Explanation

(a) (i) \(3 * \frac{2}{5} = \frac{3 + \frac{2}{5}}{2}\)

= \(\frac{\frac{17}{5}}{2}\)

= \(\frac{17}{10}\)

(ii) \(8 * y = 8\frac{1}{4}\)

\(\frac{8 + y}{2} = \frac{33}{4}\)

\(66 = 32 + 4y \implies 4y = 66 - 32 = 34\)

\(y = \frac{34}{4} = 8.5\)

(b) \(\overrightarrow{AP} = \frac{1}{2}(\overrightarrow{AB})\)

= \(\frac{1}{2} \begin{pmatrix} -4 \\ 6 \end{pmatrix}\)

= \(\begin{pmatrix} -2 \\ 3 \end{pmatrix}\)

Midpoint = \(\overrightarrow{AP} + \overrightarrow{PC} = \overrightarrow{AC}\)

\(\overrightarrow{PC} = \overrightarrow{AC} - \overrightarrow{AP}\)

= \(\begin{pmatrix} 3 \\ -8 \end{pmatrix} - \begin{pmatrix} -2 \\ 3 \end{pmatrix}\)

= \(\begin{pmatrix} 5 \\ -11 \end{pmatrix}\)

\(\overrightarrow{CP} = -1 \times \begin{pmatrix} 5 \\ -11 \end{pmatrix}\)

= \(\begin{pmatrix} -5 \\ 11 \end{pmatrix}\)

Explanation

(a) \(s = t^{2} - 6t + 5\)

Scale : On time- axis, 2 cm rep 2 s

On displacement, 2 cm rep 2m.

(b)(i) Velocity = \(\frac{\mathrm d s}{\mathrm d t} = Gradient\)

Velocity = 0 m/s when t = 3s

(ii) Average velocity = \(\frac{\text{total costume}}{\text{total time}}\)

\(\frac{5 + 4 + (4 - 3)}{4}\)

= \(\frac{10}{4} = 2.5 m/s\).

(iii) Total distance covered in the interval \(0 \leq x \leq 5\)

= 5 + 4 + 4 = 13m.

Correct Answer: B. 50 sin 54°

Correct Answer: A. 58.5%

Explanation

Number of students that scoredat most 5 marks = 12 + 1 + 9 + 6 + 3 = 31

% of student = \(\frac{31}{53} \times 100%\) = 58.5%

Correct Answer: D. \(6\frac{2}{3}\)

Correct Answer: C. 8π cm²

Explanation

a. the diagram above solution to question a.

bi. Using the sine rule

\(\frac{sinP}{XY} = \frac{sinY}{PX}\)

= \(\frac{sin93°}{180} = \frac{sinY}{60}\)

cross multiply

\(sinY\times180 = sin93°\times60\)

sinY = 0.3329

Y = \(sin^{-1}0.3329\)

Y = 19.44°

the angle between 30° and 19.44° at Y = 30° - 19.44° = 10.56°

Therefore, the bearing X from Y = 360° - 10.56° = 349°( to 3 sig. figure)

bii. Using cosine rule

Let a = |PY|, b = |PX| and c = |XY|

\(a^2 = b^2 + c^2 - 2bcCos A\) where A = 67.56°

\(a^2 = 60^2 + 180^2 - 2(60)(180)cos 67.56^0\)

\(a2 = 3600 + 32400 - 8245.06\)

\(a^2 = 27754.94^0\)

a = \(\sqrt{27754.94}\)

a = 166.6 m

∴ |PY| = 167 m (to 3 s.f)

Correct Answer: D. \(\frac{-7}{6}\)

Explanation

Expanding \((x+1)(x-2) = x^{2} - 2x + x - 2 = x^{2} - x - 2\)

\(\int_{-1}^{0} (x^{2} - x - 2) \mathrm{d}x = [\frac{x^{3}}{3} - \frac{x^{2}}{2} - 2x]_{-1}^{0}\)

= \([\frac{0}{3} - \frac{0}{2} - 2\times0 - (\frac{-1^{3}}{3} - \frac{-1^{2}}{2} - 2\times-1)]\)

= \(0 + \frac{1}{3} + \frac{1}{2} - 2 = \frac{-7}{6}\)

Note: This can also be solved using integration by parts.

\(\int uv \mathrm{d}x = u\int v \mathrm{d}x - \int u'(\int v \mathrm{d}x)\mathrm{d}x\).

Correct Answer: C. -3

Explanation

\(y = 4x^{3} + kx^{2} - 6x + 4\)

\(\frac{\mathrm d y}{\mathrm d x} = 12x^{2} + 2kx - 6\)

At P(1, m)

\(\frac{\mathrm d y}{\mathrm d x} = 12 + 2k - 6 = 0\) (parallel to the x- axis)

\(6 + 2k = 0 \implies k = -3\)

Correct Answer: D. \(\frac{4x - 5y + xy}{2xy}\)

Explanation

\(\frac{3x - y}{xy} - \frac{2x + 3y}{2xy} + \frac{1}{2}\)

= \(\frac{2(3x - y) - 1(2x + 3y) + xy}{2xy}\)

= \(\frac{6x - 2y - 2x - 3y + xy}{2xy}\)

= \(\frac{4x - 5y + xy}{2xy}\)