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Grade 12 Mathematics entrance exam, 2016 E.C. (2024)

Real questions from the Grade 12 university entrance exam in Mathematics, as students sat it in 2016 E.C. Every question comes with the correct answer and a worked explanation.

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Natural science stream
Who sits it
2016 E.C. (2024)
Exam year
52 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 1

Let $p$ and $q$ be propositions. Which one of the following propositions is equivalent to the proposition $\neg p \lor (p \land q)$ ?

A. $\neg p \lor \neg q$
B. $p \Rightarrow q$
C. $\neg p \land q$
D. $\neg p \Rightarrow q$

Show the answer and explanation

Answer: B $p \Rightarrow q$

Why

Using the distributive law, we have

\neg p \lor (p \land q) \equiv (\neg p \lor p) \land (\neg p \lor q) \equiv \text{True} \land (\neg p \lor q) \equiv \neg p \lor q

, which is equivalent to the conditional statement

p \Rightarrow q

.

Question 2

Which one of the following is NOT true about the function $f(x) = \tan^{-1} x$ ?

A.The line $x = -\frac{\pi}{2}$ is a vertical asymptote to the graph of $f$ .
B.The graph of $f$ is the image of the graph of $y = \tan x$ after it is reflected with respect to the line $y = x$ .
C.The graph of $f$ intersects the $x$ -axis at $(0, 0)$ .
D.The line $y = \frac{\pi}{2}$ is a horizontal asymptote to the graph of $f$ .

Show the answer and explanation

Answer: A The line $x = -\frac{\pi}{2}$ is a vertical asymptote to the graph of $f$ .

Why

The domain of

f(x) = \tan^{-1} x

is the entire set of real numbers

\mathbb{R}

, and it has horizontal asymptotes at

y = \pm \frac{\pi}{2}

but no vertical asymptotes. The line

x = -\frac{\pi}{2}

is actually a vertical asymptote for

y = \tan x

, not for its inverse.

Question 3

Which one of the following statements is true?

A.If $x$ and $y$ are integers and $x$ divides $y$ , then $y$ divides $x$ .
B.If $x$ is an even integer and $y$ is an odd integer, then $x^2 + y^2$ is odd.
C.If $x$ and $y$ are even integers, then $x + y$ is an odd integer.
D.If $x$ and $y$ are positive integers, then $\frac{x}{y}$ is a positive integer.

Show the answer and explanation

Answer: B If $x$ is an even integer and $y$ is an odd integer, then $x^2 + y^2$ is odd.

Why

If

x

is even, then

x^2

is even; if

y

is odd, then

y^2

is odd. The sum of an even number and an odd number,

x^2 + y^2

, is always odd.

Question 4

Let $p$ , $q$ , and $r$ be propositions. If $p$ is True , $q$ is False , and $r$ is False , which one of the following propositions has truth value True ?

A. $(p \land \neg q) \Leftrightarrow (p \lor r)$
B. $(q \land r) \Leftrightarrow (p \land \neg r)$
C. $p \Rightarrow (q \lor r)$
D. $(p \lor q) \Rightarrow r$

Show the answer and explanation

Answer: A $(p \land \neg q) \Leftrightarrow (p \lor r)$

Why

Substituting the truth values, we have

p \land \neg q \equiv \text{True} \land \text{True} \equiv \text{True}

, and

p \lor r \equiv \text{True} \lor \text{False} \equiv \text{True}

. Thus,

(p \land \neg q) \Leftrightarrow (p \lor r) \equiv \text{True} \Leftrightarrow \text{True}

, which is True.

Question 5

Which one of the following is the equation of the tangent line to the graph of $f(x) = 2x^3 - 4x^2 + 5x + 3$ at the point $(0, 3)$ ?

A. $3x + 5y - 3 = 0$
B. $3x - 5y - 3 = 0$
C. $5x - y + 3 = 0$
D. $5x + y + 3 = 0$

Show the answer and explanation

Answer: C $5x - y + 3 = 0$

Why

The derivative of

f(x)

is

f'(x) = 6x^2 - 8x + 5

, which gives a tangent slope of

f'(0) = 5

at

x = 0

. The equation of the tangent line is

y - 3 = 5(x - 0)

, which simplifies to

5x - y + 3 = 0

.

Question 6

Let $v = \vec{AB}$ and $w = \vec{CD}$ be vectors in the plane, where $A(1, -1)$ , $B(2, -2)$ , $C(-1, 3)$ , and $D(4, -3)$ . Which one of the following is equal to $v - w$ ?

A. $5i - 6j$
B. $i - 15j$
C. $-4i + 5j$
D. $4i - 5j$

Show the answer and explanation

Answer: C $-4i + 5j$

Why

The vector

v = B - A = (1, -1) = i - j

and

w = D - C = (5, -6) = 5i - 6j

. Subtracting these gives

v - w = (i - j) - (5i - 6j) = -4i + 5j

.

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Mathematics papers from other years

2018 E.C. (2026)2017 E.C. (2025)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 2016 E.C. Mathematics entrance exam questions?

Yes. They come from the national university entrance exam in Mathematics that students sat in 2016 E.C. (2024). Our set holds 52 of its questions, each with its answer and an explanation that points back to the textbook.

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Grade 12 Mathematics entrance exam, 2016 E.C. (2024)

Try questions from this paper

Let ppp and qqq be propositions. Which one of the following propositions is equivalent to the proposition ¬p∨(p∧q)\neg p \lor (p \land q)¬p∨(p∧q)?

Which one of the following is NOT true about the function f(x)=tan⁡−1xf(x) = \tan^{-1} xf(x)=tan−1x?

Which one of the following statements is true?

Let ppp, qqq, and rrr be propositions. If ppp is True , qqq is False , and rrr is False , which one of the following propositions has truth value True ?

Which one of the following is the equation of the tangent line to the graph of f(x)=2x3−4x2+5x+3f(x) = 2x^3 - 4x^2 + 5x + 3f(x)=2x3−4x2+5x+3 at the point (0,3)(0, 3)(0,3)?

Let v=AB⃗v = \vec{AB}v=AB and w=CD⃗w = \vec{CD}w=CD be vectors in the plane, where A(1,−1)A(1, -1)A(1,−1), B(2,−2)B(2, -2)B(2,−2), C(−1,3)C(-1, 3)C(−1,3), and D(4,−3)D(4, -3)D(4,−3). Which one of the following is equal to v−wv - wv−w?

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Mathematics papers from other years

Revise Mathematics before you sit this paper

Let $p$ and $q$ be propositions. Which one of the following propositions is equivalent to the proposition $\neg p \lor (p \land q)$ ?

Which one of the following is NOT true about the function $f(x) = \tan^{-1} x$ ?

Let $p$ , $q$ , and $r$ be propositions. If $p$ is True , $q$ is False , and $r$ is False , which one of the following propositions has truth value True ?

Which one of the following is the equation of the tangent line to the graph of $f(x) = 2x^3 - 4x^2 + 5x + 3$ at the point $(0, 3)$ ?

Let $v = \vec{AB}$ and $w = \vec{CD}$ be vectors in the plane, where $A(1, -1)$ , $B(2, -2)$ , $C(-1, 3)$ , and $D(4, -3)$ . Which one of the following is equal to $v - w$ ?