IEB · Grade 12 · Mathematics IEB-style Grade 12 Mathematics Paper 1: Practice Set 2
In the print dialog, choose Save as PDF. IEB-style Grade 12 Mathematics Paper 1: Practice Set 2 Question 1 1.1 [3] Solve for $x$ : $x (x - 3) = 10$ 1.2 [4] Solve for $x$ , correct to two decimal places: $2 x^{2} + 3 x - 7 = 0$ 1.3 [3] Solve for $x$ : $x (x + 4) > 12$ 1.4 [2] Simplify without a calculator: $\frac{48 - 12}{3}$ Question 2 2.1 [3] Calculate $k = 1 \sum 20 (4 k - 1)$ . 2.2.1 [3] Given the quadratic pattern $5; 8; 13; 20; \dots$ Determine $T_{n}$ . 2.2.2 [2] Which term of the pattern is 404? 2.3 [3] Prove that the sum of the first $n$ terms of an arithmetic series is $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ . Question 3 The hyperbola $y = \frac{a}{x + p} + q$ has asymptotes $x = - 2$ and $y = 1$ , and passes through $(0; 3)$ . 3.1 [3] Determine $a$ , $p$ and $q$ . 3.2 [2] Determine the $x$ -intercept. 3.3 [2] Determine the axis of symmetry with a negative gradient. 3.4 [3] For which values of $x$ is $\frac{4}{x + 2} + 1 \geq 3$ ? 3.5 [2] Write down the equation of the reflection of the hyperbola in the $y$ -axis. Question 4 4.1 [5] Determine $f^{'} (x)$ from first principles if $f (x) = x^{2} - 4 x + 1$ . 4.2 [4] Determine $D_{x} [\frac{( 2 x - 1 ) ^{2}}{x}]$ . 4.3.1 [2] A ball is kicked straight up. Its height after $t$ seconds is $h (t) = 25 t - 5 t^{2}$ metres. Calculate its velocity after 1 second. 4.3.2 [3] Calculate the velocity of the ball when it hits the ground. Question 5 Given $f (x) = lo g_{3} x$ . 5.1 [1] Write down $f^{- 1} (x)$ . 5.2 [1] $g (x) = f (x - 2) + 1$ . Write down the domain of $g$ . 5.3 [3] Determine the $x$ -intercept of $g$ . 5.4 [4] Solve for $x$ : $lo g_{3} x + lo g_{3} (x - 2) = 1$ 5.5 [2] For which values of $x$ is $f (x) < 2$ ? Question 6 The parabola $f$ has a turning point at $(- 1; 8)$ and passes through $(1; 0)$ . The line $g (x) = - 2 x + 2$ also passes through $(1; 0)$ . 6.1 [3] Show that $f (x) = - 2 x^{2} - 4 x + 6$ . 6.2 [3] Determine the coordinates of the other point of intersection of $f$ and $g$ . 6.3 [3] Write down the range of $f$ , and the turning point of $h (x) = f (x - 2) - 8$ . What does the turning point tell you about $h$ ? 6.4 [3] For which values of $k$ will $f (x) = k$ have one positive and one negative root? Question 7 7.1 [4] 150 Grade 12 learners were asked whether they belong to a gym. 70 are boys and 28 of the boys belong to a gym. 60 learners in total belong to a gym. Are being a boy and belonging to a gym independent? Show calculations. 7.2 [4] A bag contains 5 red and 3 green sweets. Two are taken without replacement. Calculate the probability that at least one is green. 7.3.1 [1] Seven friends sit in a row at the movies. In how many ways can they sit? 7.3.2 [3] In how many ways can they sit if three particular friends must sit together? 7.3.3 [2] What is the probability that those three friends sit together in a random arrangement? Question 8 8.1 [3] Calculate the nominal annual rate, compounded quarterly, that is equivalent to an effective rate of 12% p.a. 8.2 [5] A bakery buys an oven for R420 000. It will be replaced in 5 years. The price of ovens rises at 6,5% p.a., and the old oven depreciates at 15% p.a. on the reducing-balance method and will be sold. A sinking fund earns 9% p.a. compounded monthly, with monthly payments starting one month from now and ending at the replacement date. Calculate the monthly payment. 8.3 [4] A retiree invests R2 500 000 at 7,2% p.a. compounded monthly and withdraws R25 000 at the end of every month. How many full withdrawals can she make? 8.4 [3] A loan of R180 000 is taken at 13,5% p.a. compounded monthly. It is repaid with 48 equal monthly payments, starting one month after the loan is granted. Calculate the balance outstanding immediately after the 20th payment. Question 9 Given $f (x) = - x^{3} + 3 x^{2} + 9 x - 5$ . 9.1 [4] Determine the coordinates of the turning points of $f$ . 9.2 [2] Determine the coordinates of the point of inflection. 9.3 [3] Determine the equation of the tangent to $f$ at its point of inflection. 9.4 [3] For which values of $x$ is $f$ increasing, and for which values is $f^{'} (x) < 9$ ? 9.5 [3] The graph of $f$ is shifted vertically to give $y = f (x) + k$ . Determine the values of $k$ for which the new graph has exactly two $x$ -intercepts. Question 10 10.1 [5] Solve for $x$ : $(x^{2} - x)^{2} - 8 (x^{2} - x) + 12 = 0$ 10.2 [4] Solve for $x$ : $3^{2 x} - 4 \cdot 3^{x} + 3 = 0$ 10.3 [4] Prove that the roots of $x^{2} - (k + 3) x + 3 k = 0$ are real for all real values of $k$ , and rational when $k$ is rational. Question 11 11.1 [4] How many terms of $2 + 7 + 12 + \dots$ must be added to give 1 428? 11.2 [6] The second term of a convergent geometric series is 12 and its sum to infinity is 64. Determine the possible values of the first term and the common ratio. 11.3 [4] How many terms of $3 + 6 + 12 + \dots$ must be added for the sum to exceed 10 000? Question 12 12.1 [7] A closed cylindrical tin must hold 1 000 cm³. Determine the radius that uses the least metal, and calculate the minimum surface area. Give answers to two decimal places.
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IEB-style Grade 12 Mathematics Paper 1: Practice Set 2

Question 1

1.1 [3]

Solve for $x$ : $x (x - 3) = 10$

1.2 [4]

Solve for $x$ , correct to two decimal places: $2 x^{2} + 3 x - 7 = 0$

1.3 [3]

Solve for $x$ : $x (x + 4) > 12$

1.4 [2]

Simplify without a calculator: $\frac{48 - 12}{3}$

Question 2

2.1 [3]

Calculate $k = 1 \sum 20 (4 k - 1)$ .

2.2.1 [3]

Given the quadratic pattern $5; 8; 13; 20; \dots$ Determine $T_{n}$ .

2.2.2 [2]

Which term of the pattern is 404?

2.3 [3]

Prove that the sum of the first $n$ terms of an arithmetic series is $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ .

Question 3

The hyperbola $y = \frac{a}{x + p} + q$ has asymptotes $x = - 2$ and $y = 1$ , and passes through $(0; 3)$ .

3.1 [3]

Determine $a$ , $p$ and $q$ .

3.2 [2]

Determine the $x$ -intercept.

3.3 [2]

Determine the axis of symmetry with a negative gradient.

3.4 [3]

For which values of $x$ is $\frac{4}{x + 2} + 1 \geq 3$ ?

3.5 [2]

Write down the equation of the reflection of the hyperbola in the $y$ -axis.

Question 4

4.1 [5]

Determine $f^{'} (x)$ from first principles if $f (x) = x^{2} - 4 x + 1$ .

4.2 [4]

Determine $D_{x} [\frac{( 2 x - 1 ) ^{2}}{x}]$ .

4.3.1 [2]

A ball is kicked straight up. Its height after $t$ seconds is $h (t) = 25 t - 5 t^{2}$ metres. Calculate its velocity after 1 second.

4.3.2 [3]

Calculate the velocity of the ball when it hits the ground.

Question 5

Given $f (x) = lo g_{3} x$ .

5.1 [1]

Write down $f^{- 1} (x)$ .

5.2 [1]

$g (x) = f (x - 2) + 1$ . Write down the domain of $g$ .

5.3 [3]

Determine the $x$ -intercept of $g$ .

5.4 [4]

Solve for $x$ : $lo g_{3} x + lo g_{3} (x - 2) = 1$

5.5 [2]

For which values of $x$ is $f (x) < 2$ ?

Question 6

The parabola $f$ has a turning point at $(- 1; 8)$ and passes through $(1; 0)$ . The line $g (x) = - 2 x + 2$ also passes through $(1; 0)$ .

6.1 [3]

Show that $f (x) = - 2 x^{2} - 4 x + 6$ .

6.2 [3]

Determine the coordinates of the other point of intersection of $f$ and $g$ .

6.3 [3]

Write down the range of $f$ , and the turning point of $h (x) = f (x - 2) - 8$ . What does the turning point tell you about $h$ ?

6.4 [3]

For which values of $k$ will $f (x) = k$ have one positive and one negative root?

Question 7

7.1 [4]

150 Grade 12 learners were asked whether they belong to a gym. 70 are boys and 28 of the boys belong to a gym. 60 learners in total belong to a gym. Are being a boy and belonging to a gym independent? Show calculations.

7.2 [4]

A bag contains 5 red and 3 green sweets. Two are taken without replacement. Calculate the probability that at least one is green.

7.3.1 [1]

Seven friends sit in a row at the movies. In how many ways can they sit?

7.3.2 [3]

In how many ways can they sit if three particular friends must sit together?

7.3.3 [2]

What is the probability that those three friends sit together in a random arrangement?

Question 8

8.1 [3]

Calculate the nominal annual rate, compounded quarterly, that is equivalent to an effective rate of 12% p.a.

8.2 [5]

A bakery buys an oven for R420 000. It will be replaced in 5 years. The price of ovens rises at 6,5% p.a., and the old oven depreciates at 15% p.a. on the reducing-balance method and will be sold. A sinking fund earns 9% p.a. compounded monthly, with monthly payments starting one month from now and ending at the replacement date.

Calculate the monthly payment.

8.3 [4]

A retiree invests R2 500 000 at 7,2% p.a. compounded monthly and withdraws R25 000 at the end of every month. How many full withdrawals can she make?

8.4 [3]

A loan of R180 000 is taken at 13,5% p.a. compounded monthly. It is repaid with 48 equal monthly payments, starting one month after the loan is granted. Calculate the balance outstanding immediately after the 20th payment.

Question 9

Given $f (x) = - x^{3} + 3 x^{2} + 9 x - 5$ .

9.1 [4]

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

9.2 [2]

Determine the coordinates of the point of inflection.

9.3 [3]

Determine the equation of the tangent to $f$ at its point of inflection.

9.4 [3]

For which values of $x$ is $f$ increasing, and for which values is $f^{'} (x) < 9$ ?

9.5 [3]

The graph of $f$ is shifted vertically to give $y = f (x) + k$ . Determine the values of $k$ for which the new graph has exactly two $x$ -intercepts.

Question 10

10.1 [5]

Solve for $x$ : $(x^{2} - x)^{2} - 8 (x^{2} - x) + 12 = 0$

10.2 [4]

Solve for $x$ : $3^{2 x} - 4 \cdot 3^{x} + 3 = 0$

10.3 [4]

Prove that the roots of $x^{2} - (k + 3) x + 3 k = 0$ are real for all real values of $k$ , and rational when $k$ is rational.

Question 11

11.1 [4]

How many terms of $2 + 7 + 12 + \dots$ must be added to give 1 428?

11.2 [6]

The second term of a convergent geometric series is 12 and its sum to infinity is 64. Determine the possible values of the first term and the common ratio.

11.3 [4]

How many terms of $3 + 6 + 12 + \dots$ must be added for the sum to exceed 10 000?

Question 12

12.1 [7]

A closed cylindrical tin must hold 1 000 cm³. Determine the radius that uses the least metal, and calculate the minimum surface area. Give answers to two decimal places.