Math Past Paper PDF

## Sheet 1 ### 1] #### a) - $F(x) = |x| \implies$ Domain: $F \in (-\infty, \infty] $ and $R \in [0, \infty)$ #### b) - $F(x) = \frac{3}{2x - 5} \implies 2x - 5 = 0 \implies x = \frac{5}{2}$ - $R - \frac{3}{2}$ #### c) - $F(x) = \sqrt{4 + 3x} \implies 4 + 3x \ge 0 \implies x \ge \frac{-4}{3}$ - $ \...

## Sheet 1 ### 1] #### a) - $F(x) = |x| \implies$ Domain: $F \in (-\infty, \infty] $ and $R \in [0, \infty)$ #### b) - $F(x) = \frac{3}{2x - 5} \implies 2x - 5 = 0 \implies x = \frac{5}{2}$ - $R - \frac{3}{2}$ #### c) - $F(x) = \sqrt{4 + 3x} \implies 4 + 3x \ge 0 \implies x \ge \frac{-4}{3}$ - $ \implies x \in [ \frac{-4}{3}, \infty)$ ### 2) - $H(X) = \begin{cases} 2x + 2 & \text{for } x < 1\\2x - 4 & \text{for } x \ge 1 \end{cases}$ #### a) $\lim_{x \to 2}H(x)$ - $\lim_{x \to 2} H(x) = 2x + 2$ - $\lim_{x \to 2} H(x) = 2 * 2 + 2 = 6$ #### b) $\lim_{x \to 1}H(x)$ - $\lim_{x \to 1}H(x) = 2x + 2$ - $\lim_{x \to 1}H(x) = 2 * 1 + 2 = 4$ #### c) $\lim_{x \to 2}H(x)$ - $\lim_{x \to 2} H(x) = 2x - 4$ - $\lim_{x \to 2} H(x) = 2 * 2 - 4 = 0$ #### d) $\lim_{x \to -3}H(x)$ - $\lim_{x \to -3}H(x) = 2x - 4$ - $\lim_{x \to -3}H(x) = 2 * (-3) - 4 = -10$ ## Sheet 2 ### 3) #### a) - $\lim_{x \to 1} (1 - 6x)\implies (1 - 6 * 1) = -5$ #### b) - $\lim_{x \to 2} \frac{x}{x -2} \implies \frac{2}{2 - 2} = \infty$ #### c) - $\lim_{x \to 2}(x^4 + 5x^3 + x^2 - 7)\implies 16 - 40 + 4 - 7 = -27$ #### d) - $\lim_{x \to 0^+} \sqrt{x^2 - 3x + 2} \implies \sqrt{2}$ #### e) - if $\lim_{x \to 0} F(x) = \frac{1}{2}$ and $\lim_{x \to 0} g(x) = \frac{1}{2}$ #### f) - $\lim_{x \to 0} (F(x) + g(x)) \implies \frac{1}{2} + \frac{1}{2} = 1$ #### g) - $\lim_{x \to 0} (F(x) - 2g(x)) \implies \frac{1}{2} - 2(\frac{1}{2}) = -\frac{1}{2}$ #### h) - $\lim_{x \to 0} (F(x) * g(x)) \implies \frac{1}{2} * \frac{1}{2} = \frac{1}{4}$ #### i) - $\lim_{x \to 0} \frac{F(x)}{g(x)} \implies \frac{\frac{1}{2}}{\frac{1}{2}} = 1$ ### 4) - Let r(x) = $\frac{x^2 - x - 12}{(x + 3) (x + 4)}$ - Find $\lim_{x \to -3}r(x)$ - $\frac{x^2 - x - 12}{(x + 3) (x + 4)}$ - $= \frac{(x + 3)(x - 4)}{(x + 3)(x + 4)}$ - $ = x - 4$ - $\lim_{x \to -3} x - 4 = -3 - 4 = -7$ ## Sheet 3 ### 5) - Let $F(x) = \frac{x^2 - 16}{x - 4}$ - $\implies (x^2 - 16) = (x + 4)(x - 4)$ - $\frac{(x + 4)(x - 4)}{x - 4} = x + 4$ - $\lim_{x \to 4}x + 4 \implies 4 + 4 = 8 $ ### 6) #### a) - $\lim_{x \to \infty} \frac{5x + 3}{3x - 2} \implies \frac{5x + 3}{3x - 2} \implies \frac{x(5 + \frac{3}{x})}{x(3 - \frac{2}{x})} = \frac{(5 + \frac{3}{x})}{(3 - \frac{2}{x})}$ - $\implies \frac{5 + \frac{3}{\infty} }{3 - \frac{2}{\infty}} = \frac{5 + 0}{3 + 0} = \frac{5}{3}$ #### b) - $\lim_{x \to \infty} \frac{1}{x - 8} \implies \frac{1}{\infty - 8} = \frac{1}{\infty} = 0$ ### 7) #### a) - $\lim_{x \to \infty} \frac{10x + 100}{x^2 - 30} \implies \frac{x(10 + \frac{100}{x})}{x(x - \frac{30}{x})} = \frac{(10 + \frac{100}{x})}{(x - \frac{30}{x})} = \frac{(10 + \frac{100}{x})}{(1 - \frac{30}{x^2})}$ - $\frac{10 + \frac{100}{\infty}}{1 - \frac{30}{\infty}} = \frac{10 + 0}{1 - 0} = 10$ #### b) - $\lim_{x \to \infty} \frac{x^2 + x}{x^2 - 1} \implies \frac{x(x + 1)}{x(x - \frac{1}{x})} = \frac{(x + 1)}{(x - \frac{1}{x})} = \frac{(1 + \frac{1}{x})}{(1 - \frac{1}{x^2})}$ - $\frac{1 + \frac{1}{\infty}}{1 - \frac{1}{\infty}} = \frac{1 + 0}{1 - 0} = 1$ ### 8) - $F(x) = x^3$, $g(x) = 1 + x^2$ #### a) - $(F o g)(x) \implies F(g(x)) = F (1 + x^2)$ - $\implies (1 + x^2)^3$ #### b) - $(g o F)(x) \implies g(F(x)) = (x^3) g$ - $\implies g(x^3) = 1 + x^6$ ### 9) - $F(x) = \sqrt{x}$ $g(x) = x - 1$ #### a) - $(F o g)(x) \implies F(g(x)) = F(x - 1)$ - $\implies \sqrt{x - 1}$ #### b) - $(g o F)(x) \implies g(F(x)) = g(\sqrt{x})$ - $\implies \sqrt{x} - 1$ ## Sheet 4 ### 10) - $F(x) = 2x + 3$ $g(x) = -x^2 + 1$ #### a) - $(F o g)(x) \implies F(g(x)) = F (-x^2 + 1)$ - $\implies 2(-x^2 + 1) + 3$ - $\implies -2x^2 + 2 + 3$ - $\implies -2x^2 + 5$ ### 11) - $F(x) = |4x^2 + 2x - 5|$ #### a) - $F(x) = F(g(x))$ - $F(x) = \sqrt{4x^2 + 2x - 5}$ - $g(x) = 4x^2 + 2x - 5$ - $F(x) = \sqrt{x}$ #### b) - $F(x) = F(g(x)) = \sqrt{4x^2 + 2x - 5}$ - $F(x) = \sqrt{x}$ - $g(x) = 4x^2 + 2x - 5$