Facit

Sommarmatte 1

(Skillnad mellan versioner)

Nuvarande version

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

$\left\{\eqalign{ x&=n\pi\cr x&=\displaystyle \frac{\pi}{4}+\displaystyle \frac{n\pi}{2} }\right.$

$x=\displaystyle \frac{n\pi}{3}$

$\left\{\eqalign{ x&=\displaystyle \frac{\pi}{20}+\displaystyle \frac{n\pi}{2}\cr x&=-\displaystyle \frac{\pi}{30}+\displaystyle \frac{n\pi}{3} }\right.$

Svar 4.4:6

$x=n\pi$

$\left\{\eqalign{ x&=\displaystyle \frac{\pi}{4}+2n\pi\cr x&=\displaystyle \frac{\pi}{2}+n\pi\cr x&=\displaystyle \frac{3\pi}{4}+2n\pi}\right.$

$\left\{\eqalign{ x&=\displaystyle \frac{2n\pi}{3}\cr x&=\displaystyle \pi + 2n\pi\cr }\right.$

Svar 4.4:7

$\left\{ \matrix{ x=\displaystyle \frac{\pi}{6}+2n\pi\cr x=\displaystyle \frac{5\pi}{6}+2n\pi\cr x=\displaystyle \frac{3\pi}{2}+2n\pi }\right.$

$x=\pm \displaystyle \frac{\pi}{3} + 2n\pi $

$\left\{ \matrix{ x=\displaystyle \frac{\pi}{2}+2n\pi\cr x=\displaystyle \frac{\pi}{14}+\displaystyle \frac{2n\pi}{7} }\right.$

Svar 4.4:8

$\left\{\eqalign{ x&=\displaystyle \frac{\pi}{4}+2n\pi\cr x&=\displaystyle \frac{\pi}{2}+n\pi\cr x&=\displaystyle \frac{3\pi}{4}+2n\pi }\right.$

$x=\displaystyle \frac{\pi}{3}+n\pi$

$\left\{\eqalign{ x&=n\pi\cr x&=\displaystyle \frac{3\pi}{4}+n\pi }\right.$

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 <table width="100%">
 '''Svar 1.2:6'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%"><tr><td class="ntext">$\displaystyle \frac{152}{35}$</td></tr></table>
 </div>
 '''Svar 1.3:1'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 1.3:2'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 1.3:3'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 1.3:4'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 1.3:5'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 1.3:6'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 2.1:1'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 2.1:2'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 2.1:3'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 2.1:4'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 2.1:5'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 2.1:6'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 2.1:7'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 2.1:8'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 2.2:1'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 2.2:2'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 2.2:3'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 2.2:4'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 2.2:5'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 2.2:6'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 2.2:7'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 2.2:8'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 2.2:9'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 2.3:1'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 2.3:2'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 2.3:3'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 2.3:4'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 2.3:5'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 2.3:6'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 2.3:7'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 2.3:8'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 2.3:9'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 2.3:10'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 </div>
 '''Svar 3.1:1'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 3.1:2'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 3.1:3'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 3.1:4'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 3.1:5'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 3.1:6'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 3.1:7'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 3.1:8'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 3.2:1'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 3.2:2'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 3.2:3'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 3.2:4'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 3.2:5'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 3.2:6'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 3.3:1'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 3.3:2'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 3.3:3'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 3.3:4'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 3.3:5'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 3.3:6'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%" >
 <tr align="left"><td height="5px"/></tr>
 '''Svar 3.4:1'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 3.4:2'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 3.4:3'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 4.1:1'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 4.1:2'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 4.1:3'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 4.1:4'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 4.1:5'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 4.1:6'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 4.1:7'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 4.1:8'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 4.1:9'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 4.1:10'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 4.2:1'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 Facit till alla delfrågor<br \>
 <table width="100%">
 '''Svar 4.2:2'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 4.2:3'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 4.2:4'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 4.2:5'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 4.2:6'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 4.2:7'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 4.2:8'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 4.2:9'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 4.3:1'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 4.3:2'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 4.3:3'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 4.3:4'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 4.3:5'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 4.3:6'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 4.3:7'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 4.3:8'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 4.3:9'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 4.4:1'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 4.4:2'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 4.4:3'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 '''Svar 4.4:4'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%" cellpadding="10px">
 <tr>
 '''Svar 4.4:5'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 <td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
 <td class="ntext" width="50%">$x=\displaystyle \frac{n\pi}{3}$</td>
-</tr>
-<tr align="left">
 <td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
 <td class="ntext" width="50%">
 '''Svar 4.4:6'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 x&=\displaystyle \frac{3\pi}{4}+2n\pi}\right.$
 </td>
-</tr>
-<tr align="left">
 <td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
 <td class="ntext" width="50%">
 '''Svar 4.4:7'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 <td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
 <td class="ntext" width="50%">$x=\pm \displaystyle \frac{\pi}{3} + 2n\pi $</td>
-</tr>
-<tr align="left">
 <td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
 <td class="ntext" width="50%">
 '''Svar 4.4:8'''
-<div class="ovning">
+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
 <table width="100%">
 <tr align="left">
 <td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
 <td class="ntext" width="50%">$x=\displaystyle \frac{\pi}{3}+n\pi$</td>
-</tr>
-<tr align="left">
 <td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
 <td class="ntext" width="50%">
 </table>
 </div>
+<br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br>
+<br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br>
+<br><br><br><br><br><br><br><br><br>