IEB · Grade 12 · Mathematics IEB-style Grade 12 Mathematics Paper 1: Practice Set 1
In the print dialog, choose Save as PDF. IEB-style Grade 12 Mathematics Paper 1: Practice Set 1 Question 1 1.1.1 [3] Solve for $x$ : $x^{2} - 5 x - 14 = 0$ 1.1.2 [4] Solve for $x$ , correct to two decimal places: $3 x^{2} - x - 5 = 0$ 1.1.3 [3] Solve for $x$ : $x^{2} - 2 x - 15 \leq 0$ 1.2 [2] For which value of $k$ does $x^{2} + 4 x + k = 0$ have equal roots? Question 2 2.1.1 [2] Given the arithmetic sequence $4; 11; 18; \dots$ Determine $T_{n}$ . 2.1.2 [3] Calculate the sum of the first 30 terms of this sequence. 2.2.1 [3] In a geometric sequence, $T_{3} = 18$ and $T_{6} = 486$ . Determine $a$ and $r$ . 2.2.2 [2] Calculate $S_{8}$ for this sequence. 2.3 [3] Calculate $k = 1 \sum \infty 12 (\frac{1}{3})^{k - 1}$ . 2.4 [3] Given the quadratic pattern $3; 10; 21; 36; \dots$ Determine $T_{n}$ . Question 3 3.1 [5] Determine $f^{'} (x)$ from first principles if $f (x) = 3 x^{2} - x$ . 3.2 [4] Determine $\frac{d y}{d x}$ if $y = 4 x^{3} - \frac{2}{x} + x$ . 3.3 [4] Determine the equations of the two tangents to $y = x^{3} - 3 x^{2}$ that are parallel to the line $y = 9 x + 1$ . Question 4 4.1 [4] $A$ and $B$ are independent events with $P (A) = 0, 45$ and $P (B) = 0, 3$ . Calculate $P (A and B)$ , $P (A or B)$ and $P (neither)$ . Question 5 Given $f (x) = - x^{2} + 2 x + 8$ and $g (x) = x + 2$ . 5.1 [2] Determine the $x$ -intercepts of $f$ . 5.2 [2] Determine the coordinates of the turning point of $f$ . 5.3 [3] Determine the coordinates of the points of intersection of $f$ and $g$ . 5.4 [2] For which values of $x$ is $g (x) \cdot [f (x) - g (x)] \geq 0$ ? 5.5 [4] Calculate the maximum vertical distance between $f$ and $g$ for $- 2 \leq x \leq 3$ . Question 6 6.1 [2] Convert 10,5% p.a. compounded monthly to an effective annual rate. 6.2 [2] A bakkie costs R280 000 and depreciates at 15% p.a. on the reducing-balance method. Calculate its value after 4 years. 6.3 [4] Thabo needs R250 000 in 6 years' time. He saves R2 500 at the end of every month for 6 years at 8,5% p.a. compounded monthly. Show that he will not reach his target, and calculate the single amount he must deposit into the same account today to make up the shortfall. 6.4.1 [3] A home loan of R950 000 is repaid over 20 years at 11% p.a. compounded monthly. Calculate the monthly repayment. 6.4.2 [3] Calculate the outstanding balance immediately after the 96th payment. 6.5 [2] A loan of R60 000 at 12% p.a. compounded monthly is repaid at R2 000 per month. How many payments are needed? Question 7 Given $f (x) = x^{3} - 6 x^{2} + 9 x - 4$ . 7.1 [3] Show that $(x - 1)$ is a factor of $f (x)$ and factorise $f (x)$ fully. 7.2 [4] Determine the coordinates of the turning points of $f$ . 7.3 [3] Sketch the graph of $f$ , showing all intercepts and turning points. 7.4 [2] For which values of $k$ will $f (x) = k$ have three distinct real roots? Question 8 8.1 [3] Given $f (x) = 2 x^{2}$ , $x \leq 0$ . Determine $f^{- 1} (x)$ and its domain. 8.2 [3] Sketch $f$ and $f^{- 1}$ on the same set of axes. Show the line of symmetry. 8.3 [4] Given $q (x) = (\frac{1}{3})^{x}$ . Write down $q^{- 1} (x)$ and solve $q^{- 1} (x) \geq - 2$ . Question 9 Given $h (x) = \frac{3}{x - 1} + 2$ and $k (x) = 2^{x} - 4$ . 9.1 [2] Write down the equations of the asymptotes of $h$ . 9.2 [3] Determine the $x$ -intercept and the range of $k$ . 9.3 [2] Determine $k^{- 1} (x)$ in the form $y = \dots$ 9.4 [4] Determine the values of $c$ for which the line $y = - x + c$ does not meet the graph of $h$ . Question 10 10.1 [3] Determine the smallest number of terms of the arithmetic sequence $4; 11; 18; \dots$ whose sum is greater than 1 000. 10.2 [3] For which values of $x$ does $k = 1 \sum \infty (2 x - 1)^{k}$ converge? 10.3 [4] The quadratic pattern $3; 10; 21; 36; \dots$ has $T_{n} = 2 n^{2} + n$ . Between which two consecutive terms of the pattern is the difference 203? Question 11 11.1 [5] Solve for $x$ : $x + 10 - x = 4$ 11.2 [5] Solve for $x$ and $y$ : $x + y = 5$ and $\frac{1}{x} + \frac{1}{y} = \frac{5}{6}$ 11.3 [4] Show that the roots of $x^{2} - (p + 4) x + 4 p = 0$ are real for all real values of $p$ , and determine the values of $p$ for which the roots are positive and unequal. Question 12 12.1 [7] The letters of DIFFERENT are arranged in a row (the two F's are identical, and so are the two E's). (a) How many different arrangements are possible? (3) (b) Calculate the probability that an arrangement starts with D and ends with N. (4) 12.2 [4] A four-digit code is made from the digits 0 to 9 without repetition, and the first digit may not be 0. Calculate the probability that a randomly chosen code is even and greater than 5 000. Question 13 Squares of side $x$ cm are cut from the corners of a 30 cm by 30 cm sheet of cardboard. The sides are folded up to make an open box. 13.1 [8] Determine the value of $x$ that gives the largest volume, and calculate this volume.
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IEB-style Grade 12 Mathematics Paper 1: Practice Set 1

Question 1

1.1.1 [3]

Solve for $x$ : $x^{2} - 5 x - 14 = 0$

1.1.2 [4]

Solve for $x$ , correct to two decimal places: $3 x^{2} - x - 5 = 0$

1.1.3 [3]

Solve for $x$ : $x^{2} - 2 x - 15 \leq 0$

1.2 [2]

For which value of $k$ does $x^{2} + 4 x + k = 0$ have equal roots?

Question 2

2.1.1 [2]

Given the arithmetic sequence $4; 11; 18; \dots$ Determine $T_{n}$ .

2.1.2 [3]

Calculate the sum of the first 30 terms of this sequence.

2.2.1 [3]

In a geometric sequence, $T_{3} = 18$ and $T_{6} = 486$ . Determine $a$ and $r$ .

2.2.2 [2]

Calculate $S_{8}$ for this sequence.

2.3 [3]

Calculate $k = 1 \sum \infty 12 (\frac{1}{3})^{k - 1}$ .

2.4 [3]

Given the quadratic pattern $3; 10; 21; 36; \dots$ Determine $T_{n}$ .

Question 3

3.1 [5]

Determine $f^{'} (x)$ from first principles if $f (x) = 3 x^{2} - x$ .

3.2 [4]

Determine $\frac{d y}{d x}$ if $y = 4 x^{3} - \frac{2}{x} + x$ .

3.3 [4]

Determine the equations of the two tangents to $y = x^{3} - 3 x^{2}$ that are parallel to the line $y = 9 x + 1$ .

Question 4

4.1 [4]

$A$ and $B$ are independent events with $P (A) = 0, 45$ and $P (B) = 0, 3$ . Calculate $P (A and B)$ , $P (A or B)$ and $P (neither)$ .

Question 5

Given $f (x) = - x^{2} + 2 x + 8$ and $g (x) = x + 2$ .

5.1 [2]

Determine the $x$ -intercepts of $f$ .

5.2 [2]

Determine the coordinates of the turning point of $f$ .

5.3 [3]

Determine the coordinates of the points of intersection of $f$ and $g$ .

5.4 [2]

For which values of $x$ is $g (x) \cdot [f (x) - g (x)] \geq 0$ ?

5.5 [4]

Calculate the maximum vertical distance between $f$ and $g$ for $- 2 \leq x \leq 3$ .

Question 6

6.1 [2]

Convert 10,5% p.a. compounded monthly to an effective annual rate.

6.2 [2]

A bakkie costs R280 000 and depreciates at 15% p.a. on the reducing-balance method. Calculate its value after 4 years.

6.3 [4]

Thabo needs R250 000 in 6 years' time. He saves R2 500 at the end of every month for 6 years at 8,5% p.a. compounded monthly. Show that he will not reach his target, and calculate the single amount he must deposit into the same account today to make up the shortfall.

6.4.1 [3]

A home loan of R950 000 is repaid over 20 years at 11% p.a. compounded monthly. Calculate the monthly repayment.

6.4.2 [3]

Calculate the outstanding balance immediately after the 96th payment.

6.5 [2]

A loan of R60 000 at 12% p.a. compounded monthly is repaid at R2 000 per month. How many payments are needed?

Question 7

Given $f (x) = x^{3} - 6 x^{2} + 9 x - 4$ .

7.1 [3]

Show that $(x - 1)$ is a factor of $f (x)$ and factorise $f (x)$ fully.

7.2 [4]

Determine the coordinates of the turning points of $f$ .

7.3 [3]

Sketch the graph of $f$ , showing all intercepts and turning points.

7.4 [2]

For which values of $k$ will $f (x) = k$ have three distinct real roots?

Question 8

8.1 [3]

Given $f (x) = 2 x^{2}$ , $x \leq 0$ . Determine $f^{- 1} (x)$ and its domain.

8.2 [3]

Sketch $f$ and $f^{- 1}$ on the same set of axes. Show the line of symmetry.

8.3 [4]

Given $q (x) = (\frac{1}{3})^{x}$ . Write down $q^{- 1} (x)$ and solve $q^{- 1} (x) \geq - 2$ .

Question 9

Given $h (x) = \frac{3}{x - 1} + 2$ and $k (x) = 2^{x} - 4$ .

9.1 [2]

Write down the equations of the asymptotes of $h$ .

9.2 [3]

Determine the $x$ -intercept and the range of $k$ .

9.3 [2]

Determine $k^{- 1} (x)$ in the form $y = \dots$

9.4 [4]

Determine the values of $c$ for which the line $y = - x + c$ does not meet the graph of $h$ .

Question 10

10.1 [3]

Determine the smallest number of terms of the arithmetic sequence $4; 11; 18; \dots$ whose sum is greater than 1 000.

10.2 [3]

For which values of $x$ does $k = 1 \sum \infty (2 x - 1)^{k}$ converge?

10.3 [4]

The quadratic pattern $3; 10; 21; 36; \dots$ has $T_{n} = 2 n^{2} + n$ . Between which two consecutive terms of the pattern is the difference 203?

Question 11

11.1 [5]

Solve for $x$ : $x + 10 - x = 4$

11.2 [5]

Solve for $x$ and $y$ : $x + y = 5$ and $\frac{1}{x} + \frac{1}{y} = \frac{5}{6}$

11.3 [4]

Show that the roots of $x^{2} - (p + 4) x + 4 p = 0$ are real for all real values of $p$ , and determine the values of $p$ for which the roots are positive and unequal.

Question 12

12.1 [7]

The letters of DIFFERENT are arranged in a row (the two F's are identical, and so are the two E's).

(a) How many different arrangements are possible? (3)

(b) Calculate the probability that an arrangement starts with D and ends with N. (4)

12.2 [4]

A four-digit code is made from the digits 0 to 9 without repetition, and the first digit may not be 0. Calculate the probability that a randomly chosen code is even and greater than 5 000.

Question 13

Squares of side $x$ cm are cut from the corners of a 30 cm by 30 cm sheet of cardboard. The sides are folded up to make an open box.

13.1 [8]

Determine the value of $x$ that gives the largest volume, and calculate this volume.