Correct Answer: B. 150°

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

Explanation

For independent events \(P(X \cap Y) = P(X) \times P(Y)\)

\(\frac{2}{15} = \frac{1}{5} \times P(Y)\)

\(P(Y) = \frac{2}{15} ÷ \frac{1}{5} = \frac{2}{3}\)

Explanation

(a) \(2^{3y + 2} - 7(2^{2y + 2}) - 31(2^{y}) - 8 = 0\)

\(\equiv (2^{2})(2^{y})^{3} - 7(2^{2})(2^{y})^{2} - 31(2^{y}) - 8 = 0\)

Let \(2^{y} = x\),

\(4x^{3} - 28x^{2} - 31x - 8 = 0\)

If x = 8, \(4(8^{3}) - 28(8^{2}) - 31(8) - 8 = 2048 - 1792 - 248 - 8\)

= \(0\)

\(\therefore (x - 8) \text{is a factor}\)

Dividing \(\frac{4x^{3} - 28x^{2} - 31x - 8}{x - 8}\), we get

= \(4x^{2} + 4x + 1 \)

= \((2x + 1)^{2}\)

Therefore, \(x = -\frac{1}{2}, -\frac{1}{2}, 8\)

Recall, \(x = 2^{y}\)

\(2^{y} = -\frac{1}{2}\) has no real solution.

\(2^{y} = 8 = 2^{3} \implies y = 3\)

(b) \(\int (\sqrt{x^{2} + 1}) xdx\)

Let \(u^{2} = x^{2} + 1; 2\frac{\mathrm d u}{\mathrm d x} = 2x\)

\(\mathrm d u = x \mathrm d x\)

\(\therefore \int (\sqrt{x^{2} + 1}) xdx = \int (\sqrt{u^{2}}) \mathrm d u\)

= \(\int u \mathrm d u\)

= \(\frac{u^{2}}{2} + c\)

= \(\frac{x^{2} + 1}{2} + c\)

Correct Answer: A. \(11.6cm^2\)

Explanation

\(\theta = 115° , radius = 3.4cm^2\)

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

= \(\frac{115}{360} \times \frac{22}{7} \times 3.4\times 3.4\)

= \(\frac{29246.4}{2520}\)

= \(11.6cm^2\)

Correct Answer: B. 21°

Explanation

Sin (5x

Correct Answer: C. \(\frac{1}{9}\)

Explanation

\(\frac{3x^3}{(3x)^3}\) = \(\frac{3 \times x^3}{3^3 \times x^3}\)

= \(\frac{3 \times x^3}{3 \times 3 \times 3 \times x^3}\)

= \(\frac{1}{3^2}\)

= \(\frac{1}{9}\)

Correct Answer: C. 4x - r

Explanation

3x - (p - x) - ( r - p)

= 3x - p + x - r + p

= 4x - r

Explanation

\(a \Delta b = a^{3} - b^{3}\)

\((\sqrt{3} + \sqrt{2}) \Delta (\sqrt{3} - \sqrt{2}) = (\sqrt{3} + \sqrt{2})^{3} - (\sqrt{3} - \sqrt{2})^{3}\)

\((\sqrt{3} + \sqrt{2})^{3} = (\sqrt{3})^{3} + 3(\sqrt{3})^{2}(\sqrt{2}) + 3(\sqrt{3})(\sqrt{2})^{2} + (\sqrt{2})^{3}\)

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

= \(9\sqrt{3} + 11\sqrt{2}\)

\((\sqrt{3} - \sqrt{2})^{3} = (\sqrt{3})^{3} + 3(\sqrt{3})^{2}(-\sqrt{2}) + 3(\sqrt{3})(-\sqrt{2})^{2} + (-\sqrt{2})^{3}\)

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

= \(9\sqrt{3} - 11\sqrt{2}\)

\((\sqrt{3} + \sqrt{2}) \Delta (\sqrt{3} - \sqrt{2}) = 9\sqrt{3} + 11\sqrt{2} - 9\sqrt{3} + 11\sqrt{3}\)

= \(22\sqrt{2}\)

Correct Answer: C. 17:20

Correct Answer: D. -3

Explanation

\(\begin{pmatrix} 1 & -3 \\ 1 & 4 \end{pmatrix} \begin{pmatrix} -6 \\ P \end{pmatrix} = \begin{pmatrix} 3 \\ -26 \end{pmatrix}\)

\(\implies (1 \times -6) + (-3 \times P) = 3\)

\(-6 - 3P = 3 \implies -3P = 9\)

\(P = -3\)