REVISION FOR SEMESTER 1 EXAMINATION CHAPTER 1: LINEAR PROGRAMMING 01 A factory makes two types of lock, standard and large, on a particular day. On that day: the maximum number of standard locks that the factory can make is 100; the maximum number of large locks that the factory can make is 80; the factory must make at least 60 locks in total the factory must make more large locks than standard locks. Each standard lock requires 2 screws and each large lock requires 8 screws and, on the day, the factory must use at least 320 screws. On that day, the factory makes x standard locks and y large locks. (a) Translate the conditions above as inequalities, in simplified form. Answer: ………………………………………… [1] ………………………………………… [1] ………………………………………… [1] ………………………………………… [1] ………………………………………… [1] (b) Draw a suitable diagram and by shading the unwanted, label the region R which satisfies all the conditions. [4] (c) Each standard lock costs $1.50 to make and each large lock costs $3 to make. The manager of the factory wishes to minimize the cost of making the locks. Determine the values of x and y that gives the minimum cost. Hence, find this minimum cost. Answer: ………………………………………… [3] NAME: ___________________________ CLASS: _____________
Transcript
REVISION FOR SEMESTER 1 EXAMINATION CHAPTER 1: LINEAR PROGRAMMING 01 A factory makes two types of lock, standard and large, on a particular day. On that day: the maximum number of standard locks that the factory can make is 100; the maximum number of large locks that the factory can make is 80; the factory must make at least 60 locks in total the factory must make more large locks than standard locks. Each standard lock requires 2 screws and each large lock requires 8 screws and, on the day, the factory must use at least 320 screws. On that day, the factory makes x standard locks and y large locks.
(a) Translate the conditions above as inequalities, in simplified form.
Answer: ………………………………………… [1]
………………………………………… [1]
………………………………………… [1]
………………………………………… [1]
………………………………………… [1]
(b) Draw a suitable diagram and by shading the unwanted, label the region R which satisfies all the conditions. [4]
(c) Each standard lock costs $1.50 to make and each large lock costs $3 to make. The manager of the factory wishes to minimize the cost of making the locks. Determine the values of x and y that gives the minimum cost. Hence, find this minimum cost.
02 Each year, farmer Giles buys some goats, pigs and sheep. He must buy at least 110 animals.
(a) This year, Giles buys x goats, y pigs and 30 sheep. Show that this information will simplify to x + y > 80. [1]
(b) He must buy at least as many pigs as goats. Express this information as an inequality in simplest form.
Answer: ………………………………………… [1]
(c) The total of the number of pigs and the number of sheep that he buys must not be greater than 150. Express this information as an inequality in simplest form.
Answer: ………………………………………… [1]
(d) Each goat costs $16, each pig costs $8 and each sheep costs $24. He has $3120 to spend on the animals. Express the information above as an inequality and show that it simplifies to 2x + y < 300. [1]
(e) On the grid provided, show the graphs of the inequalities in part (a), (b), (c) and (d). [4]
(f) At the end of the year, Giles sells all of the animals. He makes a profit of $70 on each goat, $30 on each pig and $50 on each sheep. Giles wishes to maximize his total profit, $P. Find Giles’ maximum profit for this year and the number of each animal he must buy to obtain this maximum profit.
Answer: ………………………………………… [3]
CHAPTER 2: FUNCTIONS 03 0 0 0 Figure 1 Figure 2 Figure 3 0 0 0 Figure 4 Figure 5 Figure 6 Which of the above could be the graph of
(a) Answer: ………………………………………… [1]
(b)
Answer: ………………………………………… [1]
(c) Answer: ………………………………………… [1]
xy 1-=
xy -=1
12 += xy
04 The variables x and y are connected by the equation 𝑦 = #
$(2 – x)(5+ x).
The table below shows some values of x and the corresponding values of y.
(a) Find the values of a and b.
Answer: ………………………………………… [1]
(b) Calculate the coordinates of the minimum point of the curve.
Answer: ………………………………………… [1]
(c) Using a scale of 2 cm to 1 unit on each axis, draw the graph of the equation y = #$ (2 – x)(5+ x). [4]
(d) Use your graph to find the value/s of
(i) y when x = 0.8
Answer: ………………………………………… [1]
(ii) x when y = 1.5
Answer: ………………………………………… [1]
(e) Construct a straight line to solve the equation (2 – x)(5+ x) = – 4x + 8.
Answer: ………………………………………… [1]
(f) Write the range of values of x in (2 – x)(5+ x) > – 4x + 8
Answer: ………………………………………… [1]
x -5 -4 -3 -2 -1 0 1 2 3 y 0 3 a 6 6 5 3 0 b
05 The variable 𝑥 and 𝑦 are connected by the equation
𝑦 =𝑥$
6 +12𝑥 − 6
Some corresponding values, correct to one decimal place, are given in the following table.
(b) Using a scale of 2 cm to represent 1 unit on each axis, draw a horizontal 𝑥-axis for 0 ≤ 𝑥 ≤ 8 and a
vertical 𝑦-axis for −1 ≤ 𝑦 ≤ 7. On the same axes, plot the points given on the table and join them with a smooth curve.
(c) By drawing a tangent, find the gradient of the curve at the point (1.5, 2.4).
Answer: ………………………………………… [1]
(d) By drawing a straight line on the axes, solve 0
1
2+ #$
0= 7
Answer: ………………………………………… [1]
(e) On the same axes, draw the graphs of the straight line 𝑦 = 03.
(f) Using the graph (e), find the values of 𝑥 in the range 1 ≤ 𝑥 ≤ 7 for which 01
2+ #$
0− 6 ≤ 0
3.
Answer: ………………………………………… [1]
06 Given that f(x) = 2x – 3 g(x) = #
05#+ 2 h(x) = 3x
(i) Work out f(4).
Answer: ………………………………………… [1] (ii) Work out this fh(−1)
Answer: ………………………………………… [2] (iii) Find f-1(x), the inverse of f(x).
Answer: ………………………………………… [2] (iv) Find ff(x) in its simplest form.
Answer: ………………………………………… [2]
(v) Show that the equation f(x) = g(x) simplifies to 2x2 – 3x – 6 = 0. [3] (vi) Solve the equation 2x2 – 3x – 6 = 0. Give your answers correct to 2 decimal places. Show all your working.
Answer: ………………………………………… [4] 07 f(x) = x + $
0 – 3, x ≠ 0 g(x) = 0
$− 5
Find
(a) fg(18)
Answer: ………………………………………… [2]
(b) g-1(x)
Answer: ………………………………………… [2]
08 Below is the graph of 𝑦 = −4(20).
(i) Draw a line tangent to the curve at x = -2
(ii) Find the gradient of the curve at this point.
Answer……….……….…………………………… [2]
CHAPTER 3: PROBABILITY 09 There are 10 blue balls, some red balls and some yellow balls in a bag. When one ball is selected at random, the probability that it is blue is #
2 and the probability that it is yellow is #
;.
(a) Find the number of red balls.
Answer……….……….…………………………… [2]
(b) When x red balls are added to the bag, the probability of selecting a yellow ball decreased by <##;
. Calculate x.
Answer……….……….…………………………… [3] 10 A bag contains balls that are red, blue, green or yellow.
• The number of red balls is x – 3. • The number of blue balls is 2x. • The number of green balls is 7. • The number of yellow balls is 5x.
A counter is chosen at random. The probability it is green is =
;$.
Work out the probability it is red.
Answer……….……….…………………………… [3]
11 A bag contains 15 identical discs. There are 8 red, 4 blue and 3 white discs. A disc is picked out at random and not replaced. A second disc is then picked out of random and not replaced. A tree diagram below shows the possible outcomes. 1st Disc 2nd Disc
> =
#3? Red
> 3
#3?
Red Blue q > @
#;? White
> @
#3? Red
> 3
#;? > <
#3?
Blue Blue > <
#3?
White 𝒑 > @
#3? Red
𝒓 White Blue s White
(a) Calculate the value of p, q, r and s.
Answer: p =………….…… q =…………………….
r =….……..…...… s =……………………. [3]
(b) Expressing your answers as a fraction in its lowest terms, find the probability that
(i) both discs will be red,
Answer………….………………………………. [1]
(ii) one disc will be red and the other is blue,
Answer………….………………………………. [2]
(iii) no white disc will be drawn.
Answer………….………………………………. [2]
CHAPTER 4: STATISTICS 12 200 candidates sat for a Mathematics examination. The marks of the candidates are shown in the table below.
Marks (m) 0 < m ≤ 40 40 < m ≤ 50 50 < m ≤ 70 70 < m ≤ 100
Frequency 76 38 56 30
(a) Draw the histogram of the data and label the axes. [3]
(b) Charlize has a different frequency table.
Marks(m) Frequency
0 < m ≤ 10 10
10 < m ≤ 20 14
20 < m ≤ 30 22
30 < m ≤ 40 30
40 < m≤ 50 42
50 < m ≤ 60 32
60 < m ≤ 70 24
70 < m ≤ 80 14
80 < m ≤ 90 8
90 < m ≤ 100 4
Answer this question on the grid provided on the next page. Taking 1 cm to represent 10 marks on 𝑥-axis and 1 cm to represent 10 candidates on the 𝑦- axis. Draw the cumulative frequency graph of the table.
(c) Use the graph to find
(i) median mark,
Answer………….…………………………… [1] (ii) interquartile range,
Answer………….…………………………… [2]
(iii) the 40th percentile
Answer………….…………………………… [2]
(iv) the number of candidates who scored at least 75 marks in mathematics examination.
Answer………….……………………………. [2]
CHAPTER 5: VECTORS 13 OPMQ is a parallelogram and O is the origin. 𝑂𝑃EEEEE⃗ = p and 𝑂𝑄EEEEEE⃗ = q. L is on PQ so that PL : LQ = 2:1.
Find the following vectors in terms of p and q. Write your answers in their simplest form.
(i) 𝑃𝑄EEEEE⃗
Answer……….……….…………………………… [1]
(ii) 𝑃𝐿EEEE⃗
Answer……….……….…………………………… [1]
(iii) 𝑀𝐿EEEEEE⃗
Answer……….……….…………………………… [2]
14 In the figure below, OABC is a parallelogram. AC is extended to D such that
AC: CD= 1:2. OB intersects AC at E. It is given that = a and = c.
(a) Express each of the following, in terms of a and/or c.
(i)
Answer……….……….…………………………… [2] (ii)
Answer……….……….…………………………… [2]
(iii)
Answer……….……….…………………………… [2]
(iv)
Answer……….……….…………………………… [2] (v)
Answer……….……….…………………………… [2]
OA OC
OE
CD
OD
DB
CE
(b) OC produced cuts BD at F. Given that = hc and = k , find the value of h and of k.
Answer……….……….…………………………… [4] (c) Find the value of
(i) ,
Answer……….……….…………………………… [1]
(ii)
Answer……….……….…………………………… [2]
OF BF BD
FDBF
,CDFAreaofODCAreaofDD
15
Given that u = , v = and w = , find
(a) ,
Answer……….……….…………………………… [2]
(b) u + 3v
Answer……….……….…………………………… [2]
(a) Given that the vector w is parallel to the vector u, calculate the value of p.
Answer……….……….…………………………… [2] 16 In the diagram, 𝐷𝐶EEEEE⃗ = p, 𝐴𝐵EEEEE⃗ = 4p, 𝐴𝐷EEEEE⃗ = 3p and M is the midpoint of BC. Express the following vectors in terms of p and/or q.
(a) 𝐴𝐶EEEEE⃗
Answer……….……….…………………………… [1]
(b) 𝐵𝑀EEEEEE⃗
Answer……….……….…………………………… [2]
÷÷ø
öççè
æ68
÷÷ø
öççè
æ- 34
÷÷ø
öççè
æp16
u
D C
A B
p
3q
4p
M
17
In the diagram, = =2 and =2 . = and = .
(a) Express, as simply as possible, in terms of p and/or q,
(i)
Answer ....………………………………………… [1] (ii)
Answer ....…………………………………………. [2] (iii)
Answer ....…………………..……………………… [1]
(iv)
Answer ....………………………………………..… [2]
(b) Hence write down a fact about O, P and T.
Answer ....……………………………………….………….. [1]
OQ QRQS , PQ ST RS OP p PQ q
OQ
RS
OS
OT
18
Relative to an origin O, the position vectors of the points A and B are > 5−12?
and > 8−8? respectively.
Find
(i) 𝐴𝐵EEEEE⃗ ,
Answer……….……….…………………………… [2] (ii) N𝐴𝐵EEEEE⃗ N,
Answer……….……….…………………………… [2]
(iii) the unit vector in the direction of AB in the form xi + yj.
Answer……….……….…………………………… [2]
19
Given that A and B are the points (4, -2) and (-3, 5) respectively and 𝐵𝐶EEEEE⃗ = >64?.
(a) Express 𝐴𝐵EEEEE⃗ as a column vector.
Answer……….……….…………………………… [2] (b) Find N𝐴𝐵EEEEE⃗ N.
Answer……….……….…………………………… [2] (c) Find the coordinates of C.
Answer……….……….…………………………… [2]
20
The vector k = >2 − 𝑚4− 𝑚? has a magnitude of 6 units. (a) Form an equation in m and show that it simplifies to 𝑚$ − 6𝑚 − 8 = 0
[2] (b) Hence, find the values of m.
Answer……….……….…………………………… [2] 21
If a = >37?, b = > 2−4? and c = >−3−4?, find (a)
(i) 2a
Answer……….……….…………………………… [1] (ii) b – c
Answer……….……….…………………………… [2] (b)
(i) |𝐚 − 𝐛|
Answer……….……….…………………………… [2] (ii) |𝐛 − 𝐜|
Answer……….……….…………………………… [2]
22 In the diagram, BC = 4BD and DA = 5DX. M is the midpoint of AC. 𝐵𝐷EEEEEE⃗ = 𝒂 and 𝐶𝑀EEEEEE⃗ = 2𝒃.
(a) Express, as simply as possible, in terms of a and/or b,
(i) 𝐷𝐶EEEEE⃗ ,
Answer……….……….…………………………… [1]
(ii) 𝐷𝐴EEEEE⃗ ,
Answer……….……….…………………………… [1] (iii) 𝐷𝑋EEEEE⃗ .
Answer……….……….…………………………… [1]
(b) Show that 𝐵𝑋EEEEE⃗ = 3;(2𝒂 + 𝒃).
[2]
(c) Express 𝐵𝑀EEEEEE⃗ as simply as possible, in terms of a and b.
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