NATIONAL PATENT DIPLOMA
NORTH AMERICA 2021
Duration of the test: 2 h 00 – 100 points
Exercise 1 (26 points)
For each of the following six statements, indicate on the copy whether it is true or false.
Remember that each answer must be justified.
1) Consider the function f defined by f (x) = 3x – 7
Assertion n° l : ” the image by f of the number -1 is 2 “.
2) Consider the expression E = (x – 5)(x + 1).
Assertion #2: “The expression E has the expanded and reduced form “.
3) n is a positive integer.
Assertion n° 3 : ” when n is equal to 5, the number is a prime number”.
4) A six-sided die numbered 1-6 was rolled 15 times and the frequencies were noted
of appearance in the table below:
Assertion n° 4 : ” the frequency of appearance of 6 is 0 “.
5) Consider a triangle RAS rectangular in S.
The side [AS] measures 80 cm and the angle measures 26°.
Assertion n° 5 : “the segment [RS] measures approximately 164 cm “.
6) A rectangle ABCD is 160 cm long and 95 cm wide.
Assertion #6: The diagonals of this rectangle measure exactly 186 cm.
Exercise 2 (21 points)
One athlete completed a triathlon with a total length of 12.9 kilometers.
The three events take place in the following order:
Between two events, the athlete must change equipment on site.
The graph below represents the distance travelled (expressed in kilometers) by
the athlete, based on the athlete’s run time (expressed in minutes) during the triathlon.
The point M has abscissa 42 and ordinate 10.4.
Using the table above or by reading the graph above with the precision that it allows, answer the following questions, justifying the approach.
1) After how long did the athlete stop to perform her first
change of equipment ?
2) What is the length, expressed in kilometers, of the cycling course?
3) How long did it take the athlete to complete the running event?
4) Which of the three events was the slowest for the athlete?
5) Equipment changes between events are considered part of the
Is the athlete’s average speed over the entire triathlon greater than 14
Exercise 3 (16 points)
In this exercise, no justification is required.
A square ABCD has been constructed.
We constructed the point O on the line (DB), outside the segment [DB] and such that: OB = AB.
The point H is the symmetric of D with respect to O.
The figure below was obtained by using the same rotation of center O and angle 45° several times .
The resulting figure is symmetrical about the axis (DB) and about the point O.
1)Give two different squares, images of each other by the axial symmetry of axis (DB).
2) Is square 3 the image of square 8 by central symmetry of center O?
3) Consider the rotation of center O that transforms square 1 into square 2.
What is the image of square 8 by this rotation?
4) Consider the rotation of center O that transforms the square 2 into the square 5.
Specify the image of the segment [EF] by this rotation.
Exercise 4 (16 points)
In this exercise, no justification is required.
We have a square table below divided into nine white cells of the same size
dimensions that constitute a pattern.
Four instructions A, B, C and E allow to change the aspect of some boxes, when applying these instructions. Thus:
Note: if a cell in the pattern is already black and an instruction asks to blacken it, then this cell does not change color and remains black following this instruction.
Examples: from a pattern where all the cells are white:
The sequence of instructions A – C allows to obtain this pattern:
The sequence of instructions A – C – E allows to obtain this pattern:
1) Represent the Obtained pattern with the sequence of instructions A – B.
2) Among the following four proposals, two proposals allow
to obtain the pattern below.
Proposal No. 1: A B C
Proposal #2: C E
Proposal #3: B C E C
Proposal No. 4: C A E
3) Give a sequence of instructions that produces the pattern below.
Exercise 5 (21 points)
We want to renovate a bathroom that has the shape of a rectangular parallelepiped.
It is necessary to stick wallpaper on the four walls.
We don’t stick it on the door or the window.
Here is a diagram of the bathroom, the dimensions are expressed in meters:
The following information is available:
1) Show that the surface to be covered with wallpaper is 26,4 m².
2) Calculate the price, in euros, of one square meter of wallpaper. Round to the nearest cent.
3) If we follow the seller’s advice, how much will the bathroom renovation cost?
4) On the day of purchase, a discount of 8% is granted.
What is the price to pay after discount? Round to the nearest cent.
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