Correct Answer: D. \(\frac{1}{36}\)

Explanation

P(c and e) = \(P(c) \times P(e)\)

P(3) = \(\frac{1}{6}\)

P(5) = \(\frac{1}{6}\)

P(3 and 5) = \frac{1}{6} \times \frac{1}{6} = \frac{1}{36}\)

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

Explanation

\(\frac{log √27 - log √8}{log 3 - log 2}\)

= \(\frac{log √3^3 - log √2^3}{log 3 - log 2}\)

= \(\frac{log3^{3/2} - log2^{3/2}}{log3}\)

=\(\frac{^3/_2(log 3 - log 2)}{log 3 - log 2}\)

\(\therefore\frac{3}{2}\)

Correct Answer: B. \(\frac{\sqrt{2}}{4}(\sqrt{3} - 1)\)

Explanation

\(\cos(a + b) = \cos a\cos b - \sin a\sin b\)

\(\cos75° = \cos(30 + 45) = (\cos30)(\cos45) - (\sin30)(\sin45)\)

= \((\frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{2}) - (\frac{1}{2} \times \frac{\sqrt{2}}{2})\)

= \(\frac{\sqrt{6} - \sqrt{2}}{4}\)

= \(\frac{\sqrt{2}(\sqrt{3} - 1)}{4}\)

Correct Answer: C. P = 4 and Q = -2

Explanation

\(\frac{2x}{(x + 6)(x + 3)} = \frac{P}{x + 6} + \frac{Q}{x + 3}\)

\(\frac{2x}{(x + 6)(x + 3)} = \frac{P(x + 3) + Q(x + 6)}{(x + 6)(x + 3)}\)

Comparing equations, we have

\(2x = Px + 3P + Qx + 6Q\)

\(\implies 3P + 6Q = 0 ... (1) ; P + Q = 2 .... (2)\)

From equation (1), \(3P = -6Q \implies P = -2Q\)

\(\therefore -2Q + Q = -Q = 2 \)

\(Q = -2\)

\(P = -2Q = -2(-2) = 4\)

\(P = 4, Q = -2\)

Correct Answer: A. 25

Explanation

\(f : x \to x^{2} ; g : x \to x + 3\)

\(g(2) = 2 + 3 = 5\)

\(f o g(2) = f(5) = 5^{2} = 25\)

Explanation

(a) 5 men, 3 women

Number of ways of forming the committee = \(^{8}C_{3}\)

= \(\frac{8!}{(8 - 3)! 3!}\)

= 56 ways.

(b) p(no woman) = \(\frac{^{5}C_{3}}{^{8}C_{3}}\)

= \(\frac{10}{56}\)

\(\therefore\) p( at least one woman) = 1 - p(no woman)

= \(1 - \frac{10}{56}\)

= \(\frac{46}{56}\)

= \(\frac{23}{28}\).

Correct Answer: D. 18.97\(ms^{-1}\)

Explanation

Due to conservation of energy, K.E = P.E

\(\frac{1}{2}mv^{2} = mgh\)

\(\implies m \times 10 \times 18 = \frac{1}{2}mv^{2}\)

\(v^{2} = 2 \times 10 \times 18 = 360\)

\(v = \sqrt{360} = 18.97ms^{-1}\)

Correct Answer: A. \(\frac{1}{24}\)

Explanation

The number of arrangements for the 4 letters = \(^{4}P_{4} = \frac{4!}{(4 - 4)!}\)

\(4! = 24\)

Alphabetical order is just 1 of the arrangements for the letters

= \(\frac{1}{24}\)

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. \(\frac{1}{14}(3\sqrt{2} + 4\sqrt{5} - 6\sqrt{10} - 2)\)

Explanation

\(\frac{1 - 2\sqrt{5}}{2 + 3\sqrt{2}} = (\frac{1 - 2\sqrt{5}}{2 + 3\sqrt{2}})(\frac{2 - 3\sqrt{2}}{2 - 3\sqrt{2}})\)

= \(\frac{2 - 3\sqrt{2} - 4\sqrt{5} + 6\sqrt{10}}{4 - 6\sqrt{2} + 6\sqrt{2} - 18}\)

= \(\frac{2 - 3\sqrt{2} - 4\sqrt{5} + 6\sqrt{10}}{-14}\)

= \(\frac{1}{14}(3\sqrt{2} + 4\sqrt{5} - 2 - 6\sqrt{10})\) (dividing through with the minus sign)