Further Maths — IGCSE

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

13 resources

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

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

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

28 resources

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

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

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3.1.1 · Know and use the remainder and factor theorems.

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3.2.1 · Find factors of polynomials.

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3.3.1 · Solve cubic equations.

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1.3.1 · f( x ) = 2e

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1.3.2 · f : x ↦ lg x , for x > 0

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1.3.3 · f ( x )

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1.3.4 · fg( x ) [= f(g( x )) ]

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1.3.5 · f ( x ) [= f(f( x )) ] The notation f ( x ) will not be used with trigonometric functions.

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1.4.1 · y = a sin bx + c quadratic, cubic or trigonometric.

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1.4.2 · y = a cos bx + c

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1.4.3 · y = 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.

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1.6.2 · x: • f( x ) = e • f ( x ) = 2 x

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2.5.1 · –3 < x < 4

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2.5.2 · x < 1 or x > 6

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4.1.1 · | ax + b | = c ( c ⩾ 0) expected.

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4.1.2 · | ax + b | = cx + d For algebraic methods, any valid method is acceptable.

31 resources

4.1.3 · | ax + b| = |cx + d |

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4.1.4 · | ax + bx + c | = d using algebraic or graphical methods.

47 resources

4.2.1 · k | ax + b | > c ( c ⩾ 0 ) For algebraic methods, any valid method is

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4.2.2 · k | ax + b | ⩽ c ( c > 0) acceptable.

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4.2.3 · k | ax + b | ⩽ | cx + d | where k > 0

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4.2.4 · | ax + b | ⩽ cx + d

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4.2.5 · | ax + bx + c | > d

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4.2.6 · | ax + bx + c | ⩽ d

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4.3.1 · x 3 + x 3 - 12 = 0

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4.3.2 · 3e = 12 – 5e

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4.3.3 · Candidates are expected to identify the appropriate substitution.

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4.5.1 · f( x ) ⩾ d

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4.5.2 · f( x ) > d

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4.5.3 · f( x ) ⩽ d

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4.5.4 · f( x ) < d where f( x ) is a product of three linear factors and d is a constant.

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5.1.1 · y – x + 3 = 0 and x – 3 xy + y + 19 = 0

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5.1.2 · xy = 4 and xy = 3

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5.1.3 · + = 4 and y = x – 2

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6.2.1 · Write 3 + 21g p – 1g q as a single base 10 logarithm.

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6.2.2 · Write as a natural logarithm. log e

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7.4.1 · To straight-line form unknown constants by calculating the gradient y = Ax or intercept of the transformed graph. y = Ab

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7.4.2 · From straight-line form to an equation of the form . y = Ax + B e = Ax + B y = A 1n x + B

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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

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8.2.1 · is a tangent

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8.2.2 · is a chord

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8.2.3 · does not intersect the circle.

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8.4.1 · intersect

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8.4.2 · touch

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8.4.3 · do not intersect.

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10.4.2 · 2 formulas.: • sin A + cos A = 1 • sec A = 1 + tan A • cosec A = 1 + cot A

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10.5.1 · 4 cot θ = tan θ

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10.5.2 · 2 sec θ + tan θ – 3 = 0 i + 2 cos 3 i = 0

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10.5.3 · 5 sin 3

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10.5.4 · 3 cosec 2 d i - n = 4

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10.6.1 · sin x tan x + cos x = sec x sin i 1 + cos i

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10.6.2 · + = 2 cosec θ

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11.3.1 · repetition of objects

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11.3.2 · objects arranged in a circle

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11.3.3 · both permutations and combinations, are not included.

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13.3.1 · equating like vectors scalars.

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13.3.2 · solving problems using vector geometry, with a diagram given in more complex cases.

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

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14.3.2 · 3 For example, differentiate ` 3 x + 4 j .

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14.3.3 · For trigonometric functions angles will always be in radians.

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

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14.12.1 · ( ax + b ) for any rational n For trigonometric functions angles will always be in

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14.12.2 · sin ( ax + b ) radians.

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14.12.3 · cos ( ax + b ) Solutions for indefinite integrals should include an 2 arbitrary constant.

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14.12.4 · sec ( ax + b ) ax + b

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14.12.5 · e

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14.13.1 · between a line and a curve

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14.13.2 · between two curves

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14.13.3 · a sum of two areas.

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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.

64 resources