Question Library

\((\sqrt{6} + 2)^2 - (\sqrt{6} - 2)^2\) = \([(\sqrt{6} + 2) + (\sqrt{6} - 2)][(\sqrt{6} + 2) - (\sqrt{6} - 2)]\) = \((\sqrt{6} + 2 + \sqrt{6} - 2)(\sqrt{6} + 2 - \sqrt{6} + 2)]\) = \((2\sqrt{6})(4)\) = \(8\sqrt{6}\)

P = {2,3,5} Q = {2,3,6,9} P \(\cap\) Q = {2,3}

n(f \(\cap\) v) + n(f) + n(v) + n(f \(\cap\) v) = 46 3 + 19 + 23 + x = 46 22 + 23 + x = 46 45 + x = 46 x = 46 - 45 x = 1

w = \(\frac{v(2 + cn)}{1 - cn}\) 2v + cnv = w(1 - cn) 2v + cnv = w - cnw 2v - w = -cnv - cnw Multiply through by negative sign -2v + w = cnv + cnw -2v + w = n(cv + cw) n = \(\frac{-2v + w}{cv + cw}\) n = \(\frac{1}{c}\frac{-2v + w}{v + w}\) Re-arrange... n = \(\frac{1}{c}\frac{w - 2v}{v + w}\)

Hence f(x) = 2x3 - 11x2 + 8x - 1 f(-3) = 2(-3)3 - 11(-3)2 + 8(-3) - 1 = 2(-27) - 11(9) + 8(-3) - 1 = -54 - 99 - 24 - 1 = -178

x2 - y2 = 4 .... (1) x + y = 2 .... (2) Simplify eqn (1) (x + y)(x - y) = 4 From eqn (2) x + y = 2 so substitute it into simplified eqn (1), we have 2 (x - y) = 4 therefore, x - y = 2 ... (1) x + y = 2 --------- 2x = 4 --------- x = 2, when y = 0

y \(\propto \sqrt{n}\) y = k\(\sqrt{n}\) when y = 4, n = 4 4 = k\(\sqrt{4}\) 4 = 2k k = 2 Therefore, y = 2\(\sqrt{n}\) y = 2\(\sqrt{\frac{16}{9}}\) y = 2\((\frac{4}{3})\) y = \(\frac{8}{3}\)

U \(\propto \frac{1}{V^3}\) U = \(\frac{k}{V^3}\) k = UV\(^3\) k = 81 x 2\(^3\) = 81 x 8 When V = 3, U = \(\frac{k}{V^3}\) U = \(\frac{81 \times 8}{3^3}\) U = \(\frac{81 \times 8}{27}\) = 24

\(\frac{1}{5}y + \frac{1}{5} < \frac{1}{2}y + \frac{2}{5}\) Collect like terms \(\frac{y}{5} - \frac{y}{2} < \frac{2}{5} - \frac{1}{5}\) \(\frac{2y - 5y}{10} < \frac{2 - 1}{5}\) \(\frac{-3y}{10} < \frac{1}{5}\) \(y > \frac{-2}{3}\)

(m - 3)(m - 4) < 0 (m - 3) < 0 ; (m - 4) < 0 m < 3 ; m < 4 3 < m < 4

n2 - 6n - 4 For the 3rd term, 32 - 6(3) - 4 9 - 18 -4 = -13 For the 4th term, 42 - 6(4) - 4 16 - 24 - 4 = -12 Sum of both terms -13 - 12 = -25

Sr = \(\frac{a}{1 - r}\) \(-\frac{1}{10}\) = \(\frac{1}{8} \times \frac{1}{1 - r}\) \(-\frac{1}{10}\) = \(\frac{1}{8(1 - r)}\) \(-\frac{1}{10}\) = \(\frac{1}{8 - 8r}\) cross multiply... -1(8 - 8r) = -10 -8 + 8r = -10 8r = -2 r = -1/4

p * q = pq + p - q First execute for 3 * 4 ==> 3(4) + 3 - 4 = 12 + 3 - 4 = 11 Now we execute for 2 * 11 ==> 2(11) + 2 - 11 = 22 + 2 - 11 = 13

m * n = \(\frac{mn}{2}\) Identify, e = 2 Let a \(\in\) R, then a * a\(^{-1}\) = e a * a\(^{-1}\) = 2 -5 * a\(^{-1}\) = 2 \(\frac{-5 \times a^{-1}}{2} = 2\) \(a^{-1} = \frac{2 \times 2}{-5}\) \(a^{-1} = -\frac{4}{5}\)

\(\begin{vmatrix} 5 & 3 \\ x & 2 \end{vmatrix}\) = \(\begin{vmatrix} 3 & 5 \\ 4 & 5 \end{vmatrix}\) 10 - 3x = 15 - 20 -3x = 15 - 20 - 10 -3x = -15 x = 5