Lista de exercícios do ensino médio para impressão
Find the lateral area, the total area and the volume of a circular cone that is circumscribed to a sphere of radius $\,r\,$ and whose axial cut is an equilateral triangle.

answer: $\,S_{\text lat}\,=\,6\,\pi\,r^2\,$; $\,S_{\text total}\,=\,9\,\pi\,r^2\,$; $\,V_{\text olume}\,=\,3\,\pi\,r^3\,$
×
In the following picture:
$\,\overline{PP'}\,$ is the diameter of the sphere whose center is $\,O\,$, $\;M\,$ é is the center of a intersection with a plane perpendicular to $\,\overline{PP'}\,$. Also the measures are $\,\overline{AP}\,=\,6\,cm\;$ and $\,\overline{AP'}\,=\,8\,cm\;$. Calculate the area of the cicle whose center is $\,M\,$.

answer: The area is $\,\frac{\,576\,\pi\;}{\;25\;}\;cm^2$
×
In the picture the line r is parallel to the line s . Find the value of $\,\hat{\,\alpha\,}\,$.

×
Knowing that in the picture r // s, the value of $\;\hat{\;x\;}\;$ is:
a)
90°
b)
100°
c)
110°
d)
120°
e)
None of the above.

×
Given the following picture, find the value of $\;\hat{\;x\;}\;$ :

×
In the given picture, if r // s , we can say that the value of $\,\hat{\,\alpha\,}\,$ is:
a)
90°
b)
100°
c)
110°
d)
120°
e)
22°40'

×
Determine the quadrants in which the points are located.
a)
$\,(\sqrt{2}\,;\;-\sqrt{3})\,$
b)
$\,(-\frac{1}{2}\,;\;\frac{\sqrt{2}}{2})\,$
c)
$\,(2\,-\,\sqrt{2}\,;\;1\,-\,\sqrt{2})\,$

×
Plot in the following xOy plane the points A(4; 3) , B(-2; 5) , C(-4; -2) , D(3; -4) , E(2; 0) , F(0; -3) , G$(\frac{3}{2};\;\frac{5}{2})\;$ e H($-\frac{1}{2}$; -4).

×
Solve the following inequalities:
a)
$\,(1\,-\,4x^2)\centerdot(2x^2\,+\,3)\,\gt\,0\,$
b)
$\,(2x^2\,-\,7x\,+\,6)\centerdot(2x^2\,-\,7x\,+\,5)\,\leqslant\,0\,$
c)
$\,(x^2\,-\,x\,-\,6)\centerdot(-x^2\,+\,2x\,-1)\,\gt\,0\,$
d)
$\,(x^2\,+\,x\,-\,6)\centerdot(-x^2\,-2x\,+\,3)\,\geqslant\,0\,$
e)
$\,x^3\,-\,2x^2\,-\,x\,+\,2\,\gt\,0\,$
f)
$\,2x^3\,-\,6x^2\,+\,x\,-\,3\,\leqslant\,0\,$

a)
$\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;-\frac{3}{2}\,\lt\,x\,\lt\,-\frac{1}{2}\phantom{X}{\text or}\phantom{X}0\,\lt\,x\,\lt\,\frac{1}{2}\rbrace\;$
b)
$\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;1\,\leqslant\,x\,\leqslant\,\frac{3}{2}\phantom{X}{\text or}\phantom{X}2\,\leqslant\,x\,\leqslant\,\frac{5}{2}\rbrace\;$
c)
$\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;-2\,\lt\,x\,\lt\,3\phantom{X}{\text e}\phantom{X}x\,\ne\,1\rbrace\;$
d)
$\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;x\,=\,-3\phantom{X}{\text or}\phantom{X}1\,\leqslant\,x\,\leqslant\,2\rbrace\;$
e)
$\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;-1\,\lt\,x\,\lt\,1\phantom{X}{\text or}\phantom{X}x\,\gt\,2\rbrace\;$
e)
$\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;x\,\leqslant\,3\rbrace\;$

×
What is the measure of the smaller angle between the hour hand and the minute hand on a clockface at 12:20?

×
The numbers $\phantom{X}\sqrt[\Large 4]{5}\;$, $\phantom{X}\sqrt[\Large 3]{3}\phantom{X}$ and $\phantom{X}\sqrt{2}\;\;$ are arranged:
a)
in descending order
b)
in ascending order
c)
in non-descending order
d)
the value of the last term is half of the sum of the first term and the second term
e)
none of these

×
Considering the following functions

$\phantom{XX} \operatorname{arc\,sin:}[-1, +1] \rightarrow [ - \pi / 2, \pi / 2 ]\phantom{XX}$ and $\phantom{XXX} \operatorname{arc\,cos:} [-1, + 1] \rightarrow [0, \pi[\phantom{XX}$ ,

the value of $\phantom{X}cos(\operatorname{arc\,sin} \dfrac{3}{5} + \operatorname{arc\,cos} \dfrac{4}{5})\phantom{X}$ is:

a)
$\;\dfrac{6}{25}\;$
b)
$\;\dfrac{7}{25}\;$
c)
$\;\dfrac{1}{3}\;$
d)
$\;\dfrac{2}{5}\;$
e)
$\;\dfrac{5}{12}\;$

×
After subtracting $\phantom{X}\dfrac{5}{8\;-\;3\sqrt{7}}\phantom{X}$ from $\phantom{X}\dfrac{12}{\sqrt{7} \;+\;3}\phantom{X}$ the result is:
a)
$81 - 4\sqrt{7}$
b)
$22 + 21\sqrt{7}$
c)
$-22\;-\;21\sqrt{7}$
d)
$41\sqrt{7}\;-\;81$
e)
none of these are correct.

×
The expression $\phantom{X}[\dfrac{\sqrt{a\;+\;b}\;-\;\sqrt{a}}{b}]^{-1}\phantom{X}$ , where $\;a\;$ and $\;b\;$ are positive numbers, is equivalent to:
a)
$\;\dfrac{1}{b}$
b)
$\;b$
c)
$\;\dfrac{b \; + \; \sqrt{a}}{\sqrt{a\;+\;b}}$
d)
$\;\sqrt{b}$
e)
$\;\sqrt{a \; + \; b}\; + \; \sqrt{a}$

×
What is the vertical height (altitude) of the right cone whose radius of the circular base is equal to $\;\sqrt{3}\,$ centimeters and the slant height is 5 centimeters ?

Thinking:

# The slant height of the cone is the distance from the apex to a point in the perimeter of the base. If the cone is a right cone, the length of all the slant heights are the same.

Solution:
$\,\left.\begin{array}{rcr} \mbox{slant height }\phantom{XXXX}\;\,\rightarrow\, & \;S_h\mbox{ = 5 cm }\; \\ \,\mbox{radius of the base}\phantom{XX} \rightarrow\, & R\,=\,\sqrt{3}\\ \mbox{Pythagorean theorem}\, \rightarrow\, & (S_h)^{\large 2}\,=\,H^{\large 2}\,+\,R^{\large 2}\; \\ \end{array} \right\}\;\Rightarrow\;$
$\;\Rightarrow\;5^{\large 2}\,=\,H^{\large 2}\,+\,(\sqrt{3})^{\large 2}\;\Leftrightarrow\;H\,=\,\sqrt{22} \mbox{ cm}$
the vertical height of the cone is $\,H\,=\,\sqrt{22}\,$ cm
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A machine working 6 hours each day produces 3000 hair clips in 5 days. How many hair clips will be produced if the machine is working 4 hours each day during 8 days?

×
A tap can fill a tank in 16 hours. How much time is needed for five identical taps to fill the same tank?

answer: 3 hours and 12 minutes
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A small spur gear with 30 teeth drives another gear with 25 teeth. When the first gear has made 450 revolutions, how many revolutions has the second one made?

×
A workman makes ${\small U\$\,}$4000,00 each 16 days of hard work; how much money is he going to make working hard 20 days? answer:${\small U\$\,}$5000,00
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Three points are given in the (x,y)-plane: $\;A=(1,2)\;$, $\;B=(2,-2)\;$ and $\;C=(4,3)\;$. The equation of the line passing through the point $\;A\;$ and the midpoint of the segment $\;\overline{BC}\;$ is:
a)
$3x + 4y = 11$
b)
$4x + \dfrac{7}{2}y = 11$
c)
$x + 3y = 7$
d)
$3x + 2y = 7$
e)
$x + 2y = 5$