Lista de exercícios do ensino médio para impressão
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If A is a set with 5 elements, how many are the subsets of A ?

 



answer:

If A contains k elements, so there are 2k subsets of A.

Solution:
n(A) = 5 ⇒ n[P(A)] = 25 = 32.

Obs.: n[P(A)] means number of parts of A or number of subsets of A.

The subsets of A are 32 subsets
×
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.
sphere inscribe to a cone

 



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 $\,M\,$.
esfera e secção plana

 



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 $\,\hat{\,\alpha\,}\,$.
paralelas e transversais com bicos

 



answer: α = 90°
×
Knowing that in the picture r // s, the value of $\;\hat{\;x\;}\;$ is:
parallel crossed by an right angle
a)
90°
b)
100°
c)
110°
d)
120°
e)
None of the above.

 



answer: (B)
×
Given the following picture, find the value of $\;\hat{\;x\;}\;$ :
duas paralelas cortadas por uma transversal

 



answer: 41°42'43"
×
In the given picture, if r // s , we can say that the value of $\,\hat{\,\alpha\,}\,$ is:
ângulo de 120 graus cortado por paralelas
a)
90°
b)
100°
c)
110°
d)
120°
e)
22°40'

 



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

 



answer: a)4º b)2º c)4º
×
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).
plano cartesiano x0y

 



answer:
plano cartesiano x0y com pontos marcados

×
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)
$\,2x^3\,-\,6x^2\,+\,x\,-\,3\,\leqslant\,0\,$

 



answer:
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?

 



answer: 110°
×
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

 



answer: (A)
×
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)
$\;\dfrac{5}{12}\;$

 



answer: (B)
×
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.

 



answer: (C)
×
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)
$\;\sqrt{a \; + \; b}\; + \; \sqrt{a}$

 



answer: (E)
×
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 ?

 



answer:
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.

cone de geratriz 5cm e altura raiz de 3 cm
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
×
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?

 



answer: 3200 hair clips
×
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
×
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?

 



answer: 540 revolutions
×
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
×
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)
$x + 2y = 5$

 



answer: (A)
×
The length of the two parallel sides of a right trapezoid are 6 cm and 8 cm, respectively top and bottom bases, and the altitude is 4 cm. The distance between the intersection of the two straight lines that contain the non-parallel legs and the midpoint of the major base is:
a)
$\,5\sqrt{15}\,$ cm
b)
$\,2\sqrt{19}\,$ cm
c)
$\,3\sqrt{21}\,$ cm
d)
$\,4\sqrt{17}\,$ cm
e)
none of these
 
 

 



answer: (D)
×
The lantern shown in the picture is suspended by two strings perpendicular to each other. Knowing the length of each string is 1/2 and 6/5 , respectively, what is the distance between the lantern and the ceiling?
a)
1,69
b)
1,3
c)
0,6
d)
1/2
e)
6/13
lantern suspended by two string measuring each 1/2 e 6/5, respectively

 



answer: (E)
×
If the length of the legs of a right triangle are $\,\sqrt{3}\;$ and $\;\sqrt{4}\,$, the length of the hypotenuse is:
a)
$\,\sqrt{5}\,$
b)
$\,\sqrt{7}\,$
c)
$\,\sqrt{8}\,$
d)
$\,\sqrt{9}\,$
e)
$\,\sqrt{12}\,$

 



answer: (B)
×
In the picture below, ABCD is a rectangle, $\,\overline{AB}\,=\,4\,$, $\,\overline{BC}\,=\,1\;$ and $\,\overline{DE}\,=\,\overline{EF}\,=\,\overline{FC}\;$. The length of $\,\overline{BG}\,$ is:
a)
$\,\dfrac{\sqrt{5}}{4}\,$
b)
$\,\dfrac{5}{2}\,$
c)
$\,\dfrac{9}{4}\,$
d)
$\,\dfrac{11}{4}\,$
e)
$\,\dfrac{5}{\sqrt{2}}\,$
retangle ABCD whose base side is coincident with the ABG triangle base side

 



answer: (B)
×
In the picture below, ABFG and BCDE are squares and their each one sides measures, respectively, a and b. Knowing that $\;\overline{AG}\,=\,\overline{CD}\,+\,2\;\,$ and the perimeter of the triangle ACG is 12, we can assure that a e b are both inside the interval:
a)
]1; 5[
b)
]0; 4[
c)
]2; 6[
d)
]3; 7[
e)
]4; 8[
two squares whose sides measures, respectively, a and b

 



answer: (B)
×
The measure of the two legs of the right triangle T are, respectively, 12 cm and 5 cm each, so the length of the altitude to the hypotenuse of T is:
a)
$\,\dfrac{12}{5}\,$ cm
b)
$\,\dfrac{5}{13}\,$ cm
c)
$\,\dfrac{12}{13}\,$ cm
d)
$\,\dfrac{25}{13}\,$ cm
e)
$\,\dfrac{60}{13}\,$ cm

 



answer: (E)
×
Knowing that in the right triangle ABC shown in the picture b = 1 and c = 2, find x.
a)
$\,\sqrt{2}\,$
b)
$\,\dfrac{3}{2}\,$
c)
$\,\dfrac{3\sqrt{2}}{2}\,$
d)
$\,\dfrac{2}{3}\,$
e)
$\,\dfrac{2\sqrt{2}}{3}\,$
right triangle ABC and x is the bisector of the right angle A

 



answer: (E)
×
(FUVEST - 1977)
Given that:
$\,\overline{MP}\;\bot\;s\,$;$\;\overline{MQ}\;\bot\;t\,$;$\;\overline{MQ}\;\bot\;\overline{PQ}\,$;$\;\overline{MP}\,=\,6$
The length of $\,\overline{PQ}\,$ é is:
a)
$\,3\sqrt{3}\,$
b)
$\,3\,$
c)
$\,6\sqrt{3}\,$
d)
$\,4\sqrt{3}\,$
e)
$\,2\sqrt{3}\,$
angle whose sides are the rays s and t crossed by the straight line MP perpendicular to s

 



answer: (B)
×
The picture shows the rhombus ABCD and the point A is the center of the circle that has a 4 cm length radius. Find the area of the rhombus in square centimeters.
a)
$\,4\sqrt{3}\,$
b)
$\,8\,$
c)
$\,12\,$
d)
$\,8\sqrt{3}\,$
e)
$\,12\sqrt{3}\,$
circle whose center is A with a inner rhombus ABCD

 



answer: (D)
×
(FESP - 1991) An equilateral triangle ABC is inscriibed in a circle whose radius is 6 cm length. The triangle is intercepted by a diameter MN of the circle, forming a trapezoid, as shown in the picture below. We can say that the quotient of the triangle ABC area divided by the trapezoid APQC area is:
a)
$\,\dfrac{5}{4}\,$
b)
$\,\dfrac{9}{5}\,$
c)
$\,\dfrac{9}{8}\,$
d)
$\,\dfrac{9}{4}\,$
e)
$\,\dfrac{8}{5}\,$
circle with an equilateral triangle inscribed in

 



answer: (B)
×
AB is the diameter of the circle whose center is O and the triangle ABC is inscribed in. The quotient $\,\dfrac{s}{S}\,$ where the area $\,s\,$ of the triangle ACO is divided by the area $\,S\,$ of the triangle COB is:
a)
$\,\dfrac{5}{4}\,$
b)
$\,\dfrac{4}{3}\,$
c)
$\,\dfrac{3}{4}\,$
d)
$\,1\,$
e)
$\,\dfrac{\sqrt{3}}{2}\,$
triiangle ACB inscribed in the circle whose center is O

 



answer: (D)
×
The circle as shown in the picture below, with center P and radius 2, is tangent to three sides of the rectangle ABCD. Given that the total area of the rectangle is 32, find the distance between the point P and the diagonal AC.
a)
$\,2\dfrac{\sqrt{5}}{5}\,$
b)
$\,\dfrac{\sqrt{5}}{2}\,$
c)
$\,\dfrac{\sqrt{5}}{5}\,$
d)
$\,2\sqrt{5}\,$
e)
$\,3\dfrac{\sqrt{5}}{5}\,$
retangle with an inner circle tangent to three sides

 



answer: (A)
×
How many segments can pass through two distinct points A and B? How many of them have both endpoints located at A and B?

 



answer:

Infinite segments pass through the points A and B;
Only one segment has endpoints A and B


×
How many segments are formed by 3 distinct collinear points?

 



answer: 3 segments
×
Given four distinct points A, B, C and D in a straight line, how many rays whose endpoints are A, B, C or D are there in this line?

 



answer: 8 rays
×
Knowing that O is the center of the circle, find x in the following cases:
a)

circle with the center O and two intersecting lines in O
b)
circle with center O and a tangent and a diameter drawn

 



answer: a) 125° b) 145°
×
The circles in the picture are externally tangent. The distance between the centers $\,\overline{OC}\,$ is 28 cm and the difference between their radii is 8 cm. Find the length of each radius.
two circles that are tangent to each other externally

 



answer: 18 cm and 10 cm
×
Find x in the following cases:
a) s is perpendicular to $\;\overline{AB}\,$
circle of center O with chord AB and line s perpendicular to AB
b) $\,\overline{PA}\,$ and $\,\overline{PB}\,$ are tangent segments to the circle
external point P intersection of two tangent segments to the circumference with center O

 



answer: a) 6b) 9
×
The radius of the circle C in the picture is 16 cm and the point P is located 7 cm far from the center O.
What is the distance between P and the circunference of the circle C?
circle C width center O and a point P excentric inside

 



answer: 9 cm

×
What is the radius of the following circle which center is O,
given that:
AB = 3x - 3 and
OA = x + 3.
circle which center is O and diameter is AB

 



answer: 12

×
The length of the arc $\,\stackrel \frown{AB}\,$, as in the picture, is 22 cm and O is the center of the circle. The perimeter of the circle (circumference) is:
circle with short arc AB 45 deg
a)
990 cm
b)
67 cm
c)
176 cm
d)
88 cm
e)
none of these answers

 



answer: Answer (C)
×
M is the midpoint of AB. Find $\,\overline{AB}\,$ length.
a)

segment line AB with the midpoint M
b)

segment line AP with point M and B marked

 



answer: a) 42b) 24
×
The length of AB is 17 cm. Find x in the following cases:
a)
line segment with points A P B
b)
line segment with points P B A
c)
17 cm length line segment with points A P B
d)
line segment AP with excentric point B

 



answer: Answer:
a)
10 cm
b)
4 cm
c)
7 cm
d)
14 cm

×
With M being the midpoint of $\,\overline{AB}\,$, find x:
a)
line segment AB with midpoint M
b)
line segment AB where M is the midpoint

 



answer: Answer: a) 7b)6
×
In the cases a) and b), knowing that AB = 31, what is the length of PQ?
a)
line segment APQB with AB equal 31
b)
line segment APBQ with AB equal 31

 



answer: Answer: a) 11b)32
×
(MACKENZIE - 1969) Knowing the set $\,\mathbb{A}\,=\,\lbrace\,\lbrace\,1\,\rbrace , \,\lbrace\,2\,\rbrace,\,\lbrace\,1,\,2\,\rbrace\,\rbrace\,\;$, which one of the following statements is correct?
a)
$\,\{1\}\,\notin \mathbb{A}\,$
b)
$\,\{1\}\,\subset \mathbb{A}\,$
c)
$\,\{1\}\,\cap\,\{2\}\,\not\subset \, \mathbb{A}\,$
d)
$\,2\,\in \mathbb{A}\,$
e)
$\,\{1\}\,\cup\,\{2\}\,\in \, \mathbb{A}\,$

 



answer: Answer (E)
×
(CESCEM - 1977) The set $\,\mathbb{X}\,$ is a subset of the$\,\mathbb{N}\,$ (natural numbers) and has the following elements:
12 multiples of 4
7 multiples of 6
5 multiples of 12
8 odd numbers
The number of elements of $\,\mathbb{X}\,$ are:
a)
32
b)
27
c)
24
d)
22
e)
20

 



answer: Answer (D)
×
Choose the statement that is incorrect:
a)
$5^{\large 3}\times 5^{\large 4} = 5^{\large 7}$
b)
$2^{\large 5} \div 2^{\large 3} = 2^{\large 2}$
c)
$(0,5)^{\large 2} \times (0,2)^{\large 2} = (0,1)^{\large 2} = 0,01$
d)
$(0,4)^{\large -2} \times (-5)^{\large -2} = 0,25$
e)
$\dfrac{(0,05)^{\large 3}}{5^{\large 3}}\,=\,10^{\large -3}$

 



answer: Answer (E)
×
Choose the incorrect:
a)
$5^0 = 1$
b)
$(-2,3)^0 = 1$
c)
$5^{-2}\,=\,\dfrac{1}{25}$
d)
$(-5)^{-2}\,=\,\dfrac{1}{25}$
e)
$-5^{-2}\,=\,\dfrac{1}{25}$

 



answer: Answer (E)
×
Other math tests:
working with sets
real numbers