AQA MathematicsSyllabus 6361

Core 3

Syllabus&

Past Paper questions

You may use a graphic calculatorin this module

Functions A function is a rule that generates exactly one output for every

input. A many-to-one function is where two or more inputs generate the same output (eg ). A one-to-one function is where each output is generated by only one input.

Only one-to-one functions have an inverse. If a function, , is defined for , then is the

domain of the function. The range is the set of possible y-values.

e.g. has domain and range . has domain all real values of x, and range

has domain all real values of x, and range . has domain and range all real values of y.

has domain and range

The function gf(x) ('g of f of x') is called a composite function and tells you to 'do f first and then g'. It does not mean multiply!

e.g.

Inverse Functions A function that 'undoes the effect of f is called the inverse

function of f. It is denoted by f-1. eg and The graph of f-1(x) is a reflection of the graph of f(x) in the line . The domain of the inverse function is the range of the function

and the range of the inverse function is the domain of the function.

FunctionsBasic Transformations

translates the graph

translates the graph

gives the graph a one way stretch along the x-axis, scale factor ,

gives the graph a one way stretch along the y-axis, scale factor b. reflects the graph in the y-axis. reflects the graph in the x-axis. reflects the graph in the line y = x.

To find the inverse function write the function in the form , then rearrange to get . Then replace y with x.

e.g.

Write then rearrange

Interchange x and y

1. A sketch graph for the function of g(x) is shown below.

Draw carefully, indicating the intercepts with the axes in each case, separate sketch graphs for the functions

(i) g(2x), (ii) g(-x), (iii) g-1(x). (6 marks)

2. The function f is given by f(x) = and is sketched below.

Write down an expression for f(x+3), and sketch the graph of y = f(x+3). (3 marks)

FunctionsThe modulus function,

To sketch , sketch , then reflect any part for which y is negative in the x-axis.

When solving equations and inequalities that involve a modulus function, sketch the graphs first to determine the number of solutions.

To solve , solve and

To solve identify the equations of the graphs where the intersections occur.

The entire graph of a function y = f(x) is illustrated above.

(a) Write down the domain of the function f(x).

(b) Sketch the graph of the inverse function marking appropriate values on the axes.

(c) Write down the range of (5 marks)

The graph of is shown above.

(a) Describe the transformation which maps the graph of onto the graph of

(b) Hence write down, in terms of a, an expression for f(x). (4 marks)

A sketch of the graph y = f(x), where is shown above.

Find an expression for the inverse function, and state its domain. (4 marks)

3.

4.

5.

The graph y = f(x) is shown above for f(x) =

(a) Write down an expression for f(x + 3) and sketch the graph y = f(x + 3).

The function g(x) is defined by g(x) = x – 2, x .

(b) Write down an expression for gf(x + 3) and on a new diagram sketch the graph y = gf(x + 3).

(7 marks)

7. Solve the inequality > 4, giving your answer in an exact form. (3 marks)

Exponentials and logarithms is a growth function.

At every point on the graph of the gradient is equal to .

is the inverse of

is defined only for positive values of x since is always positive. You need to be familiar with their graphs and domain and

range. The graph of is the reflection of in the line .

To solve equations of the form use ln on both sides. To solve equations of the form use e on both sides.

6.

1. (a) The diagram shows the graph of y = f(x), where

the function f is defined for all values of x by

(i) Write down the coordinates of the point where the graph intersects the y-axis.

(ii) State the range of the function f.

(iii) Find the value of f(ln6), giving your answer as a fraction.

(b) The function g is defined for all values of x by (i) Show that gf(x) =

(ii) State the range of the function gf

(iii) Sketch the graph of y = gf(x)

(iv) Show that gf(x) = 11 x = ln5

(c) A dish of water is left to cool in a room where the temperature is 10C.

At time t minutes, where , the temperature of the water is

(i) State the temperature of the water at time t = 0.

(ii) Calculate the time at which the temperature of the water reaches 11C.

2. A function is defined by

(a) Sketch the graph of y = f(x). (b) Solve the inequality f(x) < 2

3. The functions f and g are defined by

for all values of x (a) Find an expression for

(b) Evaluate gf(8)

(c) Show that gf(x) =

Laws of logs Laws of indices

x

y

1. This question is about two possible iterative formulae for finding a solution to theequation:

(a) (i) Show that is a possible iterative formula for this equation.

(ii) Use = 0.5 to find giving your answers to 4 decimal places. Continue the process to find the solution to the equation to 2 decimal places. (6 marks)

(b) An alternative iterative formula is:

(3 marks)

Iteration To show that a root of the equation lies between two

given values, and , calculate and . If they have opposite signs and the function is continuous between the two values, there must be a root.

Sequences can be generated by a simple recurrence relation of the form , where the output from one calculation is used as the input for the next.

The iterative process can be illustrated by staircase and cobweb diagrams. Sketch the graphs of and . From the initial value of x go up to and across to .

If the sequence converges to a limit, then this is the solution of the equation .

An approximate solution to an equation can be obtained by rearranging the equation into the form .

is called an iterative formula.

Copy the graph shown.

On your graph draw a cobweb diagram to

show the approximate positions of ,

and for .

2. The sequence given by:

converges to . Find giving your answers to three decimal places.State and simplify an equation satisfied by and hence find in exact form. (6 marks)

3. The iterative sequence: , with xi = 1, converges to the number .

(a) Calculate x2 and x3 and find to 3 decimal places. (3 marks)

(b) Show that is a root of the equation x10 = ex. (2 marks)

1. A table of values for the function f(x) is as shown.

Use the Simpson’s rule, with four strips, to estimate the value of (3 marks)

2. Use the mid-ordinate rule with 3 strips to estimate

giving your answer to 2 decimal places. (4 marks)

Numerical integration to find the area under a graph

Mid-ordinate rule:

Simpson’s rule gives a more accurate value:

h (sum of end ordinates + 4sum of odd ordinates + 2 sum of

other even ordinates)

Simpson's rule needs an even number of strips. To improve the estimate, increase the number of strips.

These are in the formula

book

(5 marks)

The cross-sectional area between the skateboarding surface and the ground is given by:

Use the mid-ordinate rule with three strips to find the approximate value of this area. (5 marks)

.The diagram shows the area under part of the quadrant of a circle of radius 8 units.

Angle AOB is . The point B has coordinates (3.471, 7.208), to 3 decimal places.

(a) Find an exact expression for the area of sector AOB. (2 marks)

(b) Hence, showing all your working and without using calculus, confirm that the shaded area

is 26.87 square units to 2 decimal places. (5 marks)

(c) The equation of the quadrant of the circle is y =

Use Simpson’s rule with 2 strips to estimate the shaded area, giving your answer to 2 decimal places.

(4 marks)

Use the trapezium rule, with 3 strips, to

estimate

Give your answer to 3 decimal places.

Part of the graph of is

illustrated and is used to model a cross-section of a skateboarding surface. The x-axis represents ground-level.

3.

4.

5.

CALCULUS - differentiation Standard results:

Differentiation using product and quotient rule:

and

Differentiation using chain rule:

If a tangent has gradient m then the normal has gradient

If is positive then the function is increasing; if is negative

then the function is decreasing.

A graph has a stationary point when .

To determine if it is a maximum or minimum find . Negative maximum, positive minimum.

CALCULUS - integration Integration by inspection using the standard integrals:

If you have a fraction and you can spot a derivative on top and its function on the bottom, use

Integration by parts:

Integration by substitution:

, changes the integration variable from x to u.

Use the substitution

These are in the formula book

8. Given and when r = 5,

Calculate when r = 5.

9. Find for each of the following cases

(a) (3 marks)

(b) (2 marks)

10. (a) Given that , show that (4 marks)

(b) Use the substitution , or otherwise, to find the exact value of

(6 marks)

11. (a) Find if (2 marks)

(b) Find (3 marks)

CALCULUS - integrationEvaluation of volume of revolution

The volume of a solid of revolution about the x-axis between and is given by

The volume of a solid of revolution about the y-axis between and is given by .

12. The graph below shows the region R enclosed by the curve , the x-axis and the line .

¤

x

y

Find the exact value of the area of the region R. (4 marks)

13. (a)

Calculate the volume of the solid formed when the area between the axes and

the line x = 2 is rotated through an angle of 2 radians about the x-axis. (6 marks)

(b) Use the substitution . (7 marks)

14. (a) Find (2 marks)

(b) Find (3 marks)

15. (a) Find (3 marks)

(b) Use the substitution u = 3x – 1 to find: (6 marks)

16. (a) Show that

(b) Find the exact value of the volume of the solid formed when the region between the curve with equation y = xex and the lines y = 0, x = 0 and x = 1 is rotated completely about the x-axis.

R

17. Work out:

(a) (b)

18. (a) (3 marks)

(b) By using the substitution or otherwise, find the exact value of

(5 marks)

(c) Find the exact volume of the solid formed when the region bounded by the curve y = sec x,

the line and the x- and y-axes is rotated through a complete revolution about the x-axis.

(3 marks)

19. (a) Given that show that (4 marks)

(b) Find (3 marks)

(c) Use the substitution u = 1 + tan x, or otherwise, to find the exact value of

(5 marks)

20. Use a suitable trigonometric identity to solve the equation: , for Give your answers to 1 decimal place. (7 marks)

Trig identities and graphs

, ,

. Dividing through by gives

and dividing through by gives

You need to be familiar with the graphs of , and , their domains, ranges and periods.

You need to be familiar with the inverse functions, , and as reflections of the graphs of ,

and in the line .

The domains of these functions are restricted to , and

Solving equations Check you are working in the correct mode, degrees or radians. Use identities and rearrange the equation to get an equation

involving only one trig function, often a quadratic. Solve the quadratic to get eg . Then work on each part separately.

Find the first solution, then sketch the appropriate basic graph (eg ) and look for other solutions. Check the required range for the final answers.

To solve equations like , find solutions to and

21. A piston moves up and down a vertical tube. The height h centimetres of the end of the piston after t seconds is given by the formula

h = 10 + 2.5sin 2t.

(a) By solving an appropriate equation, find the times in the first 2 seconds when the height of the piston is 12cm. Give your answers to 2 decimal places.

(5 marks)

(b) Find the speed at which the end of the piston is moving when t = 2 seconds,giving your answer

to 3 significant figures. State whether the piston is moving upwards or downwards at this time. (5 marks)

SolutionsFunctions

1. Straight line passing thriugh: (i) ((½,0) and (0,2) (ii) (-1,0) and (0,2) (iii) (2,0) and (01)

2. ; graph translated

3. (a) domain (b) (c) range

4. (a) translation (b)

5. domain: all real values of x where x ≠ 4

6. (a) (b)

7.

Exponentials and logarithms

1. (a) (i) (0, 5) (ii) f(x) > 0 (iii) (b) (i)

(ii) gf(x) > 10 (iii) (c) (i) 15C (ii) 1.6 minutes 2. (b) x > ln3

3. (a) (b) 4 (c)

Iteration

1. (a) (i) – (ii)

2.

3. (a) (b)

Numerical integration

–5x

y

–5x

y

asymptotes at x = -3 and y = -2

1 2

–1

1

2

x

y

10

(0, 15)

1.

2. I = 1.64 (2 dp)

3. I = 1.247 (3 dp)

4. I = 0.651 (3 dp)

5. (a) 32π/7 (b) 32π/7 + ½×3.471×7.208 = 26.87091…

(c)

Calculus - differentiation and integration

8.

9. (a) (b)

10. (a) - (b)

11. (a) (b)

12.

13. (a) (b)

14. (a) (b)

15. (a) (b)

16. (a) by parts twice (b)

17. (a) (b)

18. (a) (b)

(c)

19. (a) - (b) by parts with and u = ln(x)

Trig Identities

20.

21. (a) (2 dp)

(b) , i.e. 3.27 cm.s-1 (3 sf) moving downwards