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

$72$

$3$

$-125$

$\displaystyle \frac{27}{8}$

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

$0$

$-2$

$\displaystyle \frac{3}{4}$

Svar 2.3:7

$1$

$\displaystyle -\frac{7}{4}$

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

$1$

$0$

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

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

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

$1$

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

$-1$

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