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primary 6 | Maths
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i need the answers for questions 5 and 7. it would be great if you could explain it to me in detail and point out the topics tested for both the questions. Thank you.
Date Posted: 6 months ago
Views: 72
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5. Topic: Permutations and Combinations
Two possible cases:
Case 1: 2 Dots on line A, 1 Dot on line B
Fix a dot on the triangle to be any dot in line B, now consider the scenarios for the remaining 2 dots.
For leftmost dot on line A, there are 4 possible ways to select the second dot on line A.
For the second leftmost dot on line A, there are 3 possible ways to select the second dot on line A.
This process repeats until the two rightmost dots on line A, where there is 1 possible way to select the second dot on line A (since we already know the two dots must be the rightmost ones).
Hence, for a fixed dot on line B, no. of ways to form a triangle = 4+3+2+1= 10
Since there are 8 dots on line B, total no. of ways for case 1 = 8 x 10 = 80
Case 2: 1 Dot on line A, 2 dots on line B
Using a similar method as case 1, we can calculate the total no. of ways for case 2
= 5 x (7+6+5+4+3+2+1)
= 140
Total no. of ways
= 80 + 140
=220
7. Topic: Arithmetic Sequences/Sum
1²-2²= -3, 3²-4²= -7, 5²-6²= -11
There must be 2018/2 = 1009 sets of subtraction of squares
Each subtraction of squares decreases by -4 of the previous subtraction, starting from the second subtraction (3²-4²)
So 2017²-2018²=-3-4(1008)
1²-2²+3²-4²+…+2017²-2018²
=-3-3-4(1)-3-4(2)-…-3-4(1008)
=1009(-3)-4(1+2+…+1008)
=-3027-4(1008)(1+1008)/2
=-3027-2016(1009)
=-2037171
OR
Using a²-b² = (a-b)(a+b),
1²-2²+3²-4²+…+2017²-2018²
=(1-2)(1+2)+(3-4)(3+4)+…+(2017-2018)(2017+2018)
=-(3+7+…+4035)
=-[3+3+4(1)+3+4(2)+…+3+4(1008)]
=-[1009(3)+4(1+2+…+1008)]
=-3027-4(1008)(1+1008)/2
=-3027-2016(1009)
=-2037171
(No calculator tip: Use 2016x1009=2016 x 1010 -2016)
Two possible cases:
Case 1: 2 Dots on line A, 1 Dot on line B
Fix a dot on the triangle to be any dot in line B, now consider the scenarios for the remaining 2 dots.
For leftmost dot on line A, there are 4 possible ways to select the second dot on line A.
For the second leftmost dot on line A, there are 3 possible ways to select the second dot on line A.
This process repeats until the two rightmost dots on line A, where there is 1 possible way to select the second dot on line A (since we already know the two dots must be the rightmost ones).
Hence, for a fixed dot on line B, no. of ways to form a triangle = 4+3+2+1= 10
Since there are 8 dots on line B, total no. of ways for case 1 = 8 x 10 = 80
Case 2: 1 Dot on line A, 2 dots on line B
Using a similar method as case 1, we can calculate the total no. of ways for case 2
= 5 x (7+6+5+4+3+2+1)
= 140
Total no. of ways
= 80 + 140
=220
7. Topic: Arithmetic Sequences/Sum
1²-2²= -3, 3²-4²= -7, 5²-6²= -11
There must be 2018/2 = 1009 sets of subtraction of squares
Each subtraction of squares decreases by -4 of the previous subtraction, starting from the second subtraction (3²-4²)
So 2017²-2018²=-3-4(1008)
1²-2²+3²-4²+…+2017²-2018²
=-3-3-4(1)-3-4(2)-…-3-4(1008)
=1009(-3)-4(1+2+…+1008)
=-3027-4(1008)(1+1008)/2
=-3027-2016(1009)
=-2037171
OR
Using a²-b² = (a-b)(a+b),
1²-2²+3²-4²+…+2017²-2018²
=(1-2)(1+2)+(3-4)(3+4)+…+(2017-2018)(2017+2018)
=-(3+7+…+4035)
=-[3+3+4(1)+3+4(2)+…+3+4(1008)]
=-[1009(3)+4(1+2+…+1008)]
=-3027-4(1008)(1+1008)/2
=-3027-2016(1009)
=-2037171
(No calculator tip: Use 2016x1009=2016 x 1010 -2016)
thank you so much for the detailed explanation.
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