1 Mathematics - Problem set
1. Calculate A · B if
A =
1 2 0 1 2 3
, B =
"
1 −1
0 2
#
2. Calculate A2− 3A if
A =
1 0 −1
2 1 0
0 1 2
3. Calculate
a)
1 0 −1
2 1 0
1 1 −1
, b)
1 0 −1 2
1 1 −2 1
2 1 1 −1
3 2 1 0
, c)
4 1 0 3
1 1 3 −1
−1 −1 2 −2
2 0 1 3
4. Find the inverse matrix of the following matrix
A =
2 1 0
−1 0 0
−2 1 2
5. Calculate A−1· B · A if
A =
0 0 1
0 −1 2 1 −2 3
, B =
1 0 0 0 2 0 0 0 3
6. Solve the following system of linear equations using Cramer’s rule and using matrices:
a)
x + y − z = 2 3x − y + z = 6 x + y + z = 4
, b)
x + 2y − z = 1 3x − 5y + 2z = 2
−x − 2y + z = 0 , c)
3x + 2y − 4z = 5 2x + 3y − 6z = 5 5x − y + 2z = 4
, d)
x − y + z = 1
−x + y + 2z = −1 2x − y + z = 3 7. Solve the following system of linear equations:
a)
3x + y = 9 2x + y = 7 x + 3y = 5
, b)
( 2x + y − 3z = 5 x + 2y + z = 3 , c)
( x1 + x2+ x3+ x4 = 5 x1+ x2+ 2x3− x4 = 3
8. Calculate the area of the triangle with vertices A(0, 0, 2), B(2, 1, 1), C(−1, 1, 0).
9. Given are vectors u = [1,12, m] oraz v = [2, 1, 1, ]. For which values of the parameter m: a) u k v, b) u ⊥ v, c) |u| =√
2?
10. Given are lengths of vectors |u| = 3, |v| = 1, and the angle between the vectors u, v is π3. Calculate |u − 2v|.
2 11. Find the derivatives of the following functions: a) y = arctg(ln(x))+ln(arctg(x))−2, b) y = ln(2x)x2 . 12. Find the limits of the folllowing functions:
a) lim
x→∞
x5
ex, b) lim
x→0+
√x · ln(x), c) lim
x→π+(π − x) · tg(x 2).
13. Find the lowest and the highest value of the following function f (x) = sin(2x) − x w przedziale
< −π2;π2 >.
ANSWERS 1)
A · B =
1 3 0 2 2 4
2)
A2− 3A =
−2 −1 0
−2 −2 −2 2 0 −2
3) a) -2, b) 8, c) 32, 4)
A−1 = 1 2
0 −2 0
2 4 0
−1 −4 1
5)
A−1· B · A =
3 −2 2
0 2 −2
0 0 1
6)a) x = 2, y = 1, z = 1, b) no solutions, c) infinitely many solutions, d) x = 2, y = 1, z = 0
7) a) no solutions, b) x = 73(1 + t), y = 13(1 − 5t), t ∈ R, c) x1 = s, x2 = 7 − s − 3t, x3 = 2t − 2, x4 = t, s, t ∈ R.
8) 12√ 35.
9) a) m = 12, b) m = −52, c) m =
√ 3 2 . 10)√
10.
11) a) 1+ln21(x))x+ (1+x2)·arctg(x)1 , b) 2x·ln(2x)−x2·2·2x1 (ln(2x))2 . 12)a) 0,b) 0, c) 2.
13) −π2, π2.
on the interval on the interval on the interval on the interval 12. 12. 12. 12. 12. 12
12) 12)
