Grade 12 Maths

Saya Soe Moe Aung ( Education Center)
MATRICULATION EXAMINATION(SEPTEMBER PRETEST)
MATHEMATICS Time Allowed: (3) Hours
Answer All Questions. Write your answers in the answer booklet.
Section A (Each questions carries 1 mark.)
Choose the correct or the most appropriate answer of each question. Write the letter of the correct
or the most appropriate answer.
1. 𝐼𝑓 𝑧 = 3 + 4𝑖 , 𝑡ℎ𝑒𝑛 𝑧𝑧̅ =
𝐴. −25𝑖 𝐵. 25 𝐶. −25 𝐷. 25𝑖
2. 𝑀𝑜𝑑𝑢𝑙𝑢𝑠 𝑜𝑓 𝑧 𝑖𝑠
𝑧̅
𝐴. √𝑧 𝐵. √𝑧̅ 𝐶. √𝑧𝑧̅ 𝐷. √ .
𝑧
3. 𝑇ℎ𝑒 𝑐𝑎𝑟𝑡𝑒𝑠𝑖𝑎𝑛 𝑓𝑜𝑟𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑙𝑎𝑛𝑒 𝑟⃗(3𝑖̂ + 3𝑗̂ − 𝑘̂ ) = 10 𝑖𝑠

𝐴. 2𝑥 + 3𝑦 − 𝑧 = 10 𝐵. 2𝑥 + 3𝑦 − 𝑧 = √14 𝐶. 2𝑥 + 3𝑦 − 𝑧 = −√14
𝐷. 2𝑥 + 3𝑦 + 𝑧 + 10 = 0.
4. 𝑇ℎ𝑒 𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑝ℎ𝑒𝑟𝑒 𝑥 2 + 𝑦 2 + 𝑧 2 − 4𝑥 − 2𝑦 + 2𝑧 = 8 𝑖𝑠
𝐴. 14 𝐵. √14 𝐶. 8 𝐷. √8
5. 𝑇ℎ𝑒 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑃(2,2,1)𝑎𝑛𝑑 𝑄 (0,4,2) 𝑖𝑠
𝐴. 1 𝐵. 3 𝐶. 9 𝐷. √3
6. 𝑇ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 𝑜𝑓 𝑠𝑒𝑙𝑒𝑐𝑡𝑖𝑜𝑛 𝑡𝑤𝑜 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑠𝑒𝑡 {1,2, … … . . ,12} 𝑤ℎ𝑜𝑠𝑒 𝑠𝑢𝑚 𝑖𝑠 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 3 𝑖𝑠
𝐴. 66 𝐵. 16 𝐶. 6 𝐷. 22
7. 𝑇ℎ𝑒 𝑣𝑒𝑐𝑡𝑜𝑟 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 𝑡𝑜 𝑏𝑜𝑡ℎ 𝑣𝑒𝑐𝑡𝑜𝑟𝑠 2𝑖̂ + 𝑗̂ − 𝑘̂ 𝑎𝑛𝑑 2𝑖̂ − 𝑗̂ + 4𝑘̂ 𝑖𝑠
𝐴. 3𝑖̂ − 10𝑗̂ − 4𝑘̂ 𝐵. 4𝑖̂ + 3𝑘̂ 𝐶. 4𝑖̂ − 𝑗̂ − 4𝑘̂ 𝐷. 5𝑖̂ + 6𝑗̂
8. ̂
𝑇ℎ𝑒 𝑣𝑒𝑐𝑡𝑜𝑟 𝑜𝑓|𝑖̂ − 2𝑗̂ + 3𝑘|𝑖𝑠
𝐴. √2 𝐵. √6 𝐶. √10 𝐷. √14
1 1
9. 𝑇ℎ𝑒 𝑎𝑛𝑔𝑙𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡ℎ𝑒 𝑡𝑤𝑜 𝑣𝑒𝑐𝑡𝑜𝑟𝑠 ( ) 𝑎𝑛𝑑 ( ) 𝑖𝑠
−2 3
𝐴. 0° 𝐵. 45° 𝐶. 90° 𝐷. 135°
10. 𝑇ℎ𝑒 𝑣𝑒𝑐𝑡𝑜𝑟 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑙𝑎𝑛𝑒 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑖𝑛𝑔 𝑝𝑜𝑖𝑛𝑡𝑠 𝐴 , 𝐵 𝑎𝑛𝑑 𝑃 𝑤𝑖𝑡ℎ 𝑡ℎ𝑒𝑖𝑟 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 𝑣𝑒𝑐𝑡𝑜𝑟𝑠 𝑖𝑠
⃗⃗⃗⃗⃗⃗ + 𝑡1 𝑂𝐵
𝐴. 𝑟̅ = 𝑂𝐴 ⃗⃗⃗⃗⃗⃗ + 𝑡2 𝑂𝐶
⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ + 𝑡1⃗⃗⃗⃗⃗⃗
𝐵. 𝑟⃗ = 𝑂𝐵 ⃗⃗⃗⃗⃗⃗
𝐵𝐴 + 𝑡2 𝐵𝐶 ⃗⃗⃗⃗⃗⃗ + 𝑡1⃗⃗⃗⃗⃗⃗
𝐶. 𝑟⃗ = 𝑂𝐴 ⃗⃗⃗⃗⃗⃗
𝐴𝐵 + 𝑡2 𝑂𝐶
𝐷. 𝑟⃗ = ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ .
⃗⃗⃗⃗⃗⃗ + 𝑡2 𝑂𝐶
𝐴𝐵 + 𝑡1 𝑂𝐵
Section B (Each questions carries 2 marks.)
Write only the solution of each question. (There is no need to show your working)
̅̅̅̅̅̅̅
2+3𝑖
11. 𝑆𝑖𝑚𝑝𝑙𝑖𝑓𝑦 .
−4−5𝑖
12. 𝑇ℎ𝑒 𝑐𝑜𝑟𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡𝑠 𝐴 𝑎𝑛𝑑 𝐵 𝑎𝑟𝑒 (−1,3,14) 𝑎𝑛𝑑 (2𝑘, 3𝑘, 13). 𝐺𝑖𝑣𝑒𝑛 𝑡ℎ𝑎𝑡 𝑡ℎ𝑒 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝐴 𝑡𝑜 𝐵 𝑖𝑠
√163 𝑢𝑛𝑖𝑡𝑠 𝑎𝑛𝑑 𝑡ℎ𝑎𝑡 𝑘 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑔𝑒𝑟, 𝑓𝑖𝑛𝑑 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑘.
13. 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑒𝑟 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑙𝑒 𝑥 2 − 2𝑥 + 𝑦 2 + 4𝑦 − 4 = 0.

14. Find the equation of the plane containing the point (-1,3,2) and the parallel to the plane x-y=2.
15. 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑢𝑛𝑖𝑡 𝑣𝑒𝑐𝑡𝑜𝑟 𝑜𝑓 4𝑖̂ − 2𝑗̂ − 4𝑘̂ .

3 1
16. 𝐹𝑖𝑛𝑑 𝑎 𝑣𝑒𝑐𝑡𝑜𝑟 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 𝑡𝑜 (1) 𝑎𝑛𝑑 (2)
1 3
17. 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑖𝑛𝑔 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 (2,3, −4) 𝑎𝑛𝑑 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 𝑡𝑜 𝑡ℎ𝑒 𝑝𝑙𝑎𝑛𝑒 𝑥 − 𝑧 = 3.
−2 1
18. 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 𝑝 𝑎𝑛𝑑 𝑞 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 ( 𝑝 ) 𝑖𝑠 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 𝑡𝑜 (2).
𝑞 7
19. 𝐼𝑓 𝑎⃗ 𝑎𝑛𝑑 ⃗⃗⃗⃗

𝑏 𝑎𝑟𝑒 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟, |𝑎⃗| = 3, |𝑏⃗⃗| = 1. 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 (𝑎⃗ − 𝑏⃗⃗). (𝑎⃗ + 5𝑏⃗⃗).
2 −1
20. 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑎𝑛𝑔𝑙𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 (−1) 𝑎𝑛𝑑 ( 3 )
−1 2
Section C (Each questions carries 3 marks.)
21. 𝑆𝑜𝑙𝑣𝑒 𝑡ℎ𝑒 𝑞𝑢𝑒𝑠𝑡𝑖𝑜𝑛 − 2𝑥 2 + 4𝑥 − 3 = 0.
̅̅̅̅̅
𝑧 𝑧
̅̅̅
22. 𝐿𝑒𝑡 𝑧1 = 3 − 2𝑖 , 𝑧2 = −1 + 4𝑖 , 𝑠ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 ( 1) = 1.
𝑧2 𝑧2
23. 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑙𝑎𝑛𝑒 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑖𝑛𝑔 𝐴(4,2, −3), 𝐵(1, −2,4), 𝐶(−1,0,3).
24. 𝑆ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡𝑠 (1, −1,2), (3, −2,3) 𝑎𝑛𝑑 (5, −3,4)𝑎𝑟𝑒 𝑐𝑜𝑙𝑙𝑖𝑛𝑒𝑎𝑟.
25. 𝑇ℎ𝑒𝑟𝑒 𝑎𝑟𝑒 5 𝑝𝑖𝑐𝑡𝑢𝑟𝑒 𝑛𝑎𝑖𝑙𝑠 𝑜𝑛 𝑡ℎ𝑒 𝑤𝑎𝑙𝑙. 𝐼𝑓 𝑡ℎ𝑒𝑟𝑒 𝑎𝑟𝑒 8 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 𝑝𝑖𝑐𝑡𝑢𝑟𝑒𝑠 𝑎𝑛𝑑 𝑒𝑎𝑐ℎ 𝑛𝑎𝑖𝑙 𝑐𝑎𝑛 ℎ𝑜𝑙𝑑 𝑜𝑛𝑙𝑦 𝑜𝑛𝑒
𝑝𝑖𝑐𝑡𝑢𝑟𝑒, 𝑖𝑛 ℎ𝑜𝑤 𝑚𝑎𝑛𝑦 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 𝑤𝑎𝑦𝑠 𝑐𝑎𝑛 𝑡ℎ𝑒 𝑝𝑖𝑐𝑡𝑢𝑟𝑒𝑠 𝑏𝑒 ℎ𝑢𝑛𝑔 𝑜𝑛 𝑎𝑙𝑙 𝑡ℎ𝑒 𝑛𝑎𝑖𝑙𝑠.
26. 𝐼𝑓 𝑎 𝑝ℎ𝑜𝑛𝑒 𝑘𝑒𝑦𝑝𝑎𝑑 ℎ𝑎𝑠 6 𝑑𝑖𝑔𝑖𝑡 𝑝𝑎𝑠𝑠𝑤𝑜𝑟𝑑, 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑖𝑛𝑔 0,1,2,3,4,5 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑐𝑜𝑑𝑒 𝑡𝑜 𝑜𝑝𝑒𝑛 𝑡ℎ𝑒 𝑠𝑐𝑟𝑒𝑒𝑛 𝑚𝑢𝑠𝑡 𝑏𝑒 𝑎
4 𝑑𝑖𝑔𝑖𝑡 𝑐𝑜𝑑𝑒 , ℎ𝑜𝑤 𝑚𝑎𝑛𝑦 𝑐𝑜𝑑𝑒𝑠 𝑎𝑟𝑒 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑡𝑜 𝑐𝑟𝑒𝑎𝑡𝑒 𝑖𝑓 𝑡ℎ𝑒 𝑟𝑒𝑝𝑒𝑡𝑖𝑡𝑖𝑜𝑛 𝑖𝑠 𝑎𝑙𝑙𝑜𝑤𝑒𝑑?
27. 𝑈𝑠𝑒 𝑡ℎ𝑒 𝑚𝑎𝑡ℎ𝑒𝑚𝑎𝑡𝑖𝑐𝑎𝑙 𝑖𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑙𝑒 𝑡𝑜 𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 1 + 3 + 5 + ⋯ . . +(2𝑛 − 1) = 𝑛2 .
28. 𝑆ℎ𝑜𝑤 𝑏𝑦 𝑖𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑡ℎ𝑎𝑡 3𝑛 − 1 𝑖𝑠 𝑎𝑛 𝑒𝑣𝑒𝑛 𝑛𝑢𝑚𝑏𝑒𝑟 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑛.
3𝑛−1
29. 𝑈𝑠𝑒 𝑡ℎ𝑒 𝑚𝑎𝑡ℎ𝑒𝑚𝑎𝑡𝑖𝑐𝑎𝑙 𝑖𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑙𝑒 𝑡𝑜 𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 1 + 3 + 32 + ⋯ … + 3𝑛−1 = .
2
30. 𝑆ℎ𝑜𝑤 𝑏𝑦 𝑖𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑡ℎ𝑎𝑡 9𝑛 − 1 𝑖𝑠 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 8 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑛.
Section D (Each questions carries 5 marks)
31. 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑐𝑢𝑏𝑒 𝑟𝑜𝑜𝑡𝑠 𝑜𝑓 𝑧 = 1 − 𝑖.
32. 𝑆𝑜𝑙𝑣𝑒 𝑧 4 = −𝑖.
33. 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒 𝑝𝑎𝑠𝑠𝑖𝑛𝑔 𝑡ℎ𝑟𝑜𝑢𝑔ℎ 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 (3, −2, −2)𝑎𝑛𝑑 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 𝑡𝑜 𝑡ℎ𝑒 𝑝𝑙𝑎𝑛𝑒
−2𝑥 + 3𝑦 − 𝑧 = 4. 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑝𝑙𝑎𝑛𝑒.
34. 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑙𝑎𝑛𝑒 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑡𝑜 𝑡ℎ𝑒 𝑠𝑝ℎ𝑒𝑟𝑒
(𝑥 − 2)2 + (𝑦 − 1)2 + (𝑧 + 1)2 = 14 𝑎𝑡 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 (3,4,1).
35. 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑝ℎ𝑒𝑟𝑒 𝑤𝑖𝑡ℎ 𝑐𝑒𝑛𝑡𝑒𝑟 (6, −7, −3) 𝑎𝑛𝑑 𝑡𝑜𝑢𝑐ℎ𝑖𝑛𝑔 𝑡ℎ𝑒 𝑝𝑙𝑎𝑛𝑒 4𝑥 − 2𝑦 − 𝑧 = 17.
36. 𝐴 𝑟𝑒𝑔𝑠𝑖𝑡𝑟𝑎𝑡𝑖𝑜𝑛 𝑐𝑜𝑑𝑒 𝑐𝑜𝑛𝑠𝑡𝑖𝑡𝑠 𝑜𝑓 𝑡𝑤𝑜 𝑜𝑓 𝑡ℎ𝑒 15 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 𝑐𝑎𝑝𝑖𝑡𝑎𝑙 𝑙𝑒𝑡𝑡𝑒𝑟𝑠 𝐴, 𝐵, 𝐶, … … ,0, 𝑓𝑜𝑙𝑙𝑜𝑤𝑒𝑑 𝑏𝑦 𝑜𝑛𝑒
𝑜𝑓 𝑡ℎ𝑒 𝑡𝑒𝑛 𝑑𝑖𝑔𝑖𝑡𝑠 0,1,2, … … ,9 𝑓𝑜𝑟 𝑒𝑥𝑎𝑚𝑝𝑙𝑒 𝐼𝐷5. 𝐻𝑜𝑤 𝑚𝑎𝑛𝑦 𝑐𝑜𝑑𝑒𝑠 𝑎𝑟𝑒 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑡𝑜 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑒.
(a) 𝑖𝑓 𝑡ℎ𝑒 𝑟𝑒𝑝𝑒𝑡𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑒𝑡𝑡𝑒𝑟𝑠 𝑖𝑠 𝑎𝑙𝑙𝑜𝑤𝑒𝑑 ?
` (b) 𝑖𝑓 𝑡ℎ𝑒 𝑟𝑒𝑝𝑒𝑡𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑒𝑡𝑡𝑒𝑟𝑠 𝑖𝑠 𝑛𝑜𝑡 𝑎𝑙𝑙𝑜𝑤𝑒𝑑?
(𝑐)𝑖𝑓 𝑡𝑤𝑜 𝑙𝑒𝑡𝑡𝑒𝑟𝑠 𝑖𝑛 𝑐𝑜𝑑𝑒 𝑚𝑢𝑠𝑡 𝑏𝑒 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒?
(𝑑) 𝑖𝑓 𝑡𝑤𝑜 𝑙𝑒𝑡𝑡𝑒𝑟𝑠 𝑎𝑟𝑒 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡, 𝑏𝑢𝑡 𝑚𝑢𝑠𝑡 𝑏𝑒 𝑏𝑜𝑡ℎ 𝑣𝑜𝑤𝑒𝑙𝑠 𝑜𝑟 𝑏𝑜𝑡ℎ 𝑐𝑜𝑛𝑠𝑜𝑛𝑎𝑛𝑡𝑠.
(2𝑛−1)3𝑛+1 +3
37. 𝑃𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 1.3 + 2. 32 + 3. 33 +. . . +𝑛. 3𝑛 = 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑛 𝑏𝑦 𝑡ℎ𝑒 𝑢𝑠𝑒 𝑜𝑓 𝑡ℎ𝑒
4
𝑚𝑎𝑡ℎ𝑒𝑚𝑎𝑡𝑖𝑐𝑎𝑙 𝑖𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛.
38 . 𝑇ℎ𝑒 𝑝𝑜𝑖𝑛𝑡𝑠 𝐴(3,1,2), 𝐵(−1,1,5) 𝑎𝑛𝑑 𝐶(7,2,3)𝑎𝑟𝑒 𝑣𝑒𝑟𝑡𝑖𝑐𝑒𝑠 𝑜𝑓 𝑎 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑔𝑟𝑎𝑚 𝐴𝐵𝐶𝐷.
(𝑎) 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑜𝑓 𝐷.
(𝑏)𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒 𝑡ℎ𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑔𝑟𝑎𝑚.

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Saya Soe Moe Aung ( Education Center)

MATRICULATION EXAMINATION(SEPTEMBER PRETEST)

MATHEMATICS Time Allowed: (3) Hours

Answer All Questions. Write your answers in the answer booklet.

Section A (Each questions carries 1 mark.)

or the most appropriate answer.

3. 𝑇ℎ𝑒 𝑐𝑎𝑟𝑡𝑒𝑠𝑖𝑎𝑛 𝑓𝑜𝑟𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑙𝑎𝑛𝑒 𝑟⃗(3𝑖̂ + 3𝑗̂ − 𝑘̂ ) = 10 𝑖𝑠

Section B (Each questions carries 2 marks.)

√163 𝑢𝑛𝑖𝑡𝑠 𝑎𝑛𝑑 𝑡ℎ𝑎𝑡 𝑘 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑔𝑒𝑟, 𝑓𝑖𝑛𝑑 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑘.

13. 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑒𝑟 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑙𝑒 𝑥 2 − 2𝑥 + 𝑦 2 + 4𝑦 − 4 = 0.

15. 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑢𝑛𝑖𝑡 𝑣𝑒𝑐𝑡𝑜𝑟 𝑜𝑓 4𝑖̂ − 2𝑗̂ − 4𝑘̂ .

19. 𝐼𝑓 𝑎⃗ 𝑎𝑛𝑑 ⃗⃗⃗⃗

Section C (Each questions carries 3 marks.)

21. 𝑆𝑜𝑙𝑣𝑒 𝑡ℎ𝑒 𝑞𝑢𝑒𝑠𝑡𝑖𝑜𝑛 − 2𝑥 2 + 4𝑥 − 3 = 0.

27. 𝑈𝑠𝑒 𝑡ℎ𝑒 𝑚𝑎𝑡ℎ𝑒𝑚𝑎𝑡𝑖𝑐𝑎𝑙 𝑖𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑙𝑒 𝑡𝑜 𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 1 + 3 + 5 + ⋯ . . +(2𝑛 − 1) = 𝑛2 .

30. 𝑆ℎ𝑜𝑤 𝑏𝑦 𝑖𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑡ℎ𝑎𝑡 9𝑛 − 1 𝑖𝑠 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 8 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑛.

Section D (Each questions carries 5 marks)

31. 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑐𝑢𝑏𝑒 𝑟𝑜𝑜𝑡𝑠 𝑜𝑓 𝑧 = 1 − 𝑖.

32. 𝑆𝑜𝑙𝑣𝑒 𝑧 4 = −𝑖.

34. 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑙𝑎𝑛𝑒 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑡𝑜 𝑡ℎ𝑒 𝑠𝑝ℎ𝑒𝑟𝑒

(𝑥 − 2)2 + (𝑦 − 1)2 + (𝑧 + 1)2 = 14 𝑎𝑡 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 (3,4,1).

(a) 𝑖𝑓 𝑡ℎ𝑒 𝑟𝑒𝑝𝑒𝑡𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑒𝑡𝑡𝑒𝑟𝑠 𝑖𝑠 𝑎𝑙𝑙𝑜𝑤𝑒𝑑 ?

` (b) 𝑖𝑓 𝑡ℎ𝑒 𝑟𝑒𝑝𝑒𝑡𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑒𝑡𝑡𝑒𝑟𝑠 𝑖𝑠 𝑛𝑜𝑡 𝑎𝑙𝑙𝑜𝑤𝑒𝑑?

(𝑐)𝑖𝑓 𝑡𝑤𝑜 𝑙𝑒𝑡𝑡𝑒𝑟𝑠 𝑖𝑛 𝑐𝑜𝑑𝑒 𝑚𝑢𝑠𝑡 𝑏𝑒 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒?

38 . 𝑇ℎ𝑒 𝑝𝑜𝑖𝑛𝑡𝑠 𝐴(3,1,2), 𝐵(−1,1,5) 𝑎𝑛𝑑 𝐶(7,2,3)𝑎𝑟𝑒 𝑣𝑒𝑟𝑡𝑖𝑐𝑒𝑠 𝑜𝑓 𝑎 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑔𝑟𝑎𝑚 𝐴𝐵𝐶𝐷.

(𝑎) 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑜𝑓 𝐷.

(𝑏)𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒 𝑡ℎ𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑔𝑟𝑎𝑚.

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