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QUEEN’S COLLEGE

Half-Yearly Examination, 2011 – 2012

MATHEMATICS PAPER 1

Marking Scheme

Secondary 2 Date: 12– 1 – 2012

Time: 8:30 am – 9:45 am

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1. Write your class, class number in the spaces provided on this cover.

2. This paper consists of TWO sections, A and B. Section A and Section B carry 80 marks and 40 marks respectively.

3. Attempts ALL questions in this paper. Write your answer in the spaces provided in this Question-Answer Book.

4. Unless otherwise specified, all working must be clearly shown.

5. Unless otherwise specified, numerical answers should either be exact or correct to 3 significant figures.

6. The diagrams in this paper are not necessarily drawn to scale.

|Class | | |

|Class Number | | |

| |Teacher’s Use Only |

|Question No. |Max. Marks |Marks |

|Section A: Short Questions |

|1 |8 | |

|2 |7 | |

|3 |9 | |

|4 |7 | |

|5 |8 | |

|6 |8 | |

|7 |10 | |

|8 |11 | |

|9 |12 | |

|Section B: Long Questions |

|11 |20 | |

|12 |20 | |

|Total: |120 | |

|Section A: Short Questions. (80 marks) |

|1. |Find the rates of the following. Express your answers in the units stated in the bracket. | |

| | |(8 marks) |

| |(a) |Kenny finished the 100 m race in 12.5 seconds. [m/s] | |

| |(b) |The annual salary of Kate is $174 000. [$/month] | |

| |(c) |The selling price for 400 g of beef is $11.2. [$/kg] | |

| |(d) |0.5 L of water weighs 6.8 kg.(Given:1 L = 1000[pic]) [g/[pic]] | |

| | |(a) | |

| | |[pic] 1M, 1A | |

| | | | |

| | |(b) | |

| | |[pic] 1M, 1A | |

| | | | |

| | |(c) | |

| | |[pic] 1M, 1A | |

| | | | |

| | |(d) | |

| | |[pic] 1M, 1A | |

| | | | |

|2. If [pic], find the values of P, Q, R and S. (7 marks) |

| | | | |

| | |[pic] | |

| | | | |

| | | |1M:expansion, 1A |

| | | |1 M:comparing coefficients |

| | | | |

| | | | |

| | | |1A,1A,1A,1A |

|3. Let S[pic] be the surface area of a cuboid. Its length, width and height are | |

|p cm, 2q cm and 3r cm respectively. | |

| |(a) |Express S in terms of p, q and r. |(4 marks) |

| |(b) |Make r the subject of the formula in (a). |(3 marks) |

| |(c) |Hence find the value of r if S = 94, p = 5, q = 2. |(2 marks) |

| | | | |

| |(a) |[pic] |1A, 1A, 1A |

| | | |1A |

| | | | |

| |(b) |[pic] | |

| | | |1M |

| | | |1M:factorization |

| | | |1A |

| | | | |

| | | | |

| | | | |

| | | | |

| | | | |

| | | |1M |

| | | |1M:factorization |

| | | | |

| | | | |

| | | | |

| | | | |

| | | | |

| | | | |

| | | |1A |

| |(c) |[pic] | |

| | | | |

| | | |1M |

| | | | |

| | | | |

| | | | |

| | | |1A |

|Find the quotient and remainder of [pic]. Express the |

|answer in descending powers of x. (7 marks) |

| | |[pic] |1: descending |

| | | |power of x |

| | |[pic] | |

| | | | |

| | | |1: place holder |

| | | | |

| | | |1M, 1A |

| | | | |

| | | |1A |

| | |The quotient =[pic] |1A |

| | |The remainder =[pic] |1A |

| |

|5. |

| |

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| |

| |

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| |

| |

| |

| |

| |

| |

| |

|The above frequency polygon shows the lengths (in cm) of a group of metal bars produced |

|by a machine. |

| |(a) |Complete the table below. |(3 marks) |

| | |Length(cm) |

| | |5-9 |

| | |10-14 |

| | |15-19 |

| | | |

| | | |

| | |30-34 |

| | |35-39 |

| | |40-44 |

| | |45-49 |

| | | |

| | |Frequency |

| | |3 |

| | | |

| | | |

| | |13 |

| | | |

| | |8 |

| | | |

| | | |

| | | |

| | | |

| |(b) |How many metal bars are there in the group ? |(2 marks) |

| |(c) |If metal bars of lengths less than 24.5 cm are rejected, how many bars in the group are accepted ? |(3 marks) |

| |(a) |Length(cm) |1:any |

| | |5-9 |3 correct |

| | |10-14 |2:any 6 correct|

| | |15-19 |3: all correct |

| | |20-24 | |

| | |25-29 | |

| | |30-34 | |

| | |35-39 | |

| | |40-44 | |

| | |45-49 | |

| | | | |

| | |Frequency | |

| | |3 | |

| | |11 | |

| | |9 | |

| | |13 | |

| | |21 | |

| | |8 | |

| | |6 | |

| | |5 | |

| | |4 | |

| | | | |

| |(b) |No of metal bars = 3+11+9+13+21+8+6+5+4 |1M |

| | |=80 |1A |

| | | | |

| |(c) |No of metal bars = 80 – (3+11+9+13) or 21+8+6+5+4 |1M |

| | |= 44 |1A |

|6. |Simplify the following expressions: | |

| |(a) |[pic] |(4 marks) |

| |(b) |[pic] |(4 marks) |

| |(a) |[pic] | |

| | | | |

| | | |1M |

| | | | |

| | | |1: “–“, 1A |

| | | | |

| | | |1A |

| | | | |

| |(b) |[pic] | |

| | | | |

| | | | |

| | | |1M:factorization,1A |

| | | |1M: for[pic] |

| | | | |

| | | |1A |

|7. |The thickness of 500 sheets of paper of the same quality is 46.2 mm and given that the relative error is | |

| |[pic]. | |

| |(a) |Find the percentage error of the thickness of all papers. (Correct your answer to 3 sig.fig.) |(2 marks) |

| | |Find the actual range of thickness of all papers. | |

| |(b) |Find the actual range of thickness of each sheet of paper. (Correct your answer to 3 sig.fig.) |(6 marks) |

| |(c) | | |

| | | |(2 marks) |

| | | | |

| | | | |

| |(a) |[pic] | |

| | | |1M |

| | | |1A |

| | | | |

| |(b) |[pic] | |

| | | |1M:max error |

| | | | |

| | | | |

| | | |1A |

| | | | |

| | | | |

| | | | |

| | | |1A |

| | | | |

| | | |1A |

| | | | |

| | | |1M:range, 1A |

| | | | |

| |(c) |[pic] | |

| | | | |

| | | |1A, 1A |

| | | | |

|8. |Simplify the following expressions and express the answers with positive indices. | |

| |(a) |[pic] |(5 marks) |

| |(b) |[pic] |(6 marks) |

| |(a) |[pic] | |

| | | | |

| | | | |

| | | |1M |

| | | | |

| | | |1A, 1A |

| | | | |

| | | | |

| | | |1M |

| | | | |

| | | |1A |

| | | | |

| |(b) |[pic] | |

| | | | |

| | | | |

| | | |1M, 1A |

| | | | |

| | | | |

| | | |1M, 1A |

| | | | |

| | | |1A |

| | | | |

| | | | |

| | | | |

| | | |1A |

| | | | |

|9. (a) Given that 2a = 3b. Find a : b. (2 marks) |

| |(b) |Given that [pic]. Find a : b : c. |(7 marks) |

| |(c) |Given that [pic]. Find x : y. |(3 marks) |

| | | | |

| |(a) | [pic] |1M:fractional |

| | | |form |

| | | |1A |

| | | | |

| |(b) |[pic] | |

| | | |1M:set up 2 ratios |

| | | | |

| | | | |

| | | | |

| | | | |

| | | |1M:multiplication |

| | | | |

| | | | |

| | | |1A: a:b |

| | | | |

| | | | |

| | | | |

| | | |1A |

| | | | |

| | | | |

| | | |1A |

| | | | |

| | | | |

| | | | |

| | | | |

| | | |1M |

| | | |1A |

| | | | |

| |(c) |[pic] | |

| | | | |

| | | |1M |

| | | | |

| | | | |

| | | |1A |

| | | | |

| | | | |

| | | | |

| | | | |

| | | |1A |

|Section B: Long Questions. (40 marks) |

|10. (a) Expand [pic] (6 marks) |

| |(b) |If a + b = 4 and ab = 3, find the value of [pic]. |(4 marks) |

| |(c) |Factorize [pic]. |(7 marks) |

| |(d) |Using the result of (b) and (c), find the value of [pic]. |(3 marks) |

| | | | |

| |(a) |[pic] | |

| | | | |

| | |OR |1M, 2A |

| | |[pic] |1A |

| | | |2A |

| | | | |

| | | | |

| | | | |

| | | |1M, 2A |

| | | |1A |

| | | | |

| | | |2A |

| | | | |

| |(b) |By (a), | |

| | |[pic] | |

| | | |1M:using (a) |

| | | |1M:substitution |

| | | | |

| | | |2A |

| | | | |

| |(c) |[pic] | |

| | | |1M:1M:Grouping |

| | | |Te terms |

| | | |1A,1A: factor a,b |

| | | |1A:” – “ |

| | | |1M:Identity,1A1A |

| | | | |

| |(d) |[pic] | |

| | | | |

| | | |1M: using (c) |

| | | |1M:using (b) |

| | | |1A |

|11. x and y are the units digit and the tens digit of a 2-digit number P respectively. | |

| |(a) |Express P in terms x and y. |(2 marks ) |

| |(b) |Q is another 2-digit number which is formed by reversing the digits of P. Express Q in terms of x and y. |(2 marks) |

| |(c) |P is three times the sum of its digits. Find the equation involving x and y. |(4 marks) |

| |(d) |45 added to P is equal to Q. Find another equation involving x and y. |(4 marks) |

| |(e) |By solving the equations in (c) and (d) simultaneously, find the number Q. |(8 marks) |

| | | | |

| |(a) |P = 10y + x |1A+1A |

| | | | |

| |(b) |Q = 10x + y |1A+1A |

| | | | |

| |(c) |10y + x = 3(x + y) |1M, 1A |

| | |10y + x = 3x +3y | |

| | |2x – 7y = 0 |2A |

| | | | |

| |(d) |10y + x + 45 = 10x + y |1M, 1A |

| | |9x – 9y = 45 (or x – y = 5) |2A |

| |(e) |[pic][pic] | |

| | | | |

| | |Q = 72 | |

| | | |2M: correct |

| | | |method |

| | | |1A |

| | | |1A |

| | | | |

| | | |1M:substituti |

| | | |1A |

| | | | |

| | | |1A |

| | | | |

| | | |1 |

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