Correct Answer: A. \(2\frac{1}{3}\)

Explanation

\(\left(1\frac{2}{3}\right)^2 - \left(\frac{2}{3}\right)^2=\left(\frac{5}{3}\right)^2 - \left(\frac{2}{3}\right)^2\)

Difference of two squares

\(\left(\frac{5}{3}-\frac{2}{3}\right)\left(\frac{5}{3}+\frac{2}{3}\right)=\left(\frac{3}{3}\right)\left(\frac{7}{3}\right)\\

\frac{7}{3}=2\frac{1}{3}\)

Correct Answer: B. 60°

Explanation

\(2\sin^{2} \theta = 1 + \cos \theta\)

\(2 ( 1 - \cos^{2} \theta) = 1 + \cos \theta\)

\(2 - 2\cos^{2} \theta = 1 + \cos \theta\)

\(0 = 1 - 2 + \cos \theta + 2\cos^{2} \theta\)

\(2\cos^{2} \theta + \cos \theta - 1 = 0\)

Factorizing, we have

\((\cos \theta + 1)(2\cos \theta - 1) = 0\)

Note: In the range, \(0° \leq \theta \leq 90°\), all trig functions are positive, so we consider

\(2\cos \theta = 1 \implies \cos \theta = \frac{1}{2}\)

\(\theta = 60°\).

Explanation

(a) Length of the chord = 14 cm

Radius = \(\frac{3}{2} \times 14 cm = 21 cm\)

The chord AB = \(2r \sin \frac{\theta}{2}\)

\(14 = 2(21) \sin \frac{\theta}{2}\)

\(\sin \frac{\theta}{2} = \frac{14}{42} = 0.333\)

\(\frac{\theta}{2} = \sin^{-1} (0.333)\)

\(\frac{\theta}{2} = 19.469°\)

\(\theta = 19.469° \times 2 = 38.938°\)

Length of the arc = \(\frac{\theta}{360°} \times 2\pi r\)

\(\frac{38.938}{360} \times 2 \times \frac{22}{7} \times 21\)

= \(14.277 cm\)

Perimeter of the remaining portion = 22 + 12 + 12 + 5 + 3 + 14.277

= 68.277 cm

\(\approxeq\) 68.28 cm (2 decimal places).

(b) \(\frac{3}{\sqrt{3}}(\frac{2}{\sqrt{3}} - \frac{\sqrt{12}}{6}\)

= \(\frac{3}{\sqrt{3}}(\frac{12 - \sqrt{36}}{6\sqrt{3}}\)

= \(\frac{3}{\sqrt{3}}(\frac{12 - 6}{6\sqrt{3}}\)

= \(\frac{3}{\sqrt{3}}(\frac{6}{6\sqrt{3}}\)

= \(\frac{18}{6\sqrt{9}}\)

= \(\frac{18}{18}\)

= 1.

Correct Answer: A. 10x + 1

Explanation

\(\frac{5x^ 3+x^2}{x}\) → \(\frac{5x^ 3}{x} + \frac{x^2}{x}\)

5x\(^2\) + x

Then dy/dx = 10x + 1

Correct Answer: D. 2/3

Correct Answer: D. 187m

Correct Answer: D. 240°

Explanation

\(cos\theta = \frac{adj}{hyp}=\frac{300}{600}=\frac{1}{2}\\

\theta = cos^{-1}(0.5000)=60^{\circ}\)

The bearing of P from \(Q = \theta + 180 = 60 + 180 = 240^{\circ}\)

Correct Answer: D. 5

Explanation

The equation of a circle is given as: \((x - a)^{2} + (y - b)^{2} = r^{2}\)

Expanding, we have: \(x^{2} + y^{2} - 2ax - 2by + a^{2} + b^{2} = r^{2}\)

\(\implies x^{2} + y^{2} - 2ax - 2by = r^{2} - a^{2} - b^{2}\)

Comparing with the given equation: \(3x^{2} + 3y^{2} + 24x - 12y = 15\)

Making the coefficients of \(x^{2}\) and \(y^{2}\) = 1, we have

\(x^{2} + y^{2} + 8x - 4y = 5\)

\(2a = -8 \implies a = -4\)

\(2b = 4 \implies b = 2\)

\(r^{2} - a^{2} - b^{2} = 5 \implies r^{2} = 5 + (-4)^{2} + (2)^{2} = 5 + 16 + 4 = 25\)

\(\therefore r = 5\)

Correct Answer: A. \(\begin{pmatrix} -7 \\ -7\sqrt{3} \end{pmatrix}\)

Explanation

\(F = \begin{pmatrix} F_{x} \\ F_{y} \end{pmatrix} = \begin{pmatrix} F\cos \theta \\ F\sin \theta \end{pmatrix}\)

\((14N, 240°) = \begin{pmatrix} 14\cos 240 \\ 14\sin 240 \end{pmatrix}\)

= \(\begin{pmatrix} 14 \times -0.5 \\ 14 \times \frac{-\sqrt{3}}{2} \end{pmatrix}\)

= \(\begin{pmatrix} -7 \\ -7\sqrt{3} \end{pmatrix}\)

Correct Answer: B. m