Ostukorv on tühi
Problem set 2. Part I
{{exercise_number}}. We hear in the news that the prices of medicines increase by an average of \latex{ 10 }%. Is it possible that at the same time a medicine's price drops?
{{exercise_number}}. If every beetle is an insect then is there any insect which is not a beetle?
{{exercise_number}}. Five people are arriving for a meeting, any one of them knows two of the others. Illustrate the acquaintances with a graph.
{{exercise_number}}. Could the sum of two primes be \latex{ 2,003 }?
{{exercise_number}}. The intervals \latex{\left[-3;4\right[} and \latex{\left[1;6\right]} are given. Find the intersection of these intervals.
{{exercise_number}}. The table football machine gives \latex{ 10 } balls to play with for one coin. How many different results are possible if the two teams play until every ball ends up in one of the goals?
{{exercise_number}}. Plot the function \latex{f(x)=\sqrt{x+2} (x\geq -2)} on the interval \latex{\left[-2;7\right[.}
{{exercise_number}}. Which natural numbers satisfy the following: \latex{\frac{7}{7-x} \gt 0?}
{{exercise_number}}. One leg of a right angled triangle is \latex{ 10\,cm } long, the angle facing this side is \latex{ 30 }º. Determine the radius of the circumscribed circle of the triangle.
{{exercise_number}}. Find the value of \latex{ n } for which \latex{\sqrt{10^{2}-8^{2} }=3\times \sqrt[n]{32}.}
{{exercise_number}}. The points \latex{A(-3;2)} and \latex{B(1;1)} are given. Reflect point \latex{ A } to point \latex{ B }. Determine the coordinates of the reflection.
{{exercise_number}}. An obtuse angle \latex{\alpha} satisfies \latex{\cos \alpha =-\frac{3}{5}.} Determine the value of \latex{\sin \alpha} without calculating \latex{\alpha}.
Problem set 2. Part II/A
{{exercise_number}}. On the \latex{ 13 }th week there were \latex{ 2,512,387 } tickets participating the draw of the lottery (there are \latex{ 90 } numbers and \latex{ 5 } are picked at random during the draw). That week there was one winner of the grand prize and \latex{ 31 }; \latex{ 2,127 }; \latex{ 48,359 } and \latex{ 613,841 } tickets with \latex{ 4, 3, 2, 1 } correct guesses, respectively. Create a column chart for the frequency distribution of correctly guessed numbers. Compute the average number of correctly guessed numbers on the winning tickets.
{{exercise_number}}. A car consumes \latex{ 9.4 } litres in urban areas and \latex{ 7.2 } litres on highways during \latex{ 100\,km } distance on average. For how long way is \latex{ 40 } litres of petrol enough,
- if the car is only used in urban areas?
- if \latex{ 20 }% of the distance traveled happens in urban areas, the rest on highways?
{{exercise_number}}. Compute the distance between the origin and the line described by the equation \latex{4x + 3y = 12.}
{{exercise_number}}. The \latex{ 10 } euro cent and the \latex{ 50 } euro cent coins were made of the same alloy. The diameter of the \latex{ 10 } euro cent coin was \latex{ 19\,mm } while that of the \latex{ 50 } euro cent coin was \latex{ 24\,mm }; the width of the \latex{ 10 } euro cent coin was \latex{ 1.9\,mm } while that of the \latex{ 50 } euro cent coin was \latex{ 2.3\,mm }. Suppose that the real value of a coin is proportional to its weight, then how much more does the \latex{ 50 } euro cent coin worth compared to the \latex{ 10 } euro cent coin?
Problem set 2. Part II/B
{{exercise_number}}. There are red and blue marbles in a box. Picking a marble at random the probability that it is red is \latex{ 0.4 }. If we place \latex{ 10 } more blue marbles in the box, then the probability of picking red one becomes \latex{\frac{1}{3}.}
- How many red and blue marbles are there in the box?
- We draw four marbles in a row from the original box without putting them back. What is the probability that they are all blue?
- We draw four marbles in a row from the original box such that after each draw we place the marble back. What is the probability that the four drawn marbles are all blue?
{{exercise_number}}. Consider the function \latex{f(x)=a\times x^{2}-2a\times x+3.}
- How should we choose the value of the parameter \latex{ a } such that the graph of the function is tangential to the axis \latex{ x }.
- For what choices of \latex{ a } will the function have two zero points?
- For what choices of \latex{ a } will the function have a maximum?
{{exercise_number}}. A \latex{ 400 } meter long running track is designed such that two half-circles are placed to the shorter sides of a football pitch. The ratio of the sides of the rectangular pitch is \latex{ 2 } : \latex{ 3 }.
- How long are the two straight line segments of the running track?
- During a \latex{ 400 } meters long race, the runners running on the outer lanes get a head start so that each competitor runs \latex{ 400 } meters until the finish line. How many meters head start should the runner on the second lane get compared to the one on the first lane if the width of any lane is \latex{ 1\,m }?
- There are two referees standing in two opposite vertices of the rectangle. How should the sides of the rectangle be chosen such that the length of the running track is still \latex{ 400 } meters and the two referees stand as close to each other as possible?
