Facit
Sommarmatte 1
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
| a) | 1{,}167 | b) | 2{,}250 | c) | 0{,}286 | d) | 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
| a) | \displaystyle \frac{19}{100} | b) | \displaystyle \frac{1}{240} |
Svar 1.2:4
| a) | \displaystyle \frac{6}{7} | b) | \displaystyle \frac{16}{21} | c) | \displaystyle \frac{1}{6} |
Övning 1.2:5
| a) | \displaystyle \frac{105}{4} | b) | -5 | c) | \displaystyle \frac{8}{55} |
Svar 1.2:6
| \displaystyle \frac{152}{35} |
Övning 1.3:1
| a) | 72 | b) | 3 | c) | -125 | d) | \displaystyle \frac{27}{8} |
Övning 1.3:2
| a) | 2^6 | b) | 2^{-2} | c) | 2^0 |
Övning 1.3:3
| a) | 3^{-1} | b) | 3^5 | c) | 3^4 | d) | 3^{-3} | e) | 3^{-3} |
Övning 1.3:4
| a) | 4 | b) | 3 | c) | 625 |
| d) | 16 | e) | \displaystyle \frac{1}{3750} | ||
Övning 1.3:5
| a) | 2 | b) | \displaystyle \frac{1}{2} | c) | 27 |
| d) | 2209 | e) | 9 | f) | \displaystyle \frac{25}{3} |
Övning 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
| a) | \displaystyle \frac{4}{(x+3)(x+5)} | b) | \displaystyle \frac{x^4-x^3+x^2+x-1}{x^2(x-1)} | c) | \displaystyle \frac{ax(a+1-x)}{(a+1)^2} |
Svar 2.1:8
| a) | \displaystyle \frac{x}{(x+3)(x+1)} | b) | \displaystyle \frac{2(x-3)}{x} | c) | \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} |
Övning 2.2:4
| a) | -2x+y=3 |
| b) | y=-\displaystyle\frac{3}{4}x+\frac{5}{4} |
Övning 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} |
Övning 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) | ||
Övning 2.2:7
| a) |  |
b) |  |
| c) |  |
Övning 2.2:8
| a) |  |
b) |  |
| c) |  |
Övning 2.2:9
| a) | 4\, a.e. |
| b) | 5\, a.e. |
| c) | 6\, a.e. |
Övning 2.3:1
| a) | (x-1)^2-1 | b) | (x+1)^2-2 | c) | -(x-1)^2+6 | d) | \bigl(x+\frac{5}{2}\bigr)^2-\frac{13}{4} |
Övning 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. |
Övning 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. |
Övning 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. |
Övning 2.3:5
| a) | Exempelvis \ x^2+14x+49=0\,. |
| b) | 3< x<4 |
| c) | b=-5 |
Övning 2.3:6
| a) | 0 | b) | -2 | c) | \displaystyle \frac{3}{4} |
Övning 2.3:7
| a) | 1 | b) | \displaystyle -\frac{7}{4} | c) | saknar max |
Övning 2.3:8
| Se lösningen i webmaterialet när du loggat in till kursen. |
Övning 2.3:9
| a) | (-1,0)\ och \ (1,0) | b) | (2,0)\ och \ (3,0) | c) | (1,0)\ och \ (3,0) |
Övning 2.3:10
| Se lösningen i webmaterialet när du loggat in till kursen |
Svar 3.1:1
| a) | 2^{1/2} | b) | 7^{5/2} | c) | 3^{4/3} | d) | 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
| a) | \displaystyle \frac{\sqrt{3}}{3} | b) | \displaystyle \frac{7^{2/3}}{7} | c) | 3-\sqrt{7} | d) | \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
| a) | \sqrt{5}-\sqrt{7} | b) | -\sqrt{35} | c) | \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
| a) | 1 | b) | 0 | c) | -\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
| a) | x=\ln 13 | b) | x=\displaystyle\frac{\ln 2 - \ln 13}{1+\ln 3} | c) | x=\displaystyle\frac{\ln 7 - \ln 3}{1-\ln 2} |
Svar 3.4:2
| a) | \left\{ \eqalign{ x_1&=\sqrt2 \cr x_2&=-\sqrt2 } \right. | b) | x=\ln \left(\displaystyle\frac{\sqrt17}{2}-\frac{1}{2}\right) | c) | 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 | ||
