Mathematics 9709 — specification explorer

1. Quadratics

• 1.1.1carry out the process of completing the square for a quadratic polynomial ax + bx + c and use y a completed square form
• 1.1.2find the discriminant of a quadratic polynomial ax + bx + c and use the discriminant ‘repeated root’ is included.
• 1.1.3solve quadratic equations, and quadratic By factorising, completing the square and using the inequalities, in one unknown formula.
• 1.1.4solve by substitution a pair of simultaneous e.g. equations of which one is linear and one is 2 quadratic
• 1.1.5recognise and solve equations in x which are quadratic in some function of x . tan

2. Functions

• 1.2.1understand the terms function, domain, range, one-one function, inverse function and composition of functions
• 1.2.2identify the range of a given function in simple cases, and find the composition of two given functions condition that a composite function
• 1.2.3determine whether or not a given function is one-one, and find the inverse of a one-one function in simple cases
• 1.2.4illustrate in graphical terms the relation between Sketches should include an indication of the mirror a one-one function and its inverse line
• 1.2.5understand and use the transformations of the Including use of the terms ‘translation’, ‘reflection’ graph of y = f( x ) given by and ‘stretch’ in describing transformations. Questions y = f( x ) + a , y = f( x + a ) , may involve algebraic or trigonometric functions, or other graphs with given features. y = a f( x ) , y = f( ax ) and simple combinations of these.

3. Coordinate geometry

• 1.3.1find the equation of a straight line given e.g. given two points, or one point and the gradient. sufficient information
• 1.3.1a

  1 problems:

  • understand that the equation
• 1.3.2interpret and use any of the forms y = mx + c , y – y = m ( x – x ) , ax + by + c = 0 in solving
• 1.3.2a

  2 2 ( x – a ) + ( y – b ) = r represents the circle with centre ( a , b ) and radius r involving lines and circles its associated algebraic equation, and use the line relationship between points of intersection of graphs and solutions of equations.:

  • use algebraic methods to solve problems
  • understand the relationship between a graph and

4. Circular measure

• 1.4.1understand the definition of a radian, and use the relationship between radians and degrees
• 1.4.1a

  2 in solving problems concerning the arc length triangles and areas of triangles. and sector area of a circle.:

  • use the formulae s = r i and A = r i Including calculation of lengths and angles in

5. Trigonometry

• 1.5.1sketch and use graphs of the sine, cosine and Including e.g. tangent functions (for angles of any size, and using either degrees or radians)
• 1.5.2use the exact values of the sine, cosine and tangent of 30 °, 45 °, 60 °, and related angles
• 1.5.3

  use the notations sin⁻¹x, cos⁻¹x, tan⁻¹x to denote the principal values of the inverse trigonometric relations:

  • No specialised knowledge of these functions is required, but understanding of them as examples of inverse functions is expected.
• 1.5.4

  use the identities θ≡tanθ and sin²θ+cos²θ ≡ 1:

  • e.g. in proving identities, simplifying expressions and solving equations.
• 1.5.5find all the solutions of simple trigonometrical equations lying in a specified interval (general forms of solution are not included).

6. Series

• 1.6.1use the expansion of ( a + b ) , where n is a positive integer
• 1.6.2recognise arithmetic and geometric progressions
• 1.6.3use the formulae for the n th term and for the Including knowledge that numbers sum of the first n terms to solve problems involving arithmetic or geometric progressions equivalent).
• 1.6.4

  Questions may involve more than one progression. geometric progression, and the formula for the sum to infinity of a convergent geometric progression.:

  • use the condition for the convergence of a

7. Differentiation

• 1.7.1understand the gradient of a curve at a point as the limit of the gradients of a suitable sequence of chords, and use the notations d y , and d y f ′ ( x ) , f ″( x ) , 2 for first and second derivatives
• 1.7.2use the derivative of x (for any rational n ), together with constant multiples, sums and differences of functions, and of composite functions using the chain rule
• 1.7.3apply differentiation to gradients, tangents and normals, increasing and decreasing functions and rates of change
• 1.7.4locate stationary points and determine their nature, and use information about stationary points in sketching graphs.

8. Integration

• 1.8.1understand integration as the reverse process of differentiation, and integrate ( ax + b ) (for any rational n except –1), together with constant multiples, sums and differences
• 1.8.2solve problems involving the evaluation of a constant of integration
• 1.8.3evaluate definite integrals
• 1.8.4

  use definite integration to find lines parallel to the axes, or between a curve and a line or between two curves:

  • the area of a region bounded by a curve and
  • a volume of revolution about one of the axes.