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Grade 12 Mathematics entrance exam, 2017 E.C. (2025), version 1

Real questions from the Grade 12 university entrance exam in Mathematics, as students sat it in 2017 E.C. Every question comes with the correct answer and a worked explanation. The 2017 E.C. exam went out in several versions. This page holds version 1; the others are listed further down.

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  • 2017 E.C. (2025), version 1

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  • 42 questions with answers

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Answer each question yourself before opening the answer. The explanations point to the exact place in the textbook, so you know which page to reread when you miss one.

Question 1

Which one of the following defines prime number?

  1. A.It is a natural number that has an odd number of factors
  2. B.It is an odd natural number that has only two factors
  3. C.It is a natural number that has exactly two distinct factors
  4. D.It is a natural number where one of its factors is 3
Show the answer and explanation

Answer: C It is a natural number that has exactly two distinct factors

Why

A prime number is a natural number with exactly two different factors, 11 and the number itself. That is the reason 11 is not prime: it has only one factor. Option B is tempting because most primes are odd, but 22 is prime and even, so "odd" cannot belong in the definition.

Question 2

The GCF of two numbers is 9 and their LCM is 54. If one of the numbers is 18. What is the other number?

  1. A.36
  2. B.54
  3. C.27
  4. D.72
Show the answer and explanation

Answer: C 27

Why

For any two whole numbers, GCF times LCM equals the product of the numbers. So the missing number is 9×5418=48618=27\frac{9 \times 54}{18} = \frac{486}{18} = 27. Check it: the greatest common factor of 1818 and 2727 is 99, and their least common multiple is 5454. Option B, 5454, is tempting because that number already appears in the question, but 5454 is the LCM, not the second number.

Question 3

If x = 0.424424442... and y = 0.242242224... then the fraction form of x + y is equal to

  1. A.6/10
  2. B.3/4
  3. C.2/3
  4. D.4/5
Show the answer and explanation

Answer: C 2/3

Why

Neither decimal repeats, but their digits pair up. Wherever xx has a 44, yy has a 22, and wherever xx has a 22, yy has a 44. Every digit of the sum is therefore 66, so x+y=0.6666…x + y = 0.6666\ldots, and 0.6‾=230.\overline{6} = \frac{2}{3}. Option A, 610\frac{6}{10}, is what you get by stopping after one digit, but the sixes carry on forever.

Question 4

For any real numbers a, b and c. Which one of the following is true?

  1. A.If a < b then a² < b²
  2. B.If a < b then ac < bc
  3. C.If a² < b² then b < a
  4. D.If a < b and b < c then c > a
Show the answer and explanation

Answer: D If a < b and b < c then c > a

Why

Option D is the transitive property of order: if a<ba < b and b<cb < c, then a<ca < c, which is the same as saying c>ac > a. It holds for every choice of real numbers. Option B looks safe, but multiplying an inequality by a negative number flips it: 2<32 < 3 while 2(−1)>3(−1)2(-1) > 3(-1).

Question 6

Let a and b be positive integers and p and q are real numbers. Which of the following is true?

  1. A.(a+b)ᵖ = aᵖ + bᵖ
  2. B.(a)⁻ᵖ / (b)⁻ᵠ = bᵠ / aᵖ
  3. C.(a)⁻ᵖ / (b)⁻ᵠ = bᵠ / αᵖ
  4. D.aᵖ / (a)⁻ᵠ = (a)ᵖ⁻ᵠ
Show the answer and explanation

Answer: B (a)⁻ᵖ / (b)⁻ᵠ = bᵠ / aᵖ

Why

A negative exponent moves a factor across the fraction bar: a−p=1apa^{-p} = \frac{1}{a^{p}}, and 1b−q=bq\frac{1}{b^{-q}} = b^{q}. Putting the two together, a−pb−q=bqap\frac{a^{-p}}{b^{-q}} = \frac{b^{q}}{a^{p}}, which is option B. Option C states the same rule but writes αp\alpha^{p} underneath, and α\alpha is not one of the letters the question defined. Option D fails because apa−q=ap+q\frac{a^{p}}{a^{-q}} = a^{p+q}.

Question 7

Which one of the following is the rationalizing factor to rationalize the denominator of 7/(3+√5)?

  1. A.(√3+√5)/(√3+√5)
  2. B.(√3-5)/(√3-5)
  3. C.(√3-√5)/(√3-√5)
  4. D.(3-√5)/(3-√5)
Show the answer and explanation

Answer: D (3-√5)/(3-√5)

Why

To clear a denominator of the form 3+53 + \sqrt5 you multiply by its conjugate 3−53 - \sqrt5, because (3+5)(3−5)=9−5=4(3+\sqrt5)(3-\sqrt5) = 9 - 5 = 4, a whole number. Writing the factor as 3−53−5\frac{3-\sqrt5}{3-\sqrt5} keeps the value of the fraction unchanged, so option D is the rationalizing factor. Option C, with 3\sqrt3, is tempting, but the denominator holds a plain 33, and that product would leave surds behind.

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The other Mathematics papers of 2017 E.C.

2017 E.C. (2025)2017 E.C. (2025), natural stream2017 E.C. (2025), version 22017 E.C. (2025), version 4

Mathematics papers from other years

2018 E.C. (2026)2018 E.C. (2026), social stream2018 E.C. (2026), version 12018 E.C. (2026), version 22018 E.C. (2026), version 32018 E.C. (2026), version 42018 E.C. (2026), version 52018 E.C. (2026), version 62016 E.C. (2024)2016 E.C. (2024), natural stream2016 E.C. (2024), social stream2016 E.C. model exam2015 E.C. (2023)2015 E.C. (2023), natural stream2015 E.C. (2023), social stream2014 E.C. (2022)2014 E.C. (2022), social stream2013 E.C. (2021)2013 E.C. (2021), version 12011 E.C. (2019)2009 E.C. (2017)2008 E.C. (2016)2007 E.C. (2015)2006 E.C. (2014)2005 E.C. (2013)

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Are these the real 2017 E.C. Mathematics entrance exam questions?

Yes. They come from the national university entrance exam in Mathematics that students sat in 2017 E.C. (2025). Our set holds 42 of its questions, each with its answer and an explanation that points back to the textbook.

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