Correct Answer: C. 54

Correct Answer: A. 7cm

Explanation

(a) \(\overrightarrow{AB} = \begin{pmatrix} 4 \\ 5 \end{pmatrix} = 4i + 5j\)

\(\overrightarrow{AC} = \overrightarrow{AB} + \overrightarrow{BC}\)

= \(\begin{pmatrix} 4 \\ 5 \end{pmatrix} + \begin{pmatrix} -3 \\ 5 \end{pmatrix}\)

= \(\begin{pmatrix} 1 \\ 10 \end{pmatrix}\)

(i) Let \(\alpha\) be the angle between \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\).

\(\overrightarrow{AB} \cdot \overrightarrow{AC} = |AB||AC| \cos \alpha\)

\((4i + 5j) \cdot (i + 10j) = (\sqrt{4^{2} + 5^{2}})(\sqrt{1^{2} + 10^{2}}) \cos \alpha\)

\(4 + 50 = (\sqrt{41})(\sqrt{101}) \cos \alpha\)

\(\cos \alpha = \frac{54}{(\sqrt{41})(\sqrt{101})}\)

\(\cos \alpha = 0.8392\)

\(\alpha = 32.94°\)

(ii) \(\overrightarrow{AB} - \overrightarrow{BC} = \begin{pmatrix} 4 \\ 5 \end{pmatrix} - \begin{pmatrix} -3 \\ 5 \end{pmatrix}\)

=\(\begin{pmatrix} 7 \\ 0 \end{pmatrix} = 7i\)

\(\overrightarrow{AB} - \overrightarrow{BC} = 7i\)

Unit vector in the direction of 7i = \(\frac{7i}{7} = i\)

(b) \(\overrightarrow{PQ} = \begin{pmatrix} 2 \\ 8 \end{pmatrix} ; \overrightarrow{PR} = \begin{pmatrix} 11 \\ -12 \end{pmatrix}\)

\(\overrightarrow{QR} = \overrightarrow{QP} + \overrightarrow{PR} \)

= \(-\overrightarrow{PQ} + \overrightarrow{PR}\)

= \(\begin{pmatrix} -2 \\ -8 \end{pmatrix} + \begin{pmatrix} 11 \\ -12 \end{pmatrix}\)

= \(\begin{pmatrix} 9 \\ -20 \end{pmatrix}\)

\(\frac{QM}{MR} = \frac{QP + PM}{MP + PR} = \frac{-PQ + PM}{-PM + PR}\)

\(\frac{QM}{MR} = \frac{3}{7}\)

\(7QM = 3MR\)

\(7(-PQ + PM) = 3(-PM + PR)\)

\(7(-PQ) + 7PM = 3(-PM) + 3(PR)\)

\(7(\begin{pmatrix} -2 \\ -8 \end{pmatrix} - 3(\begin{pmatrix} 11 \\ -12 \end{pmatrix} = -10(PM)\)

\(\begin{pmatrix} -47 \\ -20 \end{pmatrix} = -10(PM)\)

\(PM = \begin{pmatrix} 4.7 \\ 2 \end{pmatrix}\)

Correct Answer: C. 13

Correct Answer: C. 42°

Explanation

From the diagram, ∠ QPS = x^o (Corresponding angles)

Also, ∠ QPS = ∠ PSR(Alternate angles)

x = 42^o

Correct Answer: A. \(\frac{1}{2}\) m/s

Explanation

m1u1 + m2u2 = (m1 + m2)v

m1 = 18kg, m2 = 6kg, u1 = 4ms-1, u2 = -10m/s

18(4) + 6(-10) = (18+6)v

72 - 60 = 24v

12 = 24v

v = \(\frac{1}{2}\) m/s

Correct Answer: C. 15

Correct Answer: A. log10\(^{90}\)

Explanation

1 + 2log\(_{10}^3\)

= log\(_{10}^{10} + log_{10}^{3^2}\)

= log\(_{10}^{10} + log_{10}^{9}\)

= log\(_{10}^{10 \times 90}\) = log\(_{10}^{90}\)

Correct Answer: C. 4.0 cm

Explanation

165 = 2 x \(\frac{22}{7}\) x \(\frac{7}{2}\) x h + 2 x \(\frac{22}{7}\) x \(\frac{7}{2}\) x \(\frac{7}{2}\)

165 = 22h + 77

\(\frac{22h}{22} = \frac{88}{22}\)

height = 4cm

Correct Answer: C. 17/?2

Explanation

2x\(^3\) - 4x + 2y\(^2\) - 6y - 2 = 0

Divide through by 2: x\(^2\) - 2x + y\(^2\) -3y -1 = 0

x\(^2\) -2x + y\(^2\) - 3y = 1

x\(^2\) -2x + 1 + y\(^2\) - 3y + 9/4

= 1+ 1 + 9/4

= (x- 1)\(^2\) (y - 3/2)\(^2\)

= √[17/4]

r = √17/2