Most Tawjihi maths students understand a worked solution when they read it, then freeze in front of a new problem. The fix is a routine you use every time: write what you are looking for, choose the method that produces it, show each step, then check the answer. The three examples below are original practice problems on topics from the grade 12 scientific maths book.
The four-step routine
- 1Unknown: write exactly what the question asks for (a value, a point, a function, a matrix).
- 2Method: name the rule that gives that kind of answer (first derivative test, integration, inverse matrix...).
- 3Steps: write every step; marks are given for the method, not only the final number.
- 4Check: substitute back, or test with a second method.
Example 1: local maximum and minimum
Find the local extreme values of f(x) = x³ − 3x² + 2. Unknown: the x values and the values of f there. Method: first derivative test. f′(x) = 3x² − 6x = 3x(x − 2), so the critical numbers are x = 0 and x = 2. Test the sign: for x < 0, f′ > 0; between 0 and 2, f′ < 0; for x > 2, f′ > 0. So f has a local maximum f(0) = 2 and a local minimum f(2) = 8 − 12 + 2 = −2.
Common mistake: stopping at f′(x) = 0. A critical number is only a maximum or minimum if the sign of f′ changes there, so the sign table is part of the answer.
Example 2: finding a curve from its derivative
A curve has f′(x) = 6x² − 4x and passes through (1, 5). Find f(x). Method: integrate, then use the point. f(x) = 2x³ − 2x² + c. At x = 1: 2 − 2 + c = 5, so c = 5 and f(x) = 2x³ − 2x² + 5. Check: differentiating gives 6x² − 4x, and f(1) = 5.
Common mistake: forgetting the constant c. Without it you cannot use the point, and the answer is a different curve.
Example 3: a linear system with matrices
Solve 2x + y = 5 and x − y = 1 using matrices. Coefficient matrix A = [2 1; 1 −1], so |A| = (2)(−1) − (1)(1) = −3, which is not zero, so A has an inverse: A⁻¹ = (1/−3)[−1 −1; −1 2]. Then [x; y] = A⁻¹[5; 1] = (1/−3)[−6; −3] = [2; 1]. Check: 2(2) + 1 = 5 and 2 − 1 = 1.
Common mistake: swapping the main diagonal but forgetting to change the signs of the other diagonal, or dividing by the determinant with the wrong sign.
Before calculating anything, write what you are trying to find. Half of the wrong answers in maths exams come from solving for the wrong thing.
How EXAMEV helps
The EXAMEV Scientific Mathematics Test Bank for grade 12 has over 1,000 practice questions on derivatives, integration and matrices, organised by lesson, with an explanation for every answer and progress by topic. The examples here are EXAMEV practice. For a full revision timetable, see the six-week Tawjihi plan on this blog.