Upload
shanmuganantha-kathiravelu
View
116
Download
4
Embed Size (px)
Citation preview
1
Answer all questions.[80 marks]
Instructions: 1 This question paper consists of 25 questions. 2 Answer all questions. 3 Write your answers clearly in the spaces provided in the question paper. 4 Show your working. It may help you to get marks. 5 The diagrams provided in the questions are not drawn to scale unless stated otherwise. 6 The marks allocated for each question are shown in brackets. 7 A list of formulae is provided. 8 You may use a non-programmable scientific calculator.
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 =
ADDITIONAL MATHEMATICSSPM Forecast Paper
Time: 2 hoursPAPER 1
2
3 Given the function g(x) = 3x + k and its inverse g–1(x) = mx – 5 —6
, find the value of k and of m. [3 marks]
Answer: k =
m =
4 Solve the equation x2 – 3 —2
x = 3, giving your answers correct to two decimal places. [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. [3 marks]
Answer: (a) k =
(b) h =
(c)
y
xO
(k, 9)
3
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 = 32x – 1, giving your answer correct to two decimal places. [4 marks]
Answer: x =
4
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 diagram is
x —4
– y —3
= 1, find the area of �PQR. [3 marks]
Answer: units2
12 The table shows the distribution of the sizes of a certain brand of long pants sold at a shop.
Calculate the variance. [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 π = 3.142] [3 marks]
Answer: cm
y
xO P
Q
R(2, 5)
20°O
B
C
Size Frequency
30 3
32 5
34 2
5
14 Given that y = (x + 1)(2x – 1)2, find dy—–dx
. [4 marks]
Answer: dy—–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 =
6
18 If the non-linear equation y—–k x
= p, where k and p are constants, is reduced to the linear form, Y = mX + c,
state Y, X, m and c in terms of x, y, k and/or p. [4 marks]
Answer: Y =
X =
m =
c =
19 Given that �4–1g(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. [3 marks]
Answer: units2
21 Given that a = 7j, b = 10i – 5j and c = – 4i + j, find in terms of i and j,
(a) the vector a + 1 —5
b + 2c,
(b) the unit vector in the direction of a + 1 —5
b + 2c. [3 marks]
Answer: (a)
(b)
–1 1 2Ox
y
y = x 2 – 2x + 1
7
22 Given that cot θ = p , where θ is an obtuse angle, find sin (90° – θ) 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,
she believes that the probabilities that she will be offered a post at the companies are 1 —3
at A, 2 —5
at B and 3 —7
at C.
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. [2 marks]
Answer:
xA
B
37 55
8
Instructions: 1 This question paper consists of 3 sections: Section A , Section B and Section C. 2 Answer all questions in Section A , four questions from Section B and two questions from Section C . 3 Show your working. It may help you to get marks. 4 The diagrams provided in the questions are not drawn to scale unless stated otherwise. 5 The marks allocated for each question and sub-part of a question are shown in brackets. 6 A list of formulae is provided. 7 You may use a non-programmable scientific calculator.
Section A (40 marks)
Answer all questions in this section.
ADDITIONAL MATHEMATICSSPM Forecast Paper
Time: 2 hours 30 minutesPAPER 2
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. [2 marks](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. [3 marks](c) State the coordinates of the minimum point of f(x) using the value of k found in (b)(i). [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]
9
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. [5 marks](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. [2 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, [2 marks](b) the area of the shaded region, [3 marks](c) the perimeter of the shaded region. [3 marks]
6 (a) Prove that sin2x + tan2x sin2x = tan2x. [2 marks](b) Solve the equation
3 tan θ = 2 tan (45° – θ) for 0° � θ � 360°. [5 marks]
Systolic reading (mm Hg)
Frequency
120 – 124 2
125 – 129 3
130 – 134 7
135 – 139 11
140 – 144 9
145 – 149 5
150 – 154 3
P A O Q
B
0.822 rad
10
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, of the fungus increases with time, T hours. The 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.
[6 marks](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. [4 marks]
T (hours) 1 3 5 7 9
M (mg) 0.51 0.84 1.04 1.17 1.28
y
x
K
L (0, 6)P (x, y)
M
J O
2y – x = 4
11
9 The diagram shows the triangle OAB. The straight lines AM and OK intersect at point L. It is given that O
→A = 2x, O
→B = 14y,
OM : MB = 5 : 2 and AK =
1 —4
AB.
(a) Express each of the following vectors in terms of x and y.
(i) O→
M (ii) A→
K [3 marks]
(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. [3 marks](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. [4 marks]
10 The diagram shows a shaded region bounded bythe 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. [4 marks]
(b) Calculate the volume of the solid generated when the shaded region in the diagram is revolved through 360° about the x-axis. [6 marks]
A
O
K
LM
B
2x
14y
y
P
Q
xO 2
x = 3
y = hx 2 + k
12
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. [4 marks](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. [6 marks]
10 cm
8 cm
7 cm
43°
35°
R S
Q
P
13
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.
MaterialPrice index for 2005 based
on 2003Percentage
Flour 105 50
Sugar 110 10
Egg x 20
Butter 120 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. All products must go through two processes, i.e.‘Process I’ and ‘Process II’. The table shows the time required for ‘Process I’ and ‘Process II’ foreach type of products.
In a certain production, the firm produces x units of Mew and y units of Zeta. The times available for
‘Process I’ and ‘Process II’ are 5 hours and 1
1 —6
hours respectively.
The ratio of the number of units of the Zeta produced to the number of Mew produced is not more than 4 : 3.
ProductTime (minutes)
Process I Process II
Mew 5 1
Zeta 3 1
14
(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 s–1, 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]
15
ANSWERS
Paper 1 1 (a) 1, 3
(b) {6, –1, –4, –5} 2 (a) a = 2, b = –3
(b) 3
3 m = 1 —2
, k = 5 —3
4 x = –1.14 or 2.64 5 (a) 4
(b) 9 (c) y = 9 6 m = 2, k = – 1 —
2
7 h � 5 1 —
4
8 17(2x) 9 1.87
10 1 —3
11 13 units2
12 1.96 13 44 cm 14 (2x – 1)(6x + 3) 15 225 16 16
17 – 1 —2
or 5
18 Y = lg y, X = x, m = lg k, c = lg p 19 60 20 3 units2
21 (a) –6i + 8j
(b) – 3 —5
i + 4 —5
j 22 – p —–––––
1 + p2
23 – 8 —–35
24 72 25 0.0668
Paper 2 1 �2 2 —
7, 6 —
7 �, (–2, –2)
2 (a) f(x) = 2�x + 5 —2 �
2
– 25 —–2
+ k
(b) (i) k = 44 1 —2
(ii) k � 12 1 —
2
(c) �–2 1 —2
, 32�
3 (a) 1 1 —2
or – 1 —5
(b) 187 1 —2
, 156 1 —4
4 (a) (i) 138.125 mm Hg (ii) 10.32 mm Hg (b) (i) 135.125 mm Hg (ii) 10.32 mm Hg 5 (a) 1.982 rad. (b) 53.27 cm2
(c) 36.55 cm 6 (b) 18.43°, 116.57°, 198.43°, 296.57° 7 (a)
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.2 0.4 0.6 0.8 1.0
M
log10 (T + 1)O
0.191.0 – 0.1 = 0.9
1.28 – 0.30= 0.98
Graph of M against log10 (T + 1)
(b) (i) a � 1.55, b � 1.09 (ii) 0.19 mg 8 (a) (i) y = –2x – 8 (ii) �–5 3 —
5, 3 1 —
5 � (b) x2 + 4x + y2 – 6y = 0 9 (a) (i) 10y
(ii) – 1 —2
x + 7 —2
y
(b) (i) –2px + 10py
(ii) – 3 —2
qx – 1 —2
qy
(c) p = 7 —–22
, q = 1 —–11
10 (a) h = 1, k = 4 (b) 15714 —–
15π units3
11 (a) (i) 0.2304 (ii) 0.1402 (b) (i) 0.5403 (ii) 12
16
12 (a) 27.16 cm2
(b)
Q
R S1 S
43°
10 cm7 cm
RS1 = 5.737 cm 13 (a) 130
(b) 124.85(c) RM10(d) 106.67
14 (a) 5x + 3y � 300, x + y � 70, y � 4 —
3x
(b)
(c) (i) 15 � x � 48 (ii) RM510
15 (a) 12 1 —4
m s–1 (b) 30 5 —6
m
(c) 34 m s–2
10 20 30 40 50 60 70
100
90
80
70
60
50
40
30
20
10
O
y
x
5x + 3y = 300
8x + 6y = 48
x + y = 70
Max (45, 25)
y = x43
R
48156
8