Further Maths 0606 — specification explorer

1. Recognise and use function notation. Examples include:

• 1.3.1f( x ) = 2e
• 1.3.2f : x ↦ lg x , for x > 0
• 1.3.3f ( x )
• 1.3.4fg( x ) [= f(g( x )) ]
• 1.3.5f ( x ) [= f(f( x )) ] The notation f ( x ) will not be used with trigonometric functions.

2. Understand the relationship between If f( x ) is trigonometric it will be one of the following:

• 1.4.1y = a sin bx + c quadratic, cubic or trigonometric.
• 1.4.2y = a cos bx + c
• 1.4.3y = a tan bx + c where a is a positive integer, b is a simple fraction or integer, and c is an integer. Fractions will have a denominator of 2, 3, 4, 6 or 8 only.

3. Find the inverse of a one–one function. Correct notation must be used. For example:

• 1.6.2

  x:

  • f( x ) = e
  • f ( x ) = 2 x

1. Find the solution set for quadratic inequalities Solutions should be written in the correct form. For

• 2.5.1–3 < x < 4
• 2.5.2x < 1 or x > 6

1. Know and use the remainder and factor theorems.

• 3.1.1Know and use the remainder and factor theorems.

2. Find factors of polynomials.

• 3.2.1Find factors of polynomials.

3. Solve cubic equations.

• 3.3.1Solve cubic equations.

1. Solve equations of the type For graphical solutions, an accurate graph is

• 4.1.1|ax + b| = c ( c ⩾ 0) expected.
• 4.1.2|ax + b| = cx + d For algebraic methods, any valid method is acceptable.
• 4.1.3|ax + b| = |cx + d|
• 4.1.4|ax + bx + c| = d using algebraic or graphical methods.

2. Solve graphically or algebraically inequalities For graphical solutions, an accurate graph is

• 4.2.1k |ax + b| > c ( c ⩾ 0 ) For algebraic methods, any valid method is
• 4.2.2k |ax + b| ⩽ c ( c > 0) acceptable.
• 4.2.3k |ax + b| ⩽ |cx + d| where k > 0
• 4.2.4|ax + b| ⩽ cx + d
• 4.2.5|ax + bx + c| > d
• 4.2.6|ax + bx + c| ⩽ d

3. Use substitution to form and solve a quadratic For example:

• 4.3.1x 3 + x 3 - 12 = 0
• 4.3.23e = 12 – 5e
• 4.3.3Candidates are expected to identify the appropriate substitution.

4. Solve graphically cubic inequalities of the form

• 4.5.1f( x ) ⩾ d
• 4.5.2f( x ) > d
• 4.5.3f( x ) ⩽ d
• 4.5.4f( x ) < d where f( x ) is a product of three linear factors and d is a constant.

1. Solve simultaneous equations in two For example:

• 5.1.1y – x + 3 = 0 and x – 3 xy + y + 19 = 0
• 5.1.2xy = 4 and xy = 3
• 5.1.3+ = 4 and y = x – 2

1. Know and use the laws of logarithms, For example:

• 6.2.1Write 3 + 21g p – 1g q as a single base 10 logarithm.
• 6.2.2Write as a natural logarithm. log e

1. Transform given relationships to and from For example:

• 7.4.1To straight-line form unknown constants by calculating the gradient y = Ax or intercept of the transformed graph. y = Ab
• 7.4.2From straight-line form to an equation of the form . y = Ax + B e = Ax + B y = A 1n x + B

1. Know and use the equation of a circle with Identify the centre and radius of a circle using a circle

• 8.1.2

  2 2 Formula is given in the List of formulas.:

  • ( x – a ) + ( y – b ) = r
  • x + y + 2 gx + 2 fy + c = 0

2. Solve problems involving the intersection of a Includes finding points of intersection.

• 8.2.1is a tangent
• 8.2.2is a chord
• 8.2.3does not intersect the circle.

3. Solve problems involving the intersection of Includes finding points of intersection, finding the

• 8.4.1intersect
• 8.4.2touch
• 8.4.3do not intersect.

1. Solve problems involving the arc length and sector area of a circle, including knowledge and use of radian measure.

• 9.1.1Solve problems involving the arc length and sector area of a circle, including knowledge and use of radian measure.

1. Use the relationships: Trigonometric identities are given in the List of

• 10.4.2

  2 formulas.:

  • sin A + cos A = 1
  • sec A = 1 + tan A
  • cosec A = 1 + cot A

2. Solve, for a given domain, trigonometric Includes the use of the relationships in 10.4.

• 10.5.14 cot θ = tan θ
• 10.5.22 sec θ + tan θ – 3 = 0 i + 2 cos 3 i = 0
• 10.5.35 sin 3
• 10.5.43 cosec 2 d i - n = 4

3. Prove trigonometric relationships involving the Includes the use of the relationships in 10.4.

• 10.6.1sin x tan x + cos x = sec x sin i 1 + cos i
• 10.6.1a+ cos i sin i
• 10.6.2+ = 2 cosec θ

1. Solve problems on arrangement and selection Problems will be either in an everyday context or

• 11.3.1repetition of objects
• 11.3.2objects arranged in a circle
• 11.3.3both permutations and combinations, are not included.

1. Use the binomial theorem for expansion of (a + b)ⁿ for positive integer n.

• 12.1.1Use the binomial theorem for expansion of (a + b)ⁿ for positive integer n.

2. Use the general term nCr aⁿ⁻ˣ bʳ, 0 ⩽ r ⩽ n.

• 12.2.1Use the general term nCr aⁿ⁻ˣ bʳ, 0 ⩽ r ⩽ n.

3. Recognise arithmetic and geometric progressions and understand the difference between them.

• 12.3.1Recognise arithmetic and geometric progressions and understand the difference between them.

4. Use the formulas for the nth term and for the sum of the first n terms to solve problems involving arithmetic or geometric progressions.

• 12.4.1Use the formulas for the nth term and for the sum of the first n terms to solve problems involving arithmetic or geometric progressions.

5. Use the condition for the convergence of a geometric progression, and the formula for the sum to infinity of a convergent geometric progression.

• 12.5.1Use the condition for the convergence of a geometric progression, and the formula for the sum to infinity of a convergent geometric progression.

1. Find the magnitude of a vector; add and Includes:

• 13.3.1equating like vectors scalars.
• 13.3.2solving problems using vector geometry, with a diagram given in more complex cases.

1. Use the notations

• 14.2.2J N f ′ ( x ), f ″( x ), , = KK OO

2. Know and use the derivatives of the standard Includes constant multiples, sums and composite

• 14.3.23 For example, differentiate 3 x + 4 j .
• 14.3.3For trigonometric functions angles will always be in radians.

3. Integrate sums of terms in powers of x, Solutions for indefinite integrals should include an

• 14.11.1and 1 arbitrary constant. including . x ax + b

4. Integrate functions of the form: Includes the case where n = –1 .

• 14.12.1( ax + b ) for any rational n For trigonometric functions angles will always be in
• 14.12.2sin ( ax + b ) radians.
• 14.12.3cos ( ax + b ) Solutions for indefinite integrals should include an 2 arbitrary constant.
• 14.12.4sec ( ax + b ) ax + b
• 14.12.5e

5. Evaluate definite integrals and apply Plane areas include:

• 14.13.1between a line and a curve
• 14.13.2between two curves
• 14.13.3a sum of two areas.

6. Make use of the relationships in 14.14 to draw For example:

• 14.15.3

  2 graph, its speed–time graph, its acceleration–time graph.:

  • displacement–time from a fixed point at time, t , is given by s = 3 t – 10 t
  • distance–time + 4 t + 8 for 0 ⩽ t ⩽ 3. Sketch its displacement–time
  • velocity–time
  • speed–time
  • acceleration–time.