(a)

$$ (x - 1)(x + 2) < 0 $$

$$ - 2 < x < 1 $$

(b)

$$ (x + 3)(x - 4) \le 0 $$

$$ - 3 \le x \le 4 $$

(c)

$$ (2x + 3)(x - 2) > 0 $$

$$ x < - {3 \over 2} \text{ or } x > 2 $$

(d)

$$ x(x - 5) \ge 0 $$

$$ x \le 0 \text{ or } x \ge 5 $$

(e)

\begin{align*} x^2 - 4x & \le 0 \\ x(x - 4) & \le 0 \end{align*}

$$ 0 \le x \le 4 $$

(f)

\begin{align*} x^2 - 4 & < 0 \\ x^2 - 2^2 & < 0 \\ (x + 2)(x - 2) & < 0 \phantom{00000} [a^2 - b^2 = (a + b)(a - b)] \end{align*}

$$ -2 < x < 2 $$

(a)

\begin{align} x(x - 2) & < 3 \\ x^2 - 2x & < 3 \\ x^2 - 2x - 3 & < 0 \\ (x + 1)(x - 3) & < 0 \end{align}

$$ -1 < x < 3 $$

(b)

\begin{align} x^2 & > 4x + 12 \\ x^2 - 4x - 12 & > 0 \\ (x + 2)(x - 6) & > 0 \end{align}

$$ x < -2 \text{ or } x > 6 $$

(c)

\begin{align} 4x(x + 1) & \le 3 \\ 4x^2 + 4x & \le 3 \\ 4x^2 + 4x - 3 & \le 0 \\ (2x + 3)(2x - 1) & \le 0 \end{align}

$$ -{3 \over 2} \le x \le {1 \over 2} $$

(d)

\begin{align} (1 - x)^2 & \ge 17 - 2x \\ (1)^2 - 2(1)(x) + (x)^2 & \ge 17 - 2x \\ 1 - 2x + x^2 & \ge 17 - 2x \\ 1 - 17 - 2x + 2x + x^2 & \ge 0 \\ - 16 + x^2 & \ge 0 \\ x^2 - 16 & \ge 0 \\ (x + 4)(x - 4) & \ge 0 \end{align}

$$ x \le - 4 \text{ or } x \ge 4 $$

(e)

\begin{align} (x + 2)^2 & < x(4 - x) + 40 \\ x^2 + 2(x)(2) + 2^2 & < 4x - x^2 + 40 \\ x^2 + 4x + 4 & < 4x - x^2 + 40 \\ x^2 + x^2 + 4x - 4x + 4 - 40 & < 0 \\ 2x^2 - 36 & < 0 \\ x^2 - 18 & < 0 \\ x^2 - (\sqrt{18})^2 & < 0 \\ (x + \sqrt{18})(x - \sqrt{18}) & < 0 \end{align}

$$ -\sqrt{18} < x < \sqrt{18} $$

\begin{align*} y & = d^2 - 5d + 80 \\ \\ y & \ge 104 \\ d^2 - 5d + 80 & \ge 104 \\ d^2 - 5d - 24 & \ge 0 \\ (d + 3)(d - 8) & \ge 0 \end{align*}

\begin{align*} d \le - 3 & \text{ or } d \ge 8 \\ \\ \therefore \text{Least no. of days} & = 8 \end{align*}

(i)

\begin{align*} h & = -16t^2 + 160t + 48 \\ \\ h & > 384 \phantom{00000} [\text{Height exceeds 384 m}] \\ -16t^2 + 160t + 48 & > 384 \\ -16t^2 + 160t - 336 & > 0 \\ 16t^2 - 160t + 336 & < 0 \\ t^2 - 10t + 21 & < 0 \\ (t - 3)(t - 7) & < 0 \end{align*}

$$ 3 < t < 7 $$

(ii)

\begin{align*} h & = -16t^2 + 160t + 48 \\ & = -16(t^2 - 10t) + 48 \\ & = -16 \left[ \left(t - {10 \over 2}\right)^2 - \left(10 \over 2\right)^2 \right] + 48 \\ & = -16 [ (t - 5)^2 - 25 ] + 48 \\ & = -16(t - 5)^2 + 400 + 48 \\ & = -16(t - 5)^2 + 448 \\ \\ \text{Maximum height} & = 448 \text{ m} \\ \\ \therefore \text{Object cannot reach } & 1000 \text{ m above the ground} \end{align*}

(a)

\begin{align} x^2 - px + p & = 0 \\ \\ [a = 1, b & = -p, c = p] \\ \\ b^2 - 4ac & > 0 \\ (-p)^2 - 4(1)(p) & > 0 \\ p^2 - 4p & > 0 \\ p(p - 4) & > 0 \end{align}

$$ p < 0 \text{ or } p > 4 $$

(b)

\begin{align} 9x^2 + 2px + 1 & = 0 \\ \\ [a = 9, b & = 2p, c = 1] \\ \\ b^2 - 4ac & \ge 0 \phantom{00000} [\text{2 real distinct roots or 2 real equal roots}]\\ (2p)^2 - 4(9)(1) & \ge 0 \\ 4p^2 - 36 & \ge 0 \\ p^2 - 9 & \ge 0 \\ (p + 3)(p - 3) & \ge 0 \end{align}

$$ p \le -3 \text{ or } p \ge 3 $$

(c)

\begin{align} px^2 - 2x + 2p + 1 & = 0 \\ \\ [a = p, b & = -2, c = 2p + 1] \\ \\ b^2 - 4ac & < 0 \\ (-2)^2 - 4(p)(2p + 1) & < 0 \\ 4 - 4p (2p + 1) & < 0 \\ 4 - 8p^2 - 4p & < 0 \\ -8p^2 - 4p + 4 & < 0 \\ 8p^2 + 4p - 4 & > 0 \\ 2p^2 + p - 1 & > 0 \\ (p + 1)(2p - 1) & > 0 \end{align}

$$ p < -1 \text{ or } p > {1 \over 2} $$

(d)

\begin{align} 2x^2 - 2px + p^2 + p & = 0 \\ \\ [a = 2, b & = -2p, c = p^2 + p] \\ \\ b^2 - 4ac & \ge 0 \phantom{00000} [\text{2 real distinct roots or 2 real equal roots}] \\ (-2p)^2 - 4(2)(p^2 + p) & \ge 0 \\ 4p^2 - 8(p^2 + p) & \ge 0 \\ 4p^2 - 8p^2 - 8p & \ge 0 \\ -4p^2 - 8p & \ge 0 \\ \\ 4p^2 + 8p & \le 0 \\ p^2 + 2p & \le 0 \\ p(p + 2) & \le 0 \end{align}

$$ -2 \le p \le 0 $$

\begin{align*} \text{Let } x & \text{ denote the width of the pool} \\ \\ \text{Length of pool} & = (2x + 3) \text{ m} \\ \\ \text{Area} & \le 54 \text{ m}^2 \\ x(2x + 3) & \le 54 \\ 2x^2 + 3x & \le 54 \\ 2x^2 + 3x - 54 & \le 0 \\ (x + 6)(2x - 9) & \le 0 \end{align*}

\begin{align*} -6 \le & \phantom{.} x \le 4.5 \\ \\ \text{Since width } & \text{should be at least 2 m, } \\ 2 \text{ m} \le & \text{ width} \le 4.5 \text{ m} \end{align*}

\begin{align*} x^2 + (n + 1)x + 4n & > 11 \\ x^2 + (n + 1)x + 4n - 11 & > 0 \phantom{00000} [\text{Always positive}] \\ \\ b^2 - 4ac & = (n + 1)^2 - 4(1)(4n - 11) \\ & = (n)^2 + 2(n)(1) + (1)^2 - 4(4n - 11) \\ & = n^2 + 2n + 1 - 16n + 44 \\ & = n^2 - 14n + 45 \\ \\ b^2 - 4ac & < 0 \phantom{00000} [\text{No real roots}] \\ n^2 - 14n + 45 & < 0 \\ (n - 5)(n - 9) & < 0 \end{align*}

$$ 5 < n < 9 $$

\begin{align*} k^2 x^2 - 2x + 1 & = kx \\ k^2 x^2 - kx - 2x + 1 & = 0 \\ k^2 x^2 + (-k - 2)x + 1 & = 0 \\ \\ b^2 - 4ac & = (-k - 2)^2 - 4(k^2)(1) \\ & = (-k)^2 - 2(-k)(2) + (2)^2 - 4k^2 \\ & = k^2 + 4k + 4 - 4k^2 \\ & = -3k^2 + 4k + 4 \\ \\ b^2 - 4ac & \ge 0 \phantom{00000} [\text{Two real distinct or equal roots}] \\ -3k^2 + 4k + 4 & \ge 0 \\ 3k^2 - 4k - 4 & \le 0 \\ (3k + 2)(k - 2) & \le 0 \end{align*}

\begin{align*} -{2 \over 3} & \le k \le 2 \\ \\ \text{Since } k \ne 0 \text{ (or else} & \text{ equation will not be quadratic)}, \\ -{2 \over 3} \le k < 0 & \text{ or } \phantom{.} 0 < k \le 2 \end{align*}

(a)

\begin{align} -2 < \phantom{.} & x < 4 \\ \\ \text{Working } & \text{backwards}, \\ (x + 2)(x - 4) & < 0 \\ x^2 - 4x + 2x - 8 & < 0 \\ x^2 - 2x - 8 & < 0 \\ x^2 - 2x & < 8 \\ \\ \therefore a = -2, & \phantom{.} b = 8 \end{align}

(b)

\begin{align} x < -2 \text{ or } & x > 3 \\ \\ \text{Working } & \text{backwards}, \\ (x + 2)(x - 3) & > 0 \\ x^2 - 3x + 2x - 6 & > 0 \\ x^2 - x - 6 & > 0 \\ x^2 - 6 & > x \\ 2x^2 - 12 & > 2x \\ \\ \therefore a = -12, & \phantom{.} b = 2 \end{align}

(i)

\begin{align} d & \le 80 \\ \\ \text{Since } & d = 0.15v^2 + v, \\ 0.15v^2 + v & \le 80 \\ 0.15v^2 + v - 80 & \le 0 \\ 1.5v^2 + 10v - 800 & \le 0 \phantom{0000000} [\text{Multiply by 10}] \\ \\ 3v^2 + 20v - 1600 & \le 0 \text{ (Shown)} \end{align}

(ii)

\begin{align} 3v^2 + 20v - 1600 & \le 0 \\ (3v + 80)(v - 20) & \le 0 \end{align}

$$ - {80 \over 3} \le v \le 20 $$ $$ \therefore \text{Maximum speed} = 20 \text{ m/s} $$

(i)

\begin{align*} \text{Since } & k < 0, \\ -6k & > 0 \\ - 6k + 4 & > 0 + 4 \\ 4 - 6k & > 4 \\ \\ \therefore \text{Coefficient} & \text{ of } x^2 \text{ is positive} \end{align*}

(ii)

\begin{align*} f(x) & = 0 \\ (4 - 6k)x^2 + (8 - 2k)x + 4 & = 0 \\ \\ b^2 - 4ac & = (8 - 2k)^2 - 4(4 - 6k)(4) \\ & = (8)^2 - 2(8)(2k) + (2k)^2 - (16 - 24k)(4) \\ & = 64 - 32k + 4k^2 - (64 - 96k) \\ & = 64 - 32k + 4k^2 - 64 + 96k \\ & = 4k^2 + 64k \\ \\ b^2 - 4ac & < 0 \phantom{00000} [\text{No real roots}] \\ 4k^2 + 64k & < 0 \\ 4k(k + 16) & < 0 \end{align*}

$$ -16 < k < 0 $$

(a)

\begin{align*} y & = 2x + m \phantom{00000} \text{--- (1)} \\ y & = 2x^2 - 6x + 5 \phantom{00000} \text{--- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ 2x + m & = 2x^2 - 6x + 5 \\ 0 & = 2x^2 - 8x + 5 - m \\ \\ b^2 - 4ac & = (-8)^2 - 4(2)(5 - m) \\ & = 64 - 8(5 - m) \\ & = 64 - 40 + 8m \\ & = 24 + 8m \\ \\ b^2 - 4ac & = 0 \phantom{00000} [\text{1 real root, since line is tangent to curve}] \\ 24 + 8m & = 0 \\ 8m & = -24 \\ m & = {-24 \over 8} \\ m & = -3 \end{align*}

(b)

\begin{align*} (k + 2)x^2 - 2\sqrt{2k + 1} x + 4 & = 0 \\ \\ b^2 - 4ac & = \left(2 \sqrt{2k + 1}\right)^2 - 4(k + 2)(4) \\ & = 4(2k + 1) - (4k + 8)(4) \\ & = 8k + 4 - (16k + 32) \\ & = 8k + 4 - 16k - 32 \\ & = -8k - 28 \\ \\ \text{Since } k & > 0, \\ -8k & < 0 \\ -8k - 28 & < 0 - 28 \\ -8k - 28 & < -28 \\ \implies b^2 - 4ac & < -28 \\ \\ \text{Since } b^2 - 4ac < 0 \text{ for } & k > 0, \text{ the equation has no real roots} \end{align*}

(a)

\begin{align*} 2x + 1 & \le 8 \\ 2x & \le 7 \\ x & \le {7 \over 2} \\ x & \le 3.5 \end{align*}

(b)

\begin{align*} x + 2 & \le x^2 \\ 0 & \le x^2 - x - 2 \\ 0 & \le (x + 1)(x - 2) \end{align*}

$$ x \le -1 \text{ or } x \ge 2 $$

Combine the number line with the number line from (a) and identify the overlapping region(s):

$$ x \le -1 \text{ or } 2 \le x \le 3.5$$

\begin{align} 2 < \phantom{.} & x < k \\ \\ \text{Working } & \text{backwards,} \\ (x - 2)(x - k) & < 0 \\ x^2 - kx - 2x + 2k & < 0 \\ 2x^2 - 2kx - 4x + 4k & < 0 \\ 2x^2 + (-2k - 4)x + 4k & < 0 \\ \\ \therefore y = 2x^2 + & (-2k - 4)x + 4k \\ \\ \text{By comparison with } y & = 2x^2 + px + 16, \\ \\ 4k & = 16 \\ k & = {16 \over 4} \\ & = 4 \\ \\ \\ -2k - 4 & = p \\ -2(4) - 4 & = p \\ -12 & = p \end{align}

(i)

\begin{align} 1 + 2x + {3 \over 2}(x - 1)x & = 100000 \\ 2 + 4x + 3(x - 1)x & = 200000 \\ 2 + 4x + 3x(x - 1) & = 200000 \\ 2 + 4x + 3x^2 - 3x & = 200000 \\ 3x^2 + 4x - 3x + 2 - 200000 & = 0 \\ 3x^2 + x - 199998 & = 0 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-1 \pm \sqrt{1^2 - 4(3)(-199998)} \over 2(3)} \\ & = {-1 \pm \sqrt{2399977} \over 6} \\ & = 258.0309 \text{ or } -258.3643 \\ & \approx 258.03 \text{ or } -258.36 \end{align}

(ii)

\begin{align} 1 + 2x + {3 \over 2}(x - 1)x & \le 100000 \\ & . \\ & . \phantom{000} [\text{Same steps as part (i)}] \\ & . \\ \\ 3x^2 + x - 199998 & \le 0 \end{align}
Use the solutions from (i):
$$ -258.36 \le x \le 258.03 $$

(iii) Notice the expression is the same as the expression in (i), except the variable x is replaced by n

\begin{align} \text{Since there can only be a } & \text{maximum of 100000 lines}, \\ \\ 1 + 2n + {3 \over 2}n(n - 1) & \le 100000 \\ & . \\ & . \phantom{000} [\text{Same steps as part (ii)}] \\ & . \\ \\ 3n^2 + n - 199998 & \le 0 \end{align}
$$ -258.36 \le n \le 258.03 $$ $$ \therefore \text{Maximum no. of users} = 258.03 \approx 258 $$