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Grade 12 Mathematics entrance exam, 2011 E.C. (2019)

Real questions from the Grade 12 university entrance exam in Mathematics, as students sat it in 2011 E.C. Every question comes with the correct answer and a worked explanation.

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  • 2011 E.C. (2019)

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  • 64 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 is the set of critical numbers of f(x)=38x8/3−6x2/3f(x) = \frac{3}{8}x^{8/3} - 6x^{2/3}?

  1. A.{−2,2}\{-2, 2\}
  2. B.{−1,1}\{-1, 1\}
  3. C.{−1,0,1}\{-1, 0, 1\}
  4. D.{−2,0,2}\{-2, 0, 2\}
Show the answer and explanation

Answer: D {−2,0,2}\{-2, 0, 2\}

Why

The critical numbers of a function are where its derivative is zero or undefined. By differentiating, we get f′(x)=x2−4x1/3f'(x) = \frac{x^2 - 4}{x^{1/3}}, which is zero at x=±2x = \pm 2 and undefined at x=0x = 0, all of which are in the domain of ff.

Question 2

What is the greatest lower bound of the sequence {(−1)n1n+1}n=0∞\left\{(-1)^n \frac{1}{n+1}\right\}_{n=0}^{\infty}?

  1. A.-1
  2. B.0
  3. C.−1/2-1/2
  4. D.1/21/2
Show the answer and explanation

Answer: C −1/2-1/2

Why

Listing the first few terms of the sequence yields {1,−1/2,1/3,−1/4,1/5,−1/6,…}\{1, -1/2, 1/3, -1/4, 1/5, -1/6, \ldots\}. The negative terms are increasing towards 0, with the absolute minimum and greatest lower bound being the first negative term, which is −1/2-1/2.

Question 3

Which one of the following is true about the pair of lines:3x+9y−24=03x + 9y - 24 = 0 and 4x+12y+32=04x + 12y + 32 = 0?

  1. A.perpendicular lines
  2. B.intersecting lines
  3. C.parallel and distinct lines
  4. D.representing the same lines
Show the answer and explanation

Answer: C parallel and distinct lines

Why

Both equations can be rewritten in slope-intercept form to find their slopes: y=−13x+83y = -\frac{1}{3}x + \frac{8}{3} and y=−13x−83y = -\frac{1}{3}x - \frac{8}{3}. Since they have the same slope but different yy-intercepts, they represent parallel and distinct lines.

Question 4

Which one of the following is equal to f(x)=(x+4)2f(x) = \sqrt{(x+4)^2}, for every x∈Rx \in \mathbb{R}?

  1. A.g(x)=x+4g(x) = x + 4
  2. B.g(x)=x+2g(x) = x + 2
  3. C.g(x)=∣x+4∣g(x) = |x+4|
  4. D.g(x)=∣x∣+4g(x) = |x|+4
Show the answer and explanation

Answer: C g(x)=∣x+4∣g(x) = |x+4|

Why

By definition, for any real number uu, u2=∣u∣\sqrt{u^2} = |u|. Substituting u=x+4u = x+4 gives g(x)=∣x+4∣g(x) = |x+4|.

Question 5

The center of a circle on the line y=2xy = 2x and the line x=1x = 1 is tangent to the circle at (1,6)(1,6). How long is the radius of the circle?

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

Answer: D 2

Why

The tangent line x=1x = 1 is vertical, meaning the radius connecting the center (h,2h)(h, 2h) to the tangent point (1,6)(1, 6) must be horizontal, so 2h=6  ⟹  h=32h = 6 \implies h = 3. The radius is the horizontal distance between (3,6)(3,6) and (1,6)(1,6), which is ∣3−1∣=2|3-1| = 2.

Question 6

What is the value of lim⁡n→∞(3n+2n6n)\lim_{n \to \infty} \left(\frac{3^n + 2^n}{6^n}\right)?

  1. A.1
  2. B.0
  3. C.∞\infty
  4. D.56\frac{5}{6}
Show the answer and explanation

Answer: B 0

Why

We can split the expression into lim⁡n→∞[(1/2)n+(1/3)n]\lim_{n \to \infty} \left[ (1/2)^n + (1/3)^n \right]. Since both bases are between −1-1 and 11, their limits as n→∞n \to \infty are both 0, making the overall limit 0.

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Mathematics papers from other years

2018 E.C. (2026)2017 E.C. (2025)2016 E.C. (2024)2015 E.C. (2023)2014 E.C. (2022)2013 E.C. (2021)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 2011 E.C. Mathematics entrance exam questions?

Yes. They come from the national university entrance exam in Mathematics that students sat in 2011 E.C. (2019). Our set holds 64 of its questions, each with its answer and an explanation that points back to the textbook.

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