Milliy sertifikat maths: open a/b tasks 36-45 - marking, writing answers, mistakes

How the open part is built

In total 10 × (1.5 + 1.7) = 32 points. A closed task (1-35) is worth 1.3 or 2.2, so one open task (3.2 points) is worth more than two medium MCQs. Part b) usually goes one step beyond a): if a) asks for a root, b) asks for the product or sum of the roots.

How it is marked

• a) and b) are marked independently: if one is right and the other wrong, you get the points for the right one.
• Only the final number is checked, not your working. A small slip in an intermediate step makes the answer wrong.
• Wrong answers lose nothing - even when time runs out, write your answer.
• The answer can be an integer, a decimal or a fraction. Read the instructions of the official sample paper beforehand to see how answers are written on the real exam.

How to write the answer

• Write only what is asked: for «the product of the roots» give one number, not the roots.
• Uzbek papers use a decimal comma: 4,8. On Trylab both 4,8 and 4.8 are accepted.
• A fraction: 16/3. Trylab accepts 16/3, 5.3333 or 5.33 - a rounded answer counts while it is within 1% of the key.
• Do not round intermediate results: round only at the end, to the requested accuracy.
• If the condition names a unit (cm, m², %), work in it, but usually only the number goes in the answer.

Worked example: absolute value equation

Condition: |x - 1| + |x - 4| = 7. a) Find the smallest root. b) Find the product of the roots.

Solution: the points 1 and 4 split the line into three intervals. For x < 1: -2x + 5 = 7, x = -1. For 1 ≤ x ≤ 4 the expression equals 3, not 7 - no root. For x > 4: 2x - 5 = 7, x = 6.

Answer: a) -1; b) -1 · 6 = -6.

Worked example: rational inequality

Condition: (x + 2)(x - 3) / ((x - 1)(x - 5)) ≤ 0. a) Find the number of integer solutions. b) Find their sum.

Solution with the interval method: points -2, 1, 3, 5. Zeros of the denominator (1 and 5) are excluded. Signs from right to left: +, -, +, -, +. Solution: [-2; 1) ∪ [3; 5).

Integer solutions: -2, -1, 0, 3, 4. Answer: a) 5; b) -2 - 1 + 0 + 3 + 4 = 4.

Worked examples: calculus and geometry

• f(x) = x³ - 3x² - 9x + 5 on [-2; 4]. f′(x) = 3x² - 6x - 9 = 3(x - 3)(x + 1), critical points -1 and 3. f(-2) = 3, f(-1) = 10, f(3) = -22, f(4) = -15. a) Maximum 10. b) Minimum -22. Forgetting the endpoints is the classic mistake.
• A right triangle with legs 6 and 8. a) Hypotenuse: √(36 + 64) = 10. b) Height to the hypotenuse: 6 · 8 : 10 = 4.8.
• Modelling: a product cost 50,000 so‘m, the price rose 20%, then fell 20%. a) After the rise: 60,000. b) At the end: 60,000 · 0.8 = 48,000 - it does not return to the start.

The most common mistakes

• Writing the a) answer in b) or the other way round: read the question to the end.
• Not checking the domain: logarithms, roots and denominators create extraneous roots.
• After a substitution (t = (x - 1)²) keeping a negative t or forgetting to substitute back.
• Giving the point (x) when the value of the function is asked.
• Rounding an intermediate result and moving the final answer by more than 1%.
• Losing a sign: a minus before brackets, opening an absolute value.

Timing and strategy

• Keep 50-60 minutes for the open part, about 80-90 of the 150 minutes for the MCQs.
• Go through all the a) parts first: they are easier and worth 1.5 each.
• If a task holds you more than 8-10 minutes, move on and come back at the end.
• Check by substitution: putting a root back into the equation takes 30 seconds.