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Grade 12 Mathematics entrance exam, 2016 E.C. (2024), natural stream

Real questions from the Grade 12 university entrance exam in Mathematics, as students sat it in 2016 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

  • 2016 E.C. (2024), natural stream

    Exam year

  • 51 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 1

Which one of the following is true about the Range and Interquartile Range of a given data?

  1. A.Both are measures of the variability of each item in the data.
  2. B.Both are measures of variation of the data.
  3. C.Both are affected by extreme values in the data.
  4. D.Both are considering only the middle 50% of the data.
Show the answer and explanation

Answer: B Both are measures of variation of the data.

Why

Both range and interquartile range are measures of dispersion (variation) of a dataset. Unlike the range, the interquartile range is not affected by extreme values because it only measures the spread of the middle 50% of the data.

Question 2

Which one of the following numbers is the coefficient of x6y4x^{6}y^{4} in the expansion of (x+y)10(x + y)^{10}?

  1. A.240
  2. B.1260
  3. C.210
  4. D.4860
Show the answer and explanation

Answer: C 210

Why

Using the Binomial Theorem, the coefficient of x6y4x^6y^4 in the expansion of (x+y)10(x+y)^{10} is given by (106)=10!6!4!=210\binom{10}{6} = \frac{10!}{6!4!} = 210.

Question 3

The altitude of a frustum of a regular square pyramid is 9 cm. What is the volume of the frustum if the side length of the upper base is 6 cm and the side length of the lower base is 10 cm?

  1. A.196 cm3^3
  2. B.324 cm3^3
  3. C.588 cm3^3
  4. D.108 cm3^3
Show the answer and explanation

Answer: C 588 cm3^3

Why

The volume of a frustum of a square pyramid is V=13h(A1+A2+A1A2)V = \frac{1}{3} h (A_1 + A_2 + \sqrt{A_1 A_2}). Substituting h=9h = 9, A1=62=36A_1 = 6^2 = 36, and A2=102=100A_2 = 10^2 = 100, we get V=3(36+100+60)=588 cm3V = 3(36 + 100 + 60) = 588\text{ cm}^3.

Question 4

Which one of the following intervals is the solution set of the inequality ∣3−12∣x<5|3 - \frac{1}{2}| x < 5?

  1. A.(-2,8)
  2. B.(-4,16)
  3. C.(-8,2)
  4. D.(-16,4)
Show the answer and explanation

Answer: B (-4,16)

Why

Assuming the typo in the absolute value represents ∣3−12x∣<5|3 - \frac{1}{2}x| < 5, we solve −5<3−12x<5-5 < 3 - \frac{1}{2}x < 5 to get −8<−12x<2-8 < -\frac{1}{2}x < 2, which simplifies to −4<x<16-4 < x < 16, yielding the interval (−4,16)(-4, 16).

Question 5

A sector of a circle of diameter 8cm is formed by a central angle of 45°. What is the area of the sector?

  1. A.(2π+8)(2\pi+8)cm²
  2. B.2π2\pi cm²
  3. C.π\picm²
  4. D.8π8\pi cm²
Show the answer and explanation

Answer: B 2π2\pi cm²

Why

The radius of the circle is r=4 cmr = 4\text{ cm}. The area of the sector is given by A=θ360πr2=45360π(42)=2π cm2A = \frac{\theta}{360} \pi r^2 = \frac{45}{360} \pi (4^2) = 2\pi\text{ cm}^2.

Question 8

Which one of the following statements is the polynomial division theorem?

  1. A.If ff is a non-constant polynomial and cc is a real number, then x−cx - c is a factor of f(x)f(x) if and only if f(c)=0f(c) = 0.
  2. B.If f(x)f(x) and d(x)d(x) are polynomials such that d(x)≠0d(x) \neq 0 and degree(d(x))≤degree(f(x))\text{degree}(d(x)) \leq \text{degree}(f(x)), then there exist unique polynomials q(x)q(x) and r(x)r(x) such that f(x)=d(x)q(x)+r(x)f(x) = d(x)q(x)+r(x), where r(x)=0r(x) = 0 or degree(r(x))<degree(d(x))\text{degree}(r(x)) < \text{degree}(d(x)).
  3. C.If ff is a non-constant polynomial and cc is a real number, then the remainder when f(x)f(x) divided by x−cx - c is f(c)f(c).
  4. D.Let ff is a non-constant polynomial with integer coefficients. If the rational number cd\frac{c}{d} in its lowest terms is a root of ff, then cc is a factor of the constant term of ff and dd is a factor of the leading coefficient.
Show the answer and explanation

Answer: B If f(x)f(x) and d(x)d(x) are polynomials such that d(x)≠0d(x) \neq 0 and degree(d(x))≤degree(f(x))\text{degree}(d(x)) \leq \text{degree}(f(x)), then there exist unique polynomials q(x)q(x) and r(x)r(x) such that f(x)=d(x)q(x)+r(x)f(x) = d(x)q(x)+r(x), where r(x)=0r(x) = 0 or degree(r(x))<degree(d(x))\text{degree}(r(x)) < \text{degree}(d(x)).

Why

Option B is the statement of the polynomial division theorem (also known as the division algorithm for polynomials). Options A, C, and D are the Factor Theorem, Remainder Theorem, and Rational Root Theorem, respectively.

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The other Mathematics papers of 2016 E.C.

2016 E.C. (2024)2016 E.C. (2024), social stream2016 E.C. model exam

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 42015 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 2016 E.C. Mathematics entrance exam questions?

Yes. They come from the national university entrance exam in Mathematics that students sat in 2016 E.C. (2024). Our set holds 51 of its questions, each with its answer and an explanation that points back to the textbook.

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