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Grade 12 Mathematics entrance exam, 2018 E.C. (2026), version 5

Real questions from the Grade 12 university entrance exam in Mathematics, as students sat it in 2018 E.C. Every question comes with the correct answer and a worked explanation. The 2018 E.C. exam went out in several versions. This page holds version 5; the others are listed further down.

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  • Both streams

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  • 2018 E.C. (2026), version 5

    Exam year

  • 51 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 term of the following arithmetic sequence -6, 1, 8, 15, ... is 414?

  1. A.31
  2. B.62
  3. C.61
  4. D.32
Show the answer and explanation

Answer: C 61

Why

The n-th term of an arithmetic sequence is given by an=a1+(n−1)da_n = a_1 + (n-1)d. Substituting a1=−6a_1 = -6, d=7d = 7, and an=414a_n = 414, we get 414=−6+(n−1)7414 = -6 + (n-1)7, which simplifies to n=61n = 61.

Question 2

∑_{n=4}^{17} 1/(k²+5k+6) is equal to: (Note: the index likely n, but given as k)

  1. A.11/102
  2. B.7/60
  3. C.12/85
  4. D.10/119
Show the answer and explanation

Answer: B 7/60

Why

Using partial fractions, the term can be written as 1(k+2)(k+3)=1k+2−1k+3\frac{1}{(k+2)(k+3)} = \frac{1}{k+2} - \frac{1}{k+3}. This forms a telescoping sum from k=4k=4 to 1717, which simplifies to 16−120=760\frac{1}{6} - \frac{1}{20} = \frac{7}{60}.

Question 3

Which of the following Geometric sequence is divergent?

  1. A.a_n = (-1)^{2n+1}(3/7)^n
  2. B.a_n = (-3/5)^{2n+1}
  3. C.a_n = (1/10)(5/2)^n
  4. D.a_n = 2/5^n
Show the answer and explanation

Answer: C a_n = (1/10)(5/2)^n

Why

A geometric sequence an=a1rn−1a_n = a_1 r^{n-1} is divergent if the absolute value of its common ratio ∣r∣≥1|r| \ge 1. Here, option C has r=5/2=2.5>1r = 5/2 = 2.5 > 1, making it divergent, while the others have ∣r∣<1|r| < 1 and are convergent.

Question 4

An auditorium has 30 seats in first row, 34 in second, 38 in third. If pattern continues, total seating capacity with 20 rows?

  1. A.106
  2. B.3880
  3. C.580
  4. D.1360
Show the answer and explanation

Answer: D 1360

Why

The seating capacity forms an arithmetic series with a1=30a_1 = 30, d=4d = 4, and n=20n = 20. The sum of the first 20 terms is S20=202[2(30)+(20−1)4]=10[60+76]=1360S_{20} = \frac{20}{2}[2(30) + (20-1)4] = 10[60 + 76] = 1360.

Question 7

A manufacturer wants to design an open box with square base and surface area 75 square units. What height gives maximum volume?

  1. A.5 units
  2. B.2.5 units
  3. C.7.5 units
  4. D.15 units
Show the answer and explanation

Answer: B 2.5 units

Why

An open box with a square base xx and height hh has surface area S=x2+4xh=75S = x^2 + 4xh = 75, so h=75−x24xh = \frac{75-x^2}{4x}. The volume V=x2h=75x−x34V = x^2h = \frac{75x - x^3}{4} is maximized when V′(x)=75−3x24=0V'(x) = \frac{75-3x^2}{4} = 0, giving x=5x = 5 and a height of h=75−2520=2.5h = \frac{75-25}{20} = 2.5 units.

Question 8

What is area of region enclosed between f(x)=x²-4 over [1,5]?

  1. A.5/3
  2. B.65/3
  3. C.16/3
  4. D.86/3
Show the answer and explanation

Answer: D 86/3

Why

The function f(x)=x2−4f(x) = x^2 - 4 is negative on [1,2][1, 2] and positive on [2,5][2, 5]. The total area is ∫12(4−x2)dx+∫25(x2−4)dx=[4x−x33]12+[x33−4x]25=53+27=863\int_{1}^{2} (4-x^2) dx + \int_{2}^{5} (x^2-4) dx = [4x - \frac{x^3}{3}]_1^2 + [\frac{x^3}{3} - 4x]_2^5 = \frac{5}{3} + 27 = \frac{86}{3}.

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The other Mathematics papers of 2018 E.C.

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 6

Mathematics papers from other years

2017 E.C. (2025)2017 E.C. (2025), natural stream2017 E.C. (2025), version 12017 E.C. (2025), version 22017 E.C. (2025), version 42016 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 2018 E.C. Mathematics entrance exam questions?

Yes. They come from the national university entrance exam in Mathematics that students sat in 2018 E.C. (2026). Our set holds 51 of its questions, each with its answer and an explanation that points back to the textbook.

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