Finishing the maths paper in two hours

The CDS Elementary Mathematics paper gives you 100 questions in 120 minutes. That is 72 seconds a question on average, and each wrong answer costs a third of a mark. Most candidates who know the syllabus still lose marks here. They don't run out of knowledge. They run out of time, because they spend four minutes on a geometry question early on and then rush the last twenty.

Speed in this paper is not about calculating faster. It comes from three habits: choosing the order in which you answer, knowing when to leave a question, and using the four options as information. This guide covers all three and shows six techniques on typical question types.

A three-pass plan

The first pass does two jobs. It banks the easy marks before fatigue sets in, and it shows you the whole paper, so that you know how many hard questions remain.

Topic triage

Question difficulty varies by paper, but topics differ in how long they usually take.

Use this as a guide, not a rule. A geometry question you recognise at once is a quick question. A percentage question with five conditions is slow.

Skipping rules

• If you can't see a way to start within about 30 seconds, mark it and move on.
• If you are past about 90 seconds and the end is not in sight, mark it and move on. You can return in pass 2.
• Don't skip because a question looks long. Read it first; long wording often hides a simple question.
• In pass 3, answer only if you have eliminated two options. With two options left, a guess gains about a third of a mark on average. A blind guess among four gains nothing.

See negative marking in CDS for the arithmetic behind that last rule.

Six speed techniques

1. Test the options

Question: Which value of x satisfies √(x + 5) + x = 7? Options: 2, 4, 11, 20.

• Try x = 4: √9 + 4 = 3 + 4 = 7. Answer: 4.
• Solving algebraically means squaring and solving a quadratic, then rejecting a false root. Testing takes ten seconds.

2. Put in special values

Question: sec⁴θ − tan⁴θ equals which of: (a) 1, (b) 2 sec²θ, (c) sec²θ + tan²θ, (d) 2 tan²θ?

• Put θ = 45°: sec⁴θ = 4 and tan⁴θ = 1, so the expression is 3.
• The options give 1, 4, 3 and 2. Answer: (c).
• If two options had tied, you would try a second angle such as 60°. There the expression is 16 − 9 = 7, and sec²θ + tan²θ = 4 + 3 = 7 again.

The same trick works for algebraic identities: put a = 2 and b = 1, or x = 1.

3. Use the unit digit

Question: Find the unit digit of 7⁹⁵.

• The unit digits of powers of 7 repeat in a cycle of 4: 7, 9, 3, 1.
• 95 leaves a remainder of 3 when divided by 4, so the unit digit is the third in the cycle. Answer: 3.

Unit digits also let you check a product instantly. 47 × 53 must end in 1, so if only one option ends in 1, you are done.

4. Multiply by fractions of 100

Question: Find 1,369 × 25.

• 25 = 100/4, so 1,369 × 25 = 1,36,900 ÷ 4 = 34,225.
• Similarly, × 125 = × 1,000 ÷ 8, and × 50 = × 100 ÷ 2.

5. Bound the answer

Question: √5,000 is closest to which of 68.5, 70.7, 72.1, 75.0?

• 70² = 4,900 and 71² = 5,041, so the root lies between 70 and 71. Answer: 70.7.
• Bounding like this eliminates options in mensuration and percentage questions too.

6. Work in units per day

Question: A can finish a job in 12 days and B in 18 days. How long do they take together?

• Take the job as LCM(12, 18) = 36 units. A does 3 units a day and B does 2.
• Together, 5 units a day, so 36 ÷ 5 = 7.2 days. No fractions of fractions needed.

Practising speed

• Time every practice set, even ten questions. Untimed practice builds knowledge but not pace.
• Keep a slow-question log. After each mock, list every question that took over two minutes and write the faster method beside it. Patterns appear within a few weeks.
• Drill the facts that make you fast: squares to 30, cubes to 15, the trigonometric table, Pythagorean triplets, fraction-to-percentage pairs (1/8 = 12.5%, 1/6 ≈ 16.67%, 1/12 ≈ 8.33%).
• Sit full papers in a two-hour slot, at the time of day your exam will be held. See mock tests for CDS for how to review them.

A two-minute drill

Try to finish all eight in under two minutes, without writing more than a few digits.

1. 15% of 480
2. 36 km/h in m/s
3. 99²
4. sin²30° + cos²30°
5. If a + b = 7 and ab = 12, find a² + b².
6. The area of an equilateral triangle of side 4 cm
7. The largest of 1/2, 3/5, 5/8 and 7/11
8. The sum of the interior angles of an octagon

Answers

1. 72. 10% is 48 and 5% is 24.
2. 10 m/s. 36 × 5/18.
3. 9,801. (100 − 1)² = 10,000 − 200 + 1.
4. 1. Identity, no values needed.
5. 25. 49 − 24.
6. 4√3 cm². (√3/4) × 16.
7. 7/11. The decimals are 0.5, 0.6, 0.625 and about 0.636.
8. 1,080°. 6 × 180°.

What to do next

• Use the three-pass plan in your next full mock, and note how many questions each pass covered
• Start a slow-question log and add to it after every practice set
• Learn squares to 30 and cubes to 15 this week
• Keep the maths revision sheet for the last week before the exam