Probleme propuse - Analiza matematica. MPT

1. Rezolvati ecuatiile :
−x 2 + 5x = 6
x 2 + x + 1 = 0
x + 1
x
= 3
|1 − 2x| = 3x + 7
√ x + 1 − √ 9 − x = √ 2x − 12
√ 2x − 6 + √ x + 4 = 5
√ x+1
√ x+2 = √ x−3
√ x+4
log 4
2
x−1 = log4(4 − x)
log 4 log 2 log 3(2x − 1) = 1
2
7 x+2 − 3 · 7 x+1 − 20 · 7 x = 392
5 2x−1 + 5 x−1 = 5 x + 1
8 · 4 x + 2 · 10 x = 25 x
x 2 − 4 + 6
x 2 −4
= 5
x 3 + x 2 − 3x − 6 = 0
x 4 + 3x 3 + 4x 2 + 6x + 4 = 0
2.Rezolvati inecuatiile :
|x − 1| ≤ 2
|2x + 3| > 1
−3x 2 + x − 1 ≤ 0
x+1
x−2
≥ 3x−1
x−1
4x 2 −5x−1
2x 2 −5x+3
| x−2
x2 | > 2 −1
< 1
log 1 (5 + 4x − x
2
2 ) > −3
log 9 x − log 3 5x + 1
2
> log 1 (x + 3)
3
Probleme propuse
3. Aduceti la o forma mai simpla expresiile :
(log a b + log b a + 2)(log a b + log ab b) · log b a − 1 unde a, b, ab ∈ R ∗ + \ {1}
2 2 · 3 3 · 3√ 2 5 · 3 4 · 5 12
1

2
√ √
9 − 2 14 + 7
3
9 1
1
log2 3 log + 49 2 7
4. Se considera matricele : A =
2 1
3 2
si B =
−3 2
5 −3
a) aratati ca sunt inversabile si calculati inversele lor
−2
b) rezolvati ecuatia A · X · B = C unde C =
5
4
−3
⎛
0 1
⎞
1
5. Fie matricea : A = ⎝ 1 0 1 ⎠
1 1 0
a) aratati ca verifica identitatea : A 2 − A = 2I3
b) aratati ca este inversabila si aflati A −1
6. Calculati limitele :
1 1 lim ( +
n→∞ 2 22 + 1
23 + ... + 1
2n )
1+2+3+...+n
lim
n→∞ 3n2−1 7. Rezolva urmatoarele sisteme folosind regula lui Cramer :
⎧
⎪⎨ 2x + y + z = 2
a) x + 3y + z = 5
⎪⎩
x + y + 5z = −7
⎧
⎪⎨ 2x − y + 3z = 4
b) x + y − z = 0
⎪⎩
−5x + 2y − z = −5
1 2 3
8. Rezolvati ecuatia:
2 4 − x −1
= 0
3 6 x + 9
−a 1 b
9. Consideram determinantul : ∆(a, b) =
−b a 1
unde a, b ∈ R Aflati ∆(a, b) si
1 b −a
∆(−1, 1)