Facit

Sommarmatte 1

(Skillnad mellan versioner)

Versionen från 16 juli 2007 kl. 09.13

Svar 1.1:1

a)	-7	b)	1
c)	11	d)	1

Svar 1.1:2

a)	0	b)	-1
c)	-25	d)	-19

Svar 1.1:3

a)	naturliga talen, heltalen, rationella talen	b)	heltalen, rationella talen	c)	naturliga talen, heltalen, rationella talen
d)	heltalen, rationella talen	e)	heltalen, rationella talen	f)	naturliga talen, heltalen, rationella talen
g)	rationella talen	h)	naturliga talen, heltalen, rationella talen	i)	irrationella talen
j)	naturliga talen, heltalen, rationella talen	k)	irrationella talen	l)	irrationella talen

Svar 1.1:4

a)	\displaystyle \frac{3}{5}<\frac{5}{3}<2<\frac{7}{3}
b)	\displaystyle -\frac{1}{2}<-\frac{1}{3}<-\frac{3}{10}<-\frac{1}{5}
c)	\displaystyle \frac{1}{2}<\frac{3}{5}<\frac{21}{34}<\frac{5}{8}<\frac{2}{3}

Svar 1.1:5

1{,}167

2{,}250

0{,}286

1{,}414

Svar 1.1:6

a)	Talet är rationellt och lika med \,314/100 = 157/50\,.
b)	Talet är rationellt och är lika med \,31413/9999 = 10471/3333\,.
c)	Talet är rationellt och lika med \,1999/9990\,.
d)	Talet är irrationellt.

Svar 1.2:1

a)	\displaystyle \frac{93}{28}	b)	\displaystyle \frac{3}{35}	c)	\displaystyle -\frac{7}{30}
d)	\displaystyle \frac{47}{60}	e)	\displaystyle \frac{47}{84}

Svar 1.2:2

a)	\displaystyle {30}	b)	\displaystyle {8}
c)	\displaystyle {84}	d)	\displaystyle {225}

Svar 1.2:3

\displaystyle \frac{19}{100}

\displaystyle \frac{1}{240}

Svar 1.2:4

\displaystyle \frac{6}{7}

\displaystyle \frac{16}{21}

\displaystyle \frac{1}{6}

Svar 1.2:5

\displaystyle \frac{105}{4}

-5

\displaystyle \frac{8}{55}

Svar 1.2:6

\displaystyle \frac{152}{35}

Svar 1.3:1

-125

\displaystyle \frac{27}{8}

Svar 1.3:2

2^6

2^{-2}

2^0

Svar 1.3:3

3^{-1}

3^5

3^4

3^{-3}

Svar 1.3:4

a)	4	b)	3	c)	625
d)	16	e)	\displaystyle \frac{1}{3750}

Svar 1.3:5

a)	2	b)	\displaystyle \frac{1}{2}	c)	27
d)	2209	e)	9	f)	\displaystyle \frac{25}{3}

Svar 1.3:6

a)	256^{1/3}>200^{1/3}	b)	0{,}4^{-3}>0{,}5^{-3}	c)	0{,}2^{5}>0{,}2^{7}
d)	\bigl(5^{1/3}\bigr)^{4}>400^{1/3}	e)	125^{1/2}>625^{1/3}	f)	3^{40}>2^{56}

Svar 2.1:1

a)	3x^2-3x	b)	xy+x^2y-x^3y	c)	-4x^2+x^2y^2
d)	x^3y-x^2y+x^3y^2	e)	x^2-14x+49	f)	16y^2+40y+25
g)	9x^6-6x^3y^2+y^4	h)	9x^{10}+30x^8+25x^6

Svar 2.1:2

a)	-5x^2+20	b)	10x-11
c)	54x	d)	81x^8-16
e)	2a^2+2b^2

Svar 2.1:3

a)	(x+6)(x-6)	b)	5(x+2)(x-2)	c)	(x+3)^2
d)	(x-5)^2	e)	-2x(x+3)(x-3)	f)	(4x+1)^2

Svar 2.1:4

a)	5\, framför \,x^2\,, \,3\, framför \,x
b)	2\, framför \,x^2\,, \,1\, framför \,x
\textrm{c) }	6\, framför \,x^2\,, \,2\, framför \,x

Svar 2.1:5

a)	\displaystyle \frac{1}{1-x}	b)	-\displaystyle \frac{1}{y(y+2)}
c)	3(x-2)(x-1)	d)	\displaystyle \frac{2(y+2)}{y^2+4}

Svar 2.1:6

a)	2y	b)	\displaystyle\frac{-x+12}{(x-2)(x+3)}
c)	\displaystyle\frac{b}{a(a-b)}	d)	\displaystyle\frac{a(a+b)}{4b}

Svar 2.1:7

\displaystyle \frac{4}{(x+3)(x+5)}

\displaystyle \frac{x^4-x^3+x^2+x-1}{x^2(x-1)}

\displaystyle \frac{ax(a+1-x)}{(a+1)^2}

Svar 2.1:8

\displaystyle \frac{x}{(x+3)(x+1)}

\displaystyle \frac{2(x-3)}{x}

\displaystyle \frac{x+2}{2x+3}

Svar 2.2:1

a)	x=1	b)	x=6
c)	x=-\displaystyle\frac{3}{2}	d)	x=-\displaystyle\frac{13}{3}

Svar 2.2:2

a)	x=1	b)	x=\displaystyle\frac{5}{3}
c)	x=2	d)	x=-2

Svar 2.2:3

a)	x=9
b)	x=\displaystyle\frac{7}{5}
c)	x=\displaystyle\frac{4}{5}
d)	x=\displaystyle\frac{1}{2}

Svar 2.2:4

a)	-2x+y=3
b)	y=-\displaystyle\frac{3}{4}x+\frac{5}{4}

Svar 2.2:5

a)	y=-3x+9
b)	y=-3x+1
c)	y=3x+5
d)	y=-\displaystyle \frac{1}{2}x+5
e)	k = \displaystyle\frac{8}{5}

Svar 2.2:6

a)	\bigl(-\frac{5}{3},0\bigr)	b)	(0,5)
c)	\bigl(0,-\frac{6}{5}\bigr)	d)	(12,-13)
e)	\bigl(-\frac{1}{4},\frac{3}{2}\bigr)

Svar 2.2:7

a)		b)
c)

Svar 2.2:8

a)		b)
c)

Svar 2.2:9

a)	4\, a.e.
b)	5\, a.e.
c)	6\, a.e.

Svar 2.3:1

(x-1)^2-1

(x+1)^2-2

-(x-1)^2+6

\bigl(x+\frac{5}{2}\bigr)^2-\frac{13}{4}

Svar 2.3:2

a)	\left\{ \eqalign{ x_1 &= 1 \cr x_2 &= 3\cr }\right.	b)	\left\{ \eqalign{ y_1 &= -5 \cr y_2 &= 3\cr }\right.	c)	saknar (reella) lösning
d)	\left\{ \eqalign{ x_1 &= \textstyle\frac{1}{2}\cr x_2 &= \textstyle\frac{13}{2}\cr }\right.	e)	\left\{ \eqalign{ x_1 &= -1 \cr x_2 &= \textstyle\frac{3}{5}\cr }\right.	f)	\left\{ \eqalign{ x_1 &= \textstyle\frac{4}{3}\cr x_2 &= 2\cr }\right.

Svar 2.3:3

a)	\left\{ \eqalign{ x_1 &= 0 \cr x_2 & = -3\cr }\right.	b)	\left\{ \eqalign{ x_1 &= 3 \cr x_2 & = -5\cr }\right.
c)	\left\{ \eqalign{ x_1 & = \textstyle\frac{2}{3} \cr x_2 & = -8\cr }\right.	d)	\left\{ \eqalign{ x_1 & = 0\cr x_2 & = 12\cr }\right.
e)	\left\{ \eqalign{ x_1 & = -3 \cr x_2 & = 8\cr }\right.	f)	\left\{ \eqalign{ x_1 & = 0 \cr x_2 & = 1 \cr x_3 & = 2 }\right.

Svar 2.3:4

a)	ax^2-ax-2a=0\,, där \,a\ne 0\, är en konstant.
b)	ax^2-2ax-2a=0\,, där \,a\ne 0\, är en konstant.
c)	ax^2-(3+\sqrt{3}\,)ax+3\sqrt{3}\,a=0\,, där \,a\ne 0\, är en konstant.

Svar 2.3:5

a)	Exempelvis \ x^2+14x+49=0\,.
b)	3< x<4
c)	b=-5

Svar 2.3:6

-2

\displaystyle \frac{3}{4}

Svar 2.3:7

\displaystyle -\frac{7}{4}

saknar max

Svar 2.3:8

Se lösningen i webmaterialet när du loggat in till kursen.

Svar 2.3:9

(-1,0)\ och \ (1,0)

(2,0)\ och \ (3,0)

(1,0)\ och \ (3,0)

Svar 2.3:10

Se lösningen i webmaterialet när du loggat in till kursen

Svar 3.1:1

2^{1/2}

7^{5/2}

3^{4/3}

3^{1/4}

Svar 3.1:2

a)	3	b)	3	c)	ej definierad	d)	5^{11/6}
e)	12	f)	2	g)	-5

Svar 3.1:3

a)	3	b)	\displaystyle \frac{4\sqrt{3}}{3}
c)	2\sqrt{5}	d)	2-\sqrt{2}

Svar 3.1:4

a)	0{,}4	b)	0{,}3
c)	-4\sqrt{2}	d)	2\sqrt{3}

Svar 3.1:5

\displaystyle \frac{\sqrt{3}}{3}

\displaystyle \frac{7^{2/3}}{7}

3-\sqrt{7}

\displaystyle \frac{\sqrt{17}+\sqrt{13}}{4}

Svar 3.1:6

a)	6+2\sqrt{2}+3\sqrt{5}+\sqrt{10}	b)	-\displaystyle \frac{5+4\sqrt{3}}{23}
c)	\displaystyle \frac{2}{3}\sqrt{6}+\displaystyle \frac{2}{3}\sqrt{3}-\displaystyle \frac{2}{5}\sqrt{10}-\displaystyle \frac{2}{5}\sqrt{5}	d)	\displaystyle \frac{5\sqrt{3}+7\sqrt{2}-\sqrt{6}-12}{23}

Svar 3.1:7

\sqrt{5}-\sqrt{7}

-\sqrt{35}

\sqrt{17}

Svar 3.1:8

a)	\sqrt[\scriptstyle3]6 > \sqrt[\scriptstyle3]5	b)	7 > \sqrt7
c)	\sqrt7 > 2{,}5	d)	\sqrt[\scriptstyle3]2\cdot3 > \sqrt2\bigl(\sqrt[\scriptstyle4]3\,\bigr)^3

Svar 3.2:1

x=5

Svar 3.2:2

x=1

Svar 3.2:3

\left \{ \eqalign{ x_1 & = 3 \cr x_2 & = 4\cr } \right.

Svar 3.2:4

Saknar lösning.

Svar 3.2:5

x=1

Svar 3.2:6

x=\displaystyle\frac{5}{4}

Svar 3.3:1

a)	x=3	b)	x=-1
c)	x=-2	d)	x=4

Svar 3.3:2

a)	-1	b)	4	c)	-3	d)	0
e)	2	f)	3	g)	10	h)	-2

Svar 3.3:3

a)	3	b)	-\displaystyle \frac{1}{2}	c)	-3
d)	\displaystyle \frac{7}{3}	e)	4	f)	-2
g)	1	h)	\displaystyle \frac{5}{2}

Svar 3.3:4

-\displaystyle \frac{1}{2}\lg{3}

Svar 3.3:5

a)	5	b)	0	c)	0
d)	0	e)	-2	f)	e^2

Svar 3.3:6


a)	1{,}262
b)	1{,}663
c)	4{,}762

Svar 3.4:1

x=\ln 13

x=\displaystyle\frac{\ln 2 - \ln 13}{1+\ln 3}

x=\displaystyle\frac{\ln 7 - \ln 3}{1-\ln 2}

Svar 3.4:2

\left\{ \eqalign{ x_1&=\sqrt2 \cr x_2&=-\sqrt2 } \right.

x=\ln \left(\displaystyle\frac{\sqrt17}{2}-\frac{1}{2}\right)

Saknar lösning

Svar 3.4:3

a)	x=-\,\displaystyle\frac{1}{\ln{2}}\pm\sqrt{\left(\displaystyle\frac{1}{\ln{2}}\right)^2-1}	b)	x=\displaystyle \frac{5}{2}
c)	x=1

Svar 4.1:1

a)	90^\circ\ och \ \displaystyle \frac{\pi}{2} \textrm{ rad}	b)	135^\circ\ och \ \displaystyle \frac{3\pi}{4} \textrm{ rad}
c)	-240^\circ\ och \ \displaystyle -\frac{4\pi}{3} \textrm{ rad}	d)	2910^\circ\ och \ \displaystyle \frac{97\pi}{6} \textrm{ rad}

Svar 4.1:2

\displaystyle \frac{\pi}{4}\textrm{ rad}

\displaystyle \frac{3\pi}{4}\textrm{ rad}

-\displaystyle \frac{7\pi}{20}\textrm{ rad}

\displaystyle \frac{3\pi}{2}\textrm{ rad}

Svar 4.1:3

x=50

x=5

x=15

Svar 4.1:4

a)	5 \textrm{ l.e.}
b)	\sqrt{61} \textrm{ l.e.}
c)	(2,0)

Svar 4.1:5

a)	(x-1)^2+(y-2)^2=4
b)	(x-2)^2+(y+1)^2=13

Svar 4.1:6

a)	En cirkel med radie 3 och medelpunkt i origo.	b)	En cirkel med radie \sqrt 3 och medelpunkt i punkten (1, 2).
c)	En cirkel med radie \frac{1}{3}\sqrt 10 och medelpunkt i punkten (1/3, -7/3).

Svar 4.1:7

a)	En cirkel med medelpunkt (-1, 1) och radie \sqrt 3.	b)	En cirkel med medelpunkt (0, -2) och radie 2.
c)	En cirkel med medelpunkt (1, -3) och radie \sqrt 7.
d)	Endast punkten (1, -1).

Svar 4.1:8

\displaystyle \frac{10}{\pi}\textrm{ varv }\approx 3,2 \textrm{ varv}

Svar 4.1:9

\displaystyle \frac{32\pi}{3} \textrm{ cm}^2 \approx 33,5 \textrm{ cm}^2

Svar 4.1:10

x=9 dm

Svar 4.2:1

Facit till alla delfrågor

a)	x=13\cdot\tan {27 ^\circ} \approx 6{,}62	b)	x=25\cdot\cos {32 ^\circ} \approx 21{,}2
c)	x=\displaystyle\frac{14}{\tan {40 ^\circ}} \approx 16{,}7	d)	x=\displaystyle\frac{16}{\cos {20 ^\circ}} \approx 17{,}0
e)	x=\displaystyle\frac{11}{\sin {35 ^\circ}} \approx 19{,}2	f)	x=\displaystyle\frac{19}{\tan {50 ^\circ}} \approx 15{,}9

Svar 4.2:2

a)	\tan v=\displaystyle\frac{2}{5}	b)	\sin v=\displaystyle\frac{7}{11}
c)	\cos v=\displaystyle\frac{5}{7}	d)	\sin v=\displaystyle\frac{3}{5}
e)	v=30 ^\circ	f)	\sin \displaystyle\frac{v}{2}=\displaystyle\frac{1}{3}

Svar 4.2:3

a)	-1	b)	1	c)	0
d)	0	e)	\displaystyle \frac{1}{\sqrt{2}}	f)	\displaystyle \frac{\sqrt{3}}{2}

Svar 4.2:4

a)	\displaystyle \frac{\sqrt{3}}{2}	b)	\displaystyle \frac{1}{2}	c)	-1
d)	0	e)	\displaystyle \frac{1}{\sqrt{3}}	f)	\sqrt{3}

Svar 4.2:5

-\displaystyle \frac{1}{\sqrt{2}}

\displaystyle \frac{\sqrt{3}}{2}

-1

Svar 4.2:6

x= \sqrt{3}-1

Svar 4.2:7

Älvens bredd är \ \displaystyle\frac{100}{\sqrt{3}-1} m \approx 136{,}6 m.

Svar 4.2:8

\ell\cos \gamma=a \cos \alpha - b\cos \beta

Svar 4.2:9

Avståndet är \ \sqrt{205-48\sqrt{3}} \approx 11{,}0 km.

Svar 4.3:1

v = \displaystyle \frac{9\pi}{5}

v = \displaystyle \frac{6\pi}{7}

v = \displaystyle \frac{9\pi}{7}

Svar 4.3:2

v=\displaystyle \frac{\pi}{2}

v=\displaystyle \frac{3\pi}{5}

Svar 4.3:3

a)	-a	b)	a
c)	\sqrt{1-a^2}	d)	\sqrt{1-a^2}
e)	-a	f)	\displaystyle \frac{\sqrt{3}}{2}\sqrt{1-a^2}+\displaystyle \frac{1}{2}\cdot a

Svar 4.3:4

a)	1-b^2	b)	\sqrt{1-b^2}
c)	2b\sqrt{1-b^2}	d)	2b^2-1
e)	\sqrt{1-b^2}\cdot\displaystyle \frac{1}{\sqrt{2}} + b\cdot \displaystyle \frac{1}{\sqrt{2}}	f)	b\cdot\displaystyle \frac{1}{2}+\sqrt{1-b^2}\cdot\displaystyle \frac{\sqrt{3}}{2}

Svar 4.3:5

\cos{v}=\displaystyle \frac{2\sqrt{6}}{7}\quad och \quad\tan{v}=\displaystyle \frac{5}{2\sqrt{6}}\,.

Svar 4.3:6

a)	\sin{v}=-\displaystyle \frac{\sqrt{7}}{4}\quad och \quad\tan{v}=-\displaystyle \frac{\sqrt{7}}{3}\,.
b)	\cos{v}=-\displaystyle \frac{\sqrt{91}}{10}\quad och \quad\tan{v}=-\displaystyle \frac{3}{\sqrt{91}}\,.
c)	\sin{v}=-\displaystyle \frac{3}{\sqrt{10}}\quad och \quad\cos{v}=-\displaystyle \frac{1}{\sqrt{10}}\,.

Svar 4.3:7

a)	\sin{(x+y)}=\displaystyle \frac{4\sqrt{2}+\sqrt{5}}{9}
b)	\sin{(x+y)}=\displaystyle \frac{3\sqrt{21}+8}{25}

Svar 4.3:8

Se lösningen i webmaterialet när du loggat in till kursen

Svar 4.3:9

Se lösningen i webmaterialet när du loggat in till kursen

Svar 4.4:1

a)	\displaystyle v=\frac{\pi}{6}\,, \,\displaystyle v=\frac{5\pi}{6}	b)	\displaystyle v=\frac{\pi}{3}\,, \,\displaystyle v=\frac{5\pi}{3}
c)	\displaystyle v=\frac{\pi}{2}	d)	\displaystyle v=\frac{\pi}{4}\,, \,\displaystyle v=\frac{5\pi}{4}
e)	lösning saknas	f)	\displaystyle v=\frac{11\pi}{6}\,, \,\displaystyle v=\frac{7\pi}{6}
g)	\displaystyle v=\frac{5\pi}{6}\,, \,\displaystyle v=\frac{11\pi}{6}

Svar 4.4:2

a)	\left\{\eqalign{ x&=\displaystyle\frac{\pi}{3}+2n\pi\cr x&=\displaystyle\frac{2\pi}{3}+2n\pi } \right.	b)	\left\{\eqalign{ x&=\displaystyle\frac{\pi}{3}+2n\pi\cr x&=\displaystyle\frac{5\pi}{3}+2n\pi } \right.	c)	x=n\pi
d)	\left\{\eqalign{ x&=\displaystyle\frac{\pi}{20}+\displaystyle\frac{2n\pi}{5}\cr x&=\displaystyle\frac{3\pi}{20}+\displaystyle\frac{2n\pi}{5} } \right.	e)	\left\{\eqalign{ x&=\displaystyle\frac{\pi}{30}+\displaystyle\frac{2n\pi}{5}\cr x&=\displaystyle\frac{\pi}{6}+\displaystyle\frac{2n\pi}{5}} \right.	f)	\left\{\eqalign{ x&=\displaystyle\frac{\pi}{4}+\displaystyle\frac{2n\pi}{3}\cr x&=\displaystyle\frac{5\pi}{12}+\displaystyle\frac{2n\pi}{3}} \right.

Svar 4.4:3

a)	\left\{\eqalign{ x&=\displaystyle\frac{\pi}{6}+2n\pi\cr x&=\displaystyle\frac{11\pi}{6}+2n\pi }\right.	b)	\left\{\eqalign{ x&=\displaystyle\frac{\pi}{5}+2n\pi\cr x&=\displaystyle\frac{4\pi}{5}+2n\pi }\right.
c)	\left\{\eqalign{ x&=25^\circ + n\cdot 360^\circ\cr x&=75^\circ + n\cdot 360^\circ }\right.	d)	\left\{\eqalign{ x&=5^\circ + n \cdot 120^\circ \cr x&= 55^\circ + n \cdot 120^\circ }\right.

Svar 4.4:4

v_1=50^\circ, \ \ v_2=120^\circ, \ \ v_3=230^\circ\ \ och \ \ v_4=300^\circ

Svar 4.4:5

a)	\left\{\eqalign{ x&=n\pi\cr x&=\displaystyle \frac{\pi}{4}+\displaystyle \frac{n\pi}{2} }\right.	b)	x=\displaystyle \frac{n\pi}{3}
c)	\left\{\eqalign{ x&=\displaystyle \frac{\pi}{20}+\displaystyle \frac{n\pi}{2}\cr x&=-\displaystyle \frac{\pi}{30}+\displaystyle \frac{n\pi}{3} }\right.

Svar 4.4:6

a)	x=n\pi	b)	\left\{\eqalign{ x&=\displaystyle \frac{\pi}{4}+2n\pi\cr x&=\displaystyle \frac{\pi}{2}+n\pi\cr x&=\displaystyle \frac{3\pi}{4}+2n\pi}\right.
c)	\left\{\eqalign{ x&=\displaystyle \frac{2n\pi}{3}\cr x&=\displaystyle \pi + 2n\pi\cr }\right.

Svar 4.4:7

a)	\left\{ \matrix{ x=\displaystyle \frac{\pi}{6}+2n\pi\cr x=\displaystyle \frac{5\pi}{6}+2n\pi\cr x=\displaystyle \frac{3\pi}{2}+2n\pi }\right.	b)	x=\pm \displaystyle \frac{\pi}{3} + 2n\pi
c)	\left\{ \matrix{ x=\displaystyle \frac{\pi}{2}+2n\pi\cr x=\displaystyle \frac{\pi}{14}+\displaystyle \frac{2n\pi}{7} }\right.

Svar 4.4:8

a)	\left\{\eqalign{ x&=\displaystyle \frac{\pi}{4}+2n\pi\cr x&=\displaystyle \frac{\pi}{2}+n\pi\cr x&=\displaystyle \frac{3\pi}{4}+2n\pi }\right.	b)	x=\displaystyle \frac{\pi}{3}+n\pi
c)	\left\{\eqalign{ x&=n\pi\cr x&=\displaystyle \frac{3\pi}{4}+n\pi }\right.

Den här artikeln är hämtad från http://wiki.math.se/wikis/sf0600_0701/index.php/Facit

 <td class="ntext" width="50%">1</td>
 </tr>
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 </table>
 </div>
 <td class="ntext" width="50%">-19</td>
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 </div>
 <div class="ovning">
 <table width="100%">
-<tr><td height="5px"/></tr>
 <tr align="left" valign="top">
 <td class="ntext">a)</td>
 <td class="ntext" width="33%">irrationella talen</td>
 </tr>
-<tr><td height="5px"/></tr>
 </table>
 </div>
 <td class="ntext" width="100%">\displaystyle \frac{1}{2}<\frac{3}{5}<\frac{21}{34}<\frac{5}{8}<\frac{2}{3}</td>
 </tr>
-<tr><td height="5px"/></tr>
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 </div>
 <td class="ntext" width="25%">1{,}414</td>
 </tr>
-<tr><td height="5px"/></tr>
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 </div>
 <td class="ntext" width="100%">Talet är irrationellt.</td>
 </tr>
-<tr><td height="5px"/></tr>
 </table>
 </div>
 <td class="ntext" width="33%">\displaystyle \frac{47}{84}</td>
 </tr>
-<tr><td height="5px"/></tr>
 </table>
 </div>
 <td class="ntext" width="50%">\displaystyle {225}</td>
 </tr>
-<tr><td height="5px"/></tr>
 </table>
 </div>
 <div class="ovning">
 <table width="100%">
-<tr><td height="5px"/></tr>
 <tr align="left">
 <td class="ntext">a)</td>
 <td class="ntext" width="50%">\displaystyle \frac{1}{240}</td>
 </tr>
-<tr><td height="5px"/></tr>
 </table>
 </div>
 <td class="ntext" width="33%">\displaystyle \frac{1}{6}</td>
 </tr>
-<tr><td height="5px"/></tr>
 </table>
 </div>
 <div class="ovning">
 <table width="100%">
-<tr><td height="5px"/></tr>
 <tr align="left">
 <td class="ntext">a)</td>
 <td class="ntext" width="33%">\displaystyle \frac{8}{55}</td>
 </tr>
-<tr><td height="5px"/></tr>
 </table>
 </div>
 <td class="ntext" width="25%">\displaystyle \frac{27}{8}</td>
 </tr>
-<tr><td height="5px"/></tr>
 </table>
 </div>
 <td class="ntext" width="33%">2^0</td>
 </tr>
-<tr><td height="5px"/></tr>
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 </div>
 <td class="ntext" width="20%">3^{-3}</td>
 </tr>
-<tr><td height="5px"/></tr>
 </table>
 </div>
 <td class="ntext" width="33%">\displaystyle \frac{1}{3750}</td>
 </tr>
-<tr><td height="5px"/></tr>
 </table>
 </div>
 <td class="ntext" width="33%">\displaystyle \frac{25}{3}</td>
 </tr>
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 </table>
 </div>
 <td class="ntext" width="33%">3^{40}>2^{56}</td>
 </tr>
-<tr><td height="5px"/></tr>
 </table>
 </div>
 <td class="ntext" width="33%">9x^{10}+30x^8+25x^6</td>
 </tr>
-<tr><td height="5px"/></tr>
 </table>
 </div>
 <td class="ntext" width="50%">2a^2+2b^2</td>
 </tr>
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 </table>
 </div>
 <td class="ntext" width="33%">(4x+1)^2</td>
 </tr>
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 </table>
 </div>
 <td class="ntext" width="100%">6\, framf&ouml;r \,x^2\,, \,2\, framf&ouml;r \,x</td>
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 <td class="ntext" width="33%">\displaystyle \frac{x+2}{2x+3}</td>
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 <td class="ntext" width="50%">x=-\displaystyle\frac{13}{3}</td>
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 <td class="ntext" width="50%">x=-2</td>
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 <td class="ntext" width="100%">x=\displaystyle\frac{1}{2}</td>
 </tr>
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 </div>
 <td class="ntext" width="100%">k = \displaystyle\frac{8}{5}</td>
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 <td class="ntext" width="50%">\bigl(-\frac{1}{4},\frac{3}{2}\bigr)</td>
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 <td class="ntext" width="100%">6\, a.e.</td>
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 <td class="ntext" width="25%">\bigl(x+\frac{5}{2}\bigr)^2-\frac{13}{4}</td>
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 </div>
 <div class="ovning">
 <table width="100%">
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 <td class="ntext">a)</td>
 <td class="ntext" width="33%"> \left\{ \eqalign{  x_1 &= \textstyle\frac{4}{3}\cr x_2 &= 2\cr }\right.</td>
 </tr>
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 <td class="ntext" width="50%">\left\{ \eqalign{ x_1 & = 0 \cr x_2 & = 1 \cr x_3 & = 2 }\right.   </td>
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 <td class="ntext" width="100%">ax^2-(3+\sqrt{3}\,)ax+3\sqrt{3}\,a=0\,, där \,a\ne 0\, är en konstant.</td>
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 <td class="ntext" width="100%">b=-5</td>
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 <td class="ntext" width="33%">\displaystyle \frac{3}{4}</td>
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 <td class="ntext" width="33%">saknar max</td>
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 <td class="ntext" width="33%">(1,0)\  och \ (3,0)</td>
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 <td class="ntext" width="25%">3^{1/4}</td>
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 <td></td>
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 <td class="ntext" width="50%">2-\sqrt{2}</td>
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 <td class="ntext" width="50%">2\sqrt{3}</td>
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 <td class="ntext" width="25%">\displaystyle \frac{\sqrt{17}+\sqrt{13}}{4}</td>
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 <td class="ntext" width="50%">\displaystyle \frac{5\sqrt{3}+7\sqrt{2}-\sqrt{6}-12}{23}</td>
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 <td class="ntext" width="33%">\sqrt{17}</td>
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 <td class="ntext" width="50%">\sqrt[\scriptstyle3]2\cdot3 > \sqrt2\bigl(\sqrt[\scriptstyle4]3\,\bigr)^3</td>
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 <div class="ovning">
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 <td class="ntext" width="100%">x=5</td>
 </tr>
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 </table>
 </div>
 <div class="ovning">
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 <td class="ntext" width="100%">x=1</td>
 </tr>
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 </table>
 </div>
 <div class="ovning">
 <table width="100%">
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 <td class="ntext" width="100%">\left \{ \eqalign{ x_1 & = 3 \cr x_2 & = 4\cr } \right.</td>
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 </table>
 </div>
 <div class="ovning">
 <table width="100%">
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 <td class="ntext" width="100%">Saknar l&ouml;sning.</td>
 </tr>
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 </table>
 </div>
 <div class="ovning">
 <table width="100%">
-<tr><td height="5px"/></tr>
 <tr align="left">
 <td class="ntext" width="100%">x=1</td>
 </tr>
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 </table>
 </div>
 <div class="ovning">
 <table width="100%">
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 <td class="ntext" width="100%">x=\displaystyle\frac{5}{4}</td>
 </tr>
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 </table>
 </div>
 <td class="ntext" width="50%">x=4</td>
 </tr>
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 </div>
 <td class="ntext" width="25%">-2</td>
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 <td class="ntext" width="33%">\displaystyle \frac{5}{2}</td>
 </tr>
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 </div>
 <td class="ntext" width="33%">-\displaystyle \frac{1}{2}\lg{3}</td>
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 <td class="ntext" width="33%">e^2</td>
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 <td class="ntext">c)</td>
 <td class="ntext" width="33%">x=\displaystyle\frac{\ln 7 - \ln 3}{1-\ln 2}</td>
-<tr><td height="5px"/></tr>
 </tr>
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 <td class="ntext" width="33%">Saknar l&ouml;sning</td>
 </tr>
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 </div>
 <td class="ntext" width="50%">x=1</td>
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 <td class="ntext" width="50%">2910^\circ\  och \ \displaystyle \frac{97\pi}{6} \textrm{ rad}</td>
 </tr>
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 </table>
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 <td class="ntext" width="25%">\displaystyle \frac{3\pi}{2}\textrm{ rad}</td>
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 <td class="ntext" width="33%">x=15</td>
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 <td class="ntext" width="100%">(2,0)</td>
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 <td class="ntext" width="100%">(x-2)^2+(y+1)^2=13</td>
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 <td class="ntext" width="50%">En cirkel med radie \frac{1}{3}\sqrt 10 och medelpunkt i punkten (1/3, -7/3).</td>
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 <td class="ntext" width="50%">Endast punkten (1, -1). </td>
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 </div>
 <div class="ovning">
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 <td class="ntext" width="100%">
 \displaystyle \frac{10}{\pi}\textrm{ varv }\approx 3,2 \textrm{ varv}  </td>
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 </div>
 <div class="ovning">
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 <td class="ntext" width="100%">\displaystyle \frac{32\pi}{3} \textrm{ cm}^2 \approx 33,5 \textrm{ cm}^2</td>
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 </table>
 <div class="ovning">
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 <td class="ntext" width="100%">x=9 dm</td>
 </tr>
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 </table>
 </div>
 <td class="ntext" width="50%">x=\displaystyle\frac{19}{\tan {50 ^\circ}} \approx 15{,}9</td>
 </tr>
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 </table>
 </div>
 <td class="ntext" width="50%">\sin \displaystyle\frac{v}{2}=\displaystyle\frac{1}{3}</td>
 </tr>
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 </table>
 </div>
 <td class="ntext" width="33%">\displaystyle \frac{\sqrt{3}}{2}</td>
 </tr>
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 </div>
 <td class="ntext" width="33%">\sqrt{3}</td>
 </tr>
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 </div>
 <td class="ntext" width="25%">-1</td>
 </tr>
-<tr><td height="5px"/></tr>
 </table>
 </div>
 <td class="ntext" width="100%">x= \sqrt{3}-1</td>
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-<tr><td height="5px"/></tr>
 </table>
 </div>
 <td class="ntext" width="100%">Älvens bredd är \ \displaystyle\frac{100}{\sqrt{3}-1} m \approx 136{,}6 m.</td>
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 </table>
 </div>
 <td class="ntext" width="100%">\ell\cos \gamma=a \cos \alpha - b\cos \beta </td>
 </tr>
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 </table>
 </div>
 <td class="ntext" width="100%">Avståndet är \ \sqrt{205-48\sqrt{3}} \approx 11{,}0 km.</td>
 </tr>
-<tr><td height="5px"/></tr>
 </table>
 </div>
 <td class="ntext" width="50%">v=\displaystyle \frac{3\pi}{5}</td>
 </tr>
-<tr><td height="5px"/></tr>
 </table>
 </div>
 <td class="ntext" width="50%">\displaystyle \frac{\sqrt{3}}{2}\sqrt{1-a^2}+\displaystyle \frac{1}{2}\cdot a </td>
 </tr>
-<tr><td height="5px"/></tr>
 </table>
 </div>
 <td class="ntext" width="50%">b\cdot\displaystyle \frac{1}{2}+\sqrt{1-b^2}\cdot\displaystyle \frac{\sqrt{3}}{2}</td>
 </tr>
-<tr><td height="5px"/></tr>
 </table>
 </div>
 <td class="ntext" width="100%">\cos{v}=\displaystyle \frac{2\sqrt{6}}{7}\quad och \quad\tan{v}=\displaystyle \frac{5}{2\sqrt{6}}\,.</td>
 </tr>
-<tr><td height="5px"/></tr>
 </table>
 </div>
 <td class="ntext" width="100%">\sin{v}=-\displaystyle \frac{\sqrt{7}}{4}\quad och \quad\tan{v}=-\displaystyle \frac{\sqrt{7}}{3}\,.</td>
 </tr>
-<tr><td height="5px"/></tr>
 <tr align="left">
 <td class="ntext">b)</td>
 <td class="ntext" width="100%">\cos{v}=-\displaystyle \frac{\sqrt{91}}{10}\quad och \quad\tan{v}=-\displaystyle \frac{3}{\sqrt{91}}\,.</td>
 </tr>
-<tr><td height="5px"/></tr>
 <tr align="left">
 <td class="ntext">c)</td>
 <td class="ntext" width="50%">\sin{v}=-\displaystyle \frac{3}{\sqrt{10}}\quad och \quad\cos{v}=-\displaystyle \frac{1}{\sqrt{10}}\,.</td>
 </tr>
-<tr><td height="5px"/></tr>
 </table>
 </div>
 <td class="ntext" width="100%">\sin{(x+y)}=\displaystyle \frac{3\sqrt{21}+8}{25}</td>
 </tr>
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 </table>
 </div>
 du loggat in till kursen</td>
 </tr>
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 </table>
 </div>
 du loggat in till kursen</td>
 </tr>
-<tr><td height="5px"/></tr>
 </table>
 </div>
 <td class="ntext" width="50%">\displaystyle v=\frac{5\pi}{6}\,, \,\displaystyle v=\frac{11\pi}{6}</td>
 </tr>
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 </table>
 </div>
 </td>
 </tr>
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 </table>
 </div>
 </td>
 </tr>
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 </table>
 </div>
 </td>
 </tr>
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 </table>
 </div>
 </td>
 </tr>
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 </table>
 </div>