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Grade 12 Mathematics entrance exam, 2013 E.C. (2021), version 1

Real questions from the Grade 12 university entrance exam in Mathematics, as students sat it in 2013 E.C. Every question comes with the correct answer and a worked explanation. The 2013 E.C. exam went out in several versions. This page holds version 1; the others are listed further down.

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  • 2013 E.C. (2021), version 1

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  • 36 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

Let a function f(x) = -4x^3 - 2x - 10, then which one of the following is true about f.

  1. A.f is neither even nor odd
  2. B.f is odd
  3. C.f is even
  4. D.f is both even and odd
Show the answer and explanation

Answer: A f is neither even nor odd

Why

A function is even if f(-x) = f(x) and odd if f(-x) = -f(x). For f(x) = -4x^3 - 2x - 10, we have f(-x) = 4x^3 + 2x - 10, which is neither equal to f(x) nor to -f(x).

Question 2

If (g o f)(x) = x^2 + x + 6 and f(x) = x + 2, then what is the equivalent expression for g(x)?

  1. A.x^2 - 3x + 8
  2. B.-x^2 - 3x + 6
  3. C.x^2 + 3x + 8
  4. D.x^2 - 3x - 6
Show the answer and explanation

Answer: A x^2 - 3x + 8

Why

Letting y = f(x) = x + 2 gives x = y - 2. Substituting this into (g o f)(x) = x^2 + x + 6 yields g(y) = (y-2)^2 + (y-2) + 6 = y^2 - 3y + 8, which means g(x) = x^2 - 3x + 8.

Question 3

Which one of the following is incorrect about the relation R = {(x,y): y = (x+3)/(x+10)}?

  1. A.The range of R is (-∞,1) ∪ (1,∞)
  2. B.Domain of R is (-∞,-10) ∪ (-10,∞)
  3. C.R = {(x,y): y = (x+3)/(x+10)} can be a function
  4. D.The inverse of R is {(y,x): y = (10x+3)/(x+1)}
Show the answer and explanation

Answer: D The inverse of R is {(y,x): y = (10x+3)/(x+1)}

Why

To find the inverse of y = (x+3)/(x+10), we solve for x to get x = (3-10y)/(y-1). Swapping the variables gives the inverse relation as y = (3-10x)/(x-1), which does not match option D.

Question 4

Which one of the following is correct one-to-one correspondence?

  1. A.f: ℝ → ℝ given by f(x)=log_a x, ∀x∈ℝ
  2. B.f: ℝ → ℚ given by f(x)=4^x, ∀x∈ℝ
  3. C.f: [0,∞) → [0,∞) given by f(x)=x^2, ∀x∈ℝ
  4. D.f: [1,∞) → (3,∞) given by f(x)=-3+√(x-1), ∀x∈ℝ
Show the answer and explanation

Answer: C f: [0,∞) → [0,∞) given by f(x)=x^2, ∀x∈ℝ

Why

For the domain [0, ∞) and codomain [0, ∞), the function f(x) = x^2 is strictly increasing, making it one-to-one, and its range is [0, ∞), making it onto (hence a one-to-one correspondence).

Question 6

Which one of the following is not a rational expression?

  1. A.(x^5+9)/(x^2-16)
  2. B.√((x+2)^2)/(x^2-4)
  3. C.∛((x+2)^6)/(x^2-25)
  4. D.∛((x+2)^12)/(x^2+16)
Show the answer and explanation

Answer: B √((x+2)^2)/(x^2-4)

Why

A rational expression is the quotient of two polynomials. In option B, the numerator simplifies to |x+2| because of the square root of a square, which is not a polynomial, making the entire expression non-rational.

Question 8

What is the solution set of the expression (36x-72)/(-x^3+4x) = [ (x-1)/(x+1) - (x+1)/(x-1) ] / [ (x+2)/(x^2-1) ] ?

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

Answer: B {-3,3}

Why

Simplifying both sides yields -36 / (x(x+2)) = -4x / (x+2). For x ≠ -2, this simplifies further to x^2 = 9, giving the solutions x = 3 and x = -3, both of which are valid since they do not violate any domain restrictions.

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The other Mathematics papers of 2013 E.C.

2013 E.C. (2021)

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 exam2015 E.C. (2023)2015 E.C. (2023), natural stream2015 E.C. (2023), social stream2014 E.C. (2022)2014 E.C. (2022), social stream2011 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 2013 E.C. Mathematics entrance exam questions?

Yes. They come from the national university entrance exam in Mathematics that students sat in 2013 E.C. (2021). Our set holds 36 of its questions, each with its answer and an explanation that points back to the textbook.

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