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g(x) = x2 + 3x When g(x + 1) = (x + 1)^2 + 3(x + 1) = x\(^2\) + 1 + 2x + 3x + 3 = x\(^2\) + 5x + 4 g(x + 1) - g(x) = x2 + 5x + 8 - (x\(^2\) + 3x) = x\(^2\) + 5x + 4 - x2 -3x = 2x + 4 or 2(x + 4) = 2(x + 2)

Given that 2, 6, 12, 20...? the nth term = n\(^2\) + n check: n = 1, u1 = 2 n = 2, u2 = 4 + 2 = 6 n = 3, u3 = 9 + 3 = 12 ∴ n = 4, u4 = 16 + 4 = 20

Simply add odd number starting from '3' to the next number 2 2 + 3 = 5 5 + 5 = 10 10 + 7 = 17 17 + 9 = 26 The fifth term = 26

Identity(e) : a \(\ast\) e = a m \(\ast\) e = m...(i) m \(\ast\) e = me + m + e Because m \(\ast\) e = m : m = me + m + e m - m = e(m + 1) e = \(\frac{0}{m + 1}\) e = 0

Let x represent total vol. 2 : 3 = 2 + 3 = 5 \(\frac{3}{5}\)x = 351 x = \(\frac{351 \times 5}{3}\) = 585 Volume of smaller block = \(\frac{2}{5}\) x 585 = 234.00cm\(^3\)

The diagonal of a rhombus is a line segment that joins any two non-adjacent vertices. A rhombus has two diagonals that bisect each other at right angles. i.e this splits 6cm into 3cm each AND 8cm to 4cm Using Hyp\(^2\) = adj\(^2\) + opp\(^2\) Hyp\(^2\) = 3\(^2\) + 4\(^2\) Hyp\(^2\) = 25 Hyp = 5 ∴ Length (L) is 5cm because a rhombus is a quadrilateral with 4 equal lengths

(a) Area of △PQR = \(\frac{1}{2}\) * b * 12 = 20 b = \(\frac{2 x 20}{12}\) = 3.33cm |QR| = |UT| = 3.33cm IPQ| = \sqrt(12\(^2\) + 5\(^2\))\(^2\) = 13cm IRT| = |QU| = 12cm From △RTS sin 50^\circ= \(\frac{12}{|RS|}\) |RS| = \(\frac{12}{sin 50^\circ}\) = 15.6649 cm |TS| = \(\frac{12}{tan 50^\circ}\) = 10.069cm Perimeter = 13 + 3.33 + 15.6649 + 18.399 = 50.3939cm = 50cm (nearest whole number) (b)Area= \(\frac{1}{2}\) (3.33+ 18.4004)12 6 (21.7304) = 130.3824cm\(^2\) = 130 cm\(^2\) (nearest whole number)

Area of Trapezium = 1/2(sum of parallel sides) * h 91 = \(\frac{1}{2}\) (5 + 9)h cross multiply 91 = 7h h = \(\frac{91}{7}\) h = 13cm

Length of Arc AB = \(\frac{\theta}{360}\) 2\(\pi\)r = \(\frac{48}{360}\) x 2\(\frac{22}{7}\) x \(\frac{21}{2}\) = \(\frac{4 \times 22 \times \times 3}{30}\) \(\frac{88}{10}\) = 8.8cm Perimeter = 8.8 + 2r = 8.8 + 2(10.5) = 8.8 + 21 = 29.8cm

Perimeter of a Rhombus = 4L or 2\sqrt( d1\(^2\) + d2\(^2\)) P = 2\sqrt(12\(^2\) + 5\(^2\)) P = 2\sqrt(144 + 25) P = 2\sqrt169 = 2 * 13 P = 26cm

Where l\(^2\) = h\(^2\) + r\(^2\) l\(^2\) = 11\(^2\) + 8\(^2\) l = \sqrt185 l = 13.60cm The formula of CSA of Cone is \pirl \(\frac{22}{7}\) * 8 * 13.60 = 341.979 or 341.98 (2d.p)

Volume of a cylinder = \pir\(^2\)h First convert 3m to cm by multiplying by 100 Volume of External cylinder = \(\pi \times 13^2 \times 300\) Volume of Internal cylinder = \(\pi \times 10^2 \times 300\) Hence; Volume of External cylinder - Volume of Internal cylinder Total volume (v) = \(\pi (169 - 100) \times 300\) V = \(\pi \times 69 \times 300\) V = 20700\picm\(^3\)

Angle at the centre O is twice the angle at the circumference. i.e O = 72^\circ * 2 = 144^\circ ∠OPS = 180 - 144 = 36^\circ

the form y=mx+c where m is the gradient and c is the y-intercept. the equation to gives 5y=3x - 7. divide both side by 5 the coefficient of y → \(\frac{3}{5}\)x - \(\frac{7}{5}\) comparing this with the general equation y=mx+c, you can see that m= the gradient= \(\frac{3}{5}\).

Each interior angle = \(\frac{(n - 2)180}{n}\) 140 = \(\frac{(n - 2)180}{n}\) Cross multiply: 140n = 180n - 360 40n = 360 n = 9 sides ( A Nonagon)