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Teachers' Professional License – Specialized Test: Mathematics (Level 2): free practice test

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Practice test questions with answers and explanations

Try the questions in the test above first, then open each one here to see the correct answer and why every other option is wrong.

01If \frac{0.063\times 10^{m}}{0.0007\times 10^{n}}=9\times 10^{5}, what is m-n?

Past exam question Unit: Numbers and operations

  1. A3

    90\times 10^{k}=9\times 10^{5} gives 10^{k}=10^{4}, not 10^{3}.

  2. B5

    This ignores that \frac{0.063}{0.0007}=90, not 9.

  3. C6

    This assumes \frac{0.063}{0.0007} equals 0.9.

  4. D4 ✓ Correct answer

    \frac{0.063}{0.0007}=90=9\times 10, so 9\times 10^{1+m-n}=9\times 10^{5} and m-n=4.

Reference: ETEC Mathematics Teacher Standards, Level 2 (1441 AH / 2020), specialized standard 6.4.1, indicator 2, p. 8; laws of exponents

02What value of a (where a\ne 0) gives the equation ax^{2}-12x=-9 exactly one solution?

Past exam question Unit: Algebra and real functions

  1. A-4

    At a=-4 the discriminant is 144-36(-4)=288>0, so there are two solutions.

  2. B-3

    At a=-3 the discriminant is 144+108=252>0, so there are two solutions.

  3. C4 ✓ Correct answer

    Write ax^{2}-12x+9=0; there is one solution when the discriminant 144-36a=0, so a=4.

  4. D3

    At a=3 the discriminant is 144-108=36>0, so there are two solutions.

Reference: ETEC Mathematics Teacher Standards, Level 2 (1441 AH / 2020), specialized standard 6.4.2.3, indicator 3, p. 8

03A geometric sequence has second term -6 and fifth term 162. What is its general term?

Past exam question Unit: Calculus and its applications

  1. Aa_n=2(-3)^{n-1} ✓ Correct answer

    Correct: \frac{a_5}{a_2}=r^3=\frac{162}{-6}=-27, so r=-3 and a_1=\frac{-6}{-3}=2.

  2. Ba_n=3(-2)^{n-1}

    Wrong: the second term is -6, but the fifth term is 3(-2)^4=48, not 162.

  3. Ca_n=-2(3)^{n-1}

    Wrong: the second term -6 matches, but the fifth term is -2(3)^4=-162, not 162.

  4. Da_n=2(3)^{n-1}

    Wrong: the second term is 2(3)=6, not -6; the sign of the ratio was ignored.

Reference: ETEC Mathematics Teacher Standards, Level 2 (1441 AH / 2020), specialized standard 6.4.7, indicator 5, p. 10

04In the figure, a horizontal line crosses the broken line ABTEF at C and D, where C lies on \overline{BT} and D lies on \overline{TE}. In triangle ABC, \angle A=50^\circ and \angle B=78^\circ; also \angle CDE=120^\circ and \angle E=75^\circ. What is the measure of \angle CTD (angle 3)?

Past exam question Unit: Geometry concepts and theorems

A horizontal line is crossed by a broken line that starts at the line's left end A, goes down to B, up through C to the peak T, down through D to E, then up to the line's right end F. Triangle ABC has angles 50° and 78°, the angle between DC and DE at D is 120°, and the angle at E is 75°.

  1. A68^\circ ✓ Correct answer

    Correct. The angle at C in the top triangle is 180-50-78=52^\circ (vertical angles) and the angle at D is 180-120=60^\circ, so \angle T=180-52-60=68^\circ.

  2. B52^\circ

    This is \angle ACB in triangle ABC (180-50-78), which equals the angle at C in the top triangle by vertical angles; it is not the angle at T.

  3. C60^\circ

    This is the angle at D in the top triangle (180-120), not the angle at T.

  4. D75^\circ

    This is the given angle E in the lower right triangle; it is not needed to find the angle at T.

Reference: ETEC Mathematics Teacher Standards, Level 2 (1441 AH / 2020), specialized standard 6.4.3, indicator 2, p. 9

05The figure shows four identical circles, each with diameter 10 cm, where every two neighboring circles touch. Point A is where the two upper circles touch, and point B is where the two lower circles touch. What is the shortest distance a person can travel from A to B while walking only along the circles?

Past exam question Unit: Measurement, units and applications

Four identical circles arranged in two rows and two columns, each pair of neighboring circles touching. Point A is where the two circles in the top row touch, and point B is where the two circles in the bottom row touch.

  1. A10\pi

    10\pi is a full circumference, twice the required distance.

  2. B20\pi

    20\pi is two full circumferences, much longer than the shortest route.

  3. C25\pi

    25\pi is the area of one circle, \pi\times 5^{2}, not a length measured along a circumference.

  4. D5\pi ✓ Correct answer

    The shortest route is a quarter of the upper-left circle followed by a quarter of the lower-left circle, which is half a circumference: \frac{1}{2}\times\pi\times 10=5\pi.

Reference: ETEC Mathematics Teacher Standards, Level 2 (1441 AH / 2020), specialized standard 6.4.4, indicator 3, p. 9

06The figure shows three different paths from a to b and four different paths from b to c. In how many ways can one travel from a to c and then back to a, if the return trip uses the same set of paths and may reuse the very path taken on the way out?

Past exam question Unit: Statistics and probability

Three points a, b and c in a row; three paths join a to b (an upper arc, a straight segment and a lower arc), and four paths join b to c (two upper arcs and two lower arcs).

  1. A7

    7 is 3+4, but paths in succession are multiplied, not added.

  2. B12

    12 counts only the outward trip a\to c; the return trip is not counted.

  3. C24

    24 is 12\times 2; the outward and return counts must be multiplied, not doubled.

  4. D144 ✓ Correct answer

    Outward: 3\times 4=12 ways; return: 4\times 3=12 ways; by the multiplication principle the total is 12\times 12=144.

Reference: ETEC Mathematics Teacher Standards, Level 2 (1441 AH / 2020), specialized standard 6.4.5.6, indicator 6, p. 10

07Which quadrilateral is a counterexample to the conjecture "If the diagonals of a quadrilateral are congruent, then it is a rectangle"?

Past exam question Unit: Logic and mathematical reasoning

  1. ARhombus

    The diagonals of a rhombus are congruent only when it is a square, and a square is a rectangle, so a rhombus is not a counterexample.

  2. BIsosceles trapezoid ✓ Correct answer

    Its diagonals are congruent but it is not a rectangle: it meets the hypothesis and fails the conclusion.

  3. CParallelogram

    A parallelogram whose diagonals are congruent is a rectangle, so it cannot be a counterexample.

  4. DSquare

    The diagonals of a square are congruent, but a square is a rectangle, so it meets both the hypothesis and the conclusion.

Reference: ETEC Mathematics Teacher Standards, Level 2 (1441 AH / 2020), specialized standard 6.4.6, indicator 3, p. 10

08A teacher wanted to show her students that the Pythagorean theorem applies only to right triangles, so she showed them a triangle without a right angle and applied the theorem to it to show that it fails. Which technique did the teacher use to teach the concept?

Past exam question Unit: Strategies for teaching and learning mathematics

  1. ANon-examples ✓ Correct answer

    Non-examples are cases that do not belong to the concept and are shown to mark its limits; the non-right triangle is a non-example of the condition for using the theorem.

  2. BGeneralization

    Generalization extends a rule found in some cases to other cases, whereas here the teacher shows a case where the rule does not hold.

  3. CSet induction

    Set induction is a start-of-lesson activity that grabs attention and links to prior knowledge; it is not showing a case that falls outside the concept.

  4. DExplanation

    Explanation gives the reason a rule holds, but the teacher did not give a reason; she presented a case the theorem does not cover.

Reference: ETEC Mathematics Teacher Standards, Level 2 (1441 AH / 2020), specialized standard 6.4.8, indicator 2, p. 11

09Deciding which arithmetic operation will solve the problem belongs to which of Pólya's steps?

Past exam question Unit: Problem solving and its strategies

  1. AUnderstanding the problem

    Understanding means identifying the givens and the unknown; the operation is not chosen yet.

  2. BDevising a plan ✓ Correct answer

    Choosing the operation or strategy that links the givens to the unknown is the core of devising a plan.

  3. CCarrying out the plan

    Carrying out the plan performs the operation already chosen; choosing it comes earlier.

  4. DLooking back at the solution

    Looking back checks the result after solving; the route to the solution is not chosen there.

Reference: ETEC Mathematics Teacher Standards, Level 2 (1441 AH / 2020), specialized standard 6.4.9, indicator 1, p. 11

10Teacher Amal wants to check her students' mathematical communication skill. Which activity should she focus on?

Past exam question Unit: Mathematical communication

  1. AWriting problems and explaining their solutions ✓ Correct answer

    Mathematical communication means expressing ideas and solutions in words and symbols and explaining them, and writing problems and explaining their solutions shows this skill directly.

  2. BWriting a research paper on the life of a mathematician

    This is a research activity; it does not measure how well a student expresses and justifies mathematical ideas.

  3. CWriting a summary of mathematical symbols and memorizing their meanings

    This measures memory of symbols only, not explanation or justification.

  4. DPreparing a folded brochure that collects mathematical formulas

    This is a summarising and memorising activity; it does not measure expressing mathematical ideas.

Reference: ETEC Mathematics Teacher Standards, Level 2 (1441 AH / 2020), specialized standard 6.4.10, indicator 3, p. 11

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