Solution key to mst prelim

SOLUTION KEY TO MST
PRELIM
by Juan Apolinario C.
Reyes

−10 − 5 − 5 × 6 =
Solution
−10 − 5 − 30
= −15 − 30 = −45

Solution
3
4
+
1
2
−
2
3
=
12
=
9 + 6 − 8
12
=
7
12

Solution
−1 3
70 ÷ 10 − 14 = −7 − 14 = −21

2 − 𝑥 2 + 𝑥 + 4 + 2𝑥2
= 4 + 2𝑥 − 2𝑥 − 𝑥2
+ 4 + 2𝑥2
Solution
= 8 + 𝑥2

Solution
2
3
−
4
3
=
2 − 4
3
= −
2
3
−
2
3
1
3
= −
2
3
× 3 = −2

Solution
5𝑥𝑦
𝑤𝑧
×
−5𝑥4
𝑤3
𝑦2
= −
25𝑥4+1
𝑦1−2
𝑤3−1
𝑧
= −
25𝑥5
𝑦−1
𝑤2
𝑧
= −
25𝑥5
𝑤2
𝑧𝑦

Solution
45 − 2𝑥 = 5𝑥 − 4
−5𝑥 − 2𝑥 = −4 − 45
−7𝑥 = −49
𝑥 =
49
7
= 7

2
5
𝑥 − 3𝑥 + 10 = 2𝑥 − 3
5
2
5
𝑥 − 3𝑥 + 10 = 2𝑥 − 3
Solution
2𝑥 − 15𝑥 + 50 = 10𝑥 − 15
2𝑥 − 15𝑥 − 10𝑥 = −15 − 50
−23𝑥 = −65
𝑥 =
65
23
= 2
19
23

Solution
6
3 − 2𝑥
= 7
3 − 2𝑥
6
=
1
7
3 − 2𝑥 =
6
7
−2𝑥 =
6
7
− 3 =
6 − 21
7
= −
15
7
𝑥 =
15
7
∙
1
2
=
15
14
6
3 − 2𝑥
6
=
1
7

Solution
3 − 𝑥 2𝑥 + 5 = 0
3 − 𝑥 = 0 or 2𝑥 + 5 = 0
𝑥 = 3 or 2𝑥 = −5 → 𝑥 = −
5
2
Product Rule
If AB = 0,
then
A = 0 or B =
0

Solution
6𝑥 − 5 2
+ 10 = 10
6𝑥 − 5 2
= 0
6𝑥 − 5 = 0
𝑥 =
5
6
6𝑥 = 5

𝑥2
− 18𝑥 + 81 = 0
𝑥 − 9 2
= 0
Solution
𝑥 = 9
𝑥 − 9 2
= 𝑥 − 9 𝑥 − 9
= 𝑥2
− 9𝑥 − 9𝑥 + 81
= 𝑥2
− 18𝑥 + 81

Solution
𝑝𝑥 = 𝑇
𝑝200 = 15000
𝑝 =
15000
200
=
150
2
= 75 php

Solution
𝑃𝐴 = 𝑝
1000 × 𝐴 = 1000 × 15 × 20 = 𝑝
1000 × 300 = 300000 php

Solution
𝑥2
− 8𝑦 − 9𝑥 = 22
− 8 −1 − 9 2
4 + 8 − 18 = −6

Solution
𝑥 𝑚, 𝑦 𝑚
𝑥 𝑚 =
5 + (−5)
2
= 0
𝑦 𝑚 =
7 + 11
2
= 9
𝑥 𝑚, 𝑦 𝑚 = 0, 9

Solution
𝑚 =
∆𝑦
∆𝑥
𝑚 =
11 − 4
3 − −4
=
7
7
= 1

Solution
𝑑 = 𝑥2 − 𝑥1
2 + 𝑦2 − 𝑦1
2
𝑑 = −7 2 + −7 2
𝑑 = 2 × 49 = 7 2

Solution
𝑐 = 𝑎2 + 𝑏2 = 64 + 25 𝑐 = 89 m

Solution
180° = 90° + 50°35′
+ 𝛼
𝛼 = 90° − 50°35′
𝛼 = 89°60′
− 50°35′
𝛼 = 39°25′

Solution
𝜋𝑟2
= 36𝜋 𝑟2
=36
𝑟 = 6 m

Solution
𝐶 = 2𝜋𝑟
𝐶 = 12𝜋 m

Solution
tan 𝛽 =
5
10
tan 𝛽 =
1
2

Solution
cot 𝛼 =
5
10
=
1
2
tan 𝛼 =
10
5
= 2

Solution
cos 𝛽 =
𝐴
𝐻
𝑐 = 102 + 52 = 125 = 25 × 5
𝑐 = 5 5
cos 𝛽 =
10
5 5
=
2
5
cos 𝛽 =
2
5
∙
5
5
cos 𝛽 =
2 5
5

Solution
180° = 25° + 75° + 𝛾 𝛾 = 180° − 25° − 75° = 80°

𝑎
sin 𝛼
=
𝑏
sin 𝛽
=
𝑐
sin 𝛾
𝑎
sin 25°
=
10
sin 75°

Solution
𝑑 = 𝑠 × 𝑡
𝑑 = 37 × 3 = 111 km

111 km
36 km
𝑐 = 1112 + 362 km

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