$$\begin{gathered} Given that {\log }_{a}b=2.75 and {\log }_{a}c=0.25,find the value of: \\ i\left){\log }_a\left(\frac{b}{c}\right) \\ ii\right){\log }_a{\left(bc\right)}^2 State whether the following are ODD,EVEN or neither. \\ i)f\left(\mathcal{x}\right)=3\mathcal{x}+1 \\ ii)f\left(\mathcal{x}\right)={\mathcal{x}}^4-{\mathcal{x}}^2 \\ The first term of geometric sequence is 2 and the fourth term is 128. \\ Find the common ratio. \end{gathered}$$

$$\begin{gathered} For the following sequence:36,24,16..... \\ Determine whether it is an Arithmetic or Geometric Progression, showing reasons. \\ Find \underset{\mathcal{x}\to 4}{\lim }\frac{{\mathcal{x}}^2-\mathcal{x}-12}{\mathcal{x}-4} \\ Differentiate: \\ i)4{\mathcal{x}}^2-\mathcal{x}+1 \\ ii)\frac{3}{\sqrt{\mathcal{x}}} \\ iii\left)\frac{3{\mathcal{x}}^5-2{\mathcal{x}}^3+5}{{\mathcal{x}}^2} \\ iv\right)\sqrt{3{\mathcal{x}}^2-3} \\ v)\frac{3{\mathcal{x}}^2}{5-3\mathcal{x}} \end{gathered}$$

$$\begin{gathered} If f\left(\mathcal{x}\right)=4\mathcal{x}-7, find f(a+1) \\ If \cos A = 0 \cdot 407 find A, where A is an acute angle. \end{gathered}$$

$$\begin{gathered} i)Find all the x \& y-intercepts for the curve with the equation\mathcal{y}={\mathcal{x}}^3-1. \\ ii)Neatly sketch the curve \mathcal{y}={\mathcal{x}}^3-1 , showing the information from part i). \\ iii)Find the equation of the \tan gent to the curve \mathcal{y}={\mathcal{x}}^3-1 at the point where \mathcal{x}=1. \\ If f\left(\mathcal{x}\right)=15{\mathcal{x}}^{-2}-9{\mathcal{x}}^3.find the value of f \text{'}( -1). \\ Differentiate f\left(\mathcal{x}\right)={\mathcal{x}}^2-3\mathcal{x} by first principles u\sin g the definition \\ f\text{'}\left(\mathcal{x}\right)=\frac{f(\mathcal{x}+h)-f\left(\mathcal{x}\right)}{h} \end{gathered}$$

$$\begin{gathered} Find \\ \underset{\mathcal{x}\to 4}{\lim }\frac{{\mathcal{x}}^2+\mathcal{x}-20}{\mathcal{x}-4} \\ \underset{\mathcal{x}\to \infty }{\lim }\frac{4-\mathcal{x}+2{\mathcal{x}}^2}{{\mathcal{x}}^2+7\mathcal{x}-9} \\ Differentiate the following \\ \mathcal{y}={(7-2\mathcal{x})}^6 \mathcal{y}=7{\mathcal{x}}^{-3}+4\mathcal{x}-17 \\ f\left(\mathcal{x}\right)=\mathcal{x}\sqrt{\mathcal{x}-2} \\ f\left(\mathcal{x}\right)=\frac{\mathcal{x}+4}{\mathcal{x}+5} \\ Write down a quadratic equation with roots 4 and-7. \\ If f\left(\mathcal{x}\right)=\sqrt{{\mathcal{x}}^2+16} find f\prime \left(3\right). \\ For the parabola {(\mathcal{x}+2)}^2=4\mathcal{y}-8, find the: \\ coordinates of the vertex; \\ coordinates of the focus; \\ equation of the directrix. \end{gathered}$$

$$\begin{gathered} Consider the parabola 4\mathcal{y}={\mathcal{x}}^2-4\mathcal{x} \\ i)Show, algebraically,how the parabola can be expressed in the form \\ {(\mathcal{x}-2)}^2=4(\mathcal{y}+1) \\ ii)Write down the coordinates of the vertex and the focus. \\ iii)Find the equation of the directrix. \end{gathered}$$

$$\begin{gathered} What is the equation of a quadratic with roots 1-\sqrt{2} and 1+\sqrt{2} ? \\ \left(A\right) {\mathcal{x}}^2+\sqrt{2}\mathcal{x}-1=0 \\ \left(B\right) {\mathcal{x}}^2-2\mathcal{x}-1=0 \\ \left(C\right) {\mathcal{x}}^2+2\mathcal{x}-1=0 \\ \left(D\right) {\mathcal{x}}^2-\sqrt{2}\mathcal{x}-1=0 \end{gathered}$$

$$\begin{gathered} What is the gradient of a line that is perpendicular to 4\mathcal{x}-3\mathcal{y}=0 ? \\ \left(A\right) 4 \\ \left(B\right) -4 \\ \left(C\right) \frac{4}{3} \\ \left(D\right) -\frac{3}{4} \end{gathered}$$

$$\begin{gathered} What is the derivative of {(1-2\mathcal{x})}^5 \\ \left(A\right) 5{(1-2\mathcal{x})}^4 \\ \left(B\right) -5{(1-2\mathcal{x})}^4 \\ \left(C\right) 10{(1-2\mathcal{x})}^4 \\ \left(D\right) -10{(1-2\mathcal{x})}^4 \end{gathered}$$

$$\begin{gathered} i)Sketch \mathcal{y}={\log }_a\left(\mathcal{x}+2\right), Showing the \mathcal{x} and \mathcal{y}-intersections and the asymptote. \\ ii)Hence,state the doain and range of \mathcal{y}={\log }_2\left(\mathcal{x}+2\right). \\ In a certain arthematic series the 9^{th}term is 78, the {17}^{th}term is 50.Find \\ i)The first term and the common difference. \\ ii)The sum of the first forty terma \\ iii)The value of n for which the sum of the series is first negative. \end{gathered}$$

$$\begin{gathered} If f\left(\mathcal{x}\right)={\mathcal{x}}^2-\mathcal{x} find the simplest expression for f(\mathcal{x}+h). \\ A function \mathcal{y}=f\left(\mathcal{x}\right) is defined as folows: \\ f\left(\mathcal{x}\right)=\left\{\begin{array}{l}-2, \mathcal{x}\le -1\\ \mathcal{x}+1 -1<\mathcal{x}<2\\ {\mathcal{x}}^2+1 \mathcal{x}\ge 2\end{array}\right. \\ i)Evaluate f(-1)+f(2\left) \\ ii\right)Write an expression for f(a^2+2) \end{gathered}$$

$$\begin{gathered} Use the quadratic formula to solve {\mathcal{x}}^2+4\mathcal{x}-6=0. \\ Express your answers in simplest surd form. \\ Sketch the following on separate number planes: \\ 3\mathcal{x}-2\mathcal{y}+12=0: \\ \mathcal{y}=\frac{4}{\mathcal{x}}: \\ \mathcal{y}={\mathcal{x}}^2-4\mathcal{x}+3 \end{gathered}$$

$$\begin{gathered} State the domain of the following: \\ \mathcal{y}=\frac{2}{{\mathcal{x}}^2-1} \\ \mathcal{y}=\sqrt{2-\mathcal{x}} \\ State whether the following are ODD, EVEN or NEITHER. \\ Justify your answer. \\ f\left(\mathcal{x}\right)=1-{\mathcal{x}}^2 \\ f\left(\mathcal{x}\right)=\frac{\mathcal{x}}{1+{\mathcal{x}}^2} \\ f\left(\mathcal{x}\right)=-3\mathcal{x}-{\mathcal{x}}^2 \\ f\left(\mathcal{x}\right)=-3\left|\mathcal{x}\right| \end{gathered}$$

$$\begin{gathered} What is the domain of the function \mathcal{y}=\sqrt{2-\mathcal{x}} ? \\ State whether the following functions are ODD,EVEN or NEITHER \\ i)f\left(\mathcal{x}\right)={\mathcal{x}}^2+12 \\ ii)f\left(\mathcal{x}\right)=\frac{-\mathcal{x}}{{\mathcal{x}}^2+1} \end{gathered}$$

$$\begin{gathered} What is the slope of the line \mathcal{x}+3\mathcal{y}-5=0? \\ Sketch the following showing essential features: \\ i) \mathcal{y}=\mathcal{x}+2 \\ ii) \mathcal{y}=2 \\ iii) \mathcal{xy}=2 \\ iv) \mathcal{y}=\left|\mathcal{x}-2\right| \\ For the parabola {\mathcal{x}}^2=8(\mathcal{y}-1).write down the: \\ i) co-ordinates of the vertex \\ ii) focal length \\ iii) co-ordinates of the focus \\ iv) equation of directrix \\ Find the equation of the following lines (giving your answer in general form) \\ i)through (4,6) with slope -2. \\ ii)prependicular to \mathcal{x}-2\mathcal{y}+5=0 and pas\sin g through (3,6) \end{gathered}$$

$$\begin{gathered} Differentiate f\left(\mathcal{x}\right)={\mathcal{x}}^2-6\mathcal{x}+1,find: \\ i)the equation of the axis \\ ii)the range of values of \mathcal{y} \\ Find the prependicular dis\tan ce of the point \left(4,1\right) from the line \\ 3\mathcal{x}-4\mathcal{y}+12=0. \\ Sketch the regions of the number plane defined by: \\ i){\mathcal{x}}^2+{\mathcal{y}}^2\le 16 \\ ii)\mathcal{y}>{\mathcal{x}}^2+9 \end{gathered}$$

$$\begin{gathered} Differentiate the following: \\ i)\mathcal{y}=3-\mathcal{x}+{\mathcal{x}}^2 \\ ii)\mathcal{y}=\sqrt{\mathcal{x}} \\ iii)\mathcal{y}=\frac{\mathcal{x}+1}{\mathcal{x}-2} \\ iv)\mathcal{y}={(2-3{\mathcal{x}}^2)}^5 Find the centre and radius of the circle \mathcal{x}-4\mathcal{y}+{\mathcal{y}}^2+6\mathcal{y}=0 \end{gathered}$$

$$\begin{gathered} Find T_{10} of 5+9+13+17+.... \\ If f\left(\mathcal{y}\right)=9-{\mathcal{y}}^2,Find: \\ i\left)f\left(2\right) \\ ii\right)f\left(\mathcal{y}+1\right) \\ State the domain of f\left(\mathcal{x}\right)=\sqrt{3-\mathcal{x}} \\ Find \mathcal{x} in the following: \\ i){\log }_{\mathcal{x}}36=2 \\ ii){\log }_{8}128=\mathcal{x} \end{gathered}$$

$$\begin{gathered} Find the sum of the first 9 terms of 2-1+\frac{1}{2}-\frac{1}{4}+..... \\ State whether the following functions are ODD,EVEN or NEITHER. \\ i)f\left(\mathcal{x}\right)=\frac{4}{9+{\mathcal{x}}^2} \\ ii)f\left(\mathcal{x}\right)=\frac{\mathcal{x}}{9+\mathcal{x}} \\ iii))f\left(\mathcal{x}\right)=\frac{-\mathcal{x}}{9+\mathcal{x}} \end{gathered}$$

$$\begin{gathered} Sketch the following lines on separate diagrams (showing essential features): \\ i)\mathcal{y}=\mathcal{x}+2 \\ ii)3\mathcal{x}-\mathcal{y}-6=0 \\ Find \underset{\mathcal{x}\to 4}{\lim }\frac{\mathcal{x}-4}{{\mathcal{x}}^2-16} \end{gathered}$$