## Further Mathematics | Study Mode

Question 1
A
x < -4 or x > 2
B
x < -2 or x > 4
C
-2 < x < 4
D
-4 < x < 2
Explanation.

No official Explanation yet!

Question 2
A
-1
B
-2
C
-3
D
-4
Explanation.

$$x + 3 = 0 \implies x = -3$$

Using remainder theorem, if x + 3 is a factor, f(-3) = 0.

$$f(-3) = (-3)^{3} + 3(-3)^{2} + n(-3) - 12 = 0$$

$$-27 + 27 - 3n - 12 = 0 \implies -3n = 12$$

$$n = -4$$

Question 3
A

-4

B

-1

C

3

D

4

Explanation.

$$y = 1 - 3x + 2x^{3}$$

$$\frac{\mathrm d y}{\mathrm d x} = -3 + 6x^{2}$$

At (1, 0), $$\frac{\mathrm d y}{\mathrm d x} = -3 + 6(1^{2}) = -3 + 6 = 3$$

$$y = mx - 3 \implies \frac{\mathrm d y}{\mathrm d x} = m = 3$$ (Tangent with equal gradient)

Question 4
A
$$x^{2} + y^{2} - 4x - 6y - 12 = 0$$
B
$$x^{2} + y^{2} - 4x + 6y - 12 = 0$$
C
$$x^{2} + y^{2} + 4x + 6y - 12 = 0$$
D
$$x^{2} + y^{2} + 4x - 6y - 12 = 0$$
Explanation.

Equation of a circle with centre coordinates (a, b) : $$(x - a)^{2} + (y - b)^{2} = r^{2}$$

Area of circle = $$\pi r^{2} = 25\pi cm^{2} \implies r^{2} = 25$$

$$\therefore r = 5cm$$

(a, b) = (-2, 3)

Equation: $$(x - (-2))^{2} + (y - 3)^{2} = 5^{2}$$

$$x^{2} + 4x + 4 + y^{2} - 6y + 9 = 25 \implies x^{2} + y^{2} + 4x - 6y + 13 - 25 = 0$$

= $$x^{2} + y^{2} + 4x - 6y - 12 = 0$$

Question 5
A
$$- \sqrt{3}$$
B
$$-\frac{\sqrt{3}}{2}$$
C
$$\frac{\sqrt{3}}{2}$$
D
$$\sqrt{3}$$
Explanation.

$$\sin \theta = \frac{\sqrt{3}}{2} \implies opp = \sqrt{3}; hyp= 2$$

$$adj^{2} = 2^{2} - (\sqrt{3})^{2} = 1 \implies adj = 1$$

$$\cos \theta = \frac{1}{2}$$

$$\sin 2\theta = \sin (180 - \theta) = \sin \theta = \frac{\sqrt{3}}{2}$$

$$\cos 2\theta = \cos (180 - \theta) = -\cos \theta = -\frac{1}{2}$$

$$\tan 2\theta = \frac{\sin 2\theta}{\cos 2\theta} = \frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}}$$

= $$- \sqrt{3}$$

Question 6
A
120
B
80
C
60
D
15
Explanation.

$$(2 + x)^{6}$$

$$x^{4} = ^{6}C_{2}(2^{2})(x^{4}) = 15 \times 4 = 60$$

Question 7
A
$$a * b = \frac{1}{a} + \frac{1}{b}$$
B
$$a * b = a + b - ab$$
C
$$a * b = 2a + 2b + ab$$
D
$$a * b = a - b + ab$$
Explanation.

All other options given are commutative i.e. $$a * b = b * a$$, except option D.

$$a * b = a - b + ab$$

$$b * a = b - a + ba$$

$$a - b = -(b - a) \neq b - a$$

Question 8
A
$$6 + \sqrt{7}$$
B
$$3 + \sqrt{7}$$
C
$$3 - \sqrt{7}$$
D
$$6 - \sqrt{7}$$
Explanation.

Rationalizing $$\frac{2}{3 - \sqrt{7}}$$ by multiplying through with $$3 + \sqrt{7}$$,

$$\frac{2}{3 - \sqrt{7}} \frac{(3 + \sqrt{7})}{(3 + \sqrt{7})} = \frac{6 + 2\sqrt{7}}{9 - 7}$$

= $$\frac{6 + 2\sqrt{7}}{2} = 3 + \sqrt{7}$$

Question 9
A
$$\frac{25}{4} - m$$
B
$$\frac{25}{4} - 2m$$
C
$$\frac{25}{4} + m$$
D
$$\frac{25}{4} + 2m$$
Explanation.

$$2x^{2} - 5x + m = 0$$

$$a = 2, b = -5, c = m$$

$$\alpha + \beta = \frac{-b}{a} = \frac{5}{2}$$

$$\alpha \beta = \frac{c}{a} = \frac{m}{2}$$

$$\alpha^{2} + \beta^{2} = (\alpha + \beta)^{2} - 2\alpha\beta$$

= $$(\frac{5}{2})^{2} - 2(\frac{m}{2})$$

= $$\frac{25}{4} - m$$

Question 10
A
B
-1
C
-2
D
-3
Explanation.

$$2^{x} = 0.125 = \frac{125}{1000}$$

$$2^{x} = \frac{1}{8} = 8^{-1}$$

$$2^{x} = 2^{-3}$$

$$x = -3$$

Question Map