IGCSE MathematicsSolve the equation.

1. x² − 5x + 6 = 0Copy the question out exactly. A dropped minus sign here spoils every line after it.
2. (x − 2)(x − 3) = 0Factorise. We need two numbers that multiply to +6 and add to −5.
3. x − 2 = 0  or  x − 3 = 0A product is zero only when one of its factors is zero.
4. x = 2  or  x = 3Two answers, and the question wants both.
5. 2² − 5(2) + 6 = 0, correctSubstitute an answer back in to check it.

IGCSE PhysicsA 12 V battery is connected to a 4 Ω resistor and a 2 Ω resistor in series. Find the current, and the voltage across the 4 Ω resistor.

1. R = 4 + 2 = 6 ΩResistances in series add.
2. I = V ÷ R = 12 ÷ 6Use V = IR for the whole circuit.
3. I = 2 AIn a series circuit the same current flows through both resistors.
4. V = IR = 2 × 4 = 8 VUse V = IR again, this time for the 4 Ω resistor alone.
5. 8 + 4 = 12 V, correctCheck: the 2 Ω resistor takes 4 V, and the two voltages add up to the 12 V of the battery.

IB Diploma MathematicsFind the stationary points of y = x³ − 3x² and say which kind each one is.

1. dy/dx = 3x² − 6xDifferentiate each term.
2. 3x(x − 2) = 0At a stationary point the gradient is zero.
3. x = 0  or  x = 2Each factor gives one point.
4. d²y/dx² = 6x − 6The second derivative tells a maximum from a minimum.
5. (0, 0) maximum,  (2, −4) minimumAt x = 0 the second derivative is −6, so a maximum. At x = 2 it is +6, so a minimum.

IB Diploma PhysicsA car starts from rest and accelerates at 3 m s⁻² for 4 s. How far does it travel?

1. u = 0,  a = 3 m s⁻²,  t = 4 sList what the question gives, with units.
2. s = ut + ½at²Choose the equation that links these three to distance.
3. s = 0(4) + ½(3)(4)²Substitute before doing any arithmetic.
4. s = 24 mGive the answer with its unit.
5. ½(0 + 12)(4) = 24, correctCheck another way. The final speed is 12 m s⁻¹, so the average speed is 6 m s⁻¹ for 4 s.