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Problem set 1. Part I
{{exercise_number}}. Determine the value of \latex{\left(\frac{9}{36} \right)^{-\frac{1}{2} }}
{{exercise_number}}. Find the lengths of the face diagonals and space diagonals of the cube with sides of
length \latex{ 6\,cm }.
length \latex{ 6\,cm }.
{{exercise_number}}. Decide which of the following statements are false.
\latex{ A } = If a kite is a rectangle, then it is a square.
\latex{ B } = If a quadrilateral has two right angles then it is a cyclic quadrilateral.
\latex{ C } = There exists a parallelogram such that it is a tangential quadrilateral.
\latex{ D } = If a parallelogram has a right angle then it is a rectangle.
\latex{ E } = There exists a rectangle such that it is a rhombus.
\latex{ A } = If a kite is a rectangle, then it is a square.
\latex{ B } = If a quadrilateral has two right angles then it is a cyclic quadrilateral.
\latex{ C } = There exists a parallelogram such that it is a tangential quadrilateral.
\latex{ D } = If a parallelogram has a right angle then it is a rectangle.
\latex{ E } = There exists a rectangle such that it is a rhombus.
{{exercise_number}}. The following equality is satisfied by every real number \latex{ x }.
\latex{ (x+2) \times (x + b) = x^{2} + cx+6 }
Determine the value of \latex{ b } and \latex{ c }.
{{exercise_number}}. In the triangle \latex{ ABC } the angle at vertex \latex{ C } is a right angle and the angle at \latex{ A } is \latex{ 20 }º.
Determine the angle \latex{ BDC } where \latex{ BD } is the line segment which is the intersection of the
inner angle bisector at \latex{ B } and the triangle.
Determine the angle \latex{ BDC } where \latex{ BD } is the line segment which is the intersection of the
inner angle bisector at \latex{ B } and the triangle.
{{exercise_number}}. The age of a father in years is currently three times the combined age of his two children
in years. In four years the father's age will be the double of the sum of the two children's
age. How old is the father at the moment?
in years. In four years the father's age will be the double of the sum of the two children's
age. How old is the father at the moment?
{{exercise_number}}. Find the value \latex{ a } for which the lines determined by the equations \latex{ ax + 10y = 7 } and \latex{\frac{x}{2}-4y=5} are parallel.
{{exercise_number}}. For which real numbers of the interval \latex{[-\frac{\pi }{2};\frac{\pi }{2}]} is the inequality \latex{cos\,x \lt\frac{1}{2} } is satisfied?
{{exercise_number}}. How many different five-digit integers can be constructed using the digits \latex{ 1, 1, 2, 3, 5 } ?
{{exercise_number}}. Factorize the expression \latex{ (x^{2} – y^{2}) – (x – y) }.
{{exercise_number}}. Three regular coins are tossed at the same moment. What is the probability that we end
up with exactly two heads?
up with exactly two heads?
{{exercise_number}}. Solve the inequality \latex{\left| x–5 \right|\leq8}.
Problem set 1. Part II/A
{{exercise_number}}. While traveling between two towns a car finished one third of the distance with an average speed of \latex{ 60 \,km/h } during \latex{ 48 } minutes. During the remaining part of the trip the car's average speed was \latex{ 64\,km/h }. The driver has started the trip with a full tank and at arrival he filled it up again. It cost him \latex{ 2,592 } EUR. \latex{ 1 } litre of petrol costs \latex{ 250 } EUR.
- What is the distance of the two towns?
- What was the car's average speed during the full trip?
- Between the two towns, what was the car's average petrol consumption per \latex{ 100\,km }?
{{exercise_number}}. Four vertices of a cube, which has edges of length \latex{ 10\,cm }, determine a regular tetrahedron. Compute the surface area and volume of this tetrahedron.
{{exercise_number}}. During the semester, an undergraduate student in mathematics has to fill in a couple of tests and his grade will be determined by the average of the point he achieved. By the end of the semester it turns out that if he gets \latex{ 97 } points for the last test then his average will be \latex{ 90 } points while a result of \latex{ 73 } points would mean an average of \latex{ 87 } points. How many tests were there in the semester?
{{exercise_number}}. Find the solution of the following equations on the largest possible subset of the real numbers.
- \latex{\sqrt{(X+3)^{2} }+\sqrt{(X-4)^{2} }=10.}
- \latex{\log _{2} (x+3)+\log _{2}(x-3)=\log _{2}(x+11).}
Problem set 1. Part II/B
{{exercise_number}}. One of the travel agencies in a small town is releasing a detailed map of the town. We know the following about the rectangle \latex{ ABCD } seen on the screen of their computer corresponding a rectangular park in the town: the intersection of its diagonals is \latex{M(12;6)}, the equation defining the side \latex{ AB } is \latex{y=3x} and the diagonal \latex{ AC } is parallel to the axis \latex{ x }.
- Determine the coordinates of the vertices of the rectangle.
- What is the scale of the map seen on the screen if the longer side of the park is \latex{ 180\,m } in real life and the unit of the coordinate system of the map has a size of \latex{ 1\,cm }?
{{exercise_number}}. In a sports club there are \latex{ 80 } swimmers, \latex{ 95 } athletes and \latex{ 125 } gymnasts. The percentages of girls among swimmers, athletes and gymnasts are, respectively, \latex{ 45 }%, \latex{ 20 }%, \latex{ 68 }%. During a survey three randomly chosen members of the club are asked to fill in the questionnaire.
- What is the probability that all three of them are girls?
- What is the probability that all three of them are athletes?
- What is the probability that all three of them are athlete girls?
- What is the probability that all three of them do the same sport?
{{exercise_number}}. During January, a company with \latex{ 60 } workers conducted transactions worth \latex{ 30 } million EUR. In February this turnover increased by \latex{ 15.5 }% such that the turnover per person increased by twice as many percentage points as the number of workers.
- What is the turnover conducted by the company in February?
- By how many percentage points did the company's turnover increase in February compared to January?
- How many workers were there at the company in February?
