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Grade 12 Mathematics entrance exam, 2018 E.C. (2026), version 2

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 2; the others are listed further down.

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  • 2018 E.C. (2026), version 2

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  • 56 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

What is the 10th term of Prof. Mulatu's sequence ?

  1. A.309
  2. B.118
  3. C.191
  4. D.89
Show the answer and explanation

Answer: B 118

Why

The Mulatu sequence starts with M1=4M_1 = 4 and M2=1M_2 = 1, with subsequent terms defined by Mn=Mn−1+Mn−2M_n = M_{n-1} + M_{n-2}. Generating the terms gives: 4, 1, 5, 6, 11, 17, 28, 45, 73, 118, so the 10th term is 118.

Question 2

What is the area (in square unit) of the region bounded by y=0 and the graph of f(x)=4-x² on the interval [-2,2]?

  1. A.2/3
  2. B.22/3
  3. C.8
  4. D.32/3
Show the answer and explanation

Answer: D 32/3

Why

The area is given by the definite integral ∫−22(4−x2)dx=[4x−x33]−22=(8−83)−(−8+83)=163+163=323\int_{-2}^{2} (4-x^2) dx = [4x - \frac{x^3}{3}]_{-2}^{2} = (8 - \frac{8}{3}) - (-8 + \frac{8}{3}) = \frac{16}{3} + \frac{16}{3} = \frac{32}{3} square units.

Question 3

Let f be defined on [a, b]. Then which of the following is NOT true?

  1. A.∫ₐᵇ f(x)dx = ∫ₐᶜ f(x)dx - ∫ᶜᵇ f(x)dx for any c in (a,b)
  2. B.∫ᶜᶜ f(x)dx = 0 for any c in (a,b)
  3. C.∫ₐᵇ f(x)dx = -∫ᵇₐ f(x)dx
  4. D.∫ₐᵇ k f(x)dx = k ∫ₐᵇ f(x)dx for any constant k
Show the answer and explanation

Answer: A ∫ₐᵇ f(x)dx = ∫ₐᶜ f(x)dx - ∫ᶜᵇ f(x)dx for any c in (a,b)

Why

By the interval union property of definite integrals, ∫abf(x)dx=∫acf(x)dx+∫cbf(x)dx\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx for any c∈(a,b)c \in (a,b), which makes the minus sign in option A incorrect.

Question 4

The sides of an equilateral triangle are increasing at the rate of 2 cm/sec. The rate at which the area increases when side is 10 cm is:

  1. A.10 cm²/sec
  2. B.√3 cm²/sec
  3. C.10√3 cm²/sec
  4. D.10/3 cm²/sec
Show the answer and explanation

Answer: C 10√3 cm²/sec

Why

The area of an equilateral triangle is A=34s2A = \frac{\sqrt{3}}{4}s^2. Differentiating with respect to time tt gives dAdt=32sdsdt=32(10)(2)=103 cm2/sec\frac{dA}{dt} = \frac{\sqrt{3}}{2}s\frac{ds}{dt} = \frac{\sqrt{3}}{2}(10)(2) = 10\sqrt{3}\text{ cm}^2/\text{sec}.

Question 5

A balloon which always remains spherical is being inflated by pumping in 10 cm³ of gas per second. What is the rate at which the radius is increasing when the radius is 15 cm?

  1. A.1/(30π) cm/sec
  2. B.1/(90π) cm/sec
  3. C.1/π cm/sec
  4. D.1/(9π) cm/sec
Show the answer and explanation

Answer: B 1/(90π) cm/sec

Why

The volume of a sphere is V=43πr3V = \frac{4}{3}\pi r^3, so dVdt=4πr2drdt\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}. Given dVdt=10\frac{dV}{dt} = 10 and r=15r = 15, we get 10=4π(15)2drdt10 = 4\pi (15)^2 \frac{dr}{dt}, which simplifies to drdt=10900π=190π cm/sec\frac{dr}{dt} = \frac{10}{900\pi} = \frac{1}{90\pi}\text{ cm/sec}.

Question 6

Suppose Target Population N = 54000 vouchers and sample size n = 6000; following the systematic sampling approach if 4th unit is selected as the first unit in the sample then the next unit selected in the sample would be:

  1. A.8th
  2. B.10th
  3. C.12th
  4. D.13th
Show the answer and explanation

Answer: D 13th

Why

The sampling interval is k=N/n=54000/6000=9k = N/n = 54000/6000 = 9. If the first selected unit is the 4th, the next systematic unit is 4+9=134 + 9 = 13th.

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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 32018 E.C. (2026), version 42018 E.C. (2026), version 52018 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 56 of its questions, each with its answer and an explanation that points back to the textbook.

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