Mathematics – List 1
1
Mathematics
List 1
1. Evaluate or simplify (write as a power):
− 21
4
,
9
3
√
,
3
3
5
3
8 ,
− 32
100
− 14
,
4
√ √
3
2 · 4,
,
1
√
,
3
5
q
√
3
9
3.
2. Solve the equations and inequalities:
(a) 42x+1 = 85x−2 , (b) 7·3x+1 −5x+2 = 3x+4 −5x+3 , (c) 2x ·42x ·83x = 128,
x 2x
x2 +4
x x
(d) (3 ) ·(81 ) = 9
−x
x
(e) 5 −25·5
,
(g) 2x+2 −2x+1 ≤ 2x−2 −2x−1 ,
x−1
1
(f)
= 92x ,
3
= 24,
(h) 4x +8 < 6·2x ,
(i) 32x−1 −3x−1 ≥ 2.
3. Evaluate or simplify:
log 1 36,
6
log2
log6 2 + log6 18,
√
8,
log5 9log3 5 ,
log3 2 − log9 2,
log√5 125,
log 4
3
ln 2 + log2 e,
27
,
64
log 1 eln 2
3
2
,
log 1 3 + log4 3 + log8 3.
2
4. What profit will bring, after 4 years, a 1000 zl deposit at annual interest
of 6% in case the interest is capitalized once a year. How will this change
in case of monthly capitalization?
5. You make a long term deposit of 100 zl at annual interest of 6%. How
long will it take for the value of your deposit to exceed 1000 zl in case
the capitalization is:
(a) annual, (b) monthly?
6. Suppose the nominal annual interest of a deposit is r · 100%. What is
the effective interest after one year in case the capitalization is
(a) monthly, (b) daily, (c) n times a year (at equal time periods)?
7. Solve the equations:
(b) ln2 x+3 ln x = 4,
(a) log3 (x+1) = 2,
(d) log5 x + log5 (x + 5) = 2 + log5 2,
(c) log2 x+log8 x = 12,
(e) logx 2 − log4 x +
7
= 0.
6
8. Solve the inequalities:
1
(a) log3 x < − ,
3
(b) log 1 x ≤ 2,
2
(d) log 1 x + 2 log 1 (x − 1) ≤ log 1 6,
3
9
3
(c) log2 x ≥ log2 x2 ,
(e) log2 log3
x−1
> 0.
x+1
Mathematics – List 1
2
9. Solve the systems of equations:
y
2 log3 x − log3 y = 2
xy = 36
x =9
(a)
, (b)
, (c)
.
1
10y−x = 100
xlog3 y = 16
y = log3 x + 1
10. Sketch the graphs of the functions:
(a) y = |3x − 3|,
(b) y = 2−x ,
(c) y = log2 (2x),
(d) y = ln |x|.
Hints and answers
√ √
2. a) 5/8, b) −1, c) 1/2, d) − 2, 2, e) 2, f) 1/5, g) ∅, h) (1, 2), i) [1, ∞).
4. 1000 · (1.06)4 , 1000 · (1.005)48 . 5. a) l such that 100 · (1.06)l > 1000, b) m
365
such that 100 · (1.005)m > 1000,. 6. a) (1 + r/12)12 − 1, b) (1 + r/365)
− 1,
√
3
n
−4
9
4.
8.
a)
c) (1 + r/n)
−
1.
7.
a)
8,
b)
e
,
e,
c)
2
,
d)
5,
e)
8,
1/
√
3
x ∈ (0, 1/ 3), b) x ≥ 1/4, c) x ∈ (0, 1), d) x ≥ 3, e) x < −1. 9. a)
x = 3, y = 1 or x = 6, y = 4, b) x = 9, y = 4 or x = 4, y = 9, c) x = 3, y = 2
or x = 1/9, y = −1.