IEB · Grade 11 · Mathematics IEB-style Grade 11 Mathematics Paper 1: Practice Set 2
In the print dialog, choose Save as PDF. IEB-style Grade 11 Mathematics Paper 1: Practice Set 2 Question 1 1.1.1 [3] Simplify: $\frac{3 ^{n + 2} \cdot 9 ^{n}}{2 7 ^{n}}$ 1.1.2 [2] Expand and simplify: $(3 + 2)^{2}$ 1.1.3 [2] Rationalise the denominator: $\frac{6}{3}$ 1.1.4 [3] Simplify without a calculator: $\frac{12 + 75}{27}$ 1.2.1 [2] Solve for $x$ : $x^{2} = 7 x$ 1.2.2 [3] Solve for $x$ : $(x - 2) (x + 5) = 8$ 1.2.3 [4] Solve by completing the square, correct to two decimal places: $2 x^{2} + 12 x + 5 = 0$ 1.3 [4] Solve for $x$ and $y$ : $y = 2 x - 3$ and $y = x^{2} - 2 x$ Question 2 2.1 [4] Given the pattern $3; 7; 13; 21; \dots$ Determine $T_{n}$ . 2.2 [1] Calculate $T_{30}$ . 2.3 [3] Dots are arranged in rectangles. Figures 1 to 4 have $2; 6; 12; 20$ dots. Determine a formula for the number of dots in figure $n$ . 2.4 [2] Which figure has 420 dots? Question 3 A parabola $f$ cuts the $x$ -axis at $(- 1; 0)$ and $(5; 0)$ and the $y$ -axis at $(0; - 5)$ . 3.1 [3] Show that $f (x) = x^{2} - 4 x - 5$ . 3.2 [3] Write $f$ in the form $a (x + p)^{2} + q$ and write down the turning point. 3.3 [2] Calculate the average gradient of $f$ between $x = 2$ and $x = 5$ . 3.4 [3] For which values of $x$ is $f$ decreasing, and for which values is $f (x) \geq 7$ ? Question 4 $g (x) = a \cdot b^{x} + q$ has asymptote $y = - 4$ and passes through $(0; - 2)$ and $(1; 0)$ . 4.1 [4] Determine $a$ , $b$ and $q$ . 4.2 [3] Write down the range of $g$ and the equation of the reflection of $g$ in the $x$ -axis. 4.3 [3] For which values of $x$ is $g (x) > 12$ ? 4.4 [2] Calculate the average gradient of $g$ between $x = 0$ and $x = 2$ . 4.5 [3] Sketch $g$ , showing the asymptote and intercepts. Question 5 5.1 [3] Two fair dice are rolled. Calculate the probability that the total is 8. 5.2 [4] $A$ and $B$ are mutually exclusive, with $P (A) = 0, 35$ and $P (not B) = 0, 45$ . Calculate $P (A or B)$ and $P (neither A nor B)$ . 5.3 [6] Sipho studies for a test with probability 0,7. If he studies, he passes with probability 0,8. If he does not study, he passes with probability 0,4. (a) Calculate the probability that he passes. (4) (b) Calculate the probability that he studied but failed. (2) Question 6 A hyperbola $y = \frac{a}{x + p} + q$ has asymptotes $x = 1$ and $y = - 2$ , and passes through $(3; 0)$ . 6.1 [4] Show that the equation is $y = \frac{4}{x - 1} - 2$ . 6.2 [1] Write down the $y$ -intercept. 6.3 [3] Determine the equations of the axes of symmetry. 6.4 [4] Determine the points where the line $y = x - 3$ meets the hyperbola. 6.5 [3] For which values of $x$ is $\frac{4}{x - 1} - 2 > 0$ ? Question 7 7.1 [3] A town of 120 000 people is shrinking by 1,5% per year. Calculate the population after 8 years. 7.2 [3] A machine bought for R350 000 has a book value of R140 000 after 6 years on the straight-line method. Calculate the annual rate of depreciation. 7.3 [3] Calculate the nominal annual rate, compounded half-yearly, that is equivalent to an effective rate of 11% p.a. 7.4 [6] Lindo needs R80 000 in 4 years' time. She deposits an amount $x$ now and the same amount again after 2 years. Interest is 8% p.a. compounded quarterly. Calculate $x$ . Question 8 8.1 [3] In a group of 60 learners, $2 x$ are in the choir ( $A$ ) and $x + 10$ are in the drama club ( $B$ ). 6 are in both, and 8 are in neither. Calculate $x$ . 8.2 [2] Calculate $P (A only)$ . 8.3 [2] Are $A$ and $B$ independent? Show calculations. Question 9 9.1 [5] Solve for $x$ : $(x^{2} + 2 x)^{2} - 11 (x^{2} + 2 x) + 24 = 0$ 9.2 [4] Solve for $x$ : $x + x - 6 = 0$ 9.3 [4] For which values of $m$ is the line $y = m x - 4$ a tangent to the parabola $y = x^{2}$ ? 9.4 [4] Solve for $x$ and $y$ : $3^{x} \cdot 9^{y} = 81$ and $x - y = 1$ 9.5 [5] Show that $x^{2} - 6 x + 11 \geq 2$ for all real values of $x$ . Hence determine the maximum value of $\frac{4}{x ^{2} - 6 x + 11}$ and the value of $x$ at which it occurs. Question 10 10.1 [3] For the pattern $3; 7; 13; 21; \dots$ , between which two consecutive terms is the difference 60? 10.2 [3] The dot pattern $2; 6; 12; 20; \dots$ has $T_{n} = n^{2} + n$ . Show that no figure has exactly 100 dots. 10.3 [6] Two savings plans: Plan A: R50 every week. Plan B: R2 in week 1, R4 in week 2, R6 in week 3, and so on. After how many weeks is the total in Plan B greater than in Plan A? 10.4 [3] The pattern $3; 7; 13; 21; \dots$ has $T_{n} = n^{2} + n + 1$ . Prove that every term of the pattern is odd. Question 11 The hyperbola $y = \frac{4}{x - 1} - 2$ has asymptotes $x = 1$ and $y = - 2$ . 11.1 [4] For which values of $k$ does the line $y = - x + k$ not meet the hyperbola?
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IEB-style Grade 11 Mathematics Paper 1: Practice Set 2

Question 1

1.1.1 [3]

Simplify: $\frac{3 ^{n + 2} \cdot 9 ^{n}}{2 7 ^{n}}$

1.1.2 [2]

Expand and simplify: $(3 + 2)^{2}$

1.1.3 [2]

Rationalise the denominator: $\frac{6}{3}$

1.1.4 [3]

Simplify without a calculator: $\frac{12 + 75}{27}$

1.2.1 [2]

Solve for $x$ : $x^{2} = 7 x$

1.2.2 [3]

Solve for $x$ : $(x - 2) (x + 5) = 8$

1.2.3 [4]

Solve by completing the square, correct to two decimal places: $2 x^{2} + 12 x + 5 = 0$

1.3 [4]

Solve for $x$ and $y$ : $y = 2 x - 3$ and $y = x^{2} - 2 x$

Question 2

2.1 [4]

Given the pattern $3; 7; 13; 21; \dots$ Determine $T_{n}$ .

2.2 [1]

Calculate $T_{30}$ .

2.3 [3]

Dots are arranged in rectangles. Figures 1 to 4 have $2; 6; 12; 20$ dots. Determine a formula for the number of dots in figure $n$ .

2.4 [2]

Which figure has 420 dots?

Question 3

A parabola $f$ cuts the $x$ -axis at $(- 1; 0)$ and $(5; 0)$ and the $y$ -axis at $(0; - 5)$ .

3.1 [3]

Show that $f (x) = x^{2} - 4 x - 5$ .

3.2 [3]

Write $f$ in the form $a (x + p)^{2} + q$ and write down the turning point.

3.3 [2]

Calculate the average gradient of $f$ between $x = 2$ and $x = 5$ .

3.4 [3]

For which values of $x$ is $f$ decreasing, and for which values is $f (x) \geq 7$ ?

Question 4

$g (x) = a \cdot b^{x} + q$ has asymptote $y = - 4$ and passes through $(0; - 2)$ and $(1; 0)$ .

4.1 [4]

Determine $a$ , $b$ and $q$ .

4.2 [3]

Write down the range of $g$ and the equation of the reflection of $g$ in the $x$ -axis.

4.3 [3]

For which values of $x$ is $g (x) > 12$ ?

4.4 [2]

Calculate the average gradient of $g$ between $x = 0$ and $x = 2$ .

4.5 [3]

Sketch $g$ , showing the asymptote and intercepts.

Question 5

5.1 [3]

Two fair dice are rolled. Calculate the probability that the total is 8.

5.2 [4]

$A$ and $B$ are mutually exclusive, with $P (A) = 0, 35$ and $P (not B) = 0, 45$ . Calculate $P (A or B)$ and $P (neither A nor B)$ .

5.3 [6]

Sipho studies for a test with probability 0,7. If he studies, he passes with probability 0,8. If he does not study, he passes with probability 0,4.

(a) Calculate the probability that he passes. (4)

(b) Calculate the probability that he studied but failed. (2)

Question 6

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

6.1 [4]

Show that the equation is $y = \frac{4}{x - 1} - 2$ .

6.2 [1]

Write down the $y$ -intercept.

6.3 [3]

Determine the equations of the axes of symmetry.

6.4 [4]

Determine the points where the line $y = x - 3$ meets the hyperbola.

6.5 [3]

For which values of $x$ is $\frac{4}{x - 1} - 2 > 0$ ?

Question 7

7.1 [3]

A town of 120 000 people is shrinking by 1,5% per year. Calculate the population after 8 years.

7.2 [3]

A machine bought for R350 000 has a book value of R140 000 after 6 years on the straight-line method. Calculate the annual rate of depreciation.

7.3 [3]

Calculate the nominal annual rate, compounded half-yearly, that is equivalent to an effective rate of 11% p.a.

7.4 [6]

Lindo needs R80 000 in 4 years' time. She deposits an amount $x$ now and the same amount again after 2 years. Interest is 8% p.a. compounded quarterly. Calculate $x$ .

Question 8

8.1 [3]

In a group of 60 learners, $2 x$ are in the choir ( $A$ ) and $x + 10$ are in the drama club ( $B$ ). 6 are in both, and 8 are in neither. Calculate $x$ .

8.2 [2]

Calculate $P (A only)$ .

8.3 [2]

Are $A$ and $B$ independent? Show calculations.

Question 9

9.1 [5]

Solve for $x$ : $(x^{2} + 2 x)^{2} - 11 (x^{2} + 2 x) + 24 = 0$

9.2 [4]

Solve for $x$ : $x + x - 6 = 0$

9.3 [4]

For which values of $m$ is the line $y = m x - 4$ a tangent to the parabola $y = x^{2}$ ?

9.4 [4]

Solve for $x$ and $y$ : $3^{x} \cdot 9^{y} = 81$ and $x - y = 1$

9.5 [5]

Show that $x^{2} - 6 x + 11 \geq 2$ for all real values of $x$ . Hence determine the maximum value of $\frac{4}{x ^{2} - 6 x + 11}$ and the value of $x$ at which it occurs.

Question 10

10.1 [3]

For the pattern $3; 7; 13; 21; \dots$ , between which two consecutive terms is the difference 60?

10.2 [3]

The dot pattern $2; 6; 12; 20; \dots$ has $T_{n} = n^{2} + n$ . Show that no figure has exactly 100 dots.

10.3 [6]

Two savings plans:

Plan A: R50 every week.
Plan B: R2 in week 1, R4 in week 2, R6 in week 3, and so on.

After how many weeks is the total in Plan B greater than in Plan A?

10.4 [3]

The pattern $3; 7; 13; 21; \dots$ has $T_{n} = n^{2} + n + 1$ . Prove that every term of the pattern is odd.

Question 11

The hyperbola $y = \frac{4}{x - 1} - 2$ has asymptotes $x = 1$ and $y = - 2$ .

11.1 [4]

For which values of $k$ does the line $y = - x + k$ not meet the hyperbola?