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

A model exam set in 2016 E.C. to prepare Grade 12 students for the Mathematics entrance exam, in the same format. It is practice, and it is not the entrance exam paper itself. Every question comes with the correct answer and a worked explanation.

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Both streams
Who sits it
2016 E.C. model exam
Exam year
Arada sub-city
Set by
62 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

If $x$ and $y$ are composite natural numbers and $z$ is a prime number, then which one of the following is never true?

A. $x + y = z$
B. $x + y = 2z$
C. $xz = y$
D. $xy = z^2$

Show the answer and explanation

Answer: D $xy = z^2$

Why

Since

z

is a prime number, the only factors of

z^2

are

1, z, z^2

. If

xy = z^2

where

x

and

y

are composite, they must both be greater than 1, so the only possibility is

x=z

and

y=z

, which contradicts the assumption that

z

is prime and

x,y

are composite.

Question 2

The simplified form of the expression $(\frac{5}{4})^3 (\frac{\sqrt{5}}{2})^{-4} \div (\frac{\sqrt[3]{5}}{\sqrt{2}})^6$ is:

A. $\frac{5}{2}$
B. $\frac{2}{5}$
C. $\frac{1}{5}$
D.5

Show the answer and explanation

Answer: B $\frac{2}{5}$

Why

Simplifying each term:

(\frac{5}{4})^3 = \frac{125}{64}

,

(\frac{\sqrt{5}}{2})^{-4} = (\frac{2}{\sqrt{5}})^4 = \frac{16}{25}

, and

(\frac{\sqrt[3]{5}}{\sqrt{2}})^6 = \frac{25}{8}

. Multiplying the first two terms gives

\frac{125}{64} \times \frac{16}{25} = \frac{5}{4}

, and dividing by the third term gives

\frac{5}{4} \div \frac{25}{8} = \frac{5}{4} \times \frac{8}{25} = \frac{2}{5}

.

Question 3

To rationalize the denominator $\frac{3+\sqrt{2}}{4-\sqrt{3}}$ which one of the following expression is used as the rationalizing factor?

A. $\frac{3-\sqrt{2}}{3-\sqrt{2}}$
B. $\frac{3-\sqrt{2}}{4+\sqrt{3}}$
C. $\frac{4+\sqrt{3}}{4+\sqrt{3}}$
D. $\frac{1}{4+\sqrt{3}}$

Show the answer and explanation

Answer: C $\frac{4+\sqrt{3}}{4+\sqrt{3}}$

Why

To rationalize the denominator of a fraction of the form

\frac{a}{b-\sqrt{c}}

, we multiply both the numerator and the denominator by the conjugate of the denominator, which is

b+\sqrt{c}

. Thus, the rationalizing factor is

\frac{4+\sqrt{3}}{4+\sqrt{3}}

.

Question 4

What is the value of $x$ for $2\left(\frac{243}{32}\right)^{2x} = 3\left(\frac{8}{27}\right)^{\frac{2}{3}x-1}$ ?

A. $\frac{1}{3}$
B. $\frac{4}{3}$
C. $\frac{2}{5}$
D. $\frac{1}{4}$

Show the answer and explanation

Answer: A $\frac{1}{3}$

Why

Expressing the bases in terms of

\frac{3}{2}

, the equation becomes

2\left((\frac{3}{2})^5\right)^{2x} = 3\left((\frac{3}{2})^{-3}\right)^{\frac{2}{3}x - 1}

, which simplifies to

2(\frac{3}{2})^{10x} = 3(\frac{3}{2})^{-2x + 3}

. Dividing both sides yields

(\frac{3}{2})^{12x - 3} = \frac{3}{2}

, so

12x - 3 = 1

, giving

x = \frac{1}{3}

.

Question 5

Which one of the following systems of linear equation has infinitely many solutions?

A. $\begin{cases} 3x + y = -1 \ -x - \frac{1}{3}y = \frac{1}{3} \end{cases}$
B. $\begin{cases} x + y = 6 \ 2x - 5y = 0 \end{cases}$
C. $\begin{cases} x + 3y = 2 \ 2x - 4y = 1 \end{cases}$
D. $\begin{cases} x - 2y = 5 \ -3x + 6y = 1 \end{cases}$

Show the answer and explanation

Answer: A $\begin{cases} 3x + y = -1 \ -x - \frac{1}{3}y = \frac{1}{3} \end{cases}$

Why

Multiplying the second equation in Option A,

-x - \frac{1}{3}y = \frac{1}{3}

, by

-3

gives

3x + y = -1

. This is identical to the first equation, meaning they represent the same line and therefore have infinitely many solutions.

Question 6

What is the solution set of the inequality $|3x + 6| - 2 \ge 4$ ?

A. $(-\infty, -4) \cup (0, \infty)$
B. $[-4, 0]$
C. $(-\infty, -4] \cup [0, \infty)$
D. $(-4, 0)$

Show the answer and explanation

Answer: C $(-\infty, -4] \cup [0, \infty)$

Why

The inequality

|3x + 6| - 2 \ge 4

simplifies to

|3x + 6| \ge 6

, which implies

|x + 2| \ge 2

. This means

x + 2 \ge 2

(giving

x \ge 0

) or

x + 2 \le -2

(giving

x \le -4

), which in interval notation is

(-\infty, -4] \cup [0, \infty)

.

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The other Mathematics papers of 2016 E.C.

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All Grade 12 Mathematics papers All papers of 2016 E.C.Every subject and year

Revise Mathematics before you sit this paper

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Is this the real Grade 12 entrance exam?

No. It is a model exam set in 2016 E.C. (2024) to prepare students for the entrance exam, in the same format. Our set holds 62 of its questions, each with its answer and a worked explanation.

Where are the real entrance exam papers?

The Grade 12 entrance exam papers we hold are on the exam papers page, listed by year beside the model exams.

Can I download this model exam as a PDF?

Yes, and it is free. The question paper is one file and the answer key with its worked explanations is another, so you can try it first. Print them, or keep them on your phone for revision when there is no data.

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

Try questions from this paper

If xxx and yyy are composite natural numbers and zzz is a prime number, then which one of the following is never true?

The simplified form of the expression (54)3(52)−4÷(532)6(\frac{5}{4})^3 (\frac{\sqrt{5}}{2})^{-4} \div (\frac{\sqrt[3]{5}}{\sqrt{2}})^6(45​)3(25​​)−4÷(2​35​​)6 is:

To rationalize the denominator 3+24−3\frac{3+\sqrt{2}}{4-\sqrt{3}}4−3​3+2​​ which one of the following expression is used as the rationalizing factor?

What is the value of xxx for 2(24332)2x=3(827)23x−12\left(\frac{243}{32}\right)^{2x} = 3\left(\frac{8}{27}\right)^{\frac{2}{3}x-1}2(32243​)2x=3(278​)32​x−1?

Which one of the following systems of linear equation has infinitely many solutions?

What is the solution set of the inequality ∣3x+6∣−2≥4|3x + 6| - 2 \ge 4∣3x+6∣−2≥4?

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Revise Mathematics before you sit this paper

If $x$ and $y$ are composite natural numbers and $z$ is a prime number, then which one of the following is never true?

The simplified form of the expression $(\frac{5}{4})^3 (\frac{\sqrt{5}}{2})^{-4} \div (\frac{\sqrt[3]{5}}{\sqrt{2}})^6$ is:

To rationalize the denominator $\frac{3+\sqrt{2}}{4-\sqrt{3}}$ which one of the following expression is used as the rationalizing factor?

What is the value of $x$ for $2\left(\frac{243}{32}\right)^{2x} = 3\left(\frac{8}{27}\right)^{\frac{2}{3}x-1}$ ?

What is the solution set of the inequality $|3x + 6| - 2 \ge 4$ ?