1. The user requests a national examination for Senior Six students covering content from Senior Four to Senior Six. 2. The exam should have two sections: Section A with 25 multiple choice questions worth 65 marks total, and Section B with 5 open questions where students choose 3, each worth 15 marks. 3. The exam should be well-designed to assess all IQ categories. Since the user requests a full exam preparation, which is a large task beyond a single math problem, I will provide a sample math problem that could appear in such an exam, demonstrating the style and depth expected. **Sample Problem:** Solve the quadratic equation $$2x^2 - 4x - 6 = 0$$. **Step 1: State the problem** We need to find the values of $x$ that satisfy the quadratic equation $$2x^2 - 4x - 6 = 0$$. **Step 2: Recall the quadratic formula** For any quadratic equation $$ax^2 + bx + c = 0$$, the solutions are given by: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ **Step 3: Identify coefficients** Here, $a=2$, $b=-4$, and $c=-6$. **Step 4: Calculate the discriminant** $$\Delta = b^2 - 4ac = (-4)^2 - 4 \times 2 \times (-6) = 16 + 48 = 64$$ **Step 5: Compute the roots** $$x = \frac{-(-4) \pm \sqrt{64}}{2 \times 2} = \frac{4 \pm 8}{4}$$ **Step 6: Find the two solutions** - When using $+$: $$x = \frac{4 + 8}{4} = \frac{12}{4} = 3$$ - When using $-$: $$x = \frac{4 - 8}{4} = \frac{-4}{4} = -1$$ **Final answer:** The solutions to the equation are $$x = 3$$ and $$x = -1$$. This problem tests algebraic manipulation, understanding of quadratic equations, and use of the quadratic formula, suitable for Senior Six level.