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Grade 12 Mathematics entrance exam, 2007 E.C. (2015)

Real questions from the Grade 12 university entrance exam in Mathematics, as students sat it in 2007 E.C. Every question comes with the correct answer and a worked explanation.

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  • 2007 E.C. (2015)

    Exam year

  • 62 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 equation of an ellipse with centre at (1, 4), vertices at (10, 4) and (1, 2) is :

  1. A.4(x−1)2+81(y−4)2=3244(x-1)^2 + 81(y - 4)^2 = 324
  2. B.(x−1)2+9(y−4)2=4(x-1)^2 + 9(y - 4)^2 = 4
  3. C.9(x−1)2+4(y−4)2=19(x - 1)^2 + 4(y - 4)^2 = 1
  4. D.2(x−1)2+9(y−4)2=42(x - 1)^2 + 9(y - 4)^2 = 4
Show the answer and explanation

Answer: A 4(x−1)2+81(y−4)2=3244(x-1)^2 + 81(y - 4)^2 = 324

Why

The horizontal semi-axis is a=∣10−1∣=9a = |10 - 1| = 9 and the vertical semi-axis is b=∣4−2∣=2b = |4 - 2| = 2. The standard equation is (x−1)281+(y−4)24=1\frac{(x-1)^2}{81} + \frac{(y-4)^2}{4} = 1, which simplifies to 4(x−1)2+81(y−4)2=3244(x-1)^2 + 81(y-4)^2 = 324.

Question 3

Which of the following is convergent sequence?

  1. A.1, 1/2, 1, 1/3, 1, 1/4, …
  2. B.{(−1)n}n=1∞\{(-1)^n\}_{n=1}^{\infty}
  3. C.{10109−1100n}n=1∞\left\{10^{109} - \frac{1}{100^n}\right\}_{n=1}^{\infty}
  4. D.{n+sin⁡1n}n=1∞\left\{n + \sin\frac{1}{n}\right\}_{n=1}^{\infty}
Show the answer and explanation

Answer: C {10109−1100n}n=1∞\left\{10^{109} - \frac{1}{100^n}\right\}_{n=1}^{\infty}

Why

As n→∞n \to \infty, 1100n→0\frac{1}{100^n} \to 0, so the sequence converges to the constant 1010910^{109}. The other sequences diverge by oscillation or tending to infinity.

Question 4

Let f(x)=2x(x2+1)4f(x) = 2x(x^2 + 1)^4, then which of the following is an anti- derivative of f(x)?

  1. A.2x5(x2+1)5+c\frac{2x}{5}(x^2 + 1)^5 + c
  2. B.25(x2+1)5+c\frac{2}{5}(x^2 + 1)^5 + c
  3. C.x5(x2+1)5+1\frac{x}{5}(x^2 + 1)^5 + 1
  4. D.15(x2+1)5−1\frac{1}{5}(x^2 + 1)^5 - 1
Show the answer and explanation

Answer: D 15(x2+1)5−1\frac{1}{5}(x^2 + 1)^5 - 1

Why

Using the substitution u=x2+1u = x^2 + 1 with du=2x dxdu = 2x\,dx, we find ∫2x(x2+1)4 dx=15(x2+1)5+C\int 2x(x^2 + 1)^4\,dx = \frac{1}{5}(x^2 + 1)^5 + C. Setting the constant of integration C=−1C = -1 gives the anti-derivative in option D.

Question 5

Which one of the following is an arithmetic sequence?

  1. A.3, 5, 7, 9, 11, …
  2. B.3, 6, 12, 24, 28, …
  3. C.-3, 6, -9, 12, -15, …
  4. D.1, 3, 6, 10, 15, 21, …
Show the answer and explanation

Answer: A 3, 5, 7, 9, 11, …

Why

An arithmetic sequence has a constant difference between consecutive terms; for 3, 5, 7, 9, 11, the common difference is d=2d = 2.

Question 6

Suppose that p represents the statement “He missed the tournament”, q represents the statement “He got the gold medal.” And r represents the statement “he took a trip abroad.”. then which of the following symbolic expressions represents the statement: “If He took a trip abroad and he does not miss the tournament, then he get the gold medal.”?

  1. A.(r⇒q) ¬p(r \Rightarrow q)\ \neg p
  2. B.r∧(p⇒q)r \wedge (p \Rightarrow q)
  3. C.(r∧¬p⇒q)(r \wedge \neg p \Rightarrow q)
  4. D.¬(r∨p)∨q\neg(r \vee p) \vee q
Show the answer and explanation

Answer: C (r∧¬p⇒q)(r \wedge \neg p \Rightarrow q)

Why

'He took a trip abroad' is rr, 'he does not miss the tournament' is ¬p\neg p, and their conjunction implies 'he got the gold medal' (qq), which is symbolically written as (r∧¬p⇒q)(r \wedge \neg p \Rightarrow q).

Question 8

The Ozone level (in ppb- parts per billion) on a sunny day in metropolitan area is given by formula p(t)= 80 + 12t � t2t^2 where t is time in hours and t = 0 corresponds to 9A.M. what is the rate of increase of Ozone level after 3-hrs(i.e. at 12 A.M)?

  1. A.6 ppb
  2. B.12 ppb
  3. C.107 ppb
  4. D.113 ppb
Show the answer and explanation

Answer: A 6 ppb

Why

The rate of increase is given by the derivative p′(t)=12−2tp'(t) = 12 - 2t. At t=3t = 3 hours, p′(3)=12−2(3)=6p'(3) = 12 - 2(3) = 6 ppb.

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

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Are these the real 2007 E.C. Mathematics entrance exam questions?

Yes. They come from the national university entrance exam in Mathematics that students sat in 2007 E.C. (2015). Our set holds 62 of its questions, each with its answer and an explanation that points back to the textbook.

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