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). Using the Newton-Raphson method to find the real root correct to three decimal places. [Answer : 9. Sketch, on the same coordinate axes, the graphs y = ex and y =

a = 0 , b = 1 ; 0.683 ]

2 . Show that the equation 1+x

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

0.375 ]

1 10. Find the coordinate 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. 1 1 The x-coordinate of the point of intersection of the curves y = x2 + and y = 2 , where x x x > 0, is p. Show that 0.5 < p < 1. Using the Newton-Raphson method to determine the value of p correct to three decimal places and, hence, find the point of intersection. [Answer :

(0.794 , 1.890) , minimum ; p = 0.724 , (0.724 , 1.908) ]

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

2 t

and y = 2t +

1 t

where t 6= 0. 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

[Answer :

(b) (-1 , 3) and (1 , -3) ]

x3 12. Find the coordinates of the stationary points on the curve y = 2 abd determine their x −1 nature. Sketch the curve. Determine the number of real roots of the equation x3 = k(x2 − 1), where k ∈ R, when k varies. 12

Integration

Lee Kian Keong √ √ √ 3 3 √ 3 3 (0, 0) is inflexion point , ( 3, ) is local min. , (− 3, − ) is local max. √ √ √ 2 √ 2 √ 3 3 3 3 3 3 3 3 3 3 1 real root for −

[Answer :

13. 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. [Answer :

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

x + ln(x + 1) δx ; 0.698 ] x+1

4ekt − 1 where k is a positive constant, 4ekt + 1 3 5

(a) Find the value of f (0) (b) Show that f 0 (t) > 0 (c) Show that k[1 − f (t)2 ] = 2f 0 (t) and hence show that f 00 (t) < 0 (d) Find lim f (t)

(1)

t→∞

(e) Sketch the graph of f .

x is always decreasing. x2 − 1 Determine the coordinates of the point of inflexion of the curve, and state the intervals for which the curve is concave upwards. Sketch the curve.

15. Show that the gradient of the curve y =

[Answer :

(0, 0) ; (−1, 0) ∪ (1, ∞) ]

16. 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.

8

Integration Z

e

1. Show that

ln x dx = 1. 1

Z 2. Show that 2

3

(x − 2)2 5 dx = + 4 ln x2 3

2 3

Z 3. By using suitable substitution, find

3x − 1 √ dx x+1 [Answer : Z

4. Using an appropriate substitution, evaluate

1

3

1

2(x + 1) 2 − 8(x + 1) 2 + C ]

1

x2 (1 − x) 3 dx.

0

[Answer : 13

27 ] 140

Integration

Lee Kian Keong Z

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

π 6

cos θ dθ. (3 + 2 sin θ)2 [Answer :

1 ] 24

2x + 1 Ax + B C 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 0 (x + 1)(2 − x)

6. Express

x 1 3 + ; ln 2 ] x2 + 1 2−x 2

[Answer : 7. The gradient of the tangent to a curve at any point (x, y) is given by

dy 3x − 5 = √ , where dx 2 x

x > 0. If the curve passes through the point (1, −4). (a) find the equation of the curve, (b) sketch the curve, (c) calculate the area of the region bounded by the curve and the x-axis. [Answer :

3

1

(a) y = x 2 − 5x 2 ; (c)

20 √ 5] 5

8. 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, (b) determine the coordinate of their points of intersection, (c) calculate the area of the region R bounded by the curve and the straight line, (d) find the volume of the solid formed when R is rotated through 360◦ about the x-axis. [Answer :

(b) (−1, 3) , (2, 0) ; (c)

9 108 ; (d) π ] 2 5

9. Sketch, on the same coordinate axes, the curves y = 6 − ex and y = 5e−x , and find the coordinates of the points of intersection. Calculate the area of the region bounded by the curves. Calculate the volume of the solid formed when the region is rotated through 2π radians about the x-axis. [Answer :

(ln 5, 1) ; 6 ln 5 − 8 ; π(36 ln 5 − 48) ]

10. Find x2 + x + 2 dx, x2 + 2 Z x (b) dx. x+1 e Z

(a)

[Answer :

(a) x +

1 x 1 2 ln(x + 2) + C ; (b) − x+1 + x+1 ] 2 e e

11. 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. [Answer : 14

2 ln 3 ]

Differential Equations

Lee Kian Keong

1 12. Sketch on the same coordinates axis y = x and the curve y 2 = x. Find the coordinate of the 2 points of intersection. 1 Find the area of region bounded by the line y = x and the curve y 2 = x. 2 Find the volume of the solid formed when the region is rotated through 2π radians about the y-axis. [Answer :

(0, 0) , (4, 2) ;

4 64 ; π ] 3 15

a 13. The curve y = x(b − x), where a 6= 0, has a turning point at point (2, 1). Determine the 2 values of a and b. Calculate the area of the region bounded bt the x-axis and the curve. Calculate the volume of the solid formed by revolving the region about the x-axis. [Answer :

a=

8 32 1 ,b=4; ; π ] 2 3 15

14. 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. Calculate the area of the region bounded by the curves y = −x2 + 3x and y = 2x3 − x2 − 5x. [Answer :

2

Point of intersection=(0,0), (2,2), (-2,-10) ; Area=16 units . ]

Z 15. Using trapezium rule, with five ordinates, evaluate

1

p 4 − x2 dx.

0

[Answer :

9

1.910 ]

Differential Equations 1. 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 terms of x. [Answer :

y=

ex ] 2 + x − ex

2. Find the general solution of the differential equation x

dy = y 2 − y − 2. dx [Answer :

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

dy + 2y + 2x = 0 dx

into the differential equation (1 − x)

15

du = −2u dx

y=

2 + Ax3 ] 1 − Ax3

Differential Equations

Lee Kian Keong

4. The variables t and x are connected by dx = 2t(x − 1), dt where x 6= 1. Find x in terms of t if x = 2 when t = 1. [Answer :

x=e

t2 −1

+1 ]

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

du = u2 − 2u. dx

Hence, show that the general solution of the given differential equation maybe expressed in 2x , where A is an arbitrary constant. the form y = 1 + Ax2 Find the equation of the solution curve which passes through the point (1,4) and sketch this solution curve. [Answer : 6. Using the substitution y =

y=

4x ] 2 − x2

v , show that the differential equation x2 2y dy + y2 = − dx x

may be reduced to v2 dv = − 2. dx x Hence, find the general solution of the original differential equation. [Answer :

y=

1 ] Ax2 − x

7. Show that

2 d (ln tan x) = , dx sin 2x Hence, find the solution of the differential equation (sin 2x) for which y =

dy = 2y(1 − y) dx

1 1 when x = π. Express y explicity in terms of x in your answer. 3 4 [Answer :

y=

tan x ] 2 + tan x

8. 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, 16

Differential Equations

Lee Kian Keong

1000e1000kt , 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 =

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

(b) k =

1 ln 1000

999 195

; (c) 5 ]

9. 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. (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 (b) Show that k = 4, and calculate the amount of salt in the tank at the instant overflow occurs. (c) Sketch the graph of Q against t for 0 ≤ t ≤ 20. [Answer :

(b) Q = 4 ]

10. 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 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 . 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. [Answer :

Time = 4 hours 25 minutes ]

11. 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 differential equation and sketch its solution curve. (b) At a certain instant, there is an outbreak of an epidemic of a disease. The epidemic results in no more r offspring of the fish being produced and the fish die at a rate directly 1 proportional to . There are 900 fish before the outbreak of the epidemic and only P 400 fish are alive after 6 weeks. Determine the length of time from the outbreak of the epidemic until all the fish of that species die.

17

Trigonometry

Lee Kian Keong [Answer :

(a) P = P0 e

(a−b)t

; (b) 18 weeks ]

12. 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 terms of t. 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. [Answer :

10

v=−

√ √ 1 1 [cos(πt) + 3 sin(πt)] , x = − 2 [sin(πt) − 3 cos(πt)] π π 2 4 −1 v = ± ms ; Distance = 2 m ] π π

Trigonometry

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◦ . [Answer :

◦

◦

◦

5 sin(θ − 36.9 ) ; 73.8 , 180 ]

2. Find the values of x, where 0 ≤ x ≤ π, which satisfy the equation sin3 x sec x = 2 tan x π 3π , ,π ] 4 4

[Answer :

0,

[Answer :

36.9 , 196.3 ]

θ 2t 1 − t2 3. If t = tan , show that sin θ = and cos θ = . 2 1 + t2 1 + t2 Hence, find the values θ between 0◦ and 360◦ that satisfy the equation 10 sin θ − 5 cos θ = 2

√

◦

◦

π . 2 Hence, find the values 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), r > 0 and 0 < α <

. [Answer :

2 cos

x−

π 3

;

2π 5π 11π x:

5. Starting from the formulae for sin(A + B) and cos(A + B), prove that tan(A + B) = If 2x + y =

π , show that 4

tan A + tan B 1 − tan A tan B

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 =

18

Geometry Deduction

Lee Kian Keong

6. Express cos θ + 3 sin θ in the form r cos(θ − α), where r > 0 and 0◦ < α < 90◦ [Answer :

√

◦

10 cos(θ − 71.6 ) ]

7. Find all values of x, where 0◦ < x < 360◦ , which satisfy the equation tan x + 4 cot x = 4 sec x. [Answer :

◦

◦

41.8 , 138.2 ]

8. Find, in terms of π, all the values of x between 0 and π which satisfies the equation tan x + cot x = 8 cos 2x [Answer :

π 5π 13π 17π , , , ] 24 24 24 24

9. 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 ang the angle of elevation of B from R is β. Show that h=

k tan α cos θ tan β sin(θ − φ)

√ 10. 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. [Answer :

11

◦

59.0 ]

Geometry Deduction

1. Points A and B are in 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◦

2. 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 19

Geometry Deduction

Lee Kian Keong

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

3. 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◦ If AB = AC, show that BC is parallel to XY .

4. 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.

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.

6. 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.

20

Geometry Deduction

Lee Kian Keong

7. 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. If AB = 4 cm, AC = 8 cm and BP = 3 cm, find the area of the triangle BP C. [Answer :

2

6.54 cm ]

8. Prove that an exterior angle of a cyclic quadrilateral is equal to the opposite interior angle.

In the 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. If DA = DE, ∠CF D = α and ∠BEC = 3α, determine the value of α. [Answer :

◦

α = 18 ]

9. The diagram above 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. 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 .

10. The diagram below shows the circumscribed circle of he triangle ABC.

21

Geometry Deduction

Lee Kian Keong

The tangent to the circle at A meets the line BC extended to T . The angle bisector of the angle AT B at P m AB at Q and the circle at R. Show that (a) triangles AP T and BQT are similar, (b) P T · BT = QT · AT , (c) AP = AQ.

11. 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 .

12. 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 AC meets P C produced at point R amd ∠CRD = 90◦ . (a) Show that the triangles ABD and BP C are congruent. (b) Show that ABCD is a rhombus. (c) Find the ratio CR : P C. [Answer :

22

(c) 1:2 ]

Vectors

12

Lee Kian Keong

Vectors

4 1 1. If the angle between the vectors a = and b = is 135◦ , find the value of p. 8 p [Answer :

−

1 ] 3

2. 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. (b) At what time does the interception occur? [Answer :

◦

(a) S 58.6 W ; (b) 1341 hours ]

3. 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) some 2 2 scalar λ, and express the position vector of any point on the line P Q in terms of a, b and c. (b) Find the position vector of the point of intersection of the line LM and the line P Q. [Answer :

(a)

1 1 1 1 1 b + µ(a + c − b) ; (b) a + b + c ] 2 2 4 4 4

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

2

2

2

nAB − mnBC + mCA ]

5. 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 seem like 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 terms of u, w and θ. [Answer :

−2uw(sin θ + cos θ) ]

6. Position vectors of the points P and Q relative to the origin O are 2 i and 3 i + 4j respectively. ∼ ∼ ∼ −−→ −−→ Find the angle between vector OP and vector OQ [Answer :

◦

53.1 ]

7. 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. 23

Vectors

Lee Kian Keong

−→ 1 −−→ 1 −→ −→ −−→ 1 −→ (a) Show that AG = mAB + mAC and AG = (1 − n)AB + AC. 2 2 2 2 2 Deduce that AG = AP and CG = CR. 3 3 2 (b) Show that CR meets AP and BQ at G, where CG = CR. 3

8. A force of magnitude 2p N acts along the line OA abd 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. [Answer :

◦

30 ]

9. Let u = cos φ i + sin φ j and v = cos θ i + sin θ j, where i and j are perpendicular unit vectors. Show that 1 1 |u − v| = sin (φ − θ) 2 2

10. 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 the canal. Taking y to denote the distance northwards travelled by the swimmer, show that dy v0 x2 = 1− 2 . 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 ms−1 and the speed of the swimmer is 2 ms−1 relative to the current in the canal, (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, (b) sketch the actual path taken by the swimmer. [Answer :

(a) 20 m north of O , 40 m north of B ]

11. 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. (b) Determine the value of x for which ABCD is a rectangle. [Answer :

24

(b) x = 1 ]

Data Description

Lee Kian Keong

12. 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 CP and CQ in terms of k, a and b, where AB = a and BC = b. If the points C, P and Q lie on a straight line, determine the value of k. Hence, find CP : CQ . [Answer :

√ 1 − − → − − → CP = (2k − 1)b − ka , CQ = (1 − k)b − (1 + k)a , k = √ , CP : CQ = 1 : 3 ] 3

13. The diagram above shows non-linear 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,

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

13

(b) 4b ]

Data Description

1. 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. [Answer :

52.5 ; 6.98 ]

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

25

Data Description

Lee Kian Keong 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. [Answer :

12.65 , 12.61 , 12.58 ; positively skewed ]

3. 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 (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. [Answer :

(b) 17 years 1 month , 10 months ]

4. 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. (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. [Answer :

(a) 2 , 3 ; (b) 2.7 ]

5. 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

(a) Display the data in a stemplot 26

25 50

45 46

35 23

32 40

18 20

26 29

30 46

Data Description

Lee Kian Keong

(b) Find the median and interquartile range (c) Draw a boxplot to represent the data (d) State the shape of the frequency distribution. Give a reason for your answer [Answer :

(b) 31 , 15 ; (d) positively skewed ]

6. Show that, for the numbers x1 , x2 , x3 , . . . , xn with mean x ¯, X X (x − x ¯)2 = x2 − n¯ x2 The numbers 4, 6, 12, 5, 7, 9, 5, 11, p, q, where p < q, have mean x ¯ = 6.9 and 102.9. Calculate the of p and q.

X

(x − x ¯)2 =

[Answer :

1, 9 ]

7. 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. (b) Find the mean and standard deviation. (c) Find the median and interquartile range. (d) Draw a boxplot to represent the above data. (e) State whether the mean or the median is more suitable as a representative value of the above data. Justify your answer. [Answer :

(b) 44 , 16.8 ; (c) 41 , 15 ; (e) median ]

8. 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. (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. [Answer :

(b) 12.23 , 12.50 , 12.71 ]

9. 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. [Answer :

2:3 ]

10. 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 27

50 52

38 47

53 39

50 51

43 49

56 52

51

Probability

Lee Kian Keong

(a) Construct a stemplot to represent the data. (b) Find the median and interquartile range. (c) Calculate the mean and standard deviation. (d) Draw a boxplot to represent the data. (e) Comment on the shape of the distribution and give a reason for your answer. [Answer :

(b) 50 , 6 ; (c) 48.4 , 5.97 ]

11. 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 µ g 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 (µ g m−3 ) 170 - 174 175 - 179 180 - 184 185 - 189 190 - 194 195 - 199 200 - 201

Number of days 8 11 25 22 15 7 2

(a) State what the number 11 in the table means. (b) Calculate estimates of the mean and standard deviation of the levels of the metal dust. (c) Plot a cumulative frequency curve of the above data. Hence, estimate the median and the interquartile range. (d) Find the percentage of days for which the workplace is considered unsafe. [Answer :

14

(b) 185 , 7.22 ; (c) 184.7 , 9.8 ; (d) 4.44% ]

Probability

1. There are 20 doctors and 15 engineers attending a conderence. 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 panel discussion. 855 (a) Find the probability that three doctors are selected. 2618 33 (b) Given that two women are selected, find the probability that both of them are doctors. 68

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

28

(0.64) (0.4375)

Probability

Lee Kian Keong

3. Three balls are selected at random from one blue ball, three red balls and six white balls.Find 7 the probability that all the three balls selected are of the same color. 40

4. There are 12 towels, two of which are red. If five towels are chosen at random, find the 15 probability that at least one is red. 22

5. 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 exacly one worker receives a bonus of one month’s salary. (0.495) (b) If a male worker and a demale worker are randomly chosen, find the probability that exactly one worker receives a bonus of one month’s salary. (0.604)

6. 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.(0.1909) (b) If at least one out of the two transistors chosen is defective, find the probability that both transistors are defective. (0.0476)

7. Two archers A and B take turns to shot, with archer A taking the first shot. The probabilities 1 1 and respectively. Show that the of A and B hitting the bull’s eye in each shot are 6 5 1 probability of archer A hitting the bull-eye first is . 2

1 8. The probability that it rains in a certain area is . The probability that an accident occurs 5 1 1 at a particular corner of a road in that area is if it rains and if it does not rain. Find 20 50 5 the probability that it rains if an accident occurs at the corner. 13

9. 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. 1 [Assume that a taxi may be parked at any of the parking bays.] 4

29

Discrete Probability Distributions

Lee Kian Keong

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.

7 12

13 24

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

11. 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,

(0.19)

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

15

(0.0486)

Discrete Probability Distributions

1. 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. (b) If X = Y1 + Y2 + . . . + Yn , determine E(X) and Var(X). Comment on the distribution of X. [Answer :

(b) np, npq ; binomial distribution ]

2. A discrete random variable X takes the values of 0, 1 and 2 with the probabilities of a, b and 4 5 c respectively. Given that E(X) = and Var(X) = , find the values of a, b and c. 3 9 [Answer :

a=

1 1 1 ,b= , ] 6 3 2

(a)

1 ; (b) 9 , 19 ] 14

3. 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. (b) Find E(7X − 1) and Var(7X − 1). [Answer : 4. 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 ). (b) Find P (|Y − E(Y )| ≤ 30). 30

Discrete Probability Distributions

Lee Kian Keong [Answer :

(a) 561 ; (b)

4087 ] 5050

5. 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. (a) Find the probability that, on a particular day, i. no cars are requested, ii. at least four requests for cars are received.

(0.0183) (0.5665)

(b) Calculate the expected daily income received from the rentals.

(160.93)

(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. (YES)

6. 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. [Answer :

35 ]

7. 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 ) (b) Give one reason why W is not a Poisson random variable. [Answer :

(a) -2 , 4 ]

8. 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. [Answer :

0.7619 ]

9. 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. [Answer :

0.7361 ]

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. (b) Using a suitable approximate distribution, find the probability that more than 160 survive after surgery in a random sample of 200 heart patients. [Answer :

(a) 0.3915 ; (b) 0.97 ]

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. 31

Continuous Probability Distributions

Lee Kian Keong

(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. [Answer :

(a) 8 ; (b) 0.8795 ]

12. 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. [Answer :

16

0.2558 ]

Continuous Probability Distributions

1. 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. (b) Find the probability that the repair work takes at least 15 hours. (c) Determine the expected value of X. (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 terms of X, and determine the expected total cost of repair. [Answer :

(b)

7 ; (c) 20 hours ; (d) RM 2500 ] 8

2. 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. (b) Calculate the probability that at least one of two independent observed values of X is greater than three. [Answer : 3. 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, 32

(b)

23 ] 144

Continuous Probability Distributions

Lee Kian Keong

1 and b = 2, 6 (b) Find the cumulative distribution function of X and sketch its graph (a) Show that a =

(c) Find the probability density function of X (d) Calculate the mean and standard deviation of X. [Answer :

33

(d)

5 , 1.190 ] 2

Continuous Probability Distributions

Lee Kian Keong

4. The lifespan of a species of plant is a random variable T (tens of days). The probability density function is given by 1 e− 81 t , t > 0 f (t) = 8 0, otherwise (a) Find the cumulative distribution function of T and sketch its graph. (b) Find the probability, to three decimal places, that a plant of that species randomly chosen has a lifespan of more than 20 days. (c) Calculate the expected lifespan of that species of plant. [Answer :

(b) 0.779 ; (c) 80 days ]

5. The continuous random variable X has the probability density function 4 x2 (3 − x), 0 < x < 3, f (x) = 27 0, otherwise. (a) Calculate P

X<

3 . 2

(b) Find the cumulative distribution function of X. (a)

5 ] 16

(a) b = 4 ; (c)

14 ] 5

[Answer : 6. 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. (b) Find the cumulative distribution function of X and sketch its graph. (c) Calculate E(X). [Answer : 7. The continuous random variable X has probability 1 (1 − 2x), 25 3 f (x) = (2x − 1), 25 0,

density function 1 −2 ≤ x ≤ 2 1 ≤x≤3 2 otherwise.

(a) Sketch the graph of y = f (x) (b) Given that P (0 ≤ X ≤ k) =

13 , determine the value of k. 100 [Answer :

34

(b) k =

3 ] 2

Continuous Probability Distributions

Lee Kian Keong

8. 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), (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. [Answer :

(a) 13.8 , 1.96; (b) 0.0.7814 ]

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. [Answer :

{µ : 995 ≤ µ ≤ 1050} ]

10. 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. [Answer : 11. The random variable X has a binomial distribution with parameters n = 500 and p =

0.6687 ]

1 . 2

Using a suitable approximate distribution, find P (|X − E(X)| ≤ 25). [Answer :

0.9774 ]

12. 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. (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 distributed, 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. [Answer :

(a) 0.1587 , 0.144 ; (b) -2 , 3.56 ; 0.1446 ]

13. 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. [Answer :

82 ]

14. 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.

35

Continuous Probability Distributions

Lee Kian Keong

(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 replacedfree of charge. (b) If the manufacturer specifies that less than 0.1% of the electrical instruments have to be replacedfree of charge, determine the greatest length of the gurantee period correct to the nearest month. [Answer :

36

(a) 0.82% ; (b) 25 months ]

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