Correct Answer: B. \(15a^{4}b^{2}\)

Explanation

\((a - b)^{6} = ^{6}C_{0}(a)^{6}(-b)^{0} + ^{6}C_{1}(a)^{5}(-b)^{1} + ^{6}C_{2}(a)^{4}(-b)^{2} + ...\)

Third term = \(^{6}C_{2}(a)^{4}(-b)^{2} = \frac{6!}{(6-2)! 2!}(a^4)(b^2)\)

= \(15a^{4}b^{2}\)

Correct Answer: B. \(3 + \sqrt{7}\)

Explanation

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

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

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

Correct Answer: D. \(\frac{14}{15}\)

Explanation

P(at least one green wrapper) = 1 - P(no green wrapper)

= \(1 - (\frac{2}{6} \times \frac{1}{5})\)

= \(1 - \frac{1}{15}\)

= \(\frac{14}{15}\)

Correct Answer: C. 4

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

Explanation

(\(\frac{1}{9}\))\(^{x + 2}\) = 243\(^{x - 2}\)

3\(^{-2(x + 2)}\) = 3\(^{5(x - 2)}\);

-2(x+2) = 5(x - 2)

-2x -4 = 5x - 10

-7x = -6

x = \(\frac{6}{7}\)

Correct Answer: B. 24.50°

Explanation

\(p . q = |p||q|\cos \theta\)

\(156 + 70 = (\sqrt{13^{2} + 14^{2}})(\sqrt{12^{2} + 5^{2}}) \cos \theta\)

\(226 = (\sqrt{365})(13) \cos \theta\)

\(\frac{226}{13\sqrt{365}} = \cos \theta\)

\(\cos \theta = 0.9099\)

\(\theta = 24.50°\)

Explanation

(\(^{2x + 1}_{3}\)) = \(\frac{(2x + 1)!}{3!(2x - 2)!} = \frac{(2x + 1)2x(2x - 1)}{6}\)

(2x - 1) = \(\frac{(2x - 1)!}{3!(2x - 4)!} = \frac{(2x - 1)(2x -2)(2x - 3)}{6}\) and 2(\(^x_2\)) = 2 [\(\frac{x!}{2!(x - 2)!}\)]

2[\(\frac{x(x - 1)}{2}\)] = x(x - 1)

Substitute; (\(^{2x + 1}_3\)) - (\(^{2x + 1}_3\)) - 2(\(^x_2\))

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

Multiplying through by 6 and simplifying to arrive at; 3\(x^2 - 3x + 1\)

Correct Answer: A. \({x : -5< x< 5}\)

Explanation

\(P = {x : -2 < x < 5}\) and \(Q = {x: -5 < x < 2}\)

\((P \cup Q) = {x : -5 < x < 5}\)

Explanation

(a)

(b)

(ci) The number of candidates who scored marks between 24 and 58 = 146 - 14 = 132

(cii) if 12% of the candidates passed with distinction then 88% did not pass with distinction

⇒\(88% of 200 = \frac{88}{100}\times 200=176\)

From the cumulative frequency curve, 176 corresponds to 69 marks

∴ Thelowest mark for distinction = 69 marks

Explanation

\(^9{∫}_1\) \(\frac{x(2x-3)}{√x}\) dx = \(\frac{2x^2 - 3x)}{x^{1/2}}\)

= x\(^{-1/2}\) (2x\(^2\) - 3x)

= 2x\(^{3/2}\) - 3x\(^{1/2}\)

= \(\frac{2x^{3/2}}{3/2 + 1}\) - \(\frac{3x^{1/2}}{1/2 + 1}\)

= \(\frac{2x^{3/2}}{5/2}\) - \(\frac{3x^{1/2}}{3/2}\)

= \(^9{∫}_1\) [\(\frac{4x^{3/2}}{5}\) - \(\frac{6x^{1/2}}{3}\)]

[ \(\frac{4(9)^{3/2}}{5}\) - \(\frac{6^(9){1/2}}{3}\) ] - [ \(\frac{4(9)^{1/2}}{5}\) - \(\frac{6(1)^{1/2}}{3}\) ]

= [ \(\frac{4(27)}{5}\) - \(\frac{6(3)}{3}\) - [ \(\frac{4}{5}\) - 2 ]

= [ \(\frac{108}{5}\) - 2 ] - [ \(\frac{6}{5}\) ]

= \(\frac{78}{5}\) + \(\frac{6}{5}\) = \(\frac{84}{5}\)

16\(^{4/5}\)