Correct Answer: D. 8

Explanation

\(x \propto \frac{1}{y}\)

\(x = \frac{k}{y}\)

\(\frac{2}{3} = \frac{k}{9}\)

\(3k = 18 \implies k = 6\)

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

When y = \(\frac{3}{4}\),

x = \(\frac{6}{\frac{3}{4}}\)

= \(\frac{6 \times 4}{3}\)

= 8

Correct Answer: C. 25

Explanation

circumference of the bicycle wheel

\(= 2\pi r = 2 \times \frac{22}{7} \times 42 - 264cm = 2.64m\\\)

Number of revolution

\(=\frac{66}{2.64} = 25\)

Correct Answer: C. x - 2

Explanation

\(2x^2 + 3x - 14\)

\(2x^2 + 7x - 4x - 14\)

\(x(2x + 7) - 2(2x + 7)\)

= \((x - 2)(2x + 7)\)

The other factor = (x - 2).

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

Explanation

\(\frac{1}{x} + \frac{4}{3x} - \frac{5}{6x} + 1 = 0\)

\(\frac{6 + 8 - 5 + 6x}{6x} = 0\)

\(\frac{9 + 6x}{6x} = 0 \implies 9 + 6x = 0\)

\(6x = -9 \implies x = \frac{-3}{2}\)

Correct Answer: D. \(\frac{15}{4}\)

Explanation

\(\left(\frac{16}{81}\right)^{-\frac{3}{4}}\times \sqrt{\frac{100}{81}}\\

\frac{1}{\left(\sqrt[4]{\frac{16}{81}}\right)^3}\times \frac{10}{9}=\frac{1}{\left(\frac{2}{3}\right)^3}\times\frac{10}{9}\\

=\frac{27}{8}\times \frac{10}{9}=\frac{15}{4}\)

Correct Answer: B. \(\frac{-2}{x+3}\)

Explanation

\(\frac{1}{x-3}-\frac{3(x-1)}{x^2 - 9}\\

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

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

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

Correct Answer: D. 399000

Correct Answer: B. \(\frac{rk^2+pm^2}{m^2}\)

Explanation

\(k = m\sqrt{\frac{t - p}{r}}\)

\(\frac{k}{m} = \sqrt{\frac{t - p}{r}}\)

\((\frac{k}{m})^2 = \frac{t - p}{r}\)

\(rk^2 = m^2 (t - p)\)

\(\therefore m^2 t = rk^2 + m^2 p\)

\(t = \frac{rk^2 + m^2 p}{m^2}\)

Explanation

(a) Longitude difference = 123° - 67° = 56°

(i) Distance between X and Y along the equator = \(\frac{56}{360} \times 2 \times \frac{22}{7} \times 6400 \cos 0\)

= \(\frac{56}{360} \times 2 \times \frac{22}{7} \times 6400\)

= \(6257.78km \approxeq 6258km\)

(ii) Latitude difference = 90°

Distance from the North pole = \(\frac{90}{360} \times 2 \times \frac{22}{7} \times 6400\)

= \(10057.14km \approxeq 10057km\)

(b)

\(OP^{2} = PN^{2} + ON^{2}\)

\(4^{2} + 3^{2} = 25\)

\(OP = \sqrt{25} = 5cm\)

\(OP = OR = 5cm\)

\(\Delta NOR = scalene\)

\(\frac{OR}{\sin 20} = \frac{3}{\sin < ORN}\)

\(\frac{5}{\sin 20} = \frac{3}{\sin < ORN}\)

\(\sin < ORN = \frac{3 \times \sin 20}{5}\)

\(\sin < ORN = 0.2052\)

\(< ORN = \sin^{-1} (0.2052)\)

\(< ORN = 11.84° \approxeq 12°\)

Correct Answer: B. 150°