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Grade 12 Mathematics entrance exam, 2015 E.C. (2023), natural stream

Real questions from the Grade 12 university entrance exam in Mathematics, as students sat it in 2015 E.C. Every question comes with the correct answer and a worked explanation. This is the natural stream paper. The other stream sat a different Mathematics paper the same year, listed further down.

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  • Natural science stream

    Who sits it

  • 2015 E.C. (2023), natural stream

    Exam year

  • 52 questions with answers

    In this set

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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 4

A 20m high building casts a shadow of 4m, at the same time a 35m tree casts a shadow of x m. What is the value of x?

  1. A.8m
  2. B.11m
  3. C.7m
  4. D.15m
Show the answer and explanation

Answer: C 7m

Why

Using the properties of similar triangles, the ratio of height to shadow is constant: 20/4=35/x  ⟹  5=35/x  ⟹  x=720/4 = 35/x \implies 5 = 35/x \implies x = 7m.

Question 5

A car started moving 9km to the East then moved 4 km to the North and then 12km towards West to reach its destination. How far the car it travelled from its starting point?

  1. A.25km
  2. B.12km
  3. C.4km
  4. D.5km
Show the answer and explanation

Answer: D 5km

Why

The net displacement vector from the starting point (0,0)(0,0) is (9−12)i^+4j^=−3i^+4j^(9 - 12)\hat{i} + 4\hat{j} = -3\hat{i} + 4\hat{j}. The straight-line distance is (−3)2+42=5\sqrt{(-3)^2 + 4^2} = 5 km.

Question 6

When the polynomial P(x)=3x6+5x4−7x3+2kx2+3P(x) = 3x^{6} + 5x^{4} - 7x^{3} + 2kx^{2} + 3 is divided by (x + 1), the remainder is 4. What is the value of k?

  1. A.-5
  2. B.-7
  3. C.3
  4. D.6
Show the answer and explanation

Answer: B -7

Why

By the Remainder Theorem, P(−1)=4  ⟹  3(−1)6+5(−1)4−7(−1)3+2k(−1)2+3=4  ⟹  18+2k=4  ⟹  k=−7P(-1) = 4 \implies 3(-1)^6 + 5(-1)^4 - 7(-1)^3 + 2k(-1)^2 + 3 = 4 \implies 18 + 2k = 4 \implies k = -7.

Question 8

Which of the following statement is true about f(x)=x2−6x+10f(x) = x^{2} - 6x + 10?

  1. A.f(x)≥2f(x) ≥ 2 for all x in the domain of f.
  2. B.The line x=1x = 1 is the axis of symmetry of the graph of f.
  3. C.The graph of f opens downward.
  4. D.The vertex of the graph of f is (3, 1).
Show the answer and explanation

Answer: D The vertex of the graph of f is (3, 1).

Why

Rewriting the quadratic function in vertex form gives f(x)=(x−3)2+1f(x) = (x-3)^2 + 1, which shows that the vertex is indeed (3,1)(3, 1).

Question 9

What is the area of a regular hexagon inscribed in a circle of radius 6cm?

  1. A.483 cm248\sqrt{3} \,\text{cm}^{2}
  2. B.543 cm254\sqrt{3} \,\text{cm}^{2}
  3. C.48cm 2
  4. D.363 cm236\sqrt{3} \,\text{cm}^{2}
Show the answer and explanation

Answer: B 543 cm254\sqrt{3} \,\text{cm}^{2}

Why

A regular hexagon inscribed in a circle of radius R=6R = 6 cm consists of six equilateral triangles each of side length 6 cm. Its area is 6×34(6)2=543 cm26 \times \frac{\sqrt{3}}{4} (6)^2 = 54\sqrt{3}\text{ cm}^2.

Question 10

Which of the following defines a prime number?

  1. A.It is a number that has more than two positive factors.
  2. B.It is a number whose only factor is 1.
  3. C.It is a number that has only two positive factors.
  4. D.It is a number whose only factor is itself.
Show the answer and explanation

Answer: C It is a number that has only two positive factors.

Why

By definition, a prime number is a positive integer greater than 1 that has exactly two distinct positive factors: 1 and itself.

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The other Mathematics papers of 2015 E.C.

2015 E.C. (2023)2015 E.C. (2023), social stream

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 62017 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 exam2014 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 2015 E.C. Mathematics entrance exam questions?

Yes. They come from the national university entrance exam in Mathematics that students sat in 2015 E.C. (2023). Our set holds 52 of its questions, each with its answer and an explanation that points back to the textbook.

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Can I download the Mathematics 2015 E.C. paper as a PDF?

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