Most first-time SSC aspirants open a maths book at page one, spend three weeks on a chapter that gives one question, and reach the exam without having touched the chapters that give five. The syllabus for CGL, CHSL and CPO looks huge, but the way questions are framed is repetitive. If you pick the order of chapters well and learn one clean method per chapter, maths becomes the most predictable section of the paper.
This guide does three things. It gives a chapter priority order for a beginner, it explains what each chapter actually tests, and it walks through 51 original worked examples in algebra, number system, trigonometry (including heights and distances) and simplification. Every calculation has been checked step by step. It ends with common traps and a 60-day schedule. Percentage, profit and loss, time and work and data interpretation tricks are covered in other posts on the site, so they are not repeated here.
What the maths section looks like
In SSC CGL Tier 1 the Quantitative Aptitude section has 25 questions of 2 marks each, with a 0.5 mark negative for a wrong answer, and the whole paper (all four sections) runs for 60 minutes. CHSL Tier 1 follows a similar 25-question maths section. CPO Paper I carries a quantitative section as well, with a different marking scheme. Pattern, marks and negative marking get revised from time to time, so always check the latest notification for the exam you are filling.
The practical meaning: you have roughly 12 to 15 minutes for 25 maths questions if you want to leave time for reasoning and English. That is about 30 seconds per question on average, which is why method and speed matter more than depth of theory.
Chapter priority order for a first-time aspirant
The order below is a planning suggestion based on two things: how much each chapter helps other chapters, and how quickly a beginner can become accurate in it. It is not an official weightage. Chapter-wise question counts change from shift to shift, so treat any exact counts you see online as approximate.
- Arithmetic first (percentage, ratio, averages, profit and loss, time and work, speed and distance, interest, mixtures). It is the base of everything and the easiest to score in.
- Number system and simplification (together with arithmetic, in the first two weeks). These give you calculation speed, remainders, HCF and LCM, and approximation, which every other chapter uses.
- Algebra. Identity-based questions are short and quick once the standard forms are memorised.
- Geometry (triangles, circles, quadrilaterals) and mensuration. Heavily asked but needs diagrams and practice, so it comes after the calculation chapters are comfortable.
- Trigonometry, including heights and distances. Small formula set, high accuracy once learned.
- Data interpretation. Needs percentage and ratio fluency, so it comes after arithmetic, and it is mostly practice.
- Statistics (mean, median, mode basics) and, for Tier 2 candidates, the deeper statistics syllabus.
Algebra, number system and trigonometry are small in size but rewarding. They are what the rest of this post teaches in detail.
What each chapter actually tests
| Chapter | What the question usually asks | Core skill |
|---|---|---|
| Arithmetic | Word problems with a single idea | Translating text to equations |
| Algebra | Value of an expression given a+1/a or a+b, ab | Identities, not solving equations |
| Number system | Remainder, unit digit, divisibility, HCF, LCM, factors | Cycles, prime factors, remainders |
| Geometry | Angles, similar triangles, circles, centres of a triangle | Recalling theorems quickly |
| Trigonometry | Evaluate an expression, prove an identity, find a height | Standard values, identities |
| DI | Percent change, ratio and average from a table or chart | Reading data and approximating |
| Statistics | Mean, median, mode, simple dispersion | Formula recall |
| Simplification | BODMAS, surds, squares, approximate values | Pattern spotting and rounding |
Algebra: identities that carry the chapter
SSC rarely asks you to solve a messy equation. It gives a condition such as a+1/a=5 and asks for a different expression. The tool is a small set of identities.
- (a+b)² = a²+b²+2ab and (a-b)² = a²+b²-2ab
- a²-b² = (a+b)(a-b)
- a³+b³ = (a+b)³ - 3ab(a+b) and a³-b³ = (a-b)³ + 3ab(a-b)
- (a+b+c)² = a²+b²+c² + 2(ab+bc+ca)
- a³+b³+c³ - 3abc = (a+b+c)(a²+b²+c² - ab - bc - ca). If a+b+c=0, then a³+b³+c³ = 3abc.
- If x+1/x = k, then x²+1/x² = k²-2 and x³+1/x³ = k³-3k. Same pattern with a minus sign: if x-1/x = k, then x²+1/x² = k²+2.
Worked examples: algebra
Example 1. If a+1/a = 5, find a³+1/a³.
Formula: k³-3k with k=5. 125-15 = 110.
Example 2. If x-1/x = 4, find x⁴+1/x⁴.
x²+1/x² = 4²+2 = 18. Then x⁴+1/x⁴ = 18²-2 = 324-2 = 322.
Example 3. If a+b = 10 and ab = 21, find a²+b² and a³+b³.
a²+b² = 100-42 = 58. a³+b³ = 10³ - 3×21×10 = 1000-630 = 370. (Check: the numbers are 3 and 7: 27+343 = 370.)
Example 4. If a+b+c = 12 and ab+bc+ca = 47, find a²+b²+c².
144 - 2×47 = 144-94 = 50.
Example 5. If a+b+c = 9 and a²+b²+c² = 29, find a³+b³+c³-3abc.
First ab+bc+ca = (81-29)/2 = 26. Then the expression = 9 × (29-26) = 9×3 = 27.
Example 6. Find (a-b)³+(b-c)³+(c-a)³ for a=7, b=4, c=1.
The three bases add up to zero, so the sum is 3(a-b)(b-c)(c-a) = 3×3×3×(-6) = -162. Direct check: 27+27+(-216) = -162.
Example 7. If x+1/x = 2, find x¹⁰⁰+1/x¹⁰⁰.
x²-2x+1 = 0 gives x = 1 only. So the value is 1+1 = 2.
Example 8. If x²-3x+1 = 0, find x³+1/x³.
Divide by x: x+1/x = 3. So 27-9 = 18.
Example 9. If x = 2+√3, find x²+1/x².
1/x = 1/(2+√3) = 2-√3, so x+1/x = 4. Then x²+1/x² = 16-2 = 14. (And x³+1/x³ = 64-12 = 52.)
Example 10. 2x+3y = 13 and 3x+2y = 12. Find x²+y².
Add: 5x+5y = 25, so x+y = 5. Subtract the first from the second: x-y = -1. So x = 2, y = 3. Check: 4+9 = 13 and 6+6 = 12. x²+y² = 4+9 = 13.
Example 11. Find the remainder when x³-4x²+5x-7 is divided by (x-2).
Remainder theorem: put x = 2. 8-16+10-7 = -5.
Example 12. If (x-3) is a factor of x³-2x²+kx-12, find k.
Put x = 3: 27-18+3k-12 = 0, so 3k = 3 and k = 1.
Example 13. Minimum value of x²-6x+13.
Complete the square: (x-3)²+4. Minimum is 4 at x = 3.
Example 14. For x > 0, find the minimum of 4x+9/x.
The two terms are equal when 4x = 9/x, so x² = 9/4 and x = 3/2. Value = 6+6 = 12. (AM-GM gives the same: 2√(4×9) = 12.)
Number system: five tools to master
Divisibility rules
- By 3 or 9: digit sum divisible by 3 or 9.
- By 4: last two digits divisible by 4. By 8: last three digits divisible by 8.
- By 11: (sum of digits at odd places) minus (sum of digits at even places) is 0 or a multiple of 11.
- By 7: subtract twice the last digit from the rest of the number and repeat.
Example 15. Is 23947 divisible by 7?
2394 - 2×7 = 2380. Then 238 - 0 = 238 = 7×34. Yes, divisible. (23947 = 7×3421.)
Example 16. Is 918082 divisible by 11?
From the right, odd places: 2+0+1 = 3. Even places: 8+8+9 = 25. Difference = 22, a multiple of 11. Yes. (918082 = 11×83462.)
Example 17. Find the digit * so that 52*8 is divisible by 9.
5+2+8 = 15, so * must make the sum 18. * = 3. (5238 = 9×582.)
Example 18. Find the digit * so that 4*73 is divisible by 11.
From the right, odd places: 3+*. Even places: 7+4 = 11. Difference (3+*)-11 must be 0, so * = 8. (4873 = 11×443.)
Remainders
Work with the remainder, not the number. A number just below a multiple behaves like -1, and that makes powers easy.
Example 19. Remainder when 2¹⁰⁰ is divided by 7.
2³ = 8 leaves remainder 1. 100 = 3×33+1, so 2¹⁰⁰ ≡ 2. Remainder 2.
Example 20. Remainder when 17²³ is divided by 9.
17 = 18-1, so 17 ≡ -1 (mod 9). An odd power of -1 is -1, which is 9-1 = 8.
Example 21. Remainder when 13×17×19 is divided by 5.
Remainders: 3, 2, 4. Product = 24, which leaves 4 when divided by 5.
Unit digit and cycles
Digits 2, 3, 7, 8 repeat every 4 powers. Digits 4 and 9 repeat every 2. Digits 0, 1, 5, 6 never change. Divide the power by 4 and use the remainder (treat remainder 0 as the fourth position).
Example 22. Unit digit of 7¹²³.
Cycle for 7: 7, 9, 3, 1. 123 = 4×30+3, so third position: 3.
Example 23. Unit digit of 2⁴⁵ × 3²⁰.
2⁴⁵: 45 = 4×11+1, so 2. 3²⁰: 20 is a multiple of 4, so fourth position of 3, 9, 7, 1, giving 1. Product unit digit = 2×1 = 2.
HCF and LCM
Remember: HCF × LCM = product of the two numbers (only for two numbers). For "remainder" problems, subtract the remainder first and take HCF. For "least number with remainder r", take LCM and add r.
Example 24. HCF and LCM of 36, 48, 60.
36 = 2²×3², 48 = 2⁴×3, 60 = 2²×3×5. HCF = 2²×3 = 12. LCM = 2⁴×3²×5 = 720. So HCF 12, LCM 720.
Example 25. Two numbers have HCF 12, product 2160 and sum 96. Find them.
LCM = 2160/12 = 180. Write numbers as 12a and 12b with a, b coprime and ab = 15. Options: (1,15) giving 12 and 180 (sum 192), or (3,5) giving 36 and 60 (sum 96). Answer: 36 and 60.
Example 26. Greatest number that divides 245 and 1029 leaving remainder 5 in each case.
Take HCF of 240 and 1024. 240 = 16×15 and 1024 = 16×64, and 15 and 64 are coprime. HCF = 16, which is greater than 5, so it is valid.
Example 27. Smallest number which leaves remainder 7 when divided by 12, 15 and 20.
LCM of 12, 15, 20 = 60. Number = 60+7 = 67.
Example 28. Largest 4-digit number divisible by 12, 15 and 20.
LCM = 60. 9999 = 60×166+39, so subtract 39: 9960.
Factors and trailing zeros
If N = p^a × q^b × r^c, the number of factors is (a+1)(b+1)(c+1) and the sum of factors is the product of (1+p+...+p^a) terms.
Example 29. Number of factors and sum of factors of 360.
360 = 2³×3²×5. Count = 4×3×2 = 24. Sum = (1+2+4+8)(1+3+9)(1+5) = 15×13×6 = 1170.
Example 30. Number of trailing zeros in 50!.
Count factors of 5: 50/5 = 10, 50/25 = 2. Total 12.
Trigonometry: the small set that covers most questions
Learn the values table once and never recompute it.
| Angle | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan | 0 | 1/√3 | 1 | √3 | not defined |
Memory aid: sin values go 0, 1/2, 1/√2, √3/2, 1 which is √0/2, √1/2, √2/2, √3/2, √4/2. Cos is the same list reversed.
Identities to know cold: sin²θ+cos²θ = 1, 1+tan²θ = sec²θ, 1+cot²θ = cosec²θ, tanθ = sinθ/cosθ, and complementary angles: sin(90°-θ) = cosθ, tan(90°-θ) = cotθ, sec(90°-θ) = cosecθ.
Worked examples: trigonometry
Example 31. If sinθ = 3/5 (θ acute), find secθ+tanθ.
cosθ = 4/5 from the 3-4-5 triangle. secθ = 5/4, tanθ = 3/4. Sum = 8/4 = 2.
Example 32. If sinθ+cosθ = 7/5, find sin³θ+cos³θ.
Square: 1+2sinθcosθ = 49/25, so sinθcosθ = 12/25. Using a³+b³ = (a+b)(a²+b²-ab): (7/5)(1-12/25) = (7/5)(13/25) = 91/125. Check with sinθ = 4/5, cosθ = 3/5: 64/125+27/125 = 91/125.
Example 33. If tanθ+cotθ = 3, find tan²θ+cot²θ.
Square: tan²θ+cot²θ+2 = 9, so the answer is 7.
Example 34. Evaluate 3tan²30° + 4sin²60° - 2cos²45°.
3×(1/3) + 4×(3/4) - 2×(1/2) = 1+3-1 = 3.
Example 35. Evaluate tan17° × tan38° × tan52° × tan73°.
Pair 17° with 73° and 38° with 52°. tan73° = cot17°, so tan17°×tan73° = 1. Same for the other pair. Answer 1.
Example 36. Evaluate sin20°/cos70° + cos40°/sin50°.
cos70° = sin20° and sin50° = cos40°, so each fraction is 1. Answer 2.
Example 37. Simplify (1-sin²θ)(1+tan²θ).
(cos²θ)(sec²θ) = 1.
Heights and distances
Draw the figure first. Angle of elevation is measured upward from the horizontal, angle of depression downward from the horizontal. The angle of depression from the top equals the angle of elevation from the ground point (alternate angles). Three triangles do most of the work: 45-45-90 (sides 1:1:√2), 30-60-90 (sides 1:√3:2) and 3-4-5.
Example 38. From a point 40 m from the foot of a tower, the angle of elevation of the top is 45°. Find the height.
tan45° = 1 = h/40, so h = 40 m.
Example 39. A pole's shadow is 12√3 m long when the sun's elevation is 30°. Find the pole height.
tan30° = h/(12√3), so h = 12√3 × (1/√3) = 12 m.
Example 40. A 10 m ladder leans on a wall and makes 60° with the ground. Find the height reached and the distance of the foot from the wall.
Height = 10 sin60° = 5√3 ≈ 8.66 m. Foot distance = 10 cos60° = 5 m.
Example 41. From the top of a 50 m cliff, the angle of depression of a boat is 30°. Find the distance of the boat from the foot of the cliff.
tan30° = 50/d, so d = 50√3 ≈ 86.6 m.
Example 42. The elevation of a tower's top is 60° from a point and 30° from a point 20 m farther away on the same line. Find the height.
Let the nearer distance be x. h = x tan60° = x√3, and h = (x+20) tan30° = (x+20)/√3. So 3x = x+20, x = 10, and h = 10√3 ≈ 17.32 m.
If the angles are θ and 90°-θ, then tanθ = h/9 and tan(90°-θ) = h/16 must multiply to 1. So h²/(9×16) = 1 and h = 12 m. Check: 12/9 = 4/3 and 12/16 = 3/4, product 1.
Simplification and approximation
Three habits solve this chapter. First, apply BODMAS strictly: brackets, orders (powers and roots), then division and multiplication left to right, then addition and subtraction left to right. Second, use algebraic shortcuts instead of multiplying. Third, in approximation questions, round each number to something friendly and then compare with the options. Check the gaps between options first: if they are wide, rough rounding is enough.
Example 44. Simplify 96 ÷ 8 × 3 + 12 - 4.
96÷8 = 12, then 12×3 = 36, then 36+12-4 = 44.
Example 45. Approximate 49.98% of 600.2 + 14.9² ÷ 15.1.
Round to 50% of 600 = 300, and 15²/15 = 15. Answer about 315. Exact value is 314.68, so the rounding is safe.
Example 46. Find 1/(1×2) + 1/(2×3) + 1/(3×4) + ... + 1/(9×10).
Each term is 1/n - 1/(n+1), so the sum telescopes to 1 - 1/10 = 9/10.
Example 47. Find 78² - 22² and 997 × 1003.
78²-22² = (78+22)(78-22) = 100×56 = 5600. 997×1003 = (1000-3)(1000+3) = 1000000-9 = 999991.
Example 48. Find 52².
(50+2)² = 2500+200+4 = 2704.
Example 49. Simplify √(7+4√3).
Look for (a+b√3)² = a²+3b²+2ab√3. With a=2, b=1: 4+3+4√3 = 7+4√3. So the root is 2+√3.
Example 50. Rationalise 1/(√5-√3).
Multiply by (√5+√3)/(√5+√3): the denominator becomes 5-3 = 2. Answer (√5+√3)/2.
Example 51. Find √0.0049.
0.0049 = 49/10000, so the root is 7/100 = 0.07.
Common traps
- Sign slip in identities. x-1/x squared gives +2, not -2, when you move to x²+1/x². Write the formula before you substitute.
- Dividing by zero in algebra. Dividing x²-3x+1 = 0 by x is fine because x = 0 is not a root, but always check that.
- Odd and even powers of -1. Remainder questions with (-1)ⁿ depend on whether n is odd or even.
- Cycle of 4 with remainder 0. If the power is a multiple of 4, use the fourth value in the cycle, not the first.
- HCF for remainder problems. Subtract the remainder before taking HCF, and check the answer is larger than the remainder.
- LCM question that asks for "greatest" or "largest". You may need to subtract a leftover from the biggest number of that size.
- Degrees against ratios. sinθ never exceeds 1. If you get a value above 1, the setup is wrong.
- Wrong triangle in heights and distances. Angle of depression is measured from the horizontal at the top, not from the vertical.
- BODMAS left to right. Division and multiplication have equal rank. 96÷8×3 is 36, not 4.
- Rounding too early. In approximation, round only after you have set up the full expression, and watch for subtraction where rounding errors become large.
A 60-day chapter schedule
This is a planning suggestion for a beginner studying 2 to 3 hours of maths a day. Adjust the pace if you already know some chapters.
| Days | Focus | Practice goal |
|---|---|---|
| 1-4 | Tables, squares, cubes, fractions, BODMAS | Daily speed drill of 20 calculations |
| 5-9 | Number system: divisibility, unit digit, remainders | Examples 15 to 23 redone without notes |
| 10-12 | HCF, LCM, factors | 30 mixed questions |
| 13-24 | Arithmetic: ratio, percentage, average, profit and loss, interest, mixtures | One topic every day or two, with 25 questions each |
| 25-30 | Time and work, speed and distance | Mixed set at the end of each day |
| 31-36 | Algebra: identities and factor questions | Examples 1 to 14, then 40 fresh questions |
| 37-45 | Geometry and mensuration | Draw every figure by hand |
| 46-51 | Trigonometry and heights and distances | Examples 31 to 43, then 30 fresh questions |
| 52-54 | Simplification and approximation | Timed 25-question sets |
| 55-56 | Data interpretation and statistics basics | 3 sets |
| 57-60 | Full revision and a mock test every alternate day | Review every wrong answer |
Keep a mistake log with three columns: the question type, why you missed it, and the corrected method. Revisit it every Sunday.
How to practise these chapters
For each chapter, first learn the method from the examples above, then redo them without looking. Next, attempt a timed set of 25 questions. If accuracy is below 80 percent, redo the weak sub-topic before moving on. In the last 15 days, shift from chapter practice to mixed sectional tests so that you also train in choosing which questions to skip.
FAQs
Which maths chapter should I start with for SSC? Start with arithmetic basics together with number system and simplification. They build the calculation speed that algebra, DI and geometry depend on.
How many questions does maths have in CGL Tier 1? The Quantitative Aptitude section has 25 questions of 2 marks each, according to the pattern in recent notifications. Check the latest notification for any change.
Is algebra or geometry more important? Both matter. Algebra is faster to learn, so a beginner gains marks sooner. Geometry needs longer practice, so start it once arithmetic is steady.
Do I need to mug up all trigonometry formulas? No. Know the values table, the three squared identities and the complementary angle pairs. That covers the great majority of straightforward questions.
How do I find the unit digit quickly? Use the cycle of the last digit. Divide the power by 4, use the remainder to pick the position in the cycle, and treat remainder 0 as the fourth position.
What is the fastest way to check divisibility by 7? Subtract twice the last digit from the remaining number and repeat until the number is small enough to judge.
Is negative marking applicable? Yes, the CGL Tier 1 pattern has a 0.5 mark negative for each wrong answer. Confirm the figure for your exam in the latest notification.
How much time should I give to each question? Aim for about 30 to 40 seconds on average. Skip anything that needs more than a minute at the first pass and return later.
Can I skip trigonometry if I am weak in it? It is a small chapter with predictable questions, so skipping it is rarely wise. Learn the values and the basic identities and attempt the easy ones.
How many mock tests should I take in 60 days? A common plan is one every alternate day in the last two weeks, plus sectional maths tests each week before that. Review is more important than the count.
Closing
Algebra, number system and trigonometry are small chapters that reward clean methods. Learn the identities, the cycles and the values table, work through the examples without notes, and then test yourself under time pressure. Practise with full-length mocks and sectional maths tests on Pareeksha to see which method you still fumble when the clock is running.
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