Further Maths

• Equation of a circle

  The equation of a circle with centre (a,b)and radius r is given by

  $(x−a)^2+$$(y−b)^2=$$r^2$

  When the centre of the circle is at the origin, the equation simplifies to$x^2+$$y^2=$$r^2$
• Simultaneous equations

  A set of two or more equations involving two or more variables. There should be the same number of equations as there are variables.

  Two linear simultaneous equations can be solved using the elimination method or the substitution method.

  Two simultaneous equations, one of which is linear and one quadratic, should be solved using the substitution method.
• Differentiation

  The process of finding the gradient function or derivative.
• Factor theorem

  The factor theorem states that

  • If (x−a)(x−a) is a factor of a polynomial f(x)f(x), then f(a)=0f(a)=0.

  The converse is also true:

  • If f(a)=0f(a)=0 then (x−a)(x−a) is a factor of f(x)f(x).

  This theorem is useful in factorising polynomials: if you can find a number aa such that f(a)=0f(a)=0, then you know one of the factors of the polynomial.

  The factor theorem is a special case of the remainder theorem.
• Identity matrix

  The 2 × 2 identity matrix is I=(1001)I=(1001).

  Any identity matrix is a square matrix with 1s on its leading diagonal and zeros elsewhere.

  When any matrix is pre- or post-multiplied by the identity matrix of the appropriate size, the matrix is unchanged, i.e. MI=IM=MMI=IM=M for any conformable matrix MM.
• Reflection

  A reflection is a type of transformation in which each point on the object is mapped to an image point so that the object point and image point are the same distance from the mirror line, but on opposite sides, and the line connecting the object point and image point is perpendicular to the mirror line.

  A reflection can be represented by a matrix if the mirror line passes through the origin.

  • Reflection in the xx-axis is represented by the matrix (100−1)(100−1)
  • Reflection in the yy-axis is represented by the matrix (−1001)(−1001)
  • Reflection in the line y=xy=x is represented by the matrix(0110)(0110)
  • Reflection in the line y=−xy=−x is represented by the matrix (0−1−10)(0−1−10)