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Problem set 5. Part I
{{exercise_number}}. The price of a bicycle worth \latex{ 450 } EUR is reduced by \latex{ 20 }%. How much does it cost now?
{{exercise_number}}. How many integers are there which satisfies
- \latex{-2\lt x} and \latex{x\leq 4;}
- \latex{-2\lt x} or \latex{x\leq 4?}
{{exercise_number}}. How many different numbers greater than \latex{ 20,000 } but smaller than \latex{ 30,000 } can be constructed using the digits \latex{ 0, 1, 1, 2, 3 }?
{{exercise_number}}.
- Determine the value of \latex{\log _{2}\frac{1}{2}+\log _{2}8.}
- Express \latex{\sqrt[5]{8}} as a power of \latex{ 2 }.
{{exercise_number}}. A cylinder-shaped glass with a capacity of \latex{ 3\,dl } has a height of \latex{ 10\,cm }. What is the minimum radius of a circle-shaped coaster for which it is possible to place the glass upon it such that it is still visible from any direction?
{{exercise_number}}. Solve the inequality \latex{|2x-4|\lt 6.}
{{exercise_number}}. The largest possible sphere was carved out of a cube with sides of length \latex{ 10\,cm }. Determine the volume of the sphere.
{{exercise_number}}. Find the intersections of the circle described by the equation \latex{(x-8)^{2}+(y-6)^{2}=100} and the axes.
{{exercise_number}}. In a class there are \latex{ 15 } boys and \latex{ 10 } girls. What is the probability that a student picked at random for examination will be a boy?
{{exercise_number}}. Determine the domain and image of the function \latex{x\longmapsto \sqrt{x^{2}-4 }.}
{{exercise_number}}. In a theater, there are \latex{ 15 } seats in the first row and there are \latex{ 3 } more seats in any subsequent row. In which row are there twice as many seats as in the first row?
{{exercise_number}}. What is the probability that the product of the numbers received by rolling two regular dice is odd?
Problem set 5. Part II/A
{{exercise_number}}. There are \latex{ 25 } participants in an Olympic event.
- After the qualification round how many ways can the list of the \latex{ 8 } finalists be formed?
- Considering every participant, how many possible outcomes are there for the first three places?
{{exercise_number}}. Solve the following equations on the set of real numbers:
- \latex{2^{x}\times 8=4^{x+2};}
- \latex{\sqrt{(x-2)^{2} }=\sqrt{x-2}.}
{{exercise_number}}. The whole content of a half-sphere shaped bowl with radius of \latex{ 21\,cm } is poured in a cylinder shaped container with a vertical axis and radius of \latex{ 50\,cm }. How tall will the liquid be in this container?
{{exercise_number}}. Atriangle is given by its three vertices \latex{A(4; 6), B(–1; 18), C(–8; –3).} Determine the equation of the circumscribed circle.
Problem set 5. Part II/B
{{exercise_number}}. A bank offers two types of investments. The first one is: the interest on the first \latex{ 1 } million EUR of the deposit is \latex{ 5 }%, the interest on the part between \latex{ 1 } and \latex{ 2 } million EUR is \latex{ 6.5 }% and the interest on the part above \latex{ 2 } million EUR is \latex{ 7 }%, all credited annually. The other construction offers an interest of \latex{ 1.7 }% credited every \latex{ 3 } months. Which construction is better for a deposit of \latex{ 2 } million EUR if it is deposited for \latex{ 3 } years? (Suppose that the conditions do not change during that time.)
{{exercise_number}}. The following table shows the population with age less than \latex{ 15 } years in Hungary from the given years according to recorded data.
Year
Total population aged
\latex{ 0-14 } years
\latex{ 0-14 } years
\latex{ 1997 }
\latex{ 1,797,606 }
\latex{ 1998 }
\latex{ 1,767,223 }
\latex{ 1999 }
\latex{ 1,736,811 }
\latex{ 2000 }
\latex{ 1,706,370 }
- Setting the data from year \latex{ 1997 } as a default express the following years' data as percentages. Illustrate the differences in a column chart.
- Determine the average decrease according the four years' data.
- How large population can be expected for the year \latex{ 2020 } if the previous average decrease is realized in the following years?
{{exercise_number}}. The centreline section of a square frustum shaped lamp shade is a symmetric trapezoid with an area of \latex{1,400\,cm^{2}} and parallel sides of length \latex{a=60\,cm, b=10\,cm.}
- What is the height of the lamp shade?
- Compute the volume of the shade.
- How large cloth do we need to cover the lateral surface area of the shade if we leave \latex{ 10 }% surplus for the junctions?
