B page 1(7) ENTRANCE EXAMINATION FOR THE TECHNICAL CURRICULUM

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1 B page 1(7) Instructions ENTRANCE EXAMINATION FOR THE TECHNICAL CURRICULUM Mathematics, logical deduction and physics/chemistry For problems that include calculations the result of the calculation must be written in the space provided on the answer sheet. You must write the essential parts of the calculated solution in the space provided on the answer sheet. Note that any calculations or answers written on the problem sheet or on additional paper used for calculating the problem are NOT taken into account in grading. In problems 8 through 10 there are two alternatives to choose from for each problem either physics or chemistry. The total duration of the examination is 2 hours and 15 minutes. DO NOT TURN THE PAGE BEFORE THE SUPERVISOR HAS INSTRUCTED YOU TO DO SO!

2 B page 2(7) 1 1. The volume of a pyramid can be evaluated by using the equation V = Ah where h is 3 the height of the pyramid and A is the area of the base of the pyramid. The problem is to design such a triangular pyramid where the volume is V = 72 cm 3 and the height is h = 12 cm. The base of the pyramid is a rectangular isosceles triangle. a) Determine the area of the base of the pyramid. Pay attention to the unit of your answer. (1 point) b) Draw a picture of the base triangle of the pyramid in a proper scale. (2 points) 2. In the year 2001 a product was sold in 450 g packages for 35 marks per package. In the beginning of the year 2002 the size of the package was reduced by 24 percent and the price per kilogram was increased by 26 percent. a) Determine the price of the new package in euros ( ) and cents rounded to the nearest cent when 1 equals marks. (2 points) b) By how many percentage points did the price of one package increase or decrease? Include the effect of rounding the price in your answer. Give your answer with the accuracy of two decimals. (1 point) 3. a) If the value of x is not equal to zero, x 1 is called the inverse of x. On the answer sheet write two such numbers (with their inverses) whereby the inverse of the greater number is greater than the inverse of the smaller number. (1 point) b) In each of the parts b1 through b4, two expressions - x and y - are given, and their numerical values are difficult to obtain by using a calculator. On the answer sheet, mark down in each row (b1 through b4) exactly one check mark in the right column according to whether: x > y, x = y or x < y. Even if you do not know the correct answer you should guess since in this part no elaboration is needed and the grading is based upon the following: All four parts correct: 2 points Three parts correct: 1 point b1) x = y = b2) x = y = b3) x = 12, y = 1, b4) x = y =

3 B page 3(7) 4. The figure on the answer sheet displays the level of water in a pool as time progresses. The total capacity of the pool is 1000 cubic meters. As shown in the picture, the filling of the pool started on Thursday morning at 6 o clock. On Friday morning at 9 o clock when the pool was being filled, the pump for removal of water was started by mistake. Starting from this point onwards, water was both added and removed from the pool. The pool became empty on Monday evening at 6 o clock pm (18:00). It can be assumed that the rate at which the water was added and removed were both constant. a) When would the pool have been full if the pump to remove the water had not been started? State in your answer the day of the week and the time of the day with the accuracy of one hour. You can solve the problem graphically or mathematically. Do the needed work in the figure of the answer sheet and/or in the reserved empty space. (1 point) b) How many cubic meters of water does the removing pump get rid of in 24 hours? (1 point) c) Draw the water level as a function of time in the empty graph provided on the answer sheet based upon the following: The pool is filled starting on Tuesday morning at 6 o clock with double the rate compared to what was mentioned earlier. The pump to remove the water, which runs at the same rate as before, is started by mistake on Wednesday morning at 6 o clock. From that point onwards, the water is running both in and out of the pool. When will the pool be empty or full? (1 point) 5. The arms of a scale are of equal length so that when weighing objects, one side will hang lower if the load is heavier on that side. List the weights P1 through P4 in order starting from the heaviest and ending in the lightest based upon the following figure: Grading: If you are able to list only two weights in the correct order you will get one point. If you are able to list three weights in the correct order you will get two points. If you are able to list all four weights in the right order you will get three points. If you list the weights in any incorrect order you will not get any points.

Pay attention to the direction of numbers within the face when determining your answer.

4 B page 4(7) 6. There are three different preforms given below which consist of six numbered squares so that each preform can be folded to make a cube. You have to deduce, without folding the paper, if preforms 1 through 3 can be folded to make cubes a and b. It is possible that the same preform can be folded to make 0, 1 or 2 of the cubes a and b. Pay attention to the direction of numbers within the face when determining your answer. Preform 1 Preform 2 Preform 3 Cube a Cube b Write down your answer in the table of the answer sheet by writing Y (yes) or N (no) according to whether the preform can be folded to make the cube or not. Write down your answer in each cell. Guess if you do not know the right answer since there will be no penalty for wrong answers. Grading will be done as follows: All six answers correct: 3 points Five answers correct: 2 points Four answers correct: 1 point

5 B page 5(7) 7. There are two different kinds of members in a club: - those who always tell the truth, the truthful - those whose every statement includes a lie, the liars You are visiting the club and during your visit members of the club each answer in their characteristic way - either by always telling the truth or always lying. During the conversation you can often deduce who is truthful and who is a liar. Let us practice how to deduce who is a liar and who is truthful by solving the following problem: In the hallway of the club you meet X, Y and Z. You ask X: Are you truthful or a liar? X answers something but you do not hear what he says. You ask Y: What did X say? Y answers: X said that he is a liar. Then Z says: Do not believe Y. He always lies when he is at the club. What kind of members are X, Y and Z? Solution: Neither a truthful member nor a liar at the club can say that he is a liar because then a truthful member would lie and a liar would tell the truth. Therefore, X had to say that he is truthful and Y was lying. Therefore Y is a liar. Z said that Y was lying which is true. Therefore Z is truthful. The role of X cannot be determined based upon the above conversation. Then we begin the real problem: a) In the first table, there are two club members A and B. A says We are both liars. What kind of members are A and B? b) In the second table there are three club members I, J and K. I says: J and K are truthful. You ask a follow up question about whether J really is truthful and I, who originally made the statement, answers No. What kind of members are I, J and K? c) In the third table there are three club members P, Q and R. P says We are all liars, but Q adds One of us three is truthful. What kind of members are P, Q and R? Write your answer in the table of the answer sheet by marking each member by either T for truthful or L for liar. In all the parts, a through c, the roles of all the members can definitely be determined. Even if you would not be sure of your answer it is best to guess since there will be no penalty for wrong answers. Grading: Each completely correct part a through c yields one point.

6 B page 6(7) In problems 8, 9 and 10 one has to solve EITHER part A or part B. Each part yields an equal amount of points (3 points). In each problem you are entitled to choose either part A or part B independent of what you chose in the other two problems. If you solve both parts in any of the problems, only the lower of the point scores will be added to your total score. 8A. The speed of sound in the air is proportional to the square root of the absolute temperature. At 20 C the speed of sound is 343 m/s. What is the speed of sound at 10 C? (0 C = 273 K). 8B. Apatite (Ca 5 (PO 4 ) 3 F) is used to manufacture fertilizer. How many kilograms of pure phosphorus (P) is included in 1150 kg of apatite? The table of basic elements is given on the next page. 9A. An engine burns 1.4 kg of gasoline per hour and yields 3.5 kw of power. Calculate the efficiency of the engine when the heating value of gasoline is 43 MJ/kg. 9B. Base metal zinc (Zn) reacts with concentrated hydrochloric acid (HCl) forming gaseous hydrogen and zinc chloride. Write down the chemical reaction formula and calculate the volume of the gaseous hydrogen (H 2 ) formed at STP conditions when 5,00 g of zinc reacts completely. STP conditions: pressure p = 101,3 kpa, temperature t = 0 o C. The equation of state for ideal gas: pv = nrt R = 8,314 J /(mol K) The table of the basic elements is given on the next page. 10A. A ball is thrown straight up and it falls back to the ground. Which one of the following graphical representations describes the velocity of the ball as a function of time. v v v v t t t t ( a ) ( b ) ( c ) ( d )

7 B page 7(7) 10B. Name the following organic compounds: H 3 C CH 2 CH 3 H 3 C CH = C = CH 2 H 3 C CHCl CH 2 CH 2 OH ( a ) ( b ) ( c ) ( d ) ( e ) ( f ) The periodic table of elements

Maths GCSE Langdon Park Foundation Calculator pack A

Maths GCSE Langdon Park Foundation Calculator pack A Name: Class: Date: Time: 96 minutes Marks: 89 marks Comments: Q1. The table shows how 25 students travel to school. Walk Bus Car Taxi 9 8 7 1 Draw a