Grade 11 Mathematics Revision

• This revision program is designed to assist you in revising the critical content and skills envisaged/ planned to be covered during the 1st term. The purpose is to prepare you to understand the key concepts and to provide you with an opportunity to establish the required standard and the application of the knowledge necessary to succeed in the NCS examination.
• Exponents


  𝑎𝑎𝑛𝑛 = 𝑎𝑎 × 𝑎𝑎 × 𝑎𝑎 × 𝑎𝑎 × … to 𝑛𝑛 factors of ��𝑎, where 𝑎𝑎 > 0 and 𝑛𝑛 ∈ ℕ
• Product of powers


  𝑎𝑎𝑚�� × 𝑎𝑎𝑛𝑛 = 𝑎𝑎𝑚𝑚+𝑛𝑛
• Quotient of powers


  𝑎𝑎��𝑚/𝑎𝑎𝑛𝑛 = 𝑎𝑎𝑚𝑚−𝑛𝑛
• Power of a power


  (𝑎𝑎𝑚𝑚)𝑛𝑛 = 𝑎𝑎𝑚𝑚𝑛𝑛
• Power of a product


  (𝑎𝑎𝑎𝑎)��𝑚 = 𝑎𝑎𝑚𝑚𝑎𝑎𝑚𝑚
• Negative exponent


  𝑎𝑎−𝑚𝑚 = 1/𝑎𝑎𝑚𝑚
• Negative exponent


  1/𝑎𝑎−𝑚𝑚 = 𝑎𝑎𝑚𝑚
• Exponent of 0


  𝑎𝑎0 = 1
• Surd


  When a root of a counting number is an irrational number, then the root is called a surd
• Simplifying surds


  √𝑎𝑎𝑛𝑛/𝑚𝑚 = 𝑎𝑎𝑛𝑛/𝑚𝑚
• Simplifying surds


  √𝑎𝑎����/𝑚𝑚 = √𝑎𝑎/𝑚𝑚 . √𝑎𝑎/𝑚𝑚
• You can use the laws only if the bases are the same
• Exponent laws


  • 2𝑚𝑚 . 2𝑛𝑛 = 2𝑚𝑚+𝑛𝑛
  • 2𝑚𝑚/2𝑛𝑛 = 2𝑚𝑚−𝑛𝑛
  • 10𝑚𝑚 . 2𝑚𝑚 = 20𝑚𝑚
  • 10𝑚𝑚/2𝑚𝑚 = 5𝑚𝑚
• Remember the following when solving exponential equations:
• Exponential equations with variable in exponent


  Make the bases the same on both sides, then equate the exponents
• Exponential equations with variable in base


  Use the reciprocal of the exponent on both sides
• Exponential equations with variable in base


  If 𝑥𝑥𝑚𝑚/𝑛𝑛 = 𝑎𝑎, where 𝑎𝑎 is any constant, then if 𝑚𝑚 is odd there is only one solution, if 𝑚𝑚 is even there are two solutions (one positive and one negative)
• The final answer is by no means the most important in Mathematics. Systematic, detailed and logical layout of every step of your working is the most important.
• Do not accept the fact that you are careless. Carelessness can be overcome by checking your work. It is important to check the correctness and the validity of every step of your calculations. In this way carelessness is overcome.
• Never take short cuts in Mathematics by leaving out steps in your working.
• Despair in Mathematics can destroy your Mathematics. Never give up: try again and again and … until you get it right. Continually say to yourself: I CAN!!!!!
• The more you practice the better you will become!
• Solving quadratic equations: Factorisation, Quadratic formula
• Solving simultaneous equations: Both linear - use substitution method, One linear and one quadratic - make one variable the subject in the linear equation, substitute into the quadratic equation, solve, then substitute the solution into the linear equation
• Solving equations with surds: Get term with surd alone, square both sides, solve, test answer
• Solving quadratic inequalities: Write in standard form, make coefficient of 𝑥𝑥2 positive, change direction of inequality if multiply/divide by negative, factorise, determine answer using number line or parabola
• Nature of roots of quadratic equation: ∆ = 𝑎𝑎2 - 4𝑎𝑎𝑎𝑎, ∆ < 0 (non-real/imaginary), ∆ ≥ 0 (real), ∆ > 0 and ∆ a perfect square (real, rational and unequal), ∆ > 0 and ∆ not a perfect square (real, irrational and unequal), ∆ = 0 (real, rational and equal)
• Solve for x:


  4x - 2/x = 7
• Hence solve for a:


  4(a^2 - a) - 2/(b^2-b) = 7
• Solve for x and y:

  4x+y = 2x+4 and 2y^2 - 3xy = -4
• Solve for x and y:


  (x - 2)^2 + (y - 3)^2 = 4 and x + y = 4
• Express k as a power of 5 if k = 5^x + 5^x + 5^x + ... 625 times.


  Value
• When analysing the nature of the roots of a quadratic equation, the discriminant (Δ) is used:
• Δ = b^2 - 4ac


  Discriminant formula
• Nature of the roots

  • Δ < 0: Non real/Imaginary
  • Δ ≥ 0: Real
  • Δ > 0 and Δ a perfect square: Real, rational and unequal
  • Δ > 0 and Δ not a perfect square: Real, irrational and unequal
  • Δ = 0: Real, rational and equal
• Without solving the equation, determine the nature of the roots:

  3x^2 - x + 4 = 0
• Without solving the equation, determine the nature of the roots:

  x^2 + 2x - 3 = 0
• For which values of r will 2x^2 - 2x - r = 0 have equal roots?


  Value
• For which values of h will the roots of 3x^2 - 2hx + 3 = 0 be imaginary?


  Value