Grade 11

Mathematics, four units

The four unit syllabus as it is actually examined: arithmetic and geometric sequences, trigonometry in any triangle, the differential and integral calculus of polynomials, and probability up to the binomial experiment.

8 lessons

  1. 1How do you find the first term and the common difference from two given terms?Build two equations. If the fifth and ninth terms are given, write each as the first term plus the number of differences before it, and subtract one equation from the other. The common difference falls out at once, because the gap between those terms contains exactly four of them, and substituting back gives the first term.45m
  2. 2When does an infinite geometric sequence have a finite sum?Only when the absolute value of the common ratio is less than one, that is when the ratio lies strictly between minus one and one and is not zero. Then the terms shrink fast enough and the sum converges to the first term over one minus the ratio. If the absolute value reaches one or more, the sum grows without bound.45m
  3. 3When do you use the sine rule and when the cosine rule?The data decide. Given two sides and the angle between them, or all three sides, use the cosine rule. Given a matching pair — a side and the angle opposite it — plus one more fact, use the sine rule. The short version: cosine rule when you have no complete side-and-opposite-angle pair, sine rule when you do.50m
  4. 4How do you solve a trigonometry question set in a trapezium or a pyramid?Find a triangle inside it. The examined question is almost never a bare triangle, so the first move is to draw a height or a diagonal that breaks the shape into triangles. Once there is a triangle with enough data in it, you carry on with the sine or cosine rule exactly as usual.50m
  5. 5What is the order of operations in a full investigation of a polynomial?Domain, intersections with the axes, the derivative, setting the derivative to zero, classifying each stationary point, intervals of increase and decrease, and only then the sketch. The order is not a matter of style: each stage supplies the data for the next, and the sketch comes last so that it fits the results rather than determining them.50m
  6. 6How do you find the largest area you can fence with a given length of fence?Write the area as a function of one variable. You start with two — a length and a width — and use the constraint on the total fence to express one in terms of the other. Then differentiate the area function, set it to zero, and confirm that the point you found is genuinely a maximum and not a minimum.50m
  7. 7How do you find the area enclosed between two graphs?First find where they intersect, because those points are the limits of integration. Then evaluate the definite integral of the upper function minus the lower one between those limits. The order of the subtraction is the whole point: whichever curve lies above on that interval is written first, otherwise the area comes out negative.50m
  8. 8What changes in the calculation when you draw without replacement?The denominator changes, and sometimes the numerator too. With replacement the situation resets and every draw is independent, so you multiply the same probability. Without replacement, once one ball is out there are fewer balls in total and fewer of the colour drawn, so the second probability is worked out from a new situation.50m