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tan 315^\circ = - tan (360 - 315) = - tan 45 = -1

sin 210 = - sin (210 - 180) = - sin 30 = \(-\frac{1}{2}\)

No expanded working has been attached for this question yet. The verified answer is Option D.

The ladder makes 60^\circ inclination with the wall and its foot is 5m away from the wall. Sketching the diagram, you will see that we have \(sin 60 = \frac{5}{x}\) (where x is the length of the ladder) ⇒ \(x = \frac{5}{sin60}\) \(x = \frac{10\sqrt{3}}{3}m\)

No expanded working has been attached for this question yet. The verified answer is Option C.

mode = 72º

If a die is rolled once, the probability of getting a number greater than 2 is \(\frac{2}{3}\). i.e Pr( number greater than 2) = \(\frac{\text{ no of required outcome}}{\text{total no of expected outcome}}\) = \(\frac{4}{6} = \frac{2}{3}\)

U = (1, 2, 3, 6, 7, 8, 9, 10) E = (10, 4, 6, 8, 10) F = (x : x\(^2\) = 2\(^6\), x is odd) ∴ F = \(\phi\) Since x\(^2\) = 2\(^6\) = 64 x = \(\pm 8\) which is even ∴ E ∩ F = \(\phi\) Since there are no common elements Since E ∩ F = \(\phi\), then ( E ∩ F)' = U = (1, 2, 3, 6, 7, 8, 9, 10)

The sum of angles in a quadrilateral = 360^\circ \(\therefore (p + 10) + (2p - 30) + (3p + 20) + 4p = 360\) \(10p = 360^\circ \implies p = \frac{360}{10} = 36^\circ\)

Angular difference (\(\theta\))= 25^\circ + 20^\circ = 45^\circ \(\alpha\) = common latitude = 60^\circ \(S = \frac{\theta}{360^\circ} \times 2\pi R \cos \alpha\) \(S = \frac{45^\circ}{360^\circ} \times 2 \pi \times 6400 \times \cos 60^\circ\) = \(\frac{6400\pi}{8} = 800\pi km\)

No expanded working has been attached for this question yet. The verified answer is Option A.

2x - 4y + 3 = 0 Required distance = \(\frac{(2 \times 2) + 3(-4) + 3}{\sqrt{2^2} + (-4)^2}\) = \(\frac{4 - 12 + 3}{\sqrt{20}}\) = \(\frac{-5}{-2\sqrt{5}}\) = \(\frac{\sqrt{5}}{2}\)

sin \(\theta\) = \(\frac{m}{n}\) Opp = m; Hyp = n Adj = \(\sqrt{n^{2} - m^{2}}\) \(\cos \theta = \frac{\sqrt{n^{2} - m^{2}}}{n}\) = \(\frac{\sqrt{(n + m)(n - m)}}{n}\)

\(y = x \sin x\) \(\frac{\mathrm d y}{\mathrm d x} = x \cos x + \sin x\) \(\frac{\mathrm d^{2} y}{\mathrm d x^{2}} = x (- \sin x) + \cos x + \cos x\) = \(2 \cos x - x \sin x\)

\(\int_{\frac{\pi}{12}} ^{\frac{\pi}{4}} 2 \cos 2x \mathrm {d} x\) = \([\frac{2 \sin 2x}{2}]|_{\frac{\pi}{12}} ^{\frac{\pi}{4}}\) = \(\sin 2x |_{\frac{\pi}{12}} ^{\frac{\pi}{4}}\) = \(\sin 2(\frac{\pi}{4}) - \sin 2(\frac{\pi}{12})\) = \(\sin \frac{\pi}{2} - \sin \frac{\pi}{6}\) = \(1 - \frac{1}{2} = \frac{1}{2}\)