Correct Answer: C. (n + q)

Explanation

mn - nq - n\(^2\)

Correct Answer: B. 130^o

Explanation

Draw a line from P to Q

< PQS =< PRS (angle in the sam segment)

< PQS = 50^o

Also,< QSR =< QPR(angles in the segment)

< QPR = x^o

x + y + 5= = 180(angles in a triangle)

x + y = 180 - 50

x + y = 130^o

Explanation

(a) \(m = (6i + 8j) ; n = (-8i + \frac{7}{3}j)\)

(i) \(|m| = \sqrt{6^{2} + 8^{2}} = \sqrt{36 + 64}\)

= 10 units.

Direction of m: \(\tan \alpha = \frac{8}{6} = 1.333\)

\(\alpha = 53.13°\)

\(|n| = \sqrt{(-8)^{2} + (\frac{7}{3})^{2}} = \sqrt{64 + \frac{49}{9}}\)

= \(\sqrt{\frac{625}{9}} = \frac{25}{3} units\)

Direction of n :

\(\tan \beta = \frac{\frac{7}{3}}{-8}\)

= \(-\frac{7}{24} = -0.2917\)

\(\beta = \tan^{-1} (-0.2917) = -16.26°\)

(ii) Let the angle between m and n be \(\theta\).

\(m \cdot n = |m||n| \cos \theta\)

\(\cos \theta = \frac{m \cdot n}{|m||n|}\)

\(m \cdot n = (6i + 8j) \cdot (-8i + \frac{7}{3}j)\)

= \(-48 + \frac{56}{3}\)

= \(\frac{-88}{3}\)

\(\cos \theta = \frac{-\frac{88}{3}}{10 \times \frac{25}{3}}\)

= \(\frac{-88}{250}\)

\(\cos \theta = -0.352\)

\(\theta = \cos^{-1} (-0.352) = 110.61°\)

(b) For PQ to be perpendicular to RS, the angle between them should be 90°. That is, \(\cos \theta = 0\).

\(\overrightarrow{PQ} = \begin{pmatrix} 10 \\ 4 \end{pmatrix} - \begin{pmatrix} -2 \\ 3 \end{pmatrix} = \begin{pmatrix} 12 \\ 1 \end{pmatrix}\)

\(\overrightarrow{RS} = \begin{pmatrix} 4 \\ 0 \end{pmatrix} - \begin{pmatrix} 3 \\ 12 \end{pmatrix} = \begin{pmatrix} 1 \\ -12 \end{pmatrix}\)

\(|PQ| = \sqrt{12^{2} + 1^{2}} = \sqrt{144 + 1} = \sqrt{145}\)

\(|RS| = \sqrt{1^{2} + (-12)^{2}} = \sqrt{1 + 144} = \sqrt{145}\)

\(\overrightarrow{PQ} \cdot \overrightarrow{RS} = (12i + j) \cdot (i - 12j)\)

= \(0\)

\(\overrightarrow{PQ} \cdot \overrightarrow{RS} = |PQ||RS| \cos \theta\)

\(\cos \theta = \frac{0}{(\sqrt{145})(\sqrt{145})} = 0\)

Correct Answer: C. y = x² - x - 2

Correct Answer: B. 9.30 pm

Correct Answer: C. \(\frac{3}{8}\)

Explanation

ratio of boys to girls = 5:3

total = 5 + 3 = 8

Pr(girl) = \(\frac{3}{8}\)

Explanation

(a)(i)

(ii)

(b)(i) Those who scored less and equal to 32 marks = 75

Those who scored greater and equal to 74 = 508

Therefore those that scored between 32 and 74 = 508 - 75 = 433.

(ii) 18% of 550 = 99

18% failed, the number that passed = 550 - 99 = 451.

(iii) 8% passed with distinction, so 92% did not get distinction.

92% of 550 = 506.

Lowest mark for distinction = 74.

Correct Answer: (d) 1/8, 1/3, 3/7 and 5/9

Correct Answer: B. 60\(^o\)

Explanation

3\(x^o\) + 2\(x^o\) + 4\(x^o\) + 3\(x^o\) + 6\(x^o\) = 540\(x^o\)

\(\frac{18x^o}{18} = \frac{540^o}{18}\)

\(x^o\) = 30\(^o\)

Smallest angle = 2 x 30\(^o\) = 60\(^o\)

Correct Answer: A. 9