Primary 6 PSLE Mathematics Semestral Assessment 2 (End of Year) Paper 2

Answer Key and Marking Scheme

Subject: Mathematics Primary 6
Topic: Whole Numbers
Paper: SA2 Practice Paper (Version 2)

Section A: Multiple Choice Questions (1 Mark Each)

1. (4)

• Reasoning: The number is 4,702,159. The digit 7 is in the hundred-thousands place.
• Value = $7 \times 100,000 = 700,000$.

2. (3)

• Reasoning:
  • Divisible by 4: Last two digits must be divisible by 4.
    • 3,216 (16 is div by 4) - Yes
    • 4,518 (18 is not div by 4) - No
    • 5,112 (12 is div by 4) - Yes
    • 6,324 (24 is div by 4) - Yes
  • Divisible by 9: Sum of digits must be divisible by 9.
    • 3,216: $3+2+1+6=12$ (No)
    • 5,112: $5+1+1+2=9$ (Yes)
    • 6,324: $6+3+2+4=15$ (No)
  • Only 5,112 satisfies both.

3. (3)

• Reasoning: Number: 8,456,721.
• Ten thousands digit is 5. The digit to its right (thousands) is 6.
• Since $6 \ge 5$, round up the ten thousands digit.
• $8,450,000 + 10,000 = 8,460,000$.

4. (2)

• Reasoning:
  • Divide 1,234 by 11.
  • $1234 \div 11 = 112$ remainder $2$.
  • To be divisible, the remainder must be 0.
  • We need to add $11 - 2 = 9$? Wait, let's recheck.
  • $11 \times 112 = 1232$.
  • $1234 - 1232 = 2$.
  • Next multiple is $1232 + 11 = 1243$.
  • $1243 - 1234 = 9$.
  • Correction: Let's check the options.
  • $1234 + 1 = 1235$ (Not div by 11)
  • $1234 + 2 = 1236$ (Not div by 11)
  • $1234 + 7 = 1241$ ($1241/11 = 112.8$)
  • $1234 + 9 = 1243$ ($1243/11 = 113$).
  • Answer is 9. Option (4).
  • Self-Correction during marking: The question asks for the smallest number.
  • $1234 \div 11 = 112$ R $2$.
  • Add $11 - 2 = 9$.
  • Answer is 9. Option (4).

5. (3)

• Reasoning:
  • $36 = 2^2 \times 3^2$
  • $54 = 2 \times 3^3$
  • $72 = 2^3 \times 3^2$
  • HCF takes the lowest power of common primes: $2^1 \times 3^2 = 2 \times 9 = 18$.

6. (3)

• Reasoning:
  • 51: Divisible by 3 ($5+1=6$). Not prime.
  • 57: Divisible by 3 ($5+7=12$). Not prime.
  • 61: Not divisible by 2, 3, 5, 7 ($7 \times 8=56, 7 \times 9=63$). Prime.
  • 63: Divisible by 9. Not prime.

7. (2)

• Reasoning:
  • $A = 2^3 \times 3^2 \times 5$
  • $B = 2^2 \times 3^3 \times 7$
  • LCM takes the highest power of all primes present.
  • Primes: 2, 3, 5, 7.
  • $2^3$ (from A), $3^3$ (from B), $5^1$ (from A), $7^1$ (from B).
  • LCM = $2^3 \times 3^3 \times 5 \times 7$.

8. (A)

• Reasoning:
  • Formula: $\text{Product of numbers} = \text{HCF} \times \text{LCM}$.
  • $1800 = 15 \times \text{LCM}$.
  • $\text{LCM} = 1800 / 15$.
  • $1800 / 15 = 120$.
  • Answer is 120. Option (1).

9. (2)

• Reasoning:
  • 45,000,000.
  • Move decimal point 7 places to the left to get 4.5.
  • $4.5 \times 10^7$.

10. (3)

• Reasoning:
  • Divisibility by 6 requires divisibility by both 2 and 3 (since $6 = 2 \times 3$ and 2, 3 are coprime).
  • It does not imply divisibility by 4 or 9.

Section B: Short Answer Questions (2 Marks Each)

11. 3,045,012

• Working:
  • Millions: 3
  • Hundred Thousands: 0
  • Ten Thousands: 4
  • Thousands: 5
  • Hundreds: 0
  • Tens: 1
  • Ones: 2
• Common Mistake: Writing 3,450,012 or missing the zero placeholders.

12. 11

• Working:
  • $5432 \div 13$
  • $54 \div 13 = 4$ rem $2$
  • $23 \div 13 = 1$ rem $10$
  • $102 \div 13 = 7$ rem $11$ ($13 \times 7 = 91$, $102-91=11$)
• Answer: 11

13. $2^2 \times 3^2 \times 5$

• Working:
  • $180 = 18 \times 10$
  • $18 = 2 \times 9 = 2 \times 3^2$
  • $10 = 2 \times 5$
  • Combine: $2 \times 3^2 \times 2 \times 5 = 2^2 \times 3^2 \times 5$.

14. 5

• Working:
  • $\text{LCM}(12, x) = 60$.
  • $12 = 2^2 \times 3$.
  • $60 = 2^2 \times 3 \times 5$.
  • $x$ must provide the factor 5.
  • Smallest $x$ is 5.
  • Check: $\text{LCM}(12, 5) = 60$. Correct.
  • Note: 15, 20, 60 are also possible, but 5 is the smallest.

15. $4.05 \times 10^5, 4.5 \times 10^5, 450,001$

• Working:
  • Convert all to standard numerals:
    • $4.5 \times 10^5 = 450,000$
    • $450,001 = 450,001$
    • $4.05 \times 10^5 = 405,000$
  • Order: $405,000 < 450,000 < 450,001$
  • Original forms: $4.05 \times 10^5, 4.5 \times 10^5, 450,001$.

Section C: Structured Questions

16. 40 m (3 Marks)

• Concept: Greatest possible distance between posts at equal intervals including corners is the HCF of the length and width.
• Working:
  • Find HCF of 120 and 80.
  • $120 = 40 \times 3$
  • $80 = 40 \times 2$
  • HCF is 40.
• Answer: 40 m
• Marking: 1 mark for identifying HCF method, 1 mark for correct calculation, 1 mark for unit/answer.

17. 9:00 a.m. (3 Marks)

• Concept: Time when events coincide again is the LCM of their intervals.
• Working:
  • Find LCM of 12, 15, 20.
  • $12 = 2^2 \times 3$
  • $15 = 3 \times 5$
  • $20 = 2^2 \times 5$
  • $\text{LCM} = 2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60$ minutes.
  • 60 minutes = 1 hour.
  • Start time: 8:00 a.m.
  • Next time: 8:00 a.m. + 1 hour = 9:00 a.m.
• Answer: 9:00 a.m.

18. 214 (3 Marks)

• Concept: Common Remainder / Negative Remainder logic.
• Analysis:
  • Remainder 4 when divided by 6 $\rightarrow$ Shortage of $6-4=2$.
  • Remainder 6 when divided by 8 $\rightarrow$ Shortage of $8-6=2$.
  • Remainder 7 when divided by 9 $\rightarrow$ Shortage of $9-7=2$.
  • The number is 2 less than a common multiple of 6, 8, and 9.
• Working:
  • Find LCM of 6, 8, 9.
  • $6 = 2 \times 3$
  • $8 = 2^3$
  • $9 = 3^2$
  • $\text{LCM} = 2^3 \times 3^2 = 8 \times 9 = 72$.
  • The number is of the form $72k - 2$.
  • Possible numbers:
    • $k=1: 72 - 2 = 70$
    • $k=2: 144 - 2 = 142$
    • $k=3: 216 - 2 = 214$
    • $k=4: 288 - 2 = 286$
  • The question asks for the smallest number if the total is less than 300. Wait, "Smallest number... if total is less than 300" usually implies finding the specific solution in a range or the absolute smallest. The absolute smallest positive integer is 70.
  • Let's re-read carefully: "What is the smallest number of buttons produced if the total is less than 300?" This phrasing is slightly ambiguous. It usually means "Find the smallest valid number." 70 is valid. 142 is valid. 214 is valid. 286 is valid. The smallest is 70.
  • Correction: Often in these problems, "smallest number" refers to the first positive solution. 70.
  • Let's check 70:
    • $70 \div 6 = 11$ R $4$. (Correct)
    • $70 \div 8 = 8$ R $6$. (Correct)
    • $70 \div 9 = 7$ R $7$. (Correct)
  • Answer is 70.
• Answer: 70

19. A = 2, B = 4 (4 Marks)

• Concept: Divisibility rules for 36 (must be divisible by 4 and 9).

• Working:

  • Number: $72A4B$.
  • Divisibility by 4: The last two digits $4B$ must be divisible by 4.
    • Possible values for B: 0, 4, 8. ($40, 44, 48$ are div by 4).
  • Divisibility by 9: Sum of digits must be divisible by 9.
    • Sum $= 7 + 2 + A + 4 + B = 13 + A + B$.
  • Test values of B:
    • If $B = 0$: Sum $= 13 + A$. For this to be div by 9, $13+A$ must be 18 or 27.
      • $13+A=18 \rightarrow A=5$. Number: 72540.
      • $13+A=27 \rightarrow A=14$ (Not a digit).
    • If $B = 4$: Sum $= 13 + A + 4 = 17 + A$.
      • $17+A=18 \rightarrow A=1$. Number: 72144.
      • $17+A=27 \rightarrow A=10$ (Not a digit).
    • If $B = 8$: Sum $= 13 + A + 8 = 21 + A$.
      • $21+A=27 \rightarrow A=6$. Number: 72648.
  • The question asks for "the values". Usually, there is a unique solution or specific constraints. Let's re-read. "Find the values of digits A and B." It implies a single pair. Did I miss a constraint?
  • Let's check the options or standard patterns. Often, "divisible by 36" problems have multiple solutions. However, if this is a standard P6 question, sometimes there's a constraint like "A and B are distinct" or similar. Without that, there are 3 solutions: (5,0), (1,4), (6,8).
  • Refinement for Exam Context: In many P6 contexts, if multiple answers exist, any valid pair is accepted, or the question implies the largest or smallest number. Let's assume the question allows any valid pair, but typically, exam questions are designed for one answer. Let's look at the structure $72A4B$.
  • Let's check if I made an arithmetic error.
  • $7+2+4 = 13$. Correct.
  • If the question implies a unique answer, perhaps I should provide the most "standard" one or list all. Given the format "A = __, B = __", it expects one pair.
  • Let's assume the question meant "largest possible number" or similar. If not specified, I will provide one valid pair and note others in the teaching notes.
  • Let's pick $A=1, B=4$ as a representative answer, but strictly speaking, (5,0) and (6,8) are also correct.
  • Self-Correction: To ensure a single answer for a practice key, I will modify the question in the mind of the grader to look for the smallest number formed? No, the question text is fixed. I will provide (1, 4) as the primary answer but acknowledge others in the notes.
  • Alternative: Is there a constraint on A? No.
  • Let's provide A=1, B=4.

• Answer: A = 1, B = 4 (Note: A=5, B=0 and A=6, B=8 are also mathematically valid).

20. (a) 100, (b) 21 (4 Marks)

• Concept: Square numbers pattern.
• Analysis:
  • Fig 1: $1^2 = 1$
  • Fig 2: $2^2 = 4$
  • Fig 3: $3^2 = 9$
  • Fig $n$: $n^2$ dots.
• Working (a):
  • Figure 10: $10^2 = 100$ dots.
• Working (b):
  • $n^2 = 441$.
  • $n = \sqrt{441}$.
  • We know $20^2 = 400$. $21^2 = (20+1)^2 = 400 + 40 + 1 = 441$.
  • So, $n = 21$.
• Answer (a): 100
• Answer (b): 21