(a)

\begin{align*} x^2 + y^2 & = 34 \phantom{00000} \text{--- (1)} \\ \\ y + 3x & = 14 \\ y & = 14 - 3x \phantom{00000} \text{--- (2)} \\ \\ \text{Substitute } & \text{(2) into (1),} \\ x^2 + (14 - 3x)^2 & = 34 \\ x^2 + (14)^2 - 2(14)(3x) + (3x)^2 & = 34 \\ x^2 + 196 - 84x + 9x^2 & = 34 \\ 10x^2 - 84x + 196 & = 34 \\ 10x^2 - 84x + 162 & = 0 \\ 5x^2 - 42x + 81 & = 0 \\ (x - 3)(5x - 27) & = 0 \\ \\ x - 3 = 0 \phantom{0} & \text{ or } \phantom{0} 5x - 27 = 0 \\ x = 3 \phantom{0} & \phantom{00000000} 5x = 27 \\ & \phantom{000000000} x = {27 \over 5} \\ \\ \text{Substitute } & x = 3 \text{ into (2),} \\ y & = 14 - 3(3) \\ y & = 5 \\ \\ \text{Substitute } & x = {27 \over 5} \text{ into (2),} \\ y & = 14 - 3 \left(27 \over 5\right) \\ y & = -{11 \over 5} \\ \\ \therefore x = 3, y = 5 & \text{ or } x = {27 \over 5}, y = -{11 \over 5} \end{align*}

(b)

\begin{align*} 4x^2 - 4xy + y^2 & = 1 \phantom{00000} \text{--- (1)} \\ \\ 4x + y & = 7 \\ y & = 7 - 4x \phantom{00000} \text{--- (2)} \\ \\ \text{Substitute } & \text{(2) into (1),} \\ 4x^2 - 4x(7 - 4x) + (7 - 4x)^2 & = 1 \\ 4x^2 - 28x + 16x^2 + (7)^2 - 2(7)(4x) + (4x)^2 & = 1 \\ 20x^2 - 28x + 49 - 56x + 16x^2 & = 1 \\ 36x^2 - 84x + 49 & = 1 \\ 36x^2 - 84x + 48 & = 0 \\ 3x^2 - 7x + 4 & = 0 \\ (x - 1)(3x - 4) & = 0 \\ \\ x - 1 = 0 \phantom{0} & \text{ or } \phantom{0} 3x - 4 = 0 \\ x = 1 \phantom{0} & \phantom{0000000} 3x = 4 \\ & \phantom{00000000} x = {4 \over 3} \\ \\ \text{Substitute } & x = 1 \text{ into (2),} \\ y & = 7 - 4(1) \\ y & = 3 \\ \\ \text{Substitute } & x = {4 \over 3} \text{ into (2),} \\ y & = 7 - 4\left(4 \over 3\right) \\ y & = {5 \over 3} \\ \\ \therefore x = 1, y = 3 & \text{ or } x = {4 \over 3}, y = {5 \over 3} \end{align*}

(c)

\begin{align*} 4x + 6y & = 2 \\ 4x & = 2 - 6y \\ x & = 0.5 - 1.5y \phantom{00000} \text{--- (1)} \\ \\ 3y^2 - x^2 & = 2 \phantom{00000} \text{--- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ 3y^2 - (0.5 - 1.5y)^2 & = 2 \\ 3y^2 - [ (0.5)^2 - 2(0.5)(1.5y) + (1.5y)^2 ] & = 2 \\ 3y^2 - (0.25 - 1.5y + 2.25y^2) & = 2 \\ 3y^2 - 0.25 + 1.5y - 2.25y^2 & = 2 \\ 0.75y^2 + 1.5y - 0.25 & = 2 \\ 0.75y^2 + 1.5y - 2.25 & = 0 \\ y^2 + 2y - 3 & = 0 \\ (y - 1)(y + 3) & = 0 \\ \\ y - 1 = 0 \phantom{0} & \text{ or } \phantom{0} y + 3 = 0 \\ y = 1 \phantom{0} & \phantom{0000000} y = -3 \\ \\ \text{Substitute } & y = 1 \text{ into (1),} \\ x & = 0.5 - 1.5(1) \\ x & = -1 \\ \\ \text{Substitute } & y = -3 \text{ into (1),} \\ x & = 0.5 - 1.5(-3) \\ x & = 5 \\ \\ \therefore x = -1, y = 1 & \text{ or } x = 5, y = -3 \end{align*}

\begin{align*} \text{Expression} & = ax^2 + bx + 1 \\ \\ \text{When } x = 2 & \text{ and expression} = 1, \\ a(2)^2 + b(2) + 1 & = 1 \\ 4a + 2b + 1 & = 1 \\ 4a + 2b & = 0 \\ 2b & = -4a \\ b & = -2a \phantom{00000} \text{--- (1)} \\ \\ \text{When } x = 3 & \text{ and expression} = 4, \\ a(3)^2 + b(3) + 1 & = 4 \\ 9a + 3b + 1 & = 4 \\ 9a + 3b & = 3 \phantom{00000} \text{--- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ 9a + 3(-2a) & = 3 \\ 9a - 6a & = 3 \\ 3a & = 3 \\ a & = {3 \over 3} \\ a & = 1 \\ \\ \text{Substitute } & a = 1 \text{ into (1),} \\ b & = -2(1) \\ b & = -2 \\ \\ \therefore \text{Expression} & = (1)x^2 + (-2)x + 1 \\ & = x^2 - 2x + 1 \\ \\ \text{When } & x = 4, \\ \text{Expression} & = (4)^2 - 2(4) + 1 \\ & = 9 \end{align*}

(i)

\begin{align*} y & = -x + 6 \phantom{00000} \text{--- (1)} \\ y & = 5x^2 + 6x - 3k \phantom{00000} \text{--- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ -x + 6 & = 5x^2 + 6x - 3k \\ 0 & = 5x^2 + 7x - 3k - 6 \\ 0 & = 5x^2 + 7x + (-3k - 6) \\ \\ b^2 - 4ac & = (7)^2 - 4(5)(-3k-6) \\ & = 49 - 20(-3k-6) \\ & = 49 + 60k + 120 \\ & = 60k + 169 \\ \\ b^2 - 4ac & < 0 \phantom{00000} [\text{No real roots, since line does not intersect curve}] \\ 60k + 169 & < 0 \\ 60k & < - 169 \\ k & < -{169 \over 60} \end{align*}

(ii) For line to be tangent to the curve, b² - 4ac = 0

\begin{align*} b^2 - 4ac & = 0 \\\ 60k + 169 & = 0 \\ 60k & = - 169 \\ k & = -{169 \over 60} \end{align*}

\begin{align*} \text{Let } x \text{ denote the} & \text{ radius of the smaller circle} \\ \text{Let } y \text{ denote the} & \text{ radius of the larger circle} \\ \\ 2x + 2y & = 20 \\ x + y & = 10 \\ x & = 10 - y \phantom{00000} \text{--- (1)} \\ \\ \text{Area of circle} & = \pi r^2 \\ \pi (x)^2 + \pi (y)^2 & = 52 \pi \\ x^2 + y^2 & = 52 \phantom{00000} \text{--- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ (10 - y)^2 + y^2 & = 52 \\ (10)^2 - 2(10)(y) + (y)^2 + y^2 & = 52 \\ 100 - 20y + y^2 + y^2 & = 52 \\ 2y^2 - 20y + 100 & = 52 \\ 2y^2 - 20y + 48 & = 0 \\ y^2 - 10y + 24 & = 0 \\ (y - 4)(y - 6) & = 0 \\ \\ y - 4 = 0 \phantom{0} & \text{ or } y - 6 = 0 \\ y = 4 \phantom{0} & \phantom{000000} y = 6 \\ \\ \text{Substitute } & y = 4 \text{ into (1),} \\ x & = 10 - (4) \\ x & = 6 \\ \\ \text{Substitute } & y = 6 \text{ into (1),} \\ x & = 10 - (6) \\ x & = 4 \\ \\ \therefore \text{Radius of larger circle} & = 6 \text{ cm} \\ \\ \text{Circumference of larger circle} & = 2\pi r \\ & = 2\pi (6) \\ & = 12\pi \text{ cm} \end{align*}

(i)

\begin{align*} 2y & = 3x - 1 \\ y & = 1.5x - 0.5 \phantom{00000} \text{--- (1)} \\ \\ 4x^2 + 9y^2 & = 15xy \phantom{00000} \text{--- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ 4x^2 + 9(1.5x - 0.5)^2 & = 15x(1.5x - 0.5) \\ 4x^2 + 9[ (1.5x)^2 - 2(1.5x)(0.5) + (0.5)^2 ] & = 22.5x^2 - 7.5x \\ 4x^2 + 9 (2.25x^2 - 1.5x + 0.25) & = 22.5x^2 - 7.5x \\ 4x^2 + 20.25x^2 - 13.5x + 2.25 & = 22.5x^2 - 7.5x \\ 24.25x^2 - 13.5x + 2.25 & = 22.5x^2 - 7.5x \\ 1.75x^2 - 6x + 2.25 & = 0 \\ 7x^2 - 24x + 9 & = 0 \\ (x - 3)(7x - 3) & = 0 \\ \\ x - 3 = 0 \phantom{0} & \text{ or } 7x - 3 = 0 \\ x = 3 \phantom{0} & \phantom{000000} 7x = 3 \\ & \phantom{0000000} x = {3 \over 7} \\ \\ \text{Substitute } & x = 3 \text{ into (1),} \\ y & = 1.5(3) - 0.5 \\ y & = 4 \\ \\ \text{Substitute } & x = {3 \over 7} \text{ into (1),} \\ y & = 1.5 \left(3 \over 7\right) - 0.5 \\ y & = {1 \over 7} \\ \\ \therefore x = 3, y = 4 & \text{ or } x = {3 \over 7}, y = {1 \over 7} \end{align*}

(ii)

$$ (3, 4) \text{ and } \left({3 \over 7}, {1 \over 7}\right) $$

\begin{align*} {2 \over x} + {3 \over y} & = 13 \phantom{00000} \text{--- (1)} \\ \\ 2x + 3y & = 2 \\ 2x & = 2 - 3y \\ x & = 1 - 1.5y \phantom{00000} \text{--- (2)} \\ \\ \text{Substitute } & \text{(2) into (1),} \\ {2 \over 1 - 1.5y} + {3 \over y} & = 13 \\ {2y \over y(1 - 1.5y)} + {3(1 - 1.5y) \over y(1 - 1.5y)} & = 13 \\ {2y + 3(1 - 1.5y) \over y(1 - 1.5y)} & = 13 \\ {2y + 3 - 4.5y \over y - 1.5y^2} & = 13 \\ {3 - 2.5y \over y - 1.5y^2} & = 13 \\ 3 - 2.5y & = 13(y - 1.5y^2) \\ 3 - 2.5y & = 13y - 19.5y^2 \\ 3 - 2.5y & = -19.5y^2 + 13y \\ 0 & = -19.5y^2 + 15.5y - 3 \\ 0 & = 19.5y^2 - 15.5y + 3 \\ 0 & = 39y^2 - 31y + 6 \\ 0 & = (3y - 1)(13y - 6) \\ \\ 3y - 1 = 0 \phantom{0} & \text{ or } 13y - 6 = 0 \\ 3y = 1 \phantom{0} & \phantom{000000} 13y = 6 \\ y = {1 \over 3} \phantom{.} & \phantom{00000000} y = {6 \over 13} \\ \\ \text{Substitute } & y = {1 \over 3} \text{ into (2),} \\ x & = 1 - 1.5 \left(1 \over 3\right) \\ x & = {1 \over 2} \\ \\ \text{Substitute } & x = {6 \over 13} \text{ into (2),} \\ x & = 1 - 1.5 \left(6 \over 13\right) \\ x & = {4 \over 13} \\ \\ \therefore \text{Points of intersection are } & \left({1 \over 2}, {1 \over 3} \right) \text{ and } \left({4 \over 13}, {6 \over 13}\right) \end{align*}

\begin{align*} y & = kx + 2k^2 \phantom{00000} \text{--- (1)} \\ y & = 2x^2 - 3kx + k - 10 \phantom{00000} \text{--- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ kx + 2k^2 & = 2x^2 - 3kx + k - 10 \\ 0 & = 2x^2 - 4kx - 2k^2 + k - 10 \\ 0 & = 2x^2 - 4kx + (-2k^2 + k - 10) \\ \\ b^2 - 4ac & = (-4k)^2 - 4(2)(-2k^2 + k - 10) \\ & = 16k^2 - 8(-2k^2 + k - 10) \\ & = 16k^2 + 16k^2 - 8k + 80 \\ & = 32k^2 - 8k + 80 \\ & = 32\left(k^2 - {1 \over 4}k\right) + 80 \\ & = 32 \left[ \left(k - {1 \over 8}\right)^2 - \left(1 \over 8\right)^2 \right] + 80 \\ & = 32 \left[ \left(k - {1 \over 8}\right)^2 - {1 \over 64} \right] + 80 \\ & = 32 \left(k - {1 \over 8}\right)^2 - {1 \over 2} + 80 \\ & = 32 \left(k - {1 \over 8}\right)^2 + 79{1 \over 2} \\ \\ \text{For all real} & \text{ values of } k, \\ \left(k - {1 \over 8}\right)^2 & \ge 0 \\ 32\left(k - {1 \over 8}\right)^2 & \ge 32(0) \\ 32\left(k - {1 \over 8}\right)^2 & \ge 0 \\ 32\left(k - {1 \over 8}\right)^2 + 79{1 \over 2} & \ge 79{1 \over 2} \\ \implies b^2 - 4ac & \ge 79{1 \over 2} \\ \\ \text{Since } b^2 - 4ac > 0 \text{ for all} & \text{ real values of } k, \text{ line intersects the curve for all real values of } k \end{align*}

\begin{align*} y & = 3px^2 - 12x + p \\ \\ \text{Let } & y = 0, \\ 0 & = 3px^2 - 12x + p \\ \\ b^2 - 4ac & = (-12)^2 - 4(3p)(p) \\ & = 144 - 12p^2 \\ \\ b^2 - 4ac & = 0 \phantom{00000} [\text{One real root since curve touches } x \text{-axis only once}] \\ 144 - 12p^2 & = 0 \\ -12p^2 & = -144 \\ 12p^2 & = 144 \\ p^2 & = {144 \over 12} \\ p^2 & = 12 \\ p & = \pm \sqrt{12} \\ p & = \pm \sqrt{4} \sqrt{3} \\ p & = \pm 2\sqrt{3} \\ p & = -2\sqrt{3} \text{ or } 2 \sqrt{3} \text{ (Reject, since } p \text{ is a negative number}) \end{align*}

\begin{align*} (3x + 1) + (2x + 4) + y & = 30 \\ 5x + y + 5 & = 30 \\ 5x + y & = 25 \\ y & = 25 - 5x \phantom{00000} \text{--- (1)} \\ \\ \text{By Pythagoras} & \text{ theorem,} \\ (3x + 1)^2 & = (2x + 4)^2 + y^2 \\ (3x)^2 + 2(3x)(1) + (1)^2 & = (2x)^2 + 2(2x)(4) + (4)^2 + y^2 \\ 9x^2 + 6x + 1 & = 4x^2 + 16x + 16 + y^2 \\ 5x^2 - 10x - 15 & = y^2 \phantom{00000} \text{--- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ 5x^2 - 10x - 15 & = (25 - 5x)^2 \\ 5x^2 - 10x - 15 & = (25)^2 - 2(25)(5x) + (5x)^2 \\ 5x^2 - 10x - 15 & = 625 - 250x + 25x^2 \\ 0 & = 20x^2 - 240x + 640 \\ 0 & = 5x^2 - 60x + 160 \\ 0 & = x^2 - 12x + 32 \\ 0 & = (x - 4)(x - 8) \\ \\ x - 4 = 0 \phantom{0} & \text{ or } x - 8 = 0 \\ x = 4 \phantom{0} & \phantom{000000} x = 8 \\ \\ \text{Substitute } & x = 4 \text{ into (1),} \\ y & = 25 - 5(4) \\ y & = 5 \\ \\ \text{Substitute } & x = 8 \text{ into (1),} \\ y & = 25 - 5(8) \\ y & = -15 \text{ (Reject, since } y > 0 \text{ in this context}) \\ \\ \text{Area of triangle} & = {1 \over 2} \times b \times h \\ & = {1 \over 2} \times [2(4) + 4] \times (5) \\ & = 30 \text{ cm}^2 \end{align*}

(a)(i)

\begin{align*} R_A & = R_B + 1.2 \phantom{00000} \text{--- (1)} \\ \\ {1 \over R_T} & = {1 \over R_A} + {1 \over R_B} \\ {1 \over 1.25} & = {1 \over R_A} + {1 \over R_B} \\ {4 \over 5} & = {1 \over R_A} + {1 \over R_B} \phantom{00000} \text{--- (2)} \end{align*}

(a)(ii)

\begin{align*} \text{Substitute } & \text{(1) into (2),} \\ {4 \over 5} & = {1 \over R_B + 1.2} + {1 \over R_B} \\ {4 \over 5} & = {R_B \over R_B(R_B + 1.2)} + {R_B + 1.2 \over R_B(R_B + 1.2)} \\ {4 \over 5} & = {2R_B + 1.2 \over R_B\phantom{.}^2 + 1.2R_B} \\ 4(R_B\phantom{.}^2 + 1.2R_B) & = 5(2R_B + 1.2) \\ 4R_B\phantom{.}^2 + 4.8R_B & = 10R_B + 6 \\ 4R_B \phantom{.}^2 - 5.2R_B - 6 & = 0 \\ \\ R_B & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-(-5.2) \pm \sqrt{(-5.2)^2 - 4(4)(-6)} \over 2(4)} \\ & = {5.2 \pm \sqrt{ 123.04} \over 8} \\ & = 2.0365 \text{ or } -0.73654 \text{ (Reject, since } R_B > 0) \\ & \approx 2.04 \text{ ohms} \\ \\ \text{Substitute } & R_B = 2.0365 \text{ into } (1), \\ R_A & = 2.0365 + 1.2 \\ R_A & = 3.2365 \\ R_A & \approx 3.24 \text{ ohms} \end{align*}

(b)

\begin{align*} R_B & = R_A + k \phantom{00000} \text{--- (1)} \\ \\ {4 \over 5} & = {1 \over R_A} + {1 \over R_B} \phantom{00000} \text{--- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ {4 \over 5} & = {1 \over R_A} + {1 \over R_A + k} \\ {4 \over 5} & = {R_A + k \over R_A(R_A + k)} + {R_A \over R_A(R_A + k))} \\ {4 \over 5} & = {2R_A + k \over R_A\phantom{.}^2 + kR_A} \\ 4(R_A\phantom{.}^2 + kR_A) & = 5(2R_A + k) \\ 4R_A\phantom{.}^2 + 4kR_A & = 10R_A + 5k \\ 4R_A \phantom{.}^2 + 4kR_A - 10R_A - 5k & = 0 \\ \\ b^2 - 4ac & = (4k - 10)^2 - 4(4)(-5k) \\ & = (4k)^2 - 2(4k)(10) + (10)^2 + 80k \\ & = 16k^2 - 80k + 100 + 80k \\ & = 16k^2 + 100 \\ & = 16(k)^2 + 100 \phantom{0000} [\text{Positive for all real values of } k] \\ \\ \text{Let } & k = 2, \\ 4R_A \phantom{.}^2 + 4(2)R_A - 10R_A - 5(2) & = 0 \\ 4R_A \phantom{.}^2 - 2R_A - 10 & = 0 \\ 2R_A \phantom{.}^2 - R_A - 5 & = 0 \\ \\ R_A & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-(-1) \pm \sqrt{(-1)^2 - 4(2)(-5)} \over 2(2)} \\ & = {1 \pm \sqrt{41} \over 4} \\ & = 1.8507 \text{ or } -1.3507 \text{ (Reject, since } R_A > 0) \\ & \approx 1.85 \\ \\ \text{Substitute } & R_A = 1.8507 \text{ and } k = 2 \text{ into } (1), \\ R_B & = 1.8507 + 2 \\ R_B & = 3.8507 \\ R_B & \approx 3.85 \\ \\ \therefore k =2, R_A & = 1.85, R_B = 3.85 \end{align*}

\begin{align*} \text{General equation of quadratic curve: } \phantom{0} y & = a(x - h)^2 + k \\ \\ y & = a(x - 4)^2 + 9 \phantom{000} [\text{Maximum point } (4, 9)] \\ \\ \text{Using } & (0, 1), \\ 1 & = a(0 - 4)^2 + 9 \\ 1 & = a(16) + 9 \\ 1 & = 16a + 9 \\ -8 & = 16a \\ {-8 \over 16} & = a \\ -0.5 & = a \\ \\ \therefore y & = -0.5(x - 4)^2 + 9 \phantom{000} \text{--- (1)} \\ \\ \\ \text{General equation of line: } \phantom{0} y & = mx + c \\ \\ \text{Using } & (0, 0) \phantom{00000} [\text{Line passes through origin}] \\ 0 & = m(0) + c \\ 0 & = c \\ \\ \therefore y & = mx \phantom{000} \text{--- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ mx & = -0.5(x - 4)^2 + 9 \\ mx & = -0.5[(x)^2 - 2(x)(4) + (4)^2] + 9 \\ mx & = -0.5(x^2 - 8x + 16) + 9 \\ mx & = -0.5x^2 + 4x - 8 + 9 \\ mx & = -0.5x^2 + 4x + 1 \\ 0 & = -0.5x^2 + 4x - mx + 1 \\ \\ b^2 - 4ac & = (4 - m)^2 - 4(-0.5)(1) \\ & = (4)^2 - 2(4)(m) + (m)^2 + 2 \\ & = 16 - 8m + m^2 + 2 \\ & = m^2 - 8m + 18 \\ & = \left(m - {8 \over 2}\right)^2 - \left(8 \over 2\right)^2 + 18 \\ & = (m - 4)^2 - 16 + 18 \\ & = (m - 4)^2 + 2 \phantom{00000} [\text{Positive for all values of } m] \\ \\ \text{Since } b^2 - 4ac > 0 \text{ for all real values of } m, & \text{ the line will intersect the curve at two distinct points} \\ \\ \therefore \text{Possible value of gradient, } m & = 1 \end{align*}