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

Explanation

\(2^7 = 2^{3(5-x)}\Rightarrow 7 = 3^{5-x} \Rightarrow 7 15 - 3x\\

\Rightarrow -8 = -3x \Rightarrow x = \frac{8}{3}\)

Correct Answer: C. CCCXLII

Explanation

Note that \(\alpha\) + \(\beta\) = - \(\frac{4}{3}\) and (\(\alpha - \beta\))\(^2\)

= a\(^2\) + \(\beta\)\(^2\) - 2\(\alpha\beta\) = (\(\alpha + \beta\))\(^2\) - 4\(\alpha \beta\)

Thus, substituting for \(\alpha + \beta\) and \(\alpha \beta\)

simplify to get (\(\alpha - \beta\))\(^2\) = (-\(\frac{4}{3}\))\(^2\) - 4(-\(\frac{5}{3}\))

= \(\frac{16}{9} + \frac{20}{3}\)

= \(\frac{16+60}{9}\)

= \(\frac{76}{9}\)

Taking square root of both sides

(\(\alpha\) - \(\beta\)) = \(\sqrt{\frac{76}{9}}\)

= \(\pm \frac{2\sqrt{19}}{3}\)

Correct Answer: C. -4

Explanation

\(f : x \to x^{2} - x - 6\); \(g : x \to x - 1\)

\(g(3) = 3 - 1 = 2\)

\(f(g(3)) = f(2) = 2^{2} - 2 - 6 = 4 - 2 - 6 = -4\)

Correct Answer: C. \(\begin{pmatrix} 0.8 \\ 0.6 \end{pmatrix}\)

Explanation

\(PQ = \begin{pmatrix} 7 - 3 \\ 4 - 1 \end{pmatrix}\)

\(= \begin{pmatrix} 4 \\ 3 \end{pmatrix}\)

\(\hat{n} = \frac{\overrightarrow{PQ}}{|PQ|} \)

\(|PQ| = \sqrt{4^{2} + 3^{2}} = \sqrt{25} = 5\)

\(\hat{n} = \frac{1}{5}\begin{pmatrix} 4 \\ 3 \end{pmatrix} = \begin{pmatrix} 0.8 \\ 0.6 \end{pmatrix}\)

Correct Answer: A. 2x = 3(y - 1)

Explanation

24\(_x\) = 31\(_y\)

\(2 \times x^1 + 4 \times x^0 = 3 \times y^1 + 1 \times y^0\)

\(2x + 4 = 3y + 1 \implies 2x = 3y + 1 - 4\)

\(2x = 3y - 3 = 3(y - 1)\)

Correct Answer: B. 9

Explanation

Rearrange; 2, 2, 5, 7, 9, 9, 9, 10, 10, 11, 12, 18 Median = 9

Correct Answer: D. \(110_{two}, 42_{five}, 31_{eight}\)

Explanation

Converting each number to base 10

\(110_{two} = 1 × 2^2 + 1 × 2^1 + 0 × 2^0\)

= 1 × 4 + 1 × 2 + 0 × 1

= 4 + 2 + 0

= \(6_{ten}\)

\(31_{eight} = 3 × 8^1 + 1 × 8^0\)

= 3 × 8 + 1 × 1

= 24 + 1

= \(25_{ten}\)

\(42_{five} = 4 × 5^1 + 2 × 5^0\)

= 4 × 5 + 2 × 1

= 20 + 2

= \(22_{ten}\)

Hence, \(31_{eight} > 42_{five} > 110_{two}\)

In ascending order, \(110_{two}, 42_{five}, 31_{eight}\)

Correct Answer: D. \(\frac{167}{576}\)

Explanation

\(P(Jide) = \frac{1}{12}; P(\text{not Jide}) = \frac{11}{12}\)

\(P(Atu) = \frac{1}{6}; P(\text{not Atu}) = \frac{5}{6}\)

\(P(Obu) = \frac{1}{8}; P(\text{not Obu}) = \frac{7}{8}\)

\(P(\text{only one of them}) = P(\text{Jide not Atu not Obu}) + P(\text{Atu not Jide not Obu}) + P(\text{Obu not Jide not Atu})\)

= \((\frac{1}{12} \times \frac{5}{6} \times \frac{7}{8}) + (\frac{1}{6} \times \frac{11}{12} \times \frac{7}{8}) + (\frac{1}{8} \times \frac{11}{12} \times \frac{5}{6})\)

= \(\frac{35}{576} + \frac{77}{576} + \frac{55}{576}\)

= \(\frac{167}{576}\)

Correct Answer: B. \(q \vee \sim p\)