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EXCELLENCE It is a ten-letter word = 10! Since we have repeating letters, we have to divide to remove the duplicates accordingly. There are 4 Es, 2 Cs, 2 Ls ∴ there are: \(\frac{10!}{4!2!2!}\) ways to arrange

y = xsinx \(\frac{dy}{dx}\) = \(1 \sin x + x \cos x\) = \(\sin x + x \cos x\) At x = \(\frac{\pi}{2}\) = sin\(\frac{\pi}{2}\) + \(\frac{\pi}{2} \cos {\frac{\pi}{2}}\) = 1 + \(\frac{\pi}{2}\) \times 0 = 1

t ∝ h, t = 20, h t = ? h = 60 t = kh where k is constant 20 = 50k k = \(\frac{20}{50}\) k = \(\frac{2}{5}\) when h = 60, t = ? t = \(\frac{2}{5}\) × 60 t = 24oC

The locus of points equidistant from a given straight line PQ consists of two parallel lines, one on each side of PQ, each at the same perpendicular distance from it.

If RST = 60o RXT = 2 × RST (angle at the centre twice angle at the circumference) RXT = 2 × 60 = 120o

2x + x = 180o 3x = 180o x = 60o (exterior angle of the polygon) angle = \(\frac{\text{total angle}}{\text{number of sides}}\) 60 = \(\frac{360}{n}\) n = \(\frac{360}{60}\) n = 6 sides

Let the speed of the 1st trip be x miles/hr and the speed of the 2nd trip be 3x miles/hr Speed = distance/time ∴ Time taken to cover a distance of 50 miles on the 1st trip = \(\frac{50}{xhr}\) time taken to cover a distance of 300 miles on the next trip = \(\frac{300}{3xhr}\) = \(\frac{100}{xhr}\) ∴the new time compared with the old time is twice as much

Sum of angle at a point = 360o 2x + 3x + 4x = 360 9x = 360 x = \(\frac{360}{9}\) x = 40o

\(\frac{0.00000231}{0.007}\) to standard form = \(\frac{231 \times 10^{-8}}{7 \times 10^{-3}}\) = 33 \times 10\(^{-8 − (−3)}\) = 33 \times 10\(^{− 8 + 3}\) = 33 \times 10-5 = 3.3 x 10^-4

Total angle at a point = 3600 ∴ To get the angle occupied by fertilizer we have, 40 + 50 + 80 + 70 + 30 + fertilizer(x) = 360 270 + x = 360 x = 360 - 270 x = 90 Total amount allocated to the farm = ₦ 80,000 ∴Amount allocated to the fertilizer = \(\frac{\text{fertilizer (angle) \times Total amount}}{\text{total angle}}\) = \(\frac{90}{360}\) \times 80,000 = ₦20,000

MACICITA is an eight letter word = 8! Since we have repeating letters, we have to divide to remove duplicates accordingly. There are 2A, 2C, 2I ∴ \(\frac{8!}{2! 2! 2!}\)

∫xndx = \(\frac{x_{n + 1}}{n + 1}\) ∫dx = x + k where k is constant ∫(x2 + 3x − 5)dx ∫x2 dx + ∫3xdx − ∫5dx \(\frac{2_{2 + 1}}{2 + 1}\) + \(\frac{3x^{1 + 1}}{1 + 1}\) − 5x + k \(\frac{x_3}{3}\) + \(\frac{3x_2}{2}\) − 5x + k

Find the diagram Sin 70^\circ x = 10 Sin 70^\circ = 9.3969 Hence, length of chord MN = 2x = 2 \times 9.3969 = 18.79 = 19cm (nearest cm)

Gradient of line joining points (3, 2), (1, 4) Gradient = \(\frac{\text{Change in Y}}{\text{Change in X}}\) = \(\frac{y_2 - Y_1}{x_2 - x_1}\)\) (X1, Y1) = (3, 2) (X2, Y2) = (1, 4) Gradient = \(\frac{4 − 2}{1 + 3}\) = \(\frac{2}{-2}\) = −1

\(x \propto \frac{1}{y}\), y \(\propto\) z x = \(\frac{k}{y}\) y = mz Since y = mz, x = \(\frac{k}{mz}\), where k and m are constants. Hence, x \(\propto\) \(\frac{1}{z}\)