(i)

y = x² + ax + b

The grath cuts the x at y = 0

When x = 3, y = 0

Substitute the values into the relation.

3² + 3 x a + b = 0

9 + 3a + b = 0

3a + b = -9 ----- eqn 1

The grath cuts the y axis at x = 0

When y = 6, x = 0

Substitute the values into the relation.

0² + 0 x a + b = 6

b = 6

Substitute b = 6 into ---- eqn 1 to find a

3a + 6 = -9

3a = -9 - 6

3a = -15

Divide both sides by 3

a = $\frac{-15}{3}$ = -5

(ii)

x² + ax + b = 0

a = -5 and b = 6

x² - 5x + 6 = 0

Notes:

1. Split b(-5) such that two numbers when you multiply you will get c (6) and when you add you will get b (-5)

2. The numbers are -3 and - 2

3. -5x = -3x - 2x = 0

x²-3x - 2x + 6 = 0

x(x - 3) - 2(x - 3) = 0

(x - 3)(x - 2) = 0

When x - 3 = 0, x = 3

When x - 2 = 0, x = 2

x = 2 or 3

(a)

(b)

How to find the equation of a line joining the two points

1. Find the gradient of the line joining the points
2. Use the point (x,y) with any of the point to calculate the gradient and equate it to the calculated gradient in step 1
3. Make y the subject of the equation in step 2

Gradient = $\frac{\mathrm{Change\; in\; Y}}{\mathrm{Change\; in\; X}}$ = $\frac{{\mathrm{Y}}_2-{\mathrm{Y}}_1}{{\mathrm{X}}_2-{\mathrm{X}}_1}$

M = (3,1)
y₁ = 1
x₁ = 3

M₂ = (6,-1)
y₂ = -1
x₂ = 6

Gradient = $\frac{\mathrm{-1\; -\; 1}}{\mathrm{6\; -\; 3}}$

Gradient = $\frac{-2}{3}$

Using the point (3,1)

$\frac{\mathrm{y\; -\; 1}}{\mathrm{x\; -\; 3}}$ = $\frac{-2}{3}$

Cross multiply

3(y - 1) = -2(x - 3)
3y -3 = -2x + 6
3y = -2x + 6 + 3
3y = -2x + 9

Divide both sides by 3

y = $\frac{-2}{3}$x + $\frac{9}{3}$

y = $\frac{-2}{3}$x + 3

(a)

Scale = 2 cm to 1 unit on the x axis and 2 cm to 2 units on the y axis.

(b)

Use known points on the curve and solve the values of a, b and c from the simultaneous equations.

y = ax² + bx + c

Using the point (1,0)

0 = a(1)² + b(1) + c

0 = a + b + c ---- eqn 1

Using the point (0,5)

5 = a(0)² + b(0) + c

5 = 0 + 0 + c

c = 5

Using the point (5,0)

0 = a(5)² + b(5) + c

0 = 25a + 5b + c --- eqn 2

Substituting c = 5 into --- eqn 1

0 = a + b + 5

a = -5 - b ---- eqn 3

Substituting a = -5 - b into --eqn 2

0 = 25(-5 - b) + 5b + 5

-5 = -125 - 25b + 5b

-5 + 125 = -20b

120 = -20b

Divide both sides by -20

b = $\frac{120}{-20}$ = -6

Substituting b = -6 into ---eqn 3

a = -5 - (-6)

a = -5 + 6

a = 1

(c)

x = -0.3 or 6.3

(d)

Minimum point = (3,-4)

(e)

1 ≤ x ≤ 5

(a)

y = cos x - 3 sin x

When x = 20°

y = cos 20° - 3 sin 20° = -0.08636780919 ≈ -0.1

When x = 40°

y = cos 40° - 3 sin 40° = -1.16231838594 ≈ -1.2

When x = 80°

y = cos 80° - 3 sin 80° = -2.78077508137 ≈ -2.8

When x = 100°

y = cos 100° - 3 sin 100° = -3.1280714367 ≈ -3.1

When x = 120°

y = cos 120° - 3 sin 120° = -3.09807621135 ≈ -3.1

When x = 140°

y = cos 140° - 3 sin 140° = -2.69440727218 ≈ -2.7

When x = 180°

y = cos 180° - 3 sin 180° = -1.0

(b)

Note:

1. Each small grid on the y axis = $\frac{1}{20}$ = 0.05

2. 0.1 = $\frac{0.1}{0.05}$ = 2 small grids from 0

3. 1.2 = $\frac{1.2}{0.05}$ = 24 small grids from 0

4. 2.1 = $\frac{2.1}{0.05}$ = 42 small grids from 0

5. 2.8 = $\frac{2.8}{0.05}$ = 56 small grids from 0

6. 3.1 = $\frac{3.1}{0.05}$ = 62 small grids from 0

7. 2.7 = $\frac{2.7}{0.05}$ = 54 small grids from 0

(c)

(a)

y = 4 + 3x - 2x²

When x = -4

y = 4 + 3 x (-4) - 2 x (-4)²

y = 4 - 12 - 2 x 16

y = -8 - 32

y = -40

When x = -3

y = 4 + 3 x (-3) - 2 x (-3)²

y = 4 - 9 - 2 x 9

y = -5 - 18

y = -23

When x = -1

y = 4 + 3 x (-1) - 2 x (-1)²

y = 4 - 3 - 2 x 1

y = -1 - 2

y = -3

When x = 1

y = 4 + 3 x (1) - 2 x (1)²

y = 4 + 3 - 2 x 1

y = 7 - 2

y = 5

When x = 3

y = 4 + 3 x (3) - 2 x (3)²

y = 4 + 9 - 2 x 9

y = 13 - 18

y = -5

When x = 4

y = 4 + 3 x (4) - 2 x (4)²

y = 4 + 12 - 2 x 16

y = 16 - 32

y = -16

(b)

Notes:

1. Each small grid on the y axis = $\frac{5}{10}$ = 0.5

2. 23 = $\frac{23}{0.5}$ = 46 small grids from 0

2. 3 = $\frac{3}{0.5}$ = 6 small grids from 0

4. 4 = $\frac{4}{0.5}$ = 8 small grids from 0

5. 2 = $\frac{2}{0.5}$ = 4 small grids from 0

6. 16 = $\frac{16}{0.5}$ = 32 small grids from 0

(c)

(a)

y = 7cos x - 3sin x

When x = 30°

y = 7cos 30° - 3sin 30° = 4.6

When x = 60°

y = 7cos 60° - 3sin 60° = 0.9

When x = 120°

y = 7cos 120° - 3sin 120° = -6.1

When x = 150°

y = 7cos 150° - 3sin 150° = -7.6

(b)

Notes:

1. Each grid on the x-axis = $\frac{\mathrm{30°}}{10}$ = 3°

2. Each grid on the y-axis = $\frac{2}{10}$ = 0.2

3. Number of grids from 0 = $\frac{\mathrm{Value}}{\mathrm{Grid\; value}}$

4. Number of grids from 0 (4.9) = $\frac{4.9}{0.2}$ = 24.5 (Between 24 and 25 grids)

5. Number of grids from 0 (0.9) = $\frac{0.9}{0.2}$ = 4.5 (Between 4 and 5 grids)

6. Number of grids from 0 (6.1) = $\frac{6.1}{0.2}$ = 30.5 (Between 30 and 31 grids)

7. Number of grids from 0 (7.6) = $\frac{7.6}{0.2}$ = 38 grids

(c)

To solve for an equation using a graph, express the given equation in the form of the equation that was used to draw the graph and determine where both equation intersects. The value of x at where both equations meet/intersect is the solution.

(a)

When x = 0°

5sin 0° + 9cos 0° = 9.0

When x = 60°

5sin 60° + 9cos 60° = 8.8

When x = 90°

5sin 90° + 9cos 90° = 5

When x = 150°

5sin 150° + 9cos 150° = -5.3

(b)

Notes:

1. Each grid on an axis = $\frac{\mathrm{Interval\; on\; the\; axis}}{10}$

2. Each grid on the x-axis = $\frac{\mathrm{30°}}{10}$ = 3°

3. Each grid on the y-axis = $\frac{2}{10}$ = 0.2

4. Number of grids from 0 = $\frac{\mathrm{Value}}{\mathrm{Grid\; value}}$

5. Number of grids from 0 (10.3) = $\frac{10.3}{0.2}$ = 51.5 (Between 51 and 52 grids)

6. Number of grids from 0 (8.8) = $\frac{8.8}{0.2}$ = 44 grids

7. Number of grids from 0 (5.0) = $\frac{5.0}{0.2}$ = 25 grids

8. Number of grids from 0 (0.2) = $\frac{0.2}{0.2}$ = 1 grid

9. Number of grids from 0 (5.3) = $\frac{5.3}{0.2}$ = 26.5 grids (Between 26 and 27)

(c)

(d)

When x = 45°, y = 9.8 ± 2

Note:

1. Number of grids for 45 = $\frac{45}{3}$ = 15 grids from 0

The number of grids that the x = 45° cuts the y-axis is 49 from 0

Value per grid on the x is 0.2

The y-axis value = 49 x 0.2 = 9.8

(a)

(b)

(c)

A₁$\left(\begin{array}{c}2\\ -1\end{array}\right)$ → A₃$\left(\begin{array}{c}-1\\ 2\end{array}\right)$

B₁$\left(\begin{array}{c}1\\ -4\end{array}\right)$ → B₃$\left(\begin{array}{c}-4\\ 1\end{array}\right)$

C₁$\left(\begin{array}{c}-1\\ -2\end{array}\right)$ → C₃$\left(\begin{array}{c}-2\\ -1\end{array}\right)$

Hence the single transformation that maps ∆ A₁B₁C₁ onto ∆ A₃B₃C₃ is reflection in the line y = x or y - x = 0.

(a)

y = 2(x + 2)² - 3

When x = -5

y = 2(-5 + 2)² - 3

y = 2(-3)² - 3

y = 2 x 9 - 3

y = 18 - 3

y = 15

When x = -4

y = 2(-4 + 2)² - 3

y = 2(-2)² - 3

y = 2 x 4 - 3

y = 8 - 3

y = 5

When x = -3

y = 2(-3 + 2)² - 3

y = 2(-1)² - 3

y = 2 x 1 - 3

y = 2 - 3

y = -1

When x = -2

y = 2(-2 + 2)² - 3

y = 2(0)² - 3

y = 2 x 0 - 3

y = 0 - 3

y = -3

When x = -1

y = 2(-1 + 2)² - 3

y = 2(1)² - 3

y = 2 x 1 - 3

y = 2 - 3

y = -1

When x = 0

y = 2(0 + 2)² - 3

y = 2(2)² - 3

y = 2 x 4 - 3

y = 8 - 3

y = 5

When x = 1

y = 2(1 + 2)² - 3

y = 2(3)² - 3

y = 2 x 9 - 3

y = 18 - 3

y = 15

When x = 2

y = 2(2 + 2)² - 3

y = 2(4)² - 3

y = 2 x 16 - 3

y = 32 - 3

y = 29

(b)

Notes:

1. Each grid value = $\frac{\mathrm{Interval\; on\; the\; axis}}{\mathrm{Number\; of\; grids\; per\; interval}}$

2. Each grid value on the y-axis = $\frac{5}{10}$ = 0.5

3. Number of grids = $\frac{\mathrm{Point}}{\mathrm{Grid\; value}}$

4. Number of grids on the y-axis for 1 = $\frac{1}{0.5}$ = 2

5. Number of grids on the y-axis for 3 = $\frac{3}{0.5}$ = 6

6. Number of grids on the y-axis for 29 = $\frac{29}{0.5}$ = 58

(c)

Notes:

1. Each grid value = $\frac{\mathrm{Interval\; on\; the\; axis}}{\mathrm{Number\; of\; grids\; per\; interval}}$

2. Each grid value on the x-axis = $\frac{1}{10}$ = 0.1

(d)

The values of x for which y is increasing from the graph is −2 < x ≤ 2

(a)

(b)

0G→ = 0F→ + FG→

0G→ = $\left(\begin{array}{c}-1\\ 5\end{array}\right)$ + $\left(\begin{array}{c}1\\ 3\end{array}\right)$

0G→ = $\left(\begin{array}{c}\mathrm{-1\; +\; 1}\\ \mathrm{5\; +\; 3}\end{array}\right)$

0G→ = $\left(\begin{array}{c}0\\ 8\end{array}\right)$

G = (0, 8)

0H→ = 0G→ + GH→

0H→ = $\left(\begin{array}{c}0\\ 8\end{array}\right)$ + $\left(\begin{array}{c}3\\ -1\end{array}\right)$

0H→ = $\left(\begin{array}{c}\mathrm{0\; +\; 3}\\ \mathrm{8\; +\; -1}\end{array}\right)$

0H→ = $\left(\begin{array}{c}3\\ 7\end{array}\right)$

H = (3,7)

(c)

(d)

E₁H₁ = 5 units

G₁P = 3.6 units

Height = 3 units

Area of E₁H₁G₁P = $\frac{1}{2}$(5 + 3.6) x 3 = 12.9 square units

(a)

$\frac{\mathrm{2a\; +\; -\; 4}}{2}$ = 1

$\frac{\mathrm{2a\; -\; 4}}{2}$ = 1

Multiply both sides by the denominator, 2.

2a - 4 = 1 x 2

2a - 4 = 2

2a = 2 + 4

2a = 6

Divide both sides by 2

a = $\frac{6}{2}$

a = 3

$\frac{\mathrm{3\; +\; b}}{2}$ = 2a - 1

Multiply both sides by the denominator, 2

3 + b = 2 x 2a - 1 x 2

3 + b = 4a - 2

Substituting the value of a

a = 3

3 + b = 4 x 3 - 2

3 + b = 12 - 2

3 + b = 10

b = 10 - 3

b = 7

(b)

Point Y = (2 x 3, 3)

Point Y = (6, 3)

Point Z = (-4, 7)

The points are Y(6, 3) and Z(-4, 7)

|YZ| = $\sqrt{{(-4-6)}^2+{(7-3)}^2}$

|YZ| = $\sqrt{\mathrm{100\; +\; 16}}$

|YZ| = $\sqrt{116}$

|YZ| = $\sqrt{\mathrm{4\; x\; 29}}$

|YZ| = $\sqrt{4}$ x $\sqrt{29}$

|YZ| = 2 x $\sqrt{29}$

|YZ| = 2$\sqrt{29}$ units

|YZ| = 10.77 units

Note: either 2$\sqrt{29}$ units or 10.77 units is awarded full mark