The worksheet follows the format of the National Evaluation for graduates of grade 8: Part I and Part II contain multiple-choice items (a single correct answer), while Part III contains problems that require complete solutions.
Part I – Numbers and algebraic computation
Circle the letter of the correct answer.
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The result of the computation $36 : ( - 4) + 3 \cdot ( - 2)^{2}$ is:
- a) $- 21$
- b) $- 3$
- c) $3$
- d) $21$
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If $\frac{a}{b} = \frac{3}{5}$, then the value of the ratio $\frac{2a + b}{b}$ is:
- a) $\frac{11}{5}$
- b) $\frac{7}{5}$
- c) $\frac{6}{5}$
- d) $\frac{11}{3}$
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The price of a product is $250$ lei. The price is decreased by $20\%$, then the new price is increased by $20\%$. The final price of the product is:
- a) $250$ lei
- b) $240$ lei
- c) $245$ lei
- d) $260$ lei
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The number $\sqrt{48} - \sqrt{27} + \sqrt{12}$ is equal to:
- a) $\sqrt{33}$
- b) $9\sqrt{3}$
- c) $2\sqrt{3}$
- d) $3\sqrt{3}$
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The number of elements of the set $A = \{ x \in {\mathbb{Z}}\text{ | }|2x - 1| \leq 5\}$ is:
- a) $5$
- b) $6$
- c) $7$
- d) $11$
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Consider the numbers $x = 2\sqrt{5}$, $y = 3\sqrt{2}$, $z = \sqrt{19}$ and $t = \text{4,5}$. The largest of these numbers is:
- a) $x$
- b) $y$
- c) $z$
- d) $t$
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An urn contains $6$ white balls, $8$ red balls and $10$ black balls. A ball is drawn at random. The probability that the drawn ball is red is:
- a) $\frac{1}{4}$
- b) $\frac{5}{12}$
- c) $\frac{1}{3}$
- d) $\frac{2}{3}$
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The numbers $x$ and $y$ are directly proportional to the numbers $3$ and $5$, and $x + y = 64$. The value of the number $y$ is:
- a) $40$
- b) $24$
- c) $35$
- d) $45$
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The solution of the equation $3(x - 2) - 2(x + 1) = 5$ is:
- a) $3$
- b) $- 3$
- c) $13$
- d) $9$
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Andrei claims that the number $a = \left( \sqrt{3} - 2 \right)^{2} + 4\sqrt{3}$ is a natural number. Andrei's claim is:
- a) true
- b) false
Part II – Geometry
Circle the letter of the correct answer.
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The supplement of an angle measuring $64{^\circ}$ measures:
- a) $26{^\circ}$
- b) $116{^\circ}$
- c) $126{^\circ}$
- d) $296{^\circ}$
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In triangle $ABC$, $m(\angle A) = 50{^\circ}$ and $m(\angle B) = 70{^\circ}$. The bisector of angle $ACB$ meets side $AB$ at point $D$. The measure of angle $ADC$ is equal to:
- a) $80{^\circ}$
- b) $90{^\circ}$
- c) $110{^\circ}$
- d) $100{^\circ}$
-
A right triangle has legs of $9\text{ cm}$ and $12\text{ cm}$. The length of the median to the hypotenuse is equal to:
- a) $\text{7,5 cm}$
- b) $15\text{ cm}$
- c) $6\text{ cm}$
- d) $\text{10,5 cm}$
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In triangle $ABC$, right-angled at $A$, $AB = 6\text{ cm}$ and $BC = 10\text{ cm}$. The value of $\sin C$ is:
- a) $\frac{4}{5}$
- b) $\frac{3}{4}$
- c) $\frac{3}{5}$
- d) $\frac{4}{3}$
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A rhombus has diagonals of $10\text{ cm}$ and $24\text{ cm}$. The perimeter of the rhombus is equal to:
- a) $34\text{ cm}$
- b) $120\text{ cm}$
- c) $68\text{ cm}$
- d) $52\text{ cm}$
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A trapezoid has bases of $14\text{ cm}$ and $8\text{ cm}$, and height $5\text{ cm}$. The area of the trapezoid is equal to:
- a) $110\text{ cm}^{2}$
- b) $55\text{ cm}^{2}$
- c) $70\text{ cm}^{2}$
- d) $60\text{ cm}^{2}$
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A circle has circumference $12\pi\text{ cm}$. The area of the disk bounded by this circle is equal to:
- a) $36\pi\text{ cm}^{2}$
- b) $144\pi\text{ cm}^{2}$
- c) $12\pi\text{ cm}^{2}$
- d) $24\pi\text{ cm}^{2}$
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A cube has diagonal $6\sqrt{3}\text{ cm}$. The volume of the cube is equal to:
- a) $36\text{ cm}^{3}$
- b) $108\text{ cm}^{3}$
- c) $216\text{ cm}^{3}$
- d) $648\text{ cm}^{3}$
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In cube $ABCDA'B'C'D'$, the measure of the angle formed by lines $AC$ and $A'B$ is equal to:
- a) $30{^\circ}$
- b) $45{^\circ}$
- c) $90{^\circ}$
- d) $60{^\circ}$
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A regular quadrilateral pyramid has base edge $6\text{ cm}$ and height $4\text{ cm}$. The length of the pyramid's apothem is:
- a) $\sqrt{13}\text{ cm}$
- b) $5\text{ cm}$
- c) $7\text{ cm}$
- d) $10\text{ cm}$
Part III – Problems
Write complete solutions.
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A student read a book over three days. On the first day he read $25\%$ of the total number of pages, on the second day he read $\frac{2}{3}$ of the remaining pages, and on the third day he read the last $60$ pages.
a) Is it possible for the book to have $200$ pages? Justify your answer.
b) Find the number of pages in the book.
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The number of students in a school is between $500$ and $600$. If the students are grouped in $12$s, $15$s or $18$s, there are always $7$ students left over.
a) Show that the school cannot have $583$ students.
b) Find the number of students in the school.
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A parking lot contains cars and motorcycles, $40$ vehicles in total, with $130$ wheels altogether. Each car has $4$ wheels and each motorcycle has $2$ wheels.
a) Is it possible for the parking lot to contain $30$ cars? Justify your answer.
b) Find the number of motorcycles in the parking lot.
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Consider the expression $E(x) = (x + 3)^{2} - (x - 1)(x + 1) - 2(x + 5)$, where $x$ is a real number.
a) Show that $E(x) = 4x$, for any real number $x$.
b) Prove that the number $N = E(n) + n^{2} + 4$ is the square of a natural number, for any natural number $n$.
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Consider the expression $E(x) = \left( \frac{1}{x - 1} + \frac{1}{x + 1} \right) : \frac{2x}{x^{2} + 2x + 1}$, where $x \in {\mathbb{R}} \smallsetminus \{ - 1,\ 0,\ 1\}$.
a) Show that $E(x) = \frac{x + 1}{x - 1}$, for any $x \in {\mathbb{R}} \smallsetminus \{ - 1,\ 0,\ 1\}$.
b) Find the integers $n$, $n \notin \{ - 1,\ 0,\ 1\}$, for which $E(n)$ is an integer.
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Consider the function $f:{\mathbb{R}} \rightarrow {\mathbb{R}}$, $f(x) = 2x - 4$.
a) Show that point $A(3,\ 2)$ belongs to the graph of function $f$.
b) The graph of function $f$ meets the $Ox$ axis at point $B$ and the $Oy$ axis at point $C$. Find the distance from the origin $O$ of the Cartesian coordinate system $xOy$ to line $BC$.
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Rectangle $ABCD$ has $AB = 16\text{ cm}$ and $BC = 12\text{ cm}$. Point $M$ is the midpoint of side $CD$, and lines $AM$ and $BD$ meet at point $O$.
a) Calculate the area of triangle $ABM$.
b) Show that $BO = \frac{40}{3}\text{ cm}$.
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Right trapezoid $ABCD$ has $AB \parallel CD$, $m(\angle DAB) = 90{^\circ}$, $AB = 18\text{ cm}$, $CD = 10\text{ cm}$ and $AD = 6\text{ cm}$.
a) Show that $BC = 10\text{ cm}$.
b) Find the distance from point $D$ to line $BC$.
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An aquarium has the shape of a rectangular cuboid with interior dimensions: length $60\text{ cm}$, width $40\text{ cm}$ and height $50\text{ cm}$.
a) Calculate how many liters of water fit in the aquarium.
b) $90$ liters of water are poured into the empty aquarium. Find the height to which the water rises in the aquarium.
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Regular quadrilateral pyramid $VABCD$ has base $ABCD$ with $AB = 12\text{ cm}$ and height $VO = 8\text{ cm}$, where $O$ is the center of the base.
a) Calculate the lateral surface area of the pyramid.
b) Find the distance from point $O$ to plane $(VBC)$.