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Grade 12 Mathematics entrance exam, 2017 E.C. (2025)

Real questions from the Grade 12 university entrance exam in Mathematics, as students sat it in 2017 E.C. Every question comes with the correct answer and a worked explanation.

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  • Social science stream

    Who sits it

  • 2017 E.C. (2025)

    Exam year

  • 59 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 2

Let f(x)f(x) be a polynomial of degree at least 1, g(x)g(x) be a non-zero polynomial and c∈Rc \in \mathbb{R}. Which of the following states the Factor Theorem?

  1. A.f(c)=0f(c) = 0 if and only if x−cx - c is a factor of f(x)f(x).
  2. B.For distinct real numbers aa and bb, if f(a)f(b)<0f(a)f(b) < 0, then f(x)f(x) has at least one zero between aa and bb.
  3. C.There exist polynomials q(x)q(x) and r(x)r(x) such that f(x)=g(x)q(x)+r(x)f(x) = g(x)q(x) + r(x).
  4. D.If f(x)f(x) is divided by x−cx - c, then the remainder is f(c)f(c).
Show the answer and explanation

Answer: A f(c)=0f(c) = 0 if and only if x−cx - c is a factor of f(x)f(x).

Why

The Factor Theorem states that for any polynomial f(x)f(x) of degree at least 1, x−cx - c is a factor of f(x)f(x) if and only if f(c)=0f(c) = 0.

Question 3

Let A={1,2}A = \{1,2\} and B={1,3}B = \{1,3\} be sets. Then which one of the following Cartesian products is correct?

  1. A.B×A={(1,1),(3,1),(1,2),(3,2),(2,3)}B\times A = \{(1,1),(3,1),(1,2),(3,2),(2,3)\}.
  2. B.B×B={(1,1),(1,3),(3,3),(2,3)}B\times B = \{(1,1),(1,3),(3,3),(2,3)\}.
  3. C.A×B={(1,1),(1,3),(2,1),(2,3),(3,2)}A\times B = \{(1,1),(1,3),(2,1),(2,3),(3,2)\}.
  4. D.A×A={(1,1),(2,2),(2,1),(1,2)}A\times A = \{(1,1),(2,2),(2,1),(1,2)\}.
Show the answer and explanation

Answer: D A×A={(1,1),(2,2),(2,1),(1,2)}A\times A = \{(1,1),(2,2),(2,1),(1,2)\}.

Why

For A={1,2}A = \{1,2\}, the Cartesian product A×AA \times A consists of all ordered pairs (x,y)(x, y) where both xx and yy are in AA, yielding {(1,1),(1,2),(2,1),(2,2)}\{(1,1), (1,2), (2,1), (2,2)\}. The set in option D contains exactly these elements.

Question 4

Consider the matrix (2−310−1π314)\begin{pmatrix} 2 & -3 & 1 \\ 0 & -1 & \pi \\ 3 & 1 & 4 \end{pmatrix}. What is the cofactor of entry π\pi?

  1. A.77
  2. B.−11-11
  3. C.−11π-11\pi
  4. D.−7π-7\pi
Show the answer and explanation

Answer: B −11-11

Why

The entry π\pi is at row 2, column 3, so its cofactor is (−1)2+3det⁡(2−331)=−(2⋅1−(−3)⋅3)=−(2+9)=−11(-1)^{2+3} \det \begin{pmatrix} 2 & -3 \\ 3 & 1 \end{pmatrix} = -(2 \cdot 1 - (-3) \cdot 3) = -(2 + 9) = -11.

Question 5

When converting the recurring decimal 0.25‾0.\overline{25} into a fraction using convergent geometric series, then what are the first term and the common ratio respectively?

  1. A.0.250.25 and 0.010.01
  2. B.2525 and 0.010.01
  3. C.0.250.25 and 0.10.1
  4. D.2.52.5 and 0.10.1
Show the answer and explanation

Answer: A 0.250.25 and 0.010.01

Why

The recurring decimal 0.25‾0.\overline{25} can be written as the infinite series 0.25+0.0025+0.000025+…0.25 + 0.0025 + 0.000025 + \dots, which is a geometric series with the first term a1=0.25a_1 = 0.25 and common ratio r=0.01r = 0.01.

Question 6

What are the maximum and minimum values respectively of z=3x+yz = 3x + y subject to constraints {3x+2y≤202x−y≥4x≥0,y≥0\begin{cases} 3x + 2y \leq 20 \\ 2x - y \geq 4 \\ x \geq 0, y \geq 0 \end{cases}?

  1. A.2020 and 66
  2. B.4040 and 66
  3. C.1616 and 66
  4. D.1818 and 88
Show the answer and explanation

Answer: A 2020 and 66

Why

The vertices of the feasible region defined by the constraints are (2,0)(2, 0), (20/3,0)(20/3, 0), and (4,4)(4, 4). Evaluating z=3x+yz = 3x + y at these vertices gives 66, 2020, and 1616 respectively, making the maximum value 2020 and the minimum value 66.

Question 7

What is the domain of the function f(x)=3−x2f(x) = \sqrt{3 - x^2}?

  1. A.[3,∞)[\sqrt{3}, \infty)
  2. B.[−3,3][-\sqrt{3}, \sqrt{3}]
  3. C.[−3,3][-3, 3]
  4. D.(−∞,3]∪[3,∞)(-\infty, \sqrt{3}] \cup [\sqrt{3}, \infty)
Show the answer and explanation

Answer: B [−3,3][-\sqrt{3}, \sqrt{3}]

Why

For the square root function f(x)=3−x2f(x) = \sqrt{3 - x^2} to be defined on real numbers, the radicand must be non-negative: 3−x2≥0  ⟹  x2≤33 - x^2 \ge 0 \implies x^2 \le 3, which gives the interval [−3,3][-\sqrt{3}, \sqrt{3}].

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Mathematics papers from other years

2018 E.C. (2026)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)2006 E.C. (2014)2005 E.C. (2013)

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Are these the real 2017 E.C. Mathematics entrance exam questions?

Yes. They come from the national university entrance exam in Mathematics that students sat in 2017 E.C. (2025). Our set holds 59 of its questions, each with its answer and an explanation that points back to the textbook.

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