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Grade 12 Mathematics entrance exam, 2006 E.C. (2014)

Real questions from the Grade 12 university entrance exam in Mathematics, as students sat it in 2006 E.C. Every question comes with the correct answer and a worked explanation.

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  • 2006 E.C. (2014)

    Exam year

  • 60 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

A water tank is a circular cylinder with base radius 2m and height 3m. If the tank is empty and water is pumped into it at rate of 2m32\mathrm{m}^3/min, how long does it take for the tank to be full?

  1. A.1.5 min
  2. B.3/2 π\pimin
  3. C.6π6\pi
  4. D.12 min
Show the answer and explanation

Answer: C 6π6\pi

Why

The volume of the cylinder is V=πr2h=π(22)(3)=12π m3V = \pi r^2 h = \pi (2^2)(3) = 12\pi\text{ m}^3. At a rate of 2 m3/min2\text{ m}^3/\text{min}, the time required is 12π2=6π minutes\frac{12\pi}{2} = 6\pi\text{ minutes}.

Question 2

If A=(20−112000−1)A=\begin{pmatrix} 2 & 0 & -1 \\ 1 & 2 & 0 \\ 0 & 0 & -1 \end{pmatrix} and (2A+B)T=ATA(2A + B)^{T} = A^{T}A, then which one of the following is equal to B?

  1. A.(10−2200004)\begin{pmatrix} 1 & 0 & -2 \\ 2 & 0 & 0 \\ 0 & 0 & 4 \end{pmatrix}
  2. B.(000000000)\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}
  3. C.(80−448000−4)\begin{pmatrix} 8 & 0 & -4 \\ 4 & 8 & 0 \\ 0 & 0 & -4 \end{pmatrix}
  4. D.(120000−204)\begin{pmatrix} 1 & 2 & 0 \\ 0 & 0 & 0 \\ -2 & 0 & 4 \end{pmatrix}
Show the answer and explanation

Answer: D (120000−204)\begin{pmatrix} 1 & 2 & 0 \\ 0 & 0 & 0 \\ -2 & 0 & 4 \end{pmatrix}

Why

Taking the transpose of both sides gives 2A+B=(ATA)T=ATA2A + B = (A^T A)^T = A^T A, so B=ATA−2AB = A^T A - 2A. Calculating ATA=(52−2240−202)A^T A = \begin{pmatrix} 5 & 2 & -2 \\ 2 & 4 & 0 \\ -2 & 0 & 2 \end{pmatrix} and subtracting 2A2A yields (120000−204)\begin{pmatrix} 1 & 2 & 0 \\ 0 & 0 & 0 \\ -2 & 0 & 4 \end{pmatrix}.

Question 3

Which one of the following represents a geometric sequence?

  1. A.3, 1, 1/3, 1/9, 1/27,?
  2. B.1/2, -1/3, 1/4, -1/5, 1/6,?
  3. C.1,3, 6, 10, 15,?
  4. D.-3, 6, -9, 12, -15,?
Show the answer and explanation

Answer: A 3, 1, 1/3, 1/9, 1/27,?

Why

A sequence is geometric if the ratio between consecutive terms is constant. In A, each term is multiplied by a common ratio of r=1/3r = 1/3 (3,1,1/3,1/9,…3, 1, 1/3, 1/9, \dots).

Question 4

The sequence {(n−1)(2n+1)1−n2}n=1∞\left\{\frac{(n-1)(2n+1)}{1-n^{2}}\right\}_{n=1}^{\infty} converges to:

  1. A.−∞-\infty
  2. B.-2
  3. C.0
  4. D.1
Show the answer and explanation

Answer: B -2

Why

Expanding the numerator gives 2n2−n−12n^2 - n - 1. Taking the limit as n→∞n \to \infty of 2n2−n−11−n2\frac{2n^2 - n - 1}{1 - n^2} yields the ratio of the leading coefficients, 2−1=−2\frac{2}{-1} = -2.

Question 5

If F(x) is an ant derivative of f(x)=1−2/x2f(x) = 1 - 2/x^{2} and F(1) = 0, then F(2) is equal to:

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

Answer: A 0

Why

The general antiderivative is F(x)=x+2x+CF(x) = x + \frac{2}{x} + C. Given F(1)=1+2+C=0F(1) = 1 + 2 + C = 0, we have C=−3C = -3, which gives F(2)=2+22−3=0F(2) = 2 + \frac{2}{2} - 3 = 0.

Question 6

Which one of the following expression is a polynomial expression?

  1. A.x2−3x+sin⁡xx^{2} - 3x + \sin x
  2. B.4x3+12x2−xπx2\frac{4x^{3}+12x^{2}-x}{\pi x^{2}}
  3. C.2+π1+π2\frac{2+\pi}{1+\pi^{2}}
  4. D.2−3x23+7x52+3x−12 - 3x^{\frac{2}{3}} + 7x^{\frac{5}{2}} + 3x^{-1}
Show the answer and explanation

Answer: C 2+π1+π2\frac{2+\pi}{1+\pi^{2}}

Why

A polynomial consists of terms with non-negative integer powers of the variable. Any real constant, such as 2+π1+π2\frac{2+\pi}{1+\pi^2}, is a constant polynomial (degree 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)2011 E.C. (2019)2009 E.C. (2017)2008 E.C. (2016)2007 E.C. (2015)2005 E.C. (2013)

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Are these the real 2006 E.C. Mathematics entrance exam questions?

Yes. They come from the national university entrance exam in Mathematics that students sat in 2006 E.C. (2014). Our set holds 60 of its questions, each with its answer and an explanation that points back to the textbook.

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