Solve the equation

Show clear algebraic working.

Give your answer to 3 significant figures.

(a) Complete the table of values for $\displaystyle y = 3^x - 2$ giving your answers to one decimal place.

(b) On the grid opposite, draw the graph of $\displaystyle y = 3^x - 2$ for $\displaystyle -1 \leq x \leq 2$

(c) Use your graph to obtain an estimate, to one decimal place, of the value of $\displaystyle \mathrm{log}_3 4.5$

Show clearly how you have used your graph.

(d) By drawing a straight line on your grid obtain an estimate, to one decimal place, of the root of the equation $\displaystyle \mathrm{log}_3(8-3x) - x = 0$ in the interval $\displaystyle -1 \leq x \leq 2$

Only use this grid if you need to redraw your graph.

Without using a calculator, find the value of $\displaystyle w$

Give your answer in the form $\displaystyle \frac{a+b\sqrt{6}}{c}$ where $\displaystyle a$ and $\displaystyle b$ are integers and $\displaystyle c$ is prime.

Show your working clearly.

(a) Complete the table of values for $\displaystyle y = 3 \mathrm{log}_3(x^2 - x) - 2x$ giving your answers to 2 decimal places.

(b) On the grid opposite, draw the graph of $\displaystyle y = 3 \mathrm{log}_3(x^2 - x) - 2x$ for $\displaystyle 1.3 \leq x \leq 3.5$

(c) By drawing a suitable straight line on the grid, obtain an estimate, to one decimal place, of the root of the equation

Given that $\displaystyle \frac{a+b\sqrt{5}}{6-2\sqrt{5}} = \frac{9+4\sqrt{5}}{c}$ where $\displaystyle a$, $\displaystyle b$ and $\displaystyle c$ are prime numbers, find the value of $\displaystyle a$, the value of $\displaystyle b$ and the value of $\displaystyle c$

(a) Complete the table of values for $\displaystyle y = \mathrm{log}_3(4-x) + 3$ giving your answers to 2 decimal places.

(b) On the grid opposite, draw the graph of $\displaystyle y = \mathrm{log}_3(4-x) + 3$ in the interval $\displaystyle 0 \leq x \leq 3.5$

(c) By drawing a suitable straight line on the grid, obtain an estimate,

to one decimal place, of the root of the equation $\displaystyle 3^{2x-5} - (4-x)^3 = 0$ in the interval $\displaystyle 0 \leq x \leq 3.5$

Only use this grid if you need to redraw your graph.

Solve, using an algebraic method, the simultaneous equations

(i) Solve the equation $\displaystyle 3(\mathrm{log}_a 9 + \mathrm{log}_a 27) = 1$

Give your answer in the form $\displaystyle a = b^c$ where $\displaystyle b$ is prime and $\displaystyle c$ is an integer.

(ii) Solve the equation $\displaystyle \mathrm{log}_4 p + \mathrm{log}_p 256 = -4$

(a) Find the value of $\displaystyle a$ such that $\displaystyle \mathrm{log}_a 8 = \frac{3}{4}$

(b) Show that

(c) Hence solve the equation $\displaystyle 3x \mathrm{log}_2 x - 4 \mathrm{log}_{16} 8 + 6x \mathrm{log}_4 8 - \mathrm{log}_2 x = 0$

Without using a calculator, solve the inequality $\displaystyle \sqrt{50} x - \sqrt{18} > 6x + 5$

Give your answer in an exact form with a rationalised denominator.
Show your working clearly.

The curve $\displaystyle C$ has equation $\displaystyle y = -\mathrm{log}_4(x + 4)$

(a) Using the axes below, sketch the graph of $\displaystyle C$.

Label the coordinates of the points of intersection of $\displaystyle C$ with the coordinate axes and the equation of any asymptote to $\displaystyle C$.

(b) Solve the equation $\displaystyle \mathrm{log}_{(x+4)} 256 - \mathrm{log}_4(x + 4) = 0$

(a) Complete the table of values for $\displaystyle y = \mathrm{log}_{10}(6x-1) - x$ giving your answers to 2 decimal places.

(b) On the grid opposite, draw the graph of $\displaystyle y = \mathrm{log}_{10}(6x-1) - x$ for $\displaystyle 0.25 \leq x \leq 3$

(c) By drawing a suitable straight line on the grid, obtain an estimate, to one decimal place, of the root of the equation

Only use this grid if you need to redraw your graph.

(i) Solve the equation $\displaystyle 5(\mathrm{log}_b 9 + \mathrm{log}_b 3) = 3$

(ii) Solve the equation $\displaystyle 3\mathrm{log}_3 x + 3\mathrm{log}_x 27 = 8\mathrm{log}_4 128$

Give your answers in exact form.

Figure 1 shows part of the graph of the curve with equation $\displaystyle y = x + 2^{-(4x+1)}$

By drawing a suitable straight line on the graph, obtain an estimate, to one decimal place, of the roots of the equation $\displaystyle \mathrm{log}_2(8 - 3x) + 4x = 0$ in the interval $\displaystyle -2 \leq x \leq 6$

Solve the equation

giving your answers to 3 significant figures.

Figure 1 shows the triangle $\displaystyle ABC$

Without using a calculator, find the value of $\displaystyle a$ and the value of $\displaystyle b$

Solve the simultaneous equations

A

Figure 1 shows the rectangle $\displaystyle ABCD$.

The area of the rectangle is $\displaystyle (10 + \sqrt{2}) \mathrm{cm}^2$

Find the value of $\displaystyle a$ and the value of $\displaystyle b$
Show your working clearly.

(a) Complete the table of values for $\displaystyle y = \frac{x}{2} + 6\mathrm{e}^{-2x} + 1$
giving your answers to one decimal place.

(b) On the grid opposite, draw the graph of $\displaystyle y = \frac{x}{2} + 6\mathrm{e}^{-2x} + 1$ for $\displaystyle 0 \leq x \leq 6$

(c) By drawing a suitable straight line on your graph, obtain estimates, to one decimal place, of the roots of the equation

Only use this grid if you need to redraw your graph.

Given that $\displaystyle \frac{a + 2\sqrt{5}}{3 - \sqrt{5}} = \frac{11 + b\sqrt{5}}{2}$ where $\displaystyle a$ is an integer and $\displaystyle b$ is prime,

find the value of $\displaystyle a$ and the value of $\displaystyle b$

Show your working clearly.

Solve the equation

Given that

where $\displaystyle a$ and $\displaystyle b$ are integers, find, without using a calculator, the value of $\displaystyle a$ and the value of $\displaystyle b$.

Show your working clearly.

(a) Complete the table of values for

giving your answers to 2 decimal places where appropriate.

(b) On the grid opposite, draw the graph of

(c) By drawing an appropriate straight line on the grid, obtain an estimate, to one decimal place, of the root of the equation

(d) By drawing an appropriate straight line on the grid, obtain an estimate, to one decimal place, of the root of the equation

A curve $\displaystyle C$ has equation

(a) Using the axes below, sketch the graph of $\displaystyle C$. Label the coordinates of the points of intersection of $\displaystyle C$ with the coordinate axes.

(b) Solve the equation

(c) Solve the equation

Solve the equation

Show your working clearly.

(a) Complete the table of values for

giving your answers to 2 decimal places.

(b) On the grid opposite, draw the graph of

(c) By drawing a suitable straight line on the grid, obtain an estimate, to one decimal place, of the root of the equation

Figure 3 shows a sketch of part of the curves with equations

The curves intersect at the points $\displaystyle A$ and $\displaystyle B$, as shown in Figure 3.

(a) (i) Show that the $\displaystyle x$ coordinates of $\displaystyle A$ and $\displaystyle B$ satisfy the equation

(ii) Hence show that the $\displaystyle x$ coordinate of $\displaystyle B$ is $\displaystyle \frac{1}{3}\mathrm{ln}\,9$.

(b) Find the exact area of the finite region bounded by the two curves.

Given that $\displaystyle n$ satisfies the equation

(a) find the value of $\displaystyle n$.

Given that

(b) (i) express $\displaystyle x$ in terms of $\displaystyle p$,

(ii) express $\displaystyle xy$ in terms of $\displaystyle p$.

(a) Complete the table of values for

Give your answers to 2 decimal places where appropriate.

(b) On the grid opposite, draw the graph of

(c) By drawing a suitable straight line on the grid, obtain an estimate, to one decimal place, of the root of the equation

in the interval $\displaystyle 0 \leq x \leq 5$.

Use an algebraic method to solve the simultaneous equations

Showing your working clearly, use algebra to solve the equations

(i) Solve the equation

(ii) Solve the equation

The curve $\displaystyle C_1$ has equation

The curve $\displaystyle C_2$ has equation

The curves $\displaystyle C_1$ and $\displaystyle C_2$ intersect at the point $\displaystyle A$.

(a) Find the exact coordinates of $\displaystyle A$.

The tangent at $\displaystyle A$ to $\displaystyle C_1$ intersects the $\displaystyle x$-axis at the point $\displaystyle B$.

(b) Show that the $\displaystyle x$ coordinate of $\displaystyle B$ is

The tangent at $\displaystyle A$ to $\displaystyle C_2$ intersects the $\displaystyle x$-axis at the point $\displaystyle D$.

(c) Find the area of $\displaystyle \triangle ABD$.

(a) Complete the table of values for

Give your answers to 2 decimal places where appropriate.

(b) On the grid opposite, draw the graph of

(c) Use your graph to obtain estimates, to 1 decimal place, of the roots of the equation

(d) By drawing a suitable straight line on the grid, obtain estimates, to 1 decimal place, of the roots of the equation

Solve the equation

Given that

can be written in the form

where $\displaystyle a$ and $\displaystyle b$ are integers and $\displaystyle c$ is prime, find the value of $\displaystyle a$, the value of $\displaystyle b$ and the value of $\displaystyle c$.

Show your working clearly.

(a) Solve

(b) Hence solve

Give your answers to 3 significant figures where appropriate.

The curve with equation

intersects the curve with equation

at two points.

(c) Find the coordinates of each of these two points.

Give your answers to 3 significant figures where appropriate.

(a) Solve the equation

Given that

(b) find the value of $\displaystyle k$.

(c) Show that

(a) Write down the value of

(b) Solve the equation

Give your answer in the form $\displaystyle a^b$, where $\displaystyle a$ and $\displaystyle b$ are prime numbers.

(a) Complete the table of values for $\displaystyle y=2+\ln(2x+1)$, giving your answers to 2 decimal places.

(b) On the grid opposite, draw the graph of $\displaystyle y=2+\ln(2x+1)$ for $\displaystyle 0\leqslant x\leqslant3$.

(c) By drawing an appropriate straight line on the grid, obtain an estimate, to one decimal place, of the root of the equation

in the interval $\displaystyle 0\leqslant x\leqslant3$.

(d) By drawing an appropriate straight line on the grid, obtain an estimate, to one decimal place, of the root of the equation

in the interval $\displaystyle 0\leqslant x\leqslant3$.

Solve the equation $\displaystyle 3 log3 x -$ 8 logx $\displaystyle 3 = 10$

(a) Complete the table of values for $\displaystyle y=\ln(3x+1)+2$, giving your answers to 2 decimal places.

(b) On the grid opposite, draw the graph of $\displaystyle y=\ln(3x+1)+2$ for $\displaystyle 0\leqslant x\leqslant6$.

(c) Use your graph to obtain an estimate, to 1 decimal place, for the value of $\displaystyle \ln 10.6$.

You must show clearly how you have used your graph.

(d) By drawing a straight line on the grid, obtain estimates, to 1 decimal place, for the roots of the equation

in the interval $\displaystyle 0\leqslant x\leqslant6$.