SSC Quantitative Aptitude — Complete Study Notes

Topics Covered

1. Number System 2. Percentage 3. Ratio & Proportion 4. Average & Mixture 5. SI & CI 6. Profit, Loss & Discount 7. Time & Work 8. Time, Speed & Distance 9. Algebra 10. Geometry 11. Mensuration 12. Trigonometry 13. Statistics & DI ★ Class 9-10 NCERT

Number System

Types of Numbers

Type	Definition	Example
Natural (N)	Counting numbers	1, 2, 3, …
Whole (W)	Natural + 0	0, 1, 2, …
Integer (Z)	Whole + negatives	…−2,−1,0,1,2…
Rational	p/q form, q≠0	½, 0.75, 3
Irrational	Non-terminating, non-repeating	√2, π, e
Prime	Exactly 2 factors: 1 and itself	2,3,5,7,11,13…
Composite	More than 2 factors	4,6,8,9,10…
Co-prime	HCF = 1 (need not be prime)	(8,15), (4,9)

⚠ Exam Trap

1 is neither prime nor composite. 2 is the only even prime number.

Divisibility Rules

Divisor	Rule
2	Last digit even (0,2,4,6,8)
3	Sum of digits divisible by 3
4	Last 2 digits divisible by 4
5	Last digit 0 or 5
6	Divisible by both 2 and 3
7	Double last digit, subtract from rest; if result div by 7
8	Last 3 digits divisible by 8
9	Sum of digits divisible by 9
11	(Sum of odd-position digits) − (Sum of even-position digits) = 0 or ±11
12	Divisible by both 3 and 4
25	Last 2 digits divisible by 25 (00, 25, 50, 75)

HCF & LCM

Key Formulas

HCF × LCM = Product of two numbers (for two numbers only)
LCM = (HCF × a/gcd) × b/gcd
HCF of fractions = HCF of numerators / LCM of denominators
LCM of fractions = LCM of numerators / HCF of denominators

🔑 Shortcuts

For numbers a, b: HCF always divides LCM
If two numbers are co-prime → HCF = 1, LCM = product
"Largest number that divides a, b, c leaving same remainder r" → HCF of (a−b), (b−c), (a−c)
"Smallest number divisible by a, b, c" → LCM(a,b,c)
"Bells ring together after" → LCM of their intervals

Remainders & Unit Digits

Unit Digit Cycles

2 → 2,4,8,6 (cycle of 4) | 3 → 3,9,7,1 (cycle of 4)
4 → 4,6 (cycle of 2) | 5 → always 5
6 → always 6 | 7 → 7,9,3,1 (cycle of 4)
8 → 8,4,2,6 (cycle of 4) | 9 → 9,1 (cycle of 2)
0,1 → always 0,1

💡 Trick — Unit digit of aⁿ

Divide n by cycle length. Use remainder to find position in cycle. If remainder = 0, use last number of cycle.

Remainders — Key Results

(a + b) mod n = [(a mod n) + (b mod n)] mod n
(a × b) mod n = [(a mod n) × (b mod n)] mod n
Fermat's Little Theorem: aᵖ⁻¹ ≡ 1 (mod p) when p is prime and gcd(a,p)=1

Number of Factors & Sum

Formulas

If N = aᵖ × bq × cʳ (prime factorisation)
Number of factors = (p+1)(q+1)(r+1)
Sum of factors = [(aᵖ⁺¹−1)/(a−1)] × [(b^(q+1)−1)/(b−1)] × [(c^(r+1)−1)/(c−1)]
Number of co-prime numbers < N = N × (1−1/a)(1−1/b)(1−1/c) [Euler's Totient]

Example

N = 360 = 2³ × 3² × 5¹ → Factors = (3+1)(2+1)(1+1) = 24

Percentage

Core Formulas

Essential Formulas

% = (Part/Whole) × 100
x% of y = y% of x (symmetry property)
If A is r% more than B → B is less than A by: r/(100+r) × 100 %
If A is r% less than B → B is more than A by: r/(100−r) × 100 %
Net change (successive %): a + b + ab/100
x% of y = y% of x (use whichever is easier to calculate)

Fraction ↔ Percentage Conversions (Must Memorise)

Fraction	%	Fraction	%
1/1	100%	1/8	12.5%
1/2	50%	1/9	11.11%
1/3	33.33%	1/10	10%
1/4	25%	1/11	9.09%
1/5	20%	1/12	8.33%
1/6	16.67%	1/16	6.25%
1/7	14.28%	1/20	5%

Key Patterns in SSC Questions

✓ Common Question Types

Population problems: Use successive percentage formula
Election problems: Winner% + Loser% = 100 (of valid votes)
Income-expenditure: If income ↑r%, expenditure ↑s%, savings change = ?
Mixed percentage: Weighted average approach
Price-consumption: To keep expenditure same → % reduction in consumption = r/(100+r) × 100

Ratio & Proportion

Key Concepts

Formulas

Ratio a:b → multiply/divide both by same number
If a:b = c:d → Proportion (product of means = product of extremes → ad = bc)
If a/b = c/d → a/c = b/d (alternendo); (a+b)/b = (c+d)/d (componendo)
Compound ratio of a:b and c:d → ac:bd
Duplicate ratio of a:b → a²:b²
Sub-duplicate ratio of a:b → √a:√b
Triplicate ratio of a:b → a³:b³

Partnership

Profit Sharing

Simple partnership: Profit ∝ Capital invested (time same)
Compound partnership: Profit ∝ Capital × Time
If A invests ₹a for tₐ months and B invests ₹b for t_b months:
Share ratio = a×tₐ : b×t_b

Alligation & Mixture

Alligation Rule

(Cheaper quantity):(Dearer quantity) = (Dearer price − Mean price):(Mean price − Cheaper price)

Mixture removed then refilled:
Final concentration = C × (1 − r/V)ⁿ
where C = initial concentration, r = removed each time, V = total volume, n = times repeated

💡 Quick Check

Draw the alligation cross:
Cheaper (c) ........ Dearer (d)
........... Mean (m) ...........
Ratio = (d−m):(m−c)

Average & Mixture

Core Formulas

Average = Sum of observations / Number of observations
Sum = Average × n
If one value x is replaced by y → change in average = (y−x)/n
Average of 1 to n natural numbers = (n+1)/2
Average of first n even numbers = n+1
Average of first n odd numbers = n
Average of AP = (First term + Last term)/2

🔑 Common Tricks

If average of n numbers = A and one number is excluded, new avg = A' → Excluded number = nA − (n−1)A'
If a wrong number 'a' was used instead of 'b' → New sum = Old sum + b − a → Correct average = (Old avg × n + b − a)/n
Speed average: If same distance at speed u and v → Average speed = 2uv/(u+v) (harmonic mean, NOT arithmetic mean)

Simple Interest & Compound Interest

Simple Interest

SI Formulas

SI = PRT/100
Amount = P + SI = P(1 + RT/100)
P = SI×100/(R×T) | R = SI×100/(P×T) | T = SI×100/(P×R)

Compound Interest

CI Formulas

A = P(1 + R/100)ⁿ [annual compounding]
A = P(1 + R/200)²ⁿ [half-yearly]
A = P(1 + R/400)⁴ⁿ [quarterly]
CI = A − P

CI − SI (2 years) = P(R/100)²
CI − SI (3 years) = P(R/100)²(3 + R/100)

If sum doubles in n years (CI) → it becomes 2^(t/n) times in t years
Rule of 72: Time to double ≈ 72/R years (approximate)

Key Patterns

⚠ Important

SI grows linearly; CI grows exponentially
For 2 years: CI = SI + (P × R²/10000)
Population growth formula = same as CI: P(1+r/100)ⁿ
Depreciation: A = P(1 − R/100)ⁿ
Different rates: A = P(1+R₁/100)(1+R₂/100)(1+R₃/100)

Profit, Loss & Discount

Core Formulas

Profit = SP − CP | Loss = CP − SP
Profit% = (Profit/CP) × 100 | Loss% = (Loss/CP) × 100
SP = CP × (100+P%)/100 | SP = CP × (100−L%)/100
CP = SP × 100/(100+P%) | CP = SP × 100/(100−L%)

Marked Price (MP) and Discount:
SP = MP × (100−D%)/100
Profit% (after discount) = [(1+m/100)(1−d/100) − 1] × 100

Successive discounts d₁ and d₂:
Net discount = d₁ + d₂ − d₁d₂/100

Scenario	Formula
Same SP, equal P% & L%	Always a loss: Loss% = r²/100
Same CP, equal P% & L%	No profit / no loss
Dishonest dealer (false weight)	Profit% = (True wt − False wt)/False wt × 100
x articles for ₹y, gain g%	CP per article = y/x; use gain formula to find new rate

Time & Work

Core Formulas

If A completes work in n days → A's 1 day work = 1/n
A + B together: 1 day work = 1/a + 1/b → Time = ab/(a+b)
A, B, C together: Time = abc/(ab+bc+ca)

Pipes & Cistern:
Inlet fills in 'a' hrs, outlet empties in 'b' hrs
Net filling rate = 1/a − 1/b (if 1/a > 1/b, tank fills; else empties)
Time to fill = ab/(b−a)

Work efficiency ratio ∝ 1/time ratio (inversely proportional)

🔑 LCM Method (Fastest Approach)

Assume total work = LCM of days. Convert each person's efficiency = LCM/days. Then work with whole numbers — much faster than fractions.

Example: A=12 days, B=15 days → LCM=60 (total work). A's efficiency=5 units/day, B's=4 units/day. Together=9 units/day. Time=60/9=6⅔ days.

Men × Days × Hours = Work (MDH Formula)

Work Formula

M₁ × D₁ × H₁ / W₁ = M₂ × D₂ × H₂ / W₂
(M=Men, D=Days, H=Hours/day, W=Work done)

If efficiency differs: M₁ × D₁ × H₁ × E₁ = M₂ × D₂ × H₂ × E₂

Time, Speed & Distance

Core Formula

Speed = Distance / Time
D = S × T | T = D/S
Unit conversions: 1 km/hr = 5/18 m/s | 1 m/s = 18/5 km/hr

Average Speed = Total Distance / Total Time
Same distance at speeds u and v → Average speed = 2uv/(u+v) [NOT (u+v)/2]
Relative speed (same direction) = |S₁ − S₂|
Relative speed (opposite direction) = S₁ + S₂

Trains

Train Formulas

Time to cross a pole/person = Length of train / Speed of train
Time to cross a platform = (L_train + L_platform) / Speed
Time for two trains to cross each other = (L₁ + L₂) / Relative Speed
(Use relative speed based on direction)

Boats & Streams

Boat Formulas

Downstream speed = u + v (u = boat speed, v = stream speed)
Upstream speed = u − v
u = (Downstream + Upstream)/2
v = (Downstream − Upstream)/2

Time ratio (upstream:downstream) = (u+v):(u−v) for equal distances

Circular Track & Meeting Points

💡 Key Results

Same direction on circular track: Meeting time = L / |S₁−S₂|
Opposite direction: Meeting time = L / (S₁+S₂)
First meeting at starting point (same dir): LCM of individual lap times

Algebra

Must-Know Identities

Algebraic Identities

(a+b)² = a² + 2ab + b²
(a−b)² = a² − 2ab + b²
(a+b)(a−b) = a² − b²
(a+b)³ = a³ + 3a²b + 3ab² + b³ = a³ + b³ + 3ab(a+b)
(a−b)³ = a³ − 3a²b + 3ab² − b³ = a³ − b³ − 3ab(a−b)
a³ + b³ = (a+b)(a²−ab+b²)
a³ − b³ = (a−b)(a²+ab+b²)
a³ + b³ + c³ − 3abc = (a+b+c)(a²+b²+c²−ab−bc−ca)
If a+b+c=0 → a³+b³+c³ = 3abc

Quadratic & Linear Equations

Key Results

Quadratic ax²+bx+c=0:
Roots: x = [−b ± √(b²−4ac)] / 2a
Sum of roots (α+β) = −b/a
Product of roots (αβ) = c/a
Discriminant D = b²−4ac:
D > 0 → 2 distinct real roots
D = 0 → equal real roots
D < 0 → no real roots (complex)

Important Derived Values

🔑 If x + 1/x = k, then:

x² + 1/x² = k² − 2
x³ + 1/x³ = k³ − 3k = k(k²−3)
x⁴ + 1/x⁴ = (x²+1/x²)² − 2 = (k²−2)² − 2
If x − 1/x = k: x² + 1/x² = k² + 2; x³ − 1/x³ = k³ + 3k

Geometry

Lines & Angles

Angle Properties

Vertically opposite angles are equal
Alternate interior angles are equal (parallel lines with transversal)
Co-interior (same-side interior) angles are supplementary (sum=180°)
Corresponding angles are equal
Sum of angles in triangle = 180°
Exterior angle = Sum of two non-adjacent interior angles

Triangles — Key Theorems

Triangle Properties

Pythagoras: a² + b² = c² (right triangle)
Common Pythagorean triples: (3,4,5), (5,12,13), (8,15,17), (7,24,25), (9,40,41)

Area = ½ × base × height
Area = √[s(s−a)(s−b)(s−c)] [Heron's formula, s = (a+b+c)/2]
Area = ½ ab sin C

Medians: meet at Centroid (divides each 2:1 from vertex)
Altitudes: meet at Orthocentre
Angle bisectors: meet at Incentre (centre of inscribed circle)
Perpendicular bisectors: meet at Circumcentre

Midpoint theorem: Line joining midpoints of two sides is parallel to third side and half its length

Triangle Type	Key Property
Equilateral (a,a,a)	Area = (√3/4)a²; Height = (√3/2)a; All angles = 60°
Isosceles	Two equal sides → two equal base angles
Right-angled (30-60-90)	Sides in ratio 1:√3:2
Right-angled (45-45-90)	Sides in ratio 1:1:√2

Circles

Circle Theorems

Angle subtended at centre = 2 × angle subtended at circumference (same arc)
Angle in semicircle = 90°
Angles in same segment are equal
Tangent ⊥ radius at point of contact
Two tangents from external point are equal in length
Chord properties: equal chords equidistant from centre
Intersecting chords: PA × PB = PC × PD
Secant-tangent: PT² = PA × PB

Quadrilaterals

Shape	Area	Special Property
Square (a)	a²	Diagonal = a√2
Rectangle (l,b)	l×b	Diagonal = √(l²+b²)
Parallelogram	base × height	Diagonals bisect each other
Rhombus (d₁,d₂)	d₁×d₂/2	Diagonals ⊥, side=√(d₁²+d₂²)/2
Trapezium	½(a+b)×h	One pair of parallel sides

Mensuration

2D Figures

Figure	Area	Perimeter
Circle (r)	πr²	2πr
Semicircle	πr²/2	πr + 2r
Sector (θ°)	(θ/360)πr²	2r + (θ/360)2πr
Equilateral △ (a)	(√3/4)a²	3a
Regular hexagon (a)	(3√3/2)a²	6a

3D Solids

Solid	Volume	Curved/Lateral SA	Total SA
Cube (a)	a³	4a²	6a²
Cuboid (l,b,h)	lbh	2h(l+b)	2(lb+bh+lh)
Cylinder (r,h)	πr²h	2πrh	2πr(r+h)
Cone (r,h,l)	πr²h/3	πrl	πr(r+l)
Sphere (r)	4πr³/3	—	4πr²
Hemisphere (r)	2πr³/3	2πr²	3πr²
Frustum (R,r,h,l)	πh(R²+Rr+r²)/3	πl(R+r)	π[R²+r²+l(R+r)]

💡 Slant height of cone

l = √(r² + h²)

Trigonometry

Standard Values (Must Memorise)

Angle	sin	cos	tan	cosec	sec	cot
0°	0	1	0	∞	1	∞
30°	1/2	√3/2	1/√3	2	2/√3	√3
45°	1/√2	1/√2	1	√2	√2	1
60°	√3/2	1/2	√3	2/√3	2	1/√3
90°	1	0	∞	1	∞	0

Identities & Heights & Distances

Key Identities

sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ
sin(90°−θ) = cosθ; cos(90°−θ) = sinθ; tan(90°−θ) = cotθ

Heights & Distances:
tan(angle of elevation) = Height / Distance
Angle of elevation = angle measured upward from horizontal
Angle of depression = angle measured downward from horizontal

Statistics & Data Interpretation

Measures of Central Tendency

Formulas

Mean (Arithmetic) = Σx/n = Σfx/Σf (for grouped data)
Median: Middle value when data sorted; for even n = average of n/2 and (n/2+1)th values
Mode: Most frequently occurring value
Relation: Mode = 3 × Median − 2 × Mean (empirical formula)

Standard Deviation σ = √[Σ(x−x̄)²/n]
Variance = σ²

DI — Approach Strategy

✓ DI Tips for SSC

Always read the table/graph title and units first
For % change: (New−Old)/Old × 100
For ratio questions: Don't calculate exact values — just compare fractions
Bar/Pie/Line graphs: Estimate visually first, then calculate
In pie charts: Value = (Angle/360) × Total or (Percentage/100) × Total
Practice approximation: 34% of 4826 ≈ 1/3 × 4800 = 1600

★

Class 9 & 10 NCERT Math — Formulas, Tips & Tricks

Why This Matters for SSC

SSC CGL/CHSL QA section is almost entirely based on Class 9–10 NCERT math concepts. Master these and you cover ~80% of the paper.

CLASS 9 MATHEMATICS

Ch 1 — Number System

Key Formulas

Real Numbers = Rational + Irrational
Rationalize 1/(a+√b) → multiply by (a−√b)/(a−√b)
(√a + √b)(√a − √b) = a − b
(√a + √b)² = a + b + 2√(ab)
Laws of Exponents:
aᵐ × aⁿ = aᵐ⁺ⁿ | aᵐ / aⁿ = aᵐ⁻ⁿ | (aᵐ)ⁿ = aᵐⁿ
a⁰ = 1 | a⁻ⁿ = 1/aⁿ | (ab)ⁿ = aⁿbⁿ
a^(m/n) = ⁿ√(aᵐ)

💡 Tricks

To find √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236 — memorise these for estimation
Rationalize denominators: always multiply numerator and denominator by conjugate
Cyclic order trick: √2 < √3 < √5 < √6 < √7 — useful for comparison MCQs
If x = a + √b, then 1/x = a − √b / (a²−b)

Ch 2 — Polynomials

Key Theorems & Identities

Remainder Theorem: When p(x) is divided by (x−a), remainder = p(a)
Factor Theorem: (x−a) is a factor of p(x) iff p(a) = 0

Algebraic Identities (NCERT Ch 2 & Ch 4 Class 9):
(a+b)² = a² + 2ab + b²
(a−b)² = a² − 2ab + b²
(a+b)(a−b) = a² − b²
(x+a)(x+b) = x² + (a+b)x + ab
(a+b+c)² = a²+b²+c²+2ab+2bc+2ca
(a+b)³ = a³+3a²b+3ab²+b³
(a−b)³ = a³−3a²b+3ab²−b³
a³+b³+c³−3abc = (a+b+c)(a²+b²+c²−ab−bc−ca)
Special: If a+b+c=0 → a³+b³+c³ = 3abc

💡 Factor Theorem Shortcut

To factorize p(x) = x³−6x²+11x−6:
Try p(1)=1−6+11−6=0 ✓ → (x−1) is factor
Divide to get (x−1)(x²−5x+6) = (x−1)(x−2)(x−3)

Ch 4 — Linear Equations in Two Variables

General Form

ax + by + c = 0 (a,b not both zero)
One equation → infinite solutions (represents a line)
For integer solutions: substitute x=0,1,2... to find y

Ch 6 — Lines & Angles

Angle Sum Properties

Sum of angles on straight line = 180° (Linear pair)
Vertically opposite angles are equal
Angle sum of triangle = 180°
Exterior angle = sum of 2 non-adjacent interior angles
For parallel lines (transversal):
Corresponding angles = equal
Alternate interior angles = equal
Co-interior angles = supplementary (sum 180°)

Ch 7 — Triangles (Congruence)

Criterion	Full Form	Condition
SSS	Side-Side-Side	All 3 sides equal
SAS	Side-Angle-Side	2 sides + included angle
ASA	Angle-Side-Angle	2 angles + included side
AAS	Angle-Angle-Side	2 angles + non-included side
RHS	Right-Hypotenuse-Side	Right triangle only

⚠ SSA is NOT a valid congruence criterion!

Two triangles with 2 sides and a non-included angle equal may not be congruent.

Ch 9 — Areas of Parallelograms & Triangles

Area Theorems

Parallelograms on same base and between same parallels → equal area
Triangle = half of parallelogram with same base & parallels
Median divides triangle into two triangles of equal area
If diagonals of || gm bisect each other → each diagonal divides it into 2 △ of equal area

Ch 10 — Circles

Circle Theorems (Class 9)

Equal chords are equidistant from centre
Perpendicular from centre bisects chord (and vice versa)
Angle subtended at centre = 2 × angle at circumference
Angles in same segment are equal
Angle in semicircle = 90°
Sum of opposite angles of cyclic quadrilateral = 180°

Ch 12 — Heron's Formula

Heron's Formula

Semi-perimeter s = (a+b+c)/2
Area = √[s(s−a)(s−b)(s−c)]
For equilateral △ with side a: Area = (√3/4)a²

Trick: For right triangle (3,4,5 etc.): Area = ½ × legs product

Ch 13 — Surface Area & Volume (Class 9)

Solid	LSA/CSA	TSA	Volume
Cuboid (l,b,h)	2h(l+b)	2(lb+bh+lh)	lbh
Cube (a)	4a²	6a²	a³
Cylinder (r,h)	2πrh	2πr(r+h)	πr²h
Cone (r,h,l)	πrl, l=√(r²+h²)	πr(r+l)	πr²h/3
Sphere (r)	—	4πr²	4πr³/3
Hemisphere (r)	2πr²	3πr²	2πr³/3

💡 Memory trick for Volume ratio: Cone:Sphere:Cylinder = 1:2:3 (same radius & height/diameter)

CLASS 10 MATHEMATICS

Ch 1 — Real Numbers

Euclid's Division Algorithm & FTA

Euclid's Lemma: For a = bq + r (0 ≤ r < b)
HCF(a,b) = HCF(b,r) → repeat until r=0

Fundamental Theorem of Arithmetic:
Every composite number has unique prime factorisation
Use prime factorisation to find HCF and LCM:
HCF = product of lowest powers of common primes
LCM = product of highest powers of all primes
HCF × LCM = product of two numbers

💡 Quick Check for Terminating Decimal

p/q (fully reduced) terminates iff q = 2ᵃ × 5ᵇ (only factors 2 and 5)
Otherwise it's non-terminating repeating.

Ch 2 — Polynomials

Zeros & Coefficients Relationship

For Quadratic ax²+bx+c:
α + β = −b/a (sum of zeros)
α × β = c/a (product of zeros)
Form equation: x² − (α+β)x + αβ = 0

For Cubic ax³+bx²+cx+d:
α+β+γ = −b/a
αβ+βγ+γα = c/a
αβγ = −d/a

Ch 3 — Pair of Linear Equations

Consistency Conditions

For a₁x+b₁y+c₁=0 and a₂x+b₂y+c₂=0:
a₁/a₂ ≠ b₁/b₂ → Unique solution (intersecting lines)
a₁/a₂ = b₁/b₂ = c₁/c₂ → Infinitely many solutions (coincident)
a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → No solution (parallel lines)

Solution methods:
1. Substitution | 2. Elimination | 3. Cross-multiplication
Cross-mult: x/(b₁c₂−b₂c₁) = y/(c₁a₂−c₂a₁) = 1/(a₁b₂−a₂b₁)

💡 Elimination Trick

Make coefficients of one variable equal by multiplying equations, then add/subtract to eliminate it.

Ch 4 — Quadratic Equations

Standard Form & Methods

Standard form: ax² + bx + c = 0 (a ≠ 0)

Method 1 — Factorisation: Split middle term (find p,q: p+q=b, p×q=ac)
Method 2 — Completing the square
Method 3 — Quadratic formula: x = [−b ± √(b²−4ac)] / 2a

Discriminant D = b² − 4ac:
D > 0 → 2 distinct real roots
D = 0 → 2 equal real roots (x = −b/2a)
D < 0 → no real roots

Nature-of-roots shortcut:
D is perfect square → roots are rational
D is not perfect square → roots are irrational (in conjugate pairs)

💡 Split Middle Term Method

ax² + bx + c: Find p,q such that p+q = b AND p×q = a×c
Example: 6x²+11x+3 → a×c=18, p=9, q=2 (9+2=11, 9×2=18)
= 6x²+9x+2x+3 = 3x(2x+3)+1(2x+3) = (3x+1)(2x+3)

Ch 5 — Arithmetic Progressions (AP)

AP Formulas (HEAVILY tested in SSC)

AP: a, a+d, a+2d, ... (a=first term, d=common difference)

nth term: aₙ = a + (n−1)d
Sum of n terms: Sₙ = n/2 × [2a + (n−1)d] = n/2 × (first + last)
Number of terms: n = [(last − first)/d] + 1

If three terms in AP: use (a−d), a, (a+d) — middle is always average
If four terms in AP: use (a−3d), (a−d), (a+d), (a+3d)

Sum of first n natural numbers: n(n+1)/2
Sum of first n even numbers: n(n+1)
Sum of first n odd numbers: n²

💡 Key Tricks

If Sₙ = pn² + qn → nth term aₙ = Sₙ − Sₙ₋₁ = 2pn + (q−p)
If sum of n terms = kn², then d = 2k (difference is constant)
Mean of AP = (first + last)/2 = middle term
SSC favourite: "Sum of first 20 odd numbers" = 20² = 400

Ch 6 — Triangles (Similarity)

Similarity Theorems

Basic Proportionality Theorem (Thales): If a line is drawn parallel to one side of a triangle cutting other two sides, it divides them proportionally.
If DE∥BC in △ABC → AD/DB = AE/EC

Criteria for Similarity: AA, SSS (ratio), SAS (ratio)

Properties of Similar Triangles (△ABC ~ △PQR):
AB/PQ = BC/QR = AC/PR (ratio of corresponding sides = k)
Ratio of areas = k² = (AB/PQ)²
Ratio of perimeters = k
Ratio of medians = ratio of altitudes = ratio of angle bisectors = k

Pythagoras Theorem: In right △, (hypotenuse)² = (base)² + (height)²
Converse: If a²+b²=c², then △ is right-angled at the vertex opposite c

⚠ Exam Favourite — Midpoint Theorem

Line joining midpoints of two sides of a triangle is parallel to third side and = half its length.
Reverse: Line drawn from midpoint of one side parallel to another side bisects the third side.

Ch 7 — Coordinate Geometry

Key Formulas

Distance formula: PQ = √[(x₂−x₁)² + (y₂−y₁)²]
Section formula (internal division m:n): x = (mx₂+nx₁)/(m+n), y = (my₂+ny₁)/(m+n)
Midpoint: x = (x₁+x₂)/2, y = (y₁+y₂)/2
Centroid of △: G = [(x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3]
Area of △ = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|
Points are collinear if area of △ = 0

💡 Quick Tricks

For midpoint MCQs: just use (x₁+x₂)/2 and (y₁+y₂)/2
If ratio is 1:1, it's midpoint — use midpoint formula directly
For 3 collinear points: verify area = 0 or slopes equal
Origin distance = √(x²+y²) — standard MCQ type

Ch 8 & 9 — Trigonometry & Applications

Trigonometric Identities (Class 10)

sin²A + cos²A = 1
1 + tan²A = sec²A → sec²A − tan²A = 1
1 + cot²A = cosec²A → cosec²A − cot²A = 1

Complementary angles:
sin(90°−A) = cosA | cos(90°−A) = sinA
tan(90°−A) = cotA | cot(90°−A) = tanA
sec(90°−A) = cosecA | cosec(90°−A) = secA

Important derived values:
(sinA + cosA)² = 1 + 2sinAcosA
(sinA − cosA)² = 1 − 2sinAcosA
sin⁴A + cos⁴A = 1 − 2sin²Acos²A

💡 Trig Identity Proving Trick

Convert everything to sinA and cosA. Simplify one side to match the other. Common move: 1−sin²A = cos²A and vice versa.

Ch 10 — Circles (Class 10)

Tangent Theorems

Tangent ⊥ radius at point of contact
Two tangents from same external point are equal in length
PA = PB (tangent lengths from external point P)
Angle between two tangents + angle at centre = 180°
If PA and PB are tangents from P: ∠APB + ∠AOB = 180°

Alternate Segment Theorem: Angle between tangent and chord = angle in alternate segment

⚠ SSC Favourite Tangent Question

PA and PB are tangents from P to circle with centre O. If ∠APB = 50°, then ∠AOB = 130°. If OA=5 and PA=12, then OP = √(5²+12²) = 13.

Ch 11 — Areas Related to Circles

Circle Area Formulas

Area of circle = πr²
Circumference = 2πr
Area of sector (angle θ°) = (θ/360)πr²
Arc length = (θ/360) × 2πr
Area of segment = Area of sector − Area of triangle
Area of triangle in sector = ½r²sinθ (for isosceles triangle with 2 sides = r)

Ring area = π(R²−r²) = π(R+r)(R−r)

💡 Common Values

Use π = 22/7 when r is multiple of 7; use π = 3.14 otherwise
Area of semicircle = πr²/2; Perimeter = πr + 2r = r(π+2)

Ch 13 — Statistics (Grouped Data)

Measures for Grouped Data

MEAN:
Direct method: x̄ = Σfᵢxᵢ / Σfᵢ (xᵢ = class midpoint)
Assumed mean method: x̄ = A + Σfᵢdᵢ/Σfᵢ (dᵢ = xᵢ − A)
Step deviation method: x̄ = A + (Σfᵢuᵢ/Σfᵢ) × h (uᵢ = dᵢ/h)

MEDIAN:
Median = L + [(n/2 − cf)/f] × h
where L=lower boundary of median class, cf=cumulative frequency before median class, f=frequency of median class, h=class width

MODE:
Mode = L + [(f₁−f₀)/(2f₁−f₀−f₂)] × h
where f₁=modal class frequency, f₀=frequency of class before, f₂=frequency of class after

Empirical relation: Mode = 3 Median − 2 Mean

💡 Finding Median Class

Always find n/2 first. The class whose cumulative frequency first exceeds n/2 is the median class.

Ch 14 — Probability

Probability Formulas

P(E) = Number of favourable outcomes / Total outcomes
0 ≤ P(E) ≤ 1
P(E) + P(E') = 1 → P(not E) = 1 − P(E)
P(impossible event) = 0 | P(certain event) = 1

Standard Sample Spaces:
Coin: {H,T} → n=2
Two coins: {HH,HT,TH,TT} → n=4
Die: {1,2,3,4,5,6} → n=6
Two dice: n=36 (6×6 grid)
Deck of cards: 52 total, 4 suits × 13 cards, 26 red + 26 black
Face cards = 12 (J,Q,K of each suit)

💡 Two Dice Tricks

Sum=7: 6 ways (max probability) | Sum=2 or 12: 1 way each (min probability)
Doublets (same number both dice): 6 ways → P = 6/36 = 1/6
Sum≥8: Count = 5+4+3+2+1 = 15, P = 15/36 = 5/12

SSC Exam Strategies for NCERT Math Topics

⚡ Must-Know for SSC QA

AP sum formula tested almost every exam
Quadratic roots / nature of roots
Coordinate geometry: distance, section formula
Similarity ratio → area ratio
Tangent properties from external point
Trigonometry identities (simplification)
Statistics: mean of grouped data

✓ Time-Saving Tricks

Digit sum method to verify calculations
Plug answer choices back into equations
For AP: if total terms odd, middle term = average
Quadratic by options: substitute each option
Similarity: check if answer is k times the given
Trig: use standard angle values (0°,30°,45°,60°,90°)
Probability: total outcomes first, then count favourable

NCERT Chapter	SSC Weightage	Key Formula to Remember
Arithmetic Progressions	High ★★★	aₙ = a+(n−1)d; Sₙ = n/2[2a+(n−1)d]
Quadratic Equations	High ★★★	Discriminant D=b²−4ac; roots=(-b±√D)/2a
Trigonometry	High ★★★	sin²+cos²=1; 1+tan²=sec²
Polynomials (zeros)	Medium ★★	α+β=−b/a; αβ=c/a
Coordinate Geometry	Medium ★★	Distance, Section, Midpoint
Triangles (Similarity)	Medium ★★	Area ratio = (side ratio)²
Circles (Tangent)	Medium ★★	PA=PB; Tangent ⊥ radius
Areas Related to Circles	Medium ★★	Sector area=(θ/360)πr²
Real Numbers (HCF/LCM)	Medium ★★	HCF×LCM=product of 2 numbers
Statistics	Low-Med ★	Mode=3Median−2Mean
Probability	Low ★	P(E)=favourable/total