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ADDITIONAL MATHEMATICS
SPM Forecast PaperPAPER 1Instructions: 1 2 3 4 5 6 7 8
Time: 2 hoursThis question paper consists of 25 questions. Answer all questions. Write your answers clearly in the spaces provided in the question paper. Show your working. It may help you to get marks. The diagrams provided in the questions are not drawn to scale unless stated otherwise. The marks allocated for each question are shown in brackets. A list of formulae is provided. You may use a non-programmable scientific calculator. Answer all questions. [80 marks]
1
{(3, 6), (1, 1), (3, 1), (4, 4), (6, 5)}
Based on the above ordered pairs, state (a) the objects of 1, (b) the range.
[2 marks]
Answer: (a) (b) 2 A function f is defined by f(x) = ax + b, where a and b are constants. Given that f(4) = 5 and f(2) = 7 , find (a) the value of a and of b, (b) the value of x which is mapped onto itself. [4 marks]
Answer: (a) a = b= (b) x =
1
5 3 Given the function g(x) = 3x + k and its inverse g1(x) = mx , find the value of k and of m. 6
[3 marks]
Answer: k = m= 3 Solve the equation x2 x = 3, giving your answers correct to two decimal places. 2
4
[3 marks]
Answer: x = 5 The diagram shows the graph of the quadratic function f(x) = (x 4)2 + h which has a maximum point (k, 9). State (a) the value of k, (b) the value of h, (c) the equation of the tangent to the curve at its maximum point.y (k, 9)
[3 marks]O x
Answer: (a) k = (b) h = (c)
2
6 Given that m and k are the roots of the equation (x + 1)(x 3) + m = kx, find the value of m and of k. [4 marks]
Answer: m = k= 7 If the straight line x + y 3 = 0 does not meet the curve y = x2 + 2x + h, find the range of values of h. [4 marks]
Answer: 8 Simplify 2x + 3 + 2x + 16(2x 1) in the form k(2x) , where k is a constant. [3 marks]
Answer: 9 Solve the equation 5x = 32x1
, giving your answer correct to two decimal places.
[4 marks]
Answer: x =
3
10 Solve the equation. log10 (p + 3) = 1 + log10 p.
[3 marks]
Answer: p = 11 If the equation of the straight line PQ in the x y diagram is = 1, find the area of PQR. 4 3y R(2, 5)
[3 marks]
O Q
P
x
Answer: 12 The table shows the distribution of the sizes of a certain brand of long pants sold at a shop. Calculate the variance. Size 30 32 34 Frequency 3 5 2
units2
[4 marks]
Answer: 13 The diagram shows a sector of a circle BC with centre O. Given that BOC = 20 and the length of the arc BC is 15.36 cm, find the length of OB. [Use = 3.142]B O 20 C
[3 marks]
Answer:
cm
4
dy 14 Given that y = (x + 1)(2x 1)2, find . dx
[4 marks]
dy Answer: = dx 15 The variables x and y are related by the equation x + y = 30. Find the maximum value of z if z = xy. [4 marks]
Answer: z = 16 Given the arithmetic progression 4, 2, 0, ..., calculate the number of terms that have to be taken for its sum to be 176. [4 marks]
Answer: 17 If 9x + 4, x + 2 and x 4 are three consecutive terms of a geometric progression, find the possible values of x. [4 marks]
Answer: x =
5
y 18 If the non-linear equation = p, where k and p are constants, is reduced to the linear form, Y = mX + c, kx state Y, X, m and c in terms of x, y, k and/or p. [4 marks]
Answer: Y = X= m= c= 19Given that4 1g(x)
dx = 20, find
2 13g(x)
dx +
4 2
3g(x) dx.
[2 marks]
Answer: 20 The diagram shows the curve y = x2 2x + 1. Find the area of the shaded region.y
[3 marks]
y = x 2 2x + 1 x
1
O
1
2
Answer: 21 Given that a = 7j, b = 10i 5j and c = 4i + j, find in terms of i and j, 1 (a) the vector a + b + 2c, 5 1 (b) the unit vector in the direction of a + b + 2c. 5
units2
[3 marks]
Answer: (a) (b)
6
22 Given that cot = p , where is an obtuse angle, find sin (90 ) in terms of p.
[3 marks]
Answer: 23 A lady who has just graduated attends interviews at three companies, A, B and C. Based on her own judgement, 1 2 3 she believes that the probabilities that she will be offered a post at the companies are at A, at B and at C. 3 5 7 Calculate the probability that the lady fails to get any post. [2 marks]
Answer: 24 Three boys and three girls are to be seated in a row. Calculate the total number of arrangements if all the three boys want to sit together and all the three girls also want to sit together. [2 marks]
Answer: 25 The diagram shows the graph of a probability distribution of a continuous random variable X that is normally distributed with a standard deviation of 12. The graph is symmetrical about the vertical line AB. Calculate the area of the shaded region.B
[2 marks]
37
A 55
x
Answer:
7
ADDITIONAL MATHEMATICS
SPM Forecast PaperPAPER 2Instructions: 1 2 3 4 5 6 7
Time: 2 hours 30 minutesThis question paper consists of 3 sections: Section A , Section B and Section C. Answer all questions in Section A , four questions from Section B and two questions from Section C . Show your working. It may help you to get marks. The diagrams provided in the questions are not drawn to scale unless stated otherwise. The marks allocated for each question and sub-part of a question are shown in brackets. A list of formulae is provided. You may use a non-programmable scientific calculator.
Section A (40 marks)Answer all questions in this section. 1 Find the point of intersection of the straight line 2x 3y = 2 and the curve x2 xy + y2 = 4. [5 marks]
2 A quadratic function f is defined by f(x) = 2x2 + 10x + k, where k is a constant. (a) Express f(x) in the form a(x + p)2 + q , where a, p and q are constants. (b) Find (i) the value of k if the minimum value of f(x) is 32, (ii) the range of values of k if the graph of f(x) does not meet the x-axis. (c) State the coordinates of the minimum point of f(x) using the value of k found in (b)(i).
[2 marks]
[3 marks] [1 mark]
3 The sum of the first two terms of a geometric progression is 150. The third term exceeds the second term by 45. (a) Find the two possible values of common ratio. [4 marks] (b) Find the first term of the geometric progression in (a) whose sum to infinity exists. Hence, calculate the sum to infinity. [3 marks]
8
4 Anita measures her blood pressure 40 times and the systolic readings of her blood pressure are tabulated in a frequency table as shown. (a) For the data, calculate (i) the mean, (ii) the interquartile range without drawing an ogive. (b) Due to some technical errors of the equipment, each systolic reading has to be reduced by 3 mm Hg. State (i) the new mean, (ii) the new interquartile range.
Systolic reading (mm Hg) 120 124 125 129 130 134 135 139 140 144 145 149 150 154
Frequency 2 3 7 11 9 5 3 [2 marks] [5 marks]
5 The diagram shows a semicircle PBQ, with centre O. ABO is an isosceles triangle such that BA = BO = 10 cm. Given that ABO = 0.822 radians, find (a) BOQ in radians, (b) the area of the shaded region, (c) the perimeter of the shaded region.
B
0.822 rad P A O Q
[2 marks] [3 marks] [3 marks]
6 (a) Prove that sin2 x + tan2 x sin2 x = tan2 x. (b) Solve the equation 3 tan = 2 tan (45 ) for 0
[2 marks] 360. [5 marks]
9
Section B (40 marks)Answer four questions from this section. 7 Use the graph paper provided to answer this question. In a fungus culturing experiment, the mass, M mg, T (hours) 1 3 5 7 9 of the fungus increases with time, T hours. The M (mg) 0.51 0.84 1.04 1.17 1.28 variables M and T are related by the equation 10M = a(T + 1)b, where a and b are constants. The results of the experiment are tabulated in the table. (a) Using a scale of 2 cm to 0.2 units on both axes, plot the graph of M against log10 (T + 1) . Hence, draw the line of best fit. [5 marks] (b) Using your graph in (a), find (i) the value of a and of b, (ii) the initial mass of the fungus cultured. [5 marks]
8 Solutions to this question by scale drawing will not be accepted. In the diagram, JKLM is a trapezium. The equation of the straight line JK is 2y x = 4 and the coordinates of point L are (0, 6). (a) Find (i) the equation of the straight line JM, (ii) the coordinates of point M. (b) A point P(x, y) moves along the circumference of a circle which passes through points L and J such that LJ is the diameter of the circle. Find the equation of the locus of point P.P (x, y)
y
L (0, 6)
M
K 2y x = 4 x
[6 marks]
J
O
[4 marks]
10
9 The diagram shows the triangle OAB. The straight lines AM and OK intersect at point L. It is given that OA = 2x, OB = 14y, 1 OM : MB = 5 : 2 and AK = AB. 4 (a) Express each of the following vectors in terms of x and y. (i) OM (ii) A K (b) Given that A L = pA M and K L = qK O, express (i) A L in terms of p, x and y, (ii) K L in terms of q, x and y. (c) Using the triangle law of addition involving the vectors A K, A L and L K, find the value of p and of q.
B K A L 14y 2x M
[3 marks]O
[3 marks] [4 marks]
10 The diagram shows a shaded region bounded by the straight line PQ, the straight line x = 3, the x-axis and the curve y = hx2 + k, where h and k are constants. (a) Given that the gradient of the tangent to the curve y = hx2 + k at the point (2, 8) is 4 , find the value of h and of k. (b) Calculate the volume of the solid generated when the shaded region in the diagram is revolved through 360 about the x-axis.
y P
y = hx 2 + k
[4 marks]x=3
[6 marks]O 2 x Q
11
11 (a) A survey done by a hypermarket shows that 280 out of 500 customers require a supplementary member card for their family members. If a sample of 7 customers is chosen at random, calculate the probability that (i) exactly 3 of them require a supplementary card, (ii) less than 3 of them require a supplementary card. [5 marks]
(b) The times taken by the customers of a company to settle their invoices are normally distributed with a mean of 30 days and a standard deviation of 5 days. A discount is given for every invoice which is settled in less than 22 days. Calculate (i) the probability that an invoice is settled in 28 to 36 days, (ii) the expected number of invoices which are given discounts if there are 220 invoices. [5 marks]
Section C (20 marks)Answer two questions from this section. 12 The diagram shows the triangles PQS and QRS. (a) Calculate the area of PQS. (b) Sketch and label another QRS1 that is different from the given QRS such that the lengths of QR and QS and QRS are maintained. Hence, find the length of RS1.R 8 cm P 35
Q 10 cm 43
[4 marks]
7 cm
[6 marks]S
12
13 The following table shows the price indices of the costs of the materials needed to bake a cake. The table also shows the percentage of each material needed to bake the cake. Material Flour Sugar Egg Butter Price index for 2005 based on 2003 105 110 x 120 Percentage 50 10 20 20
The composite index of the cost for baking a cake for the year 2005 based on the year 2003 is 113.5. (a) Find the value of x. [3 marks] (b) Calculate the composite index for the year 2007 based on the year 2003 if the cost of each ingredient increased by 10% from the year 2005 to the year 2007. [1 mark] (c) The unit price of butter in the year 2005 was RM2 more than its unit price in the year 2003. Find the unit price of butter in the year 2003. [2 marks] (d) The price index of flour for the year 2005 based on the year 2001 is 112. Find the price index of flour for the year 2003 based on the year 2001. [4 marks]
14 Use the graph paper provided to answer this question. A firm produces two types of products, Mew and Zeta. Time (minutes) All products must go through two processes, i.e. Product Process I Process II Process I and Process II. The table shows the time required for Process I and Process II for Mew 5 1 each type of products. Zeta 3 1 In a certain production, the firm produces x units of Mew and y units of Zeta. The times available for 1 Process I and Process II are 5 hours and 1 hours respectively. 6 The ratio of the number of units of the Zeta produced to the number of Mew produced is not more than 4 : 3.
13
(a) Write down three inequalities, other than x 0 and y 0, which satisfy the given constraints. [3 marks] (b) Hence, using a scale of 2 cm to 10 units on both axes, construct and shade the feasible region R which satisfies all the given constraints. [3 marks] (c) Using your graph in (b), find (i) the range of the number of Mew produced if the number of Zeta produced is exactly 20, (ii) the maximum profit that can be obtained by the firm if the profits from the sales of a unit of Mew and a unit of Zeta are RM8 and RM6 respectively. [4 marks]
15 Particles A and B start to move from a fixed point O at the same time. The velocity of particle A, vA m s1, is given by vA = 12 + t t 2, where t is the time in seconds after particle A has passed through point O. The displacement of particle B, sB m, is given by sB = 2t 3 7t 2 15t, where t is the time in seconds after particle B has passed through point O. Find (a) the maximum velocity of particle A, [2 marks] (b) the displacement of particle A when particle B returns to point O, [4 marks] (c) the acceleration of particle B when particle A reverses its direction. [4 marks]
14
ANSWERSPaper 1 1 (a) 1, 3 (b) {6, 1, 4, 5} 2 (a) a = 2, b = 3 (b) 3 1 5 3 m = , k = 2 3 4 x = 1.14 or 2.64 5 (a) 4 (b) 9 (c) y = 9 1 6 m = 2, k = 2 7 h 1 5 4 1 2, 32 2 1 1 (a) 1 or 2 5 1 1 (b) 187, 156 2 4 (a) (i) 138.125 mm Hg (ii) 10.32 mm Hg (b) (i) 135.125 mm Hg (ii) 10.32 mm Hg (a) 1.982 rad. (b) 53.27 cm2 (c) 36.55 cm (b) 18.43, 116.57, 198.43, 296.57 (a) Graph of M against log (T + 1) (c)10
3
4
5
6 7
M 1.4 1.2 1.0 0.8 0.6 0.4 0.2 O 0.19 0.2 1.0 0.1 = 0.91.28 0.30 = 0.98
8 17(2x) 9 1.87 1 10 3 11 13 units2 12 1.96 13 44 cm 14 (2x 1)(6x + 3) 15 225 16 16 1 17 or 5 2 18 Y = lg y, X = x, m = lg k, c = lg p 19 60 20 3 units2 21 (a) 6i + 8j 3 4 (b) i + j 5 5 p 22 1 + p2 8 23 35 24 72 25 0.0668 Paper 2 2 6 1 2, , (2, 2) 7 7 5 2 (a) f(x) = 2 x + 2 1 (b) (i) k = 44 2 1 (ii) k 12 2
0.4
0.6
0.8
1.0 log10 (T + 1)
2
25 + k 2
(b) (i) a 1.55, b 1.09 (ii) 0.19 mg 8 (a) (i) y = 2x 8 3 1 (ii) 5, 3 5 5 (b) x2 + 4x + y2 6y = 0 9 (a) (i) 10y 1 7 (ii) x + y 2 2 (b) (i) 2px + 10py 3 1 (ii) qx qy 2 2 7 1 (c) p = , q = 22 11 10 (a) h = 1, k = 4 14 (b) 157 units3 15 11 (a) (i) 0.2304 (ii) 0.1402 (b) (i) 0.5403 (ii) 12
15
12 (a) 27.16 cm2 (b)Q
(b)
y 100 90 80 70 5x + 3y = 300 y = 4x 3
10 cm
7 cm
60 50
43 R S1 S
40 30 20 10 8 O R 15 6 10 20 30 8x + 6y = 48 48 40 50 60 70 x + y = 70 x Max (45, 25)
13 (a) (b) (c) (d) 14 (a)
RS1 = 5.737 cm 130 124.85 RM10 106.67 5x + 3y 300, x + y 4 x y 3
70,
(c) (i) 15 x 1 1 15 (a) 12 m s 4 (c) 34 m s2
48 (ii) RM510 5 (b) 30 m 6
16