Page 62 - PRE-U STPM MATHEMATICS (T) TERM 1

Mathematics Term 1 STPM Answers

4
5. 1 –3 –5 2 10. (a) x = 3, y = 2, z = –1 (b) x = –2, y = 1, z = –1
(c) x = 3, y = 5, z = –2
(d) x = 3, y = 12, z = 9
4
6. 1 –2 –2 2 STPM PRACTICE 3
6
5
5
4
–3
8
–5
43
9
7. 1 –25 14 2 ; m = –5, n = 1; 1 3 5 1 2 2 1 –120 –24 2 2. (a) 1 –1 13 2 (b) 1 2
1
,
6
67
1 8 0 0 2 1 5 –7 1 2 –2 7 5
1
8. 0 8 0 , 8 –6 10 2 3. (a) (b) None
0 0 8 15 –21 –5 1 – 3
5
16
–20
4
8
–2
3
1
1
10. (a) 81 –10 2 1 2 (b) – 231 –2 –3 –2 2 (c) – 2 3 5 3 (d) – 1 1 5 –7 2
4
1
–2
–5
–5
–2
–1 2 11 –3 3
–2
1
0
1
1
1
(c) inverse does not exist (d) 31 –7 –1 0 3 2 4. x = –4, y = 9 4
1
–2
9
1
(e) – 60 1 7 –13 –3 2 5. (a) 1 –1 –4 –7 2
–29
11
–5 –2
1
13 –7 –9 1 – 2 9 1 9 4 9
13
8
5
2
–1
9
11. (a) 1 –3 3 3 17 2 (b) 1 –3 –11 12 2 (b) Q = (Q + 2Q – 8I) = – 1 9 – 4 9 – 7 9
11
0
0 3 13 1 –5 16 1 9 – 5 9 – 2 9
5
7
10
1
1
(c) 61 –8 –3 –3 2 (d) 31 –5 –11 –23 2 6. (b) det B = –4 det A = –4(p – q)(q – r)(p – r)
25
6
13
2 –1 1 –3 –6 6 7. a + d, ad – bc
–2
4
–1
1
1
13 –11
0
–60 –123
111
(e) 181 116 233 –211 2 (f) 61 –2 –4 6 2 9. 1 4 –3 –1 2
–26 –53 49 0 –3 3 –7 4 2
Exercise 3.4 –1 –1 2 –2
2
1. (a) x = 1, y = –2 (b) x = –2, y = 1 10. W = 1 –3 5 –5 ; x = 2, y = 7, z = –10
(c) x = 0, y = –1 (d) p = –1, q = –2 2 –4 3
(e) r = 4, s = 7 (f) u = 2, v = –3 11. (a) a = 3, b ≠ 5
2. (a) (–1, 2) (b) None (b) a = 3, b = 5
(c) (–1, 1) (d) (3, –1) (c) a ≠ 3, b = any real values (real numbers)
1
3. (a) x = , y = k + 1 , k ≠ 0 3 2 1 x 1
k k 12. (a) 1 9 6 4 21 2 1 2
y =
a
2
(b) x = 2 , y = – 1 ; k ≠ ±2 6 4 –2 z a
k + 2 k + 2 3 2 1 1
4. (a) x = 1, y = –1, z = –1 (b) augmented matrix 1 9 6 4 a 2 2 ;
(b) x = –1, y = 2, z = 2 6 4 –2 a
(c) x = –15, y = 8, z = 2 3 2 1 1
5. (a) unique solution, (1, –2, –2) row-echelon form 1 0 0 1 a – 3 2
2
(b) no solution 0 0 0 4a + a – 14
2
(c) infinitely many solutions
Since row 3 has all zero entries, the system of linear
3
5
6. x = , y = , z = 1 equations does not have a unique solution.
4 4 2 (c) a = , –2 ; x = t, y = – t, z = 1 where t ∈ R
7
3
1 6 0 0 2 4 2
7. AB = 0 6 0 (d) 5 a : a ≠ , a ≠ –2, a ∈ R6
7
0 0 6 4
(a) x = 1, y = 1, z = 2 1 3 1 n
(b) x = –1, y = –2, z = 2 1 13. (a) augmented matrix 1 2 3 m 13 4n 2
5
m 2n ;
z 1 2
– —
x
3
1
1
–5 ;
y =
8. A = – 10 1 1 –2 –5 2 1 2 5 row-echelon from 1 1 0 –1 m – 2 n 0 2
8
4
–1
3
– —
5 0 –5 5 2
–2 0 0 m – 11m + 28 n
3
— (b) (i) m = 4, 7 and n ≠ 0
2
(ii)
m ≠ 4, 7 and n ∈ R
0
2 1 2
8 1
1
3
y =
–1
9. B = 2 3 2 7 –4 ; x z 1 2 (c) m = 4, 7 and n = 0 ; x = –16t, y = 5t, z = t
—
–5 –1 4 4 where t ∈ R
7
—
4
281
Answers STPM Math T S1.indd 281 3/28/18 4:25 PM