1. MNOŽINY
1. Jsou dány množiny A = {2; 4; 6}, B = {4; 6}. Urcete A u B, A n B, A - B, B - A a B~.
[Au B = {2;4; 6},AnB = {4;6},A-B = {2},B-A=0, B~ = {2}]2. Jsou dány množiny A = {x E N, 6 < x < IO}, B = {x E N, 8 ::; x ::; 9}. Urcete A u B,
AnB,A-B,B-Aa B~.
[Au B = {7;8; 9},AnB = {8;9},A-B = {7},B-A = 0,B~ = {7}]3. Jsou dány množiny S = {x E N; x::; 4} a T = {x E Z; -2 < x < I}. Urcete S u T, S n T.
rSuT = {-I. 0'1. 2' 3' 4} Sn T = 0 ]~ , , , , , ,4. Jsou dány množiny W = {x E Z; -2 ::;x < 4}, A = {x E No; x < 3}, B = {3; -2}. Urcete
AuB,AnB, A~, B~.
[A u B = {- 2; O;1;2;3}, AnB = 0, A~ = {- 2;-I;3},B~ = {-I; O;I;2}]5. Urcete sjednocení a prunik množin:
a) A = {-5. O. 3' 7} B = {O. l' 2. 3}, ", , , ,
b) A = {x E Z; X ~ O}, B = {x E Z; X::;O}
c) A = {x E Z; Ixl ~ 2}, B = {x E Z; Ixl ~ I}
[A u B = {x E Z;Jxl ~I},AnB = {xE Z;lxl ~2}]
d) A = {x E Z; X < -3}, B = {x E Z; Ixl ~ I}
[AUB= {xEZ;lxl ~I},AnB = {XEZ; x <-3}]
6. Urcete doplnek množiny A v množine B:
a) A = {3; 4; 5; 6; 7}, B = {x E No; x ::; IO} [A~ = {O;1; 2; 8; 9; lO}]
b) A = {x E Z; Ix! ~ 2}, B = Z [A~ = {-I; O;I}]
c)A= {x E No;x~2},B=No [A~ = {O;I}]
d) A = {x E Z; X~ 5}, B = {x E Z; X > 3} [A~ = {4}]
e) A = {x E R; x ::;-3}, B = {x E R; Ixl ~ 3} [A~ = {xER; x ~3}]
f) A = {x E R; Ixl ::;-1 },B = {x E R; x ::;-1 } [A~ = {xER; x ::;- 1}]7. Jsou dány množiny F = {xER; -2 < x < 2}, G = {x ER; -1 ::; x ::; 2}. Urcete FuG,
FnG, F-G.[F u G = {x E R; - 2 < x ::;2} , F n G = {x E R; - 1::;x < 2}, F - G = {x E R; - 2 < x < - I}]
8. Urcete všechny podmnožiny X množiny {I; 2; 3; 4}, pro než platí:a){I;3;4}nX={I;4}
b) {3} nX=0
c) {I;3} nX= {2;4}9. Urcete všechny množiny X, pro než je A u X = B, je-li:
a)A= {x E N;x::;2},B= {x E No;x<4} [X={0;I;2;3},{0;I;3},{0;2;3},{0;3}]
b) A = Z, B = No [nelze]
c)A=0,B={0} [X= {O}]
d) A = {I}, B = {2} [nelze]10. Urcete doplnkymnožinA=N, B = {x E N;x> I},C= {I} vmnožineN.
[A~ =0,B~ ={I},C~ ={xEN;x>I}]
[AuB= {-S.0.I'2'3'7 }AnB= {0'3 }], , , , , , ,
[A u B = Z,AnB = {O}]
[X = {I; 2; 4}, {I; 4}]
[X = {I}, {2}, {4}, {I; 2}, {I; 4}, {2; 4}, {I; 2; 4}, 0]
[nelze]
2. INTERVALY
1. Dané množiny zapište jako intervaly:a) A = {x E R; -4 < x ~ O}
b) B = {x E R; x < -Ji}
c)C= {x E R; 4 ~x~ J3}S
d) D = {x E R; x > 10}
e)E= {x E R;x~2,S}
f) F = {x E R; -8 < x < -S}
g) G = {x E R; x ~ -6,8}
h) H = {x E R; Ixl ~ 17}
i)I= {x E R; Ixl ~O}
[(-4;0)]
[(- 00;- Ji)]
[(: ;F3)][(10;00)]
[(- 00; 2,S)]
[(- 8; - S)]
[(- 6,8; 00)]
[(-17; 17)]
[(- 00; 00)]
[(-~; ~)]j) J = {x E R; x2 < 7}2. Dané intervaly zapište jako množiny:
a) (-6; 2)
b) (-00; 4)
c) (O;00)
d) (-8; O)
e) (-00;-3)
f)(~; 2~)
g) (-12,S; -10)3. Urcete i) sjednocení a ii) prunik intervalu:
a) (-00; 3), (2; 00) [i)(- 00;00),ii) (2; 3)]
b) (1; 00),(2; 00) [i)(1;00),ii) (2; 00)]
c)(2; 3), (1;00) [i)(1;oo),ii)(2;3)]
d) (-3; 2), (2; 4) [i)(-3; 2)u (2; 4),ii) 0]
e)(-oo; O),(O;I) [i)(- 00;1),ii) {O}]
f) (O;1), (O; I) [i)(O;1),ii) (0;1)]4. Rozhodnete, která z následujících množin je interval, a pak príslušný interval zapište:
a) {x E Z; X> O} [není]
b) {x E R; x > O} [(0;00)]
c){xER;-I~x~3} [(-1;3)]
d) Q [není]
e) R [(- 00;00)]
f) (3; 4) u (3; 4) [(3;4)]
g) {x E R; 1 < Ixl < 2} [není]
h) {x E R; Ixl ~ S} [není]
[{x E R; - 6 ~ x ~ 2}]
[{xER;x~4}]
[{xER;x>O}]
[{x E R; - 8 < x ~ O}]
[{x E R; x < - 3}]
[{XER;~~X<Ji}][{x E R; -12,S < x < -lO}]
3. MOCNINY S CELÝM EXPONENTEM
1 15a-3b2e 16a-2b-Ie315 . 8ab-3e-7 25a6b4e-4
a 3x+4b 4x+S a x-I b x+62. .
(j a2x-3b3x-2 a2x-3b3x-1
[2.3.5-la-12eIS]
[a2X+Sb3x]
3. e2~,27n 2;)'
4.(~
J3 .
(
2aSb-2
)
-2
e-3 d2 e-4 d3
5. 0,000048. 500008000 0,00060
6. 0,00001: 1000000,01 0,001
7. 35000 :70000000,00025 0,00001
8. 0,000000009. 2700003000 0,09
( J
2
( J
38a3b-2 ge2
9. 15a-2e3 : 4a4b-4
(2 . 92
)3 84
(3)
4
10. 33.42 '272' 2"
3y2 93.2r211. I 2:lY .5 27
2n-3 3n+4 3n+S 4n-S
12. xy. x y-n+1 n-2 3-2n I-Sn
X y X Y
[2 .3-I ]
[24a -4 b 7e]
[5.10-1]
[10-11]
[5-1 .10-3]
[32.10-6]
[212.3-8.5-2a22b-16e-12]
[rl.3]
[5-3]
[X8n-2 ylln ]
( J
-3
( J
-3a-3b e-2d3
13 - .-. e-Id2 a-IbS
15.36.73
(21,10
)
2
14. 2 :9.8 180.14
(-ut .(-45Y .70215.( )
3( )
4- 60 .182. - 75
16. (x::)' =(~2r
[a6b12e3 d-3]
[23 .3.5.73]
[22.3-1.5-6.72]
~3kZ-7k]
(
aSb-4
J
-3 .(
a-2b-3
J
-2
17. e-3d 2 . e4d -S
18. 0,002.400000: 90000,00008 0,003
[a-19b6e-17 d16]
[TI .10]
4. ODMOCNINY. MOCNINY S RACIONÁLNÍM EXPONENTEM
-l. 5WX 4 Y 7 X2 21. Yxy.j;Y [
5 15
]X 4 Y 28
1?í7. 5 1
. 6. ~X5 Y (5y-23
X -4 Vx2y
. 7. I a.Vb .~a..fbVa. Jh -;;-
[2M]
[2::]
[~][2f]
[)]
[a ~b -~][2-i ]
[3-&5~]
~ 2. J3(J-š+ F6)+ J-š(J3+ F6)- F6(J3+ Fs)1 1
~ 3. 22.if4.ifl6.3212
a2B 4. ~19a11-~
16
1 1 1 1- - - -
. 5. 2.82 -7.182 +5.722 -502
~ 3 1
~ 8. (%)4.(~)8 .(%)2 .18i
(15~ '27-~ ). 9. . jij9
(
1 1
)
-2 ',~254 . 98 ?v3ifi7
(
1 1
)
-3
103.8-2
. 10. . fiiJ4
(
1 1
)
-2"~
254 . 48 ~2Vs
3
[
1
, 11. 54 (25a-2b-2)ir[(5aWir~
[2~]
. 12. -J2ab?J4a2b4 V8a3b5 ~ha5b61{j4a2b2
[5&a'b' ]
[
9 35 17
]
24 a 12b 4
[a~ ]
~ 1
. 13a. a 2 2-1
. . a a4
(
2
J
~ .~
as a3 5 a3
5. MNOHOCLENY
1. Vypocítejte s využitím vzorcu:
a)(-a+~)'
b) (O,2x+O,4yY
c)(-ab2-c3Y
d) (3x + 2y Y + (3x - 2y Y - S(X2+ y2 )
e) (9p-2Y -(2p-1Y +4p
f) (3x - 2y)3
g)(x-;y)'
h) (O,2ab+10yY
i) (2X2 y3 Z - 3xy2 Z3y
2. Delte mnohocleny:
a) (- 4Sb5 -12b3 + 36b2 ): (- 6b2 )
b) (21x3 -31x2 +39x-6):(7x-1)
. c) (X4 -7X2 -9):(x-2)
[a2-a+~]
[O,O4X2 + O,16xy + O,16y2]
[a2b4 + 2ab2 c3 + C6]
[1Ox2]
[77p2 -2Sp+3]
[27x3 - 54x2y + 36xy2 - Sy3]
[3 2 16 2 64 3
]x -4x y+-xy --y
3 27
[O,OOSa3b3+ 1,2a2b2y + 60aby2 + 1000y3]
[Sx6y9z3 -36x5y8z5 + 54x4y7z7 -27x3y6Z9]
[Sb3+ 2b - 6]
[3X2 -4X+5-~
]7x-1
[
3 2 21
]x +2x -3x-6--
x-2
[5m2+ 3m -10]
[4X2 +2X+10-~
]x-3
[
3 a+2
]a +a-3+~ a -1
[
6 5 4 3 2 11
]a -a +a -a +a -a+ +-
a+1
. d) (15m4 -m3 -m2 +41m-70):(3m2 -2m+7)
( e) (4X3 -10x2 +4x-40):(x-3)
f) (a 5 - 3a 2 + 5): (a 2 - 1)
g) (a7 +2):(a+1)
3. Rozložte na soucin:
a) 1O(y-1)m+1- yb) rs - 6r + 24 - 4s
c) 2nz + ky + kz + 2ny
d) 3ac+ 2d - 3ad - 2ce) 6a3 -15a2 + 15b2 - 6ab2
f) (I -2Y - 12Z2
g) (2c+dY -(3d -lY
h) a4-b4
i) (y-1Y-4
[(y -lX10m -1)]
[(r - 4Xs - 6)]
[(z + y X2n + k)]
[(3a - 2Xc - d)]
[3(a - bXa + bX2a - 5)]
[(I - 2 + ft XI - 2 - ft)]
[(2c+ 4d -lX2c - 2d + 1)]
[(a - b Xa + bXa2 + b2)]
[(y + 1Xy -3)]
6. ÚPRAVY VÝRAZU
. 3.
1 a+6 .(
a+3 - a-2)7a+6 a+6 a-6
, 2.( 2X2 2 + 1)
:(1-~
)y -X x-y1
m+l+-m-l1
1+-m2-1
11'4. (1+1+~ ):(I-~
)1-1 1-1
[6~a]
[y:x]
[m+ 1]
[2~t']
~ 8.
x+y x-y-+-x-y x+y1 1-+-
(x+yr (x- yr1-1
1--6 1+1.. I(1- 1)
1+1+1
, 7. (a' :b' +bHC1, + b1,}:: :::]2 b 2 2 b 2
a ~+2a 2b- a +b + a1 1 1 1-+-b a
[X2- y2 ]. 5.
[1]
[
ab 2
]a-b
[a 2 + b 2 ]b a
1 [x' ~:+ 1]~ 9.
X
x-X
x--I-x
.. 10. a-e . a3 -e3 .(I+~_I+e
): e(l+e)-a
a2 + ae+ e2 a2b -be2 a -e e be
11 2X2 -2x+2. x3 +18 . 2 . 2x -25 x -4x-5
~ 12[(
~+ 3a .a2 +ab+b2)
. 2a+b
]
.~. a - b a 3 - b3 a + b . a 2 + 2ab + b 2 a + b
r4 - S4
[(
S2
J(
2r r2
J]13. r2s2 : 1+7 1--;+7
14 X .[(
X+2
)3. X(X2 +4X+4)
]. 3x + 6 x - 2 . 3x2 -12x + 12
[a:e]
[X~5]
[a~b]
[~
]r-s
[x~2]
58. LIMITA FUNKCE
1 1. 3x + 4
, 1mx42 x2 + 1
2 I' sm x
, 1mX4~ 1+ cos x
4
[2]
[F2-1]
x2 -93, lim 2
x43 x - 2x - 3
4 1. x2 + x - 2
, 1mx4-2 x2 + 5x + 6
5 I' x + 3
, 1m ~x4-3 "1/ x + 4 - 1
6 1. 2 - .Jx - 3
, 1mx47 x2 - 49
7 1. .Jx + 13 - 2.Jx + 1
, 1mx43 X 2 - 9
[~]
[-3]
[2]
[- 516]
[- /6]
[~]
[2]
[8]
8, lim sin 2xX40 3x
9 I' 1- cos2x
, 1mX40 x2
10, lim sin 4xX40 ~"I/x+l-l
11. lim 1- cos 2x + tg 2XX40 'xsmx
12, lim 2x3 - x2 + 5X4OC> X
2+x-2
3
13,limx-3x+lX4OC> 2 - 2 3X -x
[3]
[00]
2
14, lim x -2x+5X4OC> 2x 3 2-x +4
15, lim 2x3 + 5X4-OC>x2-5
2
16, lim x + 7x - 44x44 X
26- x+8
17,~ 9-x2éYFx -3
18 I' 3tg2 x,lm-X40 2X2
[-1]
[O]
19 I' cos x - smx
, 1mX4~ cos 2x
4
[- 00]
[1;]
~E./
[%]
[~]