Correct Answer: A. 163.0

Explanation

∑fx = (160 * 4) + (161 * 6) + (162 * 3) + (163 * 7) + (164 * 8) + (165 * 9)

=640 + 966 + 486 + 1,141 + 1,312 + 1,485

= 6,030

∑f = 4 + 6 + 3 + 7 + 8 + 9

= 37

= \(\frac{∑fx}{∑f}\)

= \(\frac{6030}{37}\)

= 162.97 or 163

Explanation

(a) \(px - 2qx - 4qy + 2py\)

= \(x(p - 2q) + 2y(p - 2q)\)

= \((x + 2y)(p - 2q)\)

(b) \(U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\)

\(P = {1, 2, 4, 6, 10} ; Q = {2, 3, 6, 9}\)

\((P \cup Q) = {1, 2, 3, 4, 6, 9, 10}\)

\((P \cup Q)' = {5, 7, 8}\)

\(P' \cap Q' = {3, 5, 7, 8, 9} \cap {1, 4, 5, 7, 8, 10}\)

= \({5, 7, 8}\)

Hence, \((P \cup Q)' = P' \cap Q'\).

Correct Answer: B. \(\frac{-1}{5}, \frac{3}{2}\)

Explanation

The fraction \(\frac{ x^2 + 2 }{ 10x^2 - 13x - 3}\) is undefined when the denominator is equal to zero

\(then 10x^2 - 13x - 3 = 0\)

by factorisation, \(10x^2 - 13x - 3\) = 0 becomes \( 10x^2 - 15x +2x -3\) = 0

\(5x(2x - 3) + 1(2x - 3) = 0\)

\((5x + 1)(2x - 3) = 0\)

\(then, x = \frac{-1}{5}\) or \(\frac{3}{2}\)

Correct Answer: D. y = -3

Explanation

Let the straight line PQ = y

2x\(^2\) - 5x - 1 - y = 2x\(^2\) - 5x + 2

y = 2x\(^2\) - 5x - 1 - 2x\(^2\) + 5x - 2

y = -3

Correct Answer: d. 100

Explanation

(a) \(h = \frac{mt}{d(m + p)}\)

\(dh(m + p) = mt\)

\(dhm + dhp = mt \implies dhp = mt - dhm\)

\(dhp = m(t - dh) \implies m = \frac{dhp}{t - dh}\)

(b)

In the diagram above, < WXM = 90° (angle in a semicircle)

< WMX = 180° - (90° + 48°)

= 42°

< XMZ = 180° - 42° (angles on a straight line)

= 138°

< WYZ = 180° - 138° (opp. angles of a cyclic quadrilateral)

= 42°

(c)

(ii) From the table, the truth set of :

(I) \(3 \star n = 3; n = {5}\)

(II) \(n \star n = 3; n = { }\)

Correct Answer: A. 123000

Correct Answer: D. 6⁵/₁₂cm²

Explanation

Area of sector = \(\frac{\theta}{360°} \times \pi r^2\)

= \(\frac{60}{360} \times \frac{22}{7} \times 3.5 \times 3.5\)

= \(\frac{77}{12}\)

= \(6\frac{5}{12}\) cm\(^2\)

Explanation

(a) \(^{n + 1}C_{3} - ^{n - 1}C_{3}\)

= \(\frac{(n + 1)!}{3! (n + 1 - 3)!} - \frac{(n - 1)!}{3! (n - 1 - 3)!}\)

= \(\frac{(n + 1)!}{3! (n - 2)!} - \frac{(n - 1)!}{3! (n - 4)!}\)

= \(\frac{(n + 1)(n)(n - 1)(n - 2)!}{3! (n - 2)!} - \frac{(n - 1)(n - 2)(n - 3)(n - 4)!}{3! (n - 4)!}\)

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

= \(\frac{(n - 1)[(n + 1)(n) - (n - 2)(n - 3)]}{6}\)

= \(\frac{(n - 1)[n^{2} + n - (n^{2} - 5n + 6)}{6}\)

= \(\frac{(n - 1)[6n - 6]}{6}\)

= \(\frac{6(n - 1)^{2}}{6}\)

= \((n - 1)^{2}\)

(b) p(six) = p = \(\frac{1}{6}\), p(not six) = q = \(\frac{5}{6}\).

\((p + q)^{5} = p^{5} + 5p^{4} q + 10p^{3} q^{2} + 10p^{2} q^{3} + 5pq^{4} + q^{5}\)

(i) p( at most 2 sixes) = \(q^{5} + 5pq^{4} + 10p^{2} q^{3}\)

= \((\frac{5}{6})^{5} + 5(\frac{1}{6})(\frac{5}{6})^{4} + 10(\frac{1}{6})^{2} (\frac{5}{6})^{3}\)

= \(\frac{3125}{7776} + \frac{3125}{7776} + \frac{1250}{7776}\)

= \(\frac{7500}{7776} = 0.9645 \approxeq 0.965\) (3 d.p)

(ii) p(exactly 3 sixes) = \(10p^{3} q^{2}\)

= \(10(\frac{1}{6})^{3} (\frac{5}{6})^{2}\)

= \(\frac{250}{7776}\)

= \(0.032\) (3 d.p)

Correct Answer: D. 36.8%

Explanation

\(\frac{7}{19}\) as a percentage will be

\(\frac{7}{19}\times \frac{100}{1}\% = 36.8\%\)