Zad. 1
f(x)=ax+b
f(x) = (m-1)x+6
a=m-1
funkcja rosnaca dla a>0
m-1>0
m>1
Odp. D
Zad. 2
[tex]f(x)=-\sqrt2x+4\\0=-\sqrt2x+4\\\\\sqrt2x=4/:\sqrt2\\x=\frac4{\sqrt2}*\frac{\sqrt2}{\sqrt2}=\frac{4\sqrt2}{2}=2\sqrt2[/tex]
Odp. D.
Zad. 3
[tex]y = 2x-3\\a = 2\\[/tex]
Proste sa rownolegle wtedy, kiedy ich wspolczynnik kierunkowy a jest rowny
[tex]D(-2,1)\\1=2*(-2)+b\\1=-4+b\\1+4=b\\5=b\\y=2x+5[/tex]
Odp. C.
Zad. 4
[tex]3x-6y+7=0\\3x+7=6y /:6\\\frac{3x+7}{6}=y\\y=\frac{3}{6}x+\frac76\\y=\frac12x+\frac76\\[/tex]
a = 1/2
[tex]y=\frac12x[/tex]
Odp. A
Zad. 5
[tex]M=(3, -5)\\N=(-1,7)\\\\\left \{ {{-5=3a+b} \atop {7=-a+b/*3}} \right. \\\left \{ {{-5=3a+b} \atop {21=-3a+3b}} \right. \\-5+21=b+3b\\16=4b\\b=4\\7=-a+4\\7-4=-a\\3=-a\\a=-3\\y=-3x+4[/tex]
Odp. A
Zad. 6
[tex]k: y= -\frac14x+\frac72\\A=(-2, 4)\\a_1 = -\frac14\\a_2=?\\a_1*a_2=-1\\-\frac14*a_2=-1/*4\\-a_2=-4\\a_2=4\\\\4=4*(-2)+b\\4=-8+b\\12=b\\\\y=4x+12[/tex]
Odp. D
Zad. 7
[tex]A=(-5, 2)\\B=(3, -2)\\|AB|=\sqrt{(3+5)^2+(-2-2)^2}=\sqrt{8^2+(-4)^2}=\sqrt{64+16}=\sqrt{80}=\sqrt{4*4*5}=4\sqrt5[/tex]
[tex]Ob=3*|AB|=3*4\sqrt5=12\sqrt5[/tex]
Odp. C.
Zad. 8.
[tex]S = (-4, 7)\\Q=(17, 12)\\\\S=(\frac{x_2+x_1}{2}, \frac{y_2+y_1}{2})\\(-4,7)=(\frac{17+x_1}{2}, \frac{12+y_1}{2})\\-4=\frac{17+x_1}{2} /*2\\-8=17+x_1\\-8-17=x_1\\x_1=-25\\\\7=\frac{12+y_1}{2} /*2\\14=12+y_1\\14-12=y_1\\2=y_1\\\\P=(-25,2)[/tex]
Odp. C.
Zad. 9
[tex]|BC| = d\\d = a\sqrt2\\\\\\|BC| = \sqrt{(5+2)^2+(1-4)^2}=\sqrt{7^2+(-3)^2}=\sqrt{49+9}=\sqrt{58}[/tex]
[tex]\sqrt{58}=a\sqrt2/:\sqrt2\\a=\frac{\sqrt{58}}{\sqrt{2}}=\frac{\sqrt{58}*\sqrt2}{2}=\frac{\sqrt{116}}{2}=\sqrt{29}\\P=a^2\\P=(\sqrt{29})^2=29\\[/tex]
Odp. A
Zad. 10
[tex](4, 1)=(\frac{x_2-3}{2}; \frac{y_2+2}{2})\\4=\frac{x_2-3}{2}\\8=x_2-3\\11=x_2\\\\1=\frac{y_2+2}{2}\\2=y_2+2\\0=y_2\\B=(11, 0)\\|AB| = \sqrt{(11+3)^2+(0-2)^2}=\sqrt{14^2+(-2)^2}=\sqrt{196+4}=\sqrt{200}=\sqrt{100*2}=10\sqrt2[/tex]Odp. D.
Zad. 11
[tex]Mediana = 8\\1, 4, 7, 9, 9, 9\\Odp. D[/tex]
Zad. 12
[tex]\frac{2+4+7+8+9}{5}=\frac{2+4+7+8+9+x}{6}\\\frac{30}{5} = \frac{30+x}{6}\\6=\frac{30+x}{6}\\36=30+x\\36-30=x\\6=x[/tex]
Odp. D
Zad. 13
[tex]\frac{x-1+3x+5x+1+7x}{4}=72\\\frac{16x}{4}=72\\4x=72\\x=72:4\\x=18[/tex]
Odp. D
Zad. 14
Mozliwe wyniki:
r, r, r
r, r, o
r, o, o
o, o, o
o, o, r
o, r, r
o, r, o
r, o, r
(8 kombinacji)
Co najmniej jedna reszka:
r, r, r,
r, r, o
r, o, o
o, o, r
o, r, r
o, r, o
r, o, r
(7 kombinacji)
[tex]\frac78[/tex]
Odp. A
Zad. 15
[tex]|\Omega| = 6*6=36\\A={(5, 5)}\\|A|=1\\P(A)=\frac{|A|}{|\Omega|} = \frac{1}{36}[/tex]
Odp. D
Zad. 16
[tex]2x^2-4x>(x+3)(x-2)\\2x^2-4x>x^2-2x+3x-6\\2x^2-x^2-4x+2x-3x+6>0\\x^2-x+6>0\\\Delta=(-1)^2+4*1*6\\\Delta=1+24\\\Delta=25\\\sqrt{\Delta}=5\\x_1=\frac{1-5}{2}=\frac{-4}{2}=-2\\x_2=\frac{1+5}{2}=\frac62=3\\a>0\\[/tex]
x∈(-∞, -2)∪(3; ∞)
Zad. 17
A = (-2, 2)
B = (2, 10)
Wyznaczamy wzor prostej przechodzacej przez punkty A i B
[tex]\left \{ {{2=-2a+b} \atop {10=2a+b}} \right.\\12=2b\\b=6\\2=-2a+6\\2-6=-2a\\-4=-2a\\a=2\\\\y=2x+6[/tex]
Wyznaczamy srodek odcinka |AB|
[tex]S=(\frac{-2+2}{2}; \frac{2+10}2)\\S=(0; 6)[/tex]
Wyznaczamy wzor prostej prostopadlej do prostej y przechodzacy przez punkt S
[tex]2*a_2=-1\\a_2=-\frac12\\6=-\frac12*0+b\\6=0+b\\b=6\\y=-\frac12x+6[/tex]
Zad. 18
A=(-2, 4)
B = (6, -2)
Przez punkt C przechodzi prosta y=0, wiec C=(x, 0)
|AC| = |BC|
[tex]\sqrt{(x+2)^2+(0-4)^2} = \sqrt{(x-6)^2+(0+2)^2}\\\sqrt{x^2+4x+4+16} = \sqrt{x^2-24x+36+4}\\\sqrt{x^2+4x+20} = \sqrt{x^2-24x+40}\\x^2+4x+20=x^2-24x+40\\x^2-x^2+4x+24x+20-40=0\\28x-20=0\\28x=20/:28\\x=\frac{20}{28}=\frac{10}{14}=\frac57[/tex]
[tex]C=(\frac57; 0)[/tex]
Zad. 19
[tex]|\Omega|=6*6=36\\A = {(1,3 ), (1,6), (2,3), (2,6), (3, 3), (3,4), (3, 5), (3,6), (4, 6),(5, 6), (6,6) }\\|A| = 11\\P(A)=\frac{|A|}{|\Omega|}=\frac{11}{36}[/tex]
Zad. 20
[tex]X = \frac{2+3+5+6+8}{5}=\frac{24}{5}=4,8[/tex]
[tex]\delta=\sqrt{\frac{(2-4,8)^2+(3-4,8)^2+(5-4,8)^2+(6-4,8)^2+(8-4,8)^2}5}\\\delta=\sqrt{\frac{(-2,8)^2+(-1,8)^2+0,2^2+1,2^2+3,2^2}5}\\\delta=\sqrt{\frac{7.84+3.24+0.04+1.44+10.24}5}\\\delta=\sqrt{\frac{22.8}5}\\\delta=\sqrt{4,56}[/tex]
