Mathematics-Complete-Mastery
📐 MATHEMATICS COMPLETE MASTERY GUIDE 📚
Class 6-12 | All Formulas | 100/100 Strategy
Mathematics is the foundation of all sciences and the key to unlocking analytical thinking, problem-solving abilities, and logical reasoning that will serve you throughout life. This comprehensive mastery guide covers the complete mathematics curriculum from Class 6 to Class 12, including every formula, theorem, proof, and concept you need to excel. Whether you're struggling with basic arithmetic or tackling advanced calculus, this guide provides crystal-clear explanations, step-by-step solutions to 50+ example problems, quick calculation tricks that save precious exam time, shortcuts used by toppers, common mistakes students make (and how to avoid them), chapter-wise weightage for strategic preparation, and proven techniques to score 100/100 in mathematics. We cover Algebra (equations, polynomials, quadratic equations, sequences), Geometry (triangles, circles, coordinate geometry), Trigonometry (ratios, identities, equations), Calculus (limits, derivatives, integrals), Mensuration (area, volume formulas), Statistics & Probability, and more. Additionally, we provide an honest comparison between RD Sharma and RS Aggarwal reference books to help you choose the best resource. Master mathematics with confidence and transform it from your weakest subject into your strongest scoring opportunity!
🔢 CLASS 6-8 FOUNDATION MATHEMATICS
Building strong fundamentals in arithmetic, geometry, and basic algebra
➕ Number System & Arithmetic (Class 6-8)
1. Number Types & Properties
- Natural Numbers (N): 1, 2, 3, 4, 5... (counting numbers)
- Whole Numbers (W): 0, 1, 2, 3, 4... (natural + zero)
- Integers (Z): ...-3, -2, -1, 0, 1, 2, 3... (positive + negative + zero)
- Rational Numbers (Q): Numbers in p/q form where q ≠ 0 (Example: 1/2, 3/4, -5/7)
- Irrational Numbers: Cannot be written as p/q (Example: √2, π, √3)
- Real Numbers (R): Rational + Irrational numbers
- Prime Numbers: Numbers with only 2 factors - 1 and itself (2, 3, 5, 7, 11, 13...)
- Composite Numbers: Numbers with more than 2 factors (4, 6, 8, 9, 10...)
2. Divisibility Rules (Quick Tricks)
- By 2: Last digit is even (0, 2, 4, 6, 8)
- By 3: Sum of digits divisible by 3 (Example: 123 → 1+2+3=6, divisible by 3)
- By 4: Last two digits divisible by 4 (Example: 316 → 16 divisible by 4)
- By 5: Last digit is 0 or 5
- By 6: Divisible by both 2 and 3
- By 8: Last three digits divisible by 8
- By 9: Sum of digits divisible by 9
- By 10: Last digit is 0
- By 11: (Sum of digits at odd places) - (Sum of digits at even places) = 0 or multiple of 11
🔍 Example 1: Check if 5832 is divisible by 3, 4, and 8
By 3: 5+8+3+2 = 18, divisible by 3 ✓
By 4: Last 2 digits = 32, 32÷4 = 8 ✓
By 8: Last 3 digits = 832, 832÷8 = 104 ✓
Answer: 5832 is divisible by 3, 4, and 8
3. LCM & HCF (Greatest Importance!)
Formulas: HCF × LCM = Product of two numbers
For two numbers a and b:
HCF(a,b) × LCM(a,b) = a × b
For two numbers a and b:
HCF(a,b) × LCM(a,b) = a × b
Methods to find HCF:
- Prime Factorization: Find common prime factors, take lowest power
- Division Method: Divide larger by smaller repeatedly
Methods to find LCM:
- Prime Factorization: Take all prime factors with highest power
- Division Method: Divide by common factors until no common factor remains
🔍 Example 2: Find HCF and LCM of 24 and 36
Prime Factorization:
24 = 2³ × 3¹
36 = 2² × 3²
HCF: Take lowest powers = 2² × 3¹ = 4 × 3 = 12
LCM: Take highest powers = 2³ × 3² = 8 × 9 = 72
Verification: HCF × LCM = 12 × 72 = 864 = 24 × 36 ✓
4. Fractions, Decimals & Percentages
Conversions: Fraction to Decimal: Divide numerator by denominator
Decimal to Percentage: Multiply by 100
Percentage to Fraction: Divide by 100
Important Fraction to % conversions (Memorize!):
1/2 = 50%, 1/3 = 33.33%, 1/4 = 25%, 1/5 = 20%
1/6 = 16.67%, 1/8 = 12.5%, 1/10 = 10%
2/3 = 66.67%, 3/4 = 75%, 3/5 = 60%
Decimal to Percentage: Multiply by 100
Percentage to Fraction: Divide by 100
Important Fraction to % conversions (Memorize!):
1/2 = 50%, 1/3 = 33.33%, 1/4 = 25%, 1/5 = 20%
1/6 = 16.67%, 1/8 = 12.5%, 1/10 = 10%
2/3 = 66.67%, 3/4 = 75%, 3/5 = 60%
5. Profit, Loss & Discount
Basic Formulas: Profit = SP - CP (Selling Price - Cost Price)
Loss = CP - SP
Profit % = (Profit/CP) × 100
Loss % = (Loss/CP) × 100
SP = CP × (100 + Profit%)/100
SP = CP × (100 - Loss%)/100
Discount:
Discount = Marked Price - Selling Price
Discount % = (Discount/Marked Price) × 100
Loss = CP - SP
Profit % = (Profit/CP) × 100
Loss % = (Loss/CP) × 100
SP = CP × (100 + Profit%)/100
SP = CP × (100 - Loss%)/100
Discount:
Discount = Marked Price - Selling Price
Discount % = (Discount/Marked Price) × 100
🔍 Example 3: A shopkeeper buys an item for ₹500 and sells at 20% profit. Find SP.
CP = ₹500
Profit% = 20%
SP = CP × (100 + Profit%)/100
SP = 500 × (100 + 20)/100
SP = 500 × 120/100
SP = 500 × 1.2 = ₹600
SP = 500 × (100 + 20)/100
SP = 500 × 120/100
SP = 500 × 1.2 = ₹600
6. Simple Interest & Compound Interest
Simple Interest (SI):
SI = (P × R × T) / 100
Where: P = Principal, R = Rate%, T = Time (years)
Amount A = P + SI
Compound Interest (CI):
A = P(1 + R/100)T
CI = A - P
Special Cases:
• Half-yearly: A = P(1 + R/200)(2T)
• Quarterly: A = P(1 + R/400)(4T)
SI = (P × R × T) / 100
Where: P = Principal, R = Rate%, T = Time (years)
Amount A = P + SI
Compound Interest (CI):
A = P(1 + R/100)T
CI = A - P
Special Cases:
• Half-yearly: A = P(1 + R/200)(2T)
• Quarterly: A = P(1 + R/400)(4T)
🔍 Example 4: Find SI on ₹5000 at 10% per annum for 2 years
SI = (P × R × T) / 100
SI = (5000 × 10 × 2) / 100
SI = 100000 / 100 = ₹1000
Amount = P + SI = 5000 + 1000 = ₹6000
SI = (5000 × 10 × 2) / 100
SI = 100000 / 100 = ₹1000
Amount = P + SI = 5000 + 1000 = ₹6000
💡 Quick Trick for Percentages:
To find 15% of 80: Think (10% + 5%). 10% of 80 = 8, 5% of 80 = 4, Total = 12. Much faster than (15×80)/100!
📐 Basic Geometry & Mensuration (Class 6-8)
1. Lines, Angles & Triangles
- Angle Types: Acute (<90°), Right (=90°), Obtuse (>90°, <180°), Straight (=180°), Reflex (>180°)
- Complementary Angles: Sum = 90° (Example: 30° and 60°)
- Supplementary Angles: Sum = 180° (Example: 120° and 60°)
- Vertically Opposite Angles: Always equal
- Linear Pair: Adjacent angles on a straight line, sum = 180°
2. Triangle Properties (Very Important!)
- Angle Sum Property: Sum of all 3 angles = 180°
- Exterior Angle Property: Exterior angle = Sum of two opposite interior angles
- Types by Sides: Equilateral (all sides equal), Isosceles (2 sides equal), Scalene (all different)
- Types by Angles: Acute (all angles < 90°), Right (one angle = 90°), Obtuse (one angle > 90°)
Triangle Formulas: Area = (1/2) × base × height
Heron's Formula (when all 3 sides known):
s = (a+b+c)/2 (semi-perimeter)
Area = √[s(s-a)(s-b)(s-c)]
Pythagoras Theorem (Right triangle):
(Hypotenuse)² = (Base)² + (Height)²
h² = b² + p²
Heron's Formula (when all 3 sides known):
s = (a+b+c)/2 (semi-perimeter)
Area = √[s(s-a)(s-b)(s-c)]
Pythagoras Theorem (Right triangle):
(Hypotenuse)² = (Base)² + (Height)²
h² = b² + p²
🔍 Example 5: Find the third angle if two angles are 50° and 60°
Sum of angles = 180°
Third angle = 180° - (50° + 60°)
Third angle = 180° - 110° = 70°
Third angle = 180° - (50° + 60°)
Third angle = 180° - 110° = 70°
3. Quadrilaterals
- Sum of angles: 360° (all quadrilaterals)
- Square: All sides equal, all angles 90°, diagonals equal and perpendicular
- Rectangle: Opposite sides equal, all angles 90°, diagonals equal
- Parallelogram: Opposite sides parallel and equal, opposite angles equal
- Rhombus: All sides equal, opposite angles equal, diagonals perpendicular
- Trapezium: One pair of opposite sides parallel
4. Area & Perimeter Formulas (Memorize All!)
Square (side = a):
Perimeter = 4a
Area = a²
Diagonal = a√2
Rectangle (length = l, breadth = b):
Perimeter = 2(l + b)
Area = l × b
Diagonal = √(l² + b²)
Triangle (sides = a, b, c, height = h):
Perimeter = a + b + c
Area = (1/2) × base × height
Circle (radius = r):
Circumference = 2πr
Area = πr²
Parallelogram (base = b, height = h):
Area = base × height
Trapezium (parallel sides = a, b, height = h):
Area = (1/2) × (a + b) × h
Perimeter = 4a
Area = a²
Diagonal = a√2
Rectangle (length = l, breadth = b):
Perimeter = 2(l + b)
Area = l × b
Diagonal = √(l² + b²)
Triangle (sides = a, b, c, height = h):
Perimeter = a + b + c
Area = (1/2) × base × height
Circle (radius = r):
Circumference = 2πr
Area = πr²
Parallelogram (base = b, height = h):
Area = base × height
Trapezium (parallel sides = a, b, height = h):
Area = (1/2) × (a + b) × h
🔍 Example 6: Find area of rectangle with length 12 cm and breadth 8 cm
Area = l × b
Area = 12 × 8 = 96 cm²
Perimeter = 2(l + b)
Perimeter = 2(12 + 8) = 2 × 20 = 40 cm
Area = 12 × 8 = 96 cm²
Perimeter = 2(l + b)
Perimeter = 2(12 + 8) = 2 × 20 = 40 cm
5. Volume & Surface Area (3D Shapes)
Cube (edge = a):
Volume = a³
Total Surface Area = 6a²
Lateral Surface Area = 4a²
Cuboid (length = l, breadth = b, height = h):
Volume = l × b × h
Total Surface Area = 2(lb + bh + hl)
Lateral Surface Area = 2h(l + b)
Cylinder (radius = r, height = h):
Volume = πr²h
Curved Surface Area = 2πrh
Total Surface Area = 2πr(r + h)
Cone (radius = r, height = h, slant height = l):
Volume = (1/3)πr²h
Curved Surface Area = πrl
Total Surface Area = πr(l + r)
Slant height: l = √(r² + h²)
Sphere (radius = r):
Volume = (4/3)πr³
Surface Area = 4πr²
Hemisphere (radius = r):
Volume = (2/3)πr³
Curved Surface Area = 2πr²
Total Surface Area = 3πr²
Volume = a³
Total Surface Area = 6a²
Lateral Surface Area = 4a²
Cuboid (length = l, breadth = b, height = h):
Volume = l × b × h
Total Surface Area = 2(lb + bh + hl)
Lateral Surface Area = 2h(l + b)
Cylinder (radius = r, height = h):
Volume = πr²h
Curved Surface Area = 2πrh
Total Surface Area = 2πr(r + h)
Cone (radius = r, height = h, slant height = l):
Volume = (1/3)πr²h
Curved Surface Area = πrl
Total Surface Area = πr(l + r)
Slant height: l = √(r² + h²)
Sphere (radius = r):
Volume = (4/3)πr³
Surface Area = 4πr²
Hemisphere (radius = r):
Volume = (2/3)πr³
Curved Surface Area = 2πr²
Total Surface Area = 3πr²
⚠️ Common Mistake:
Students often confuse perimeter with area! Perimeter is the boundary length (in units like cm, m), while Area is the space covered (in square units like cm², m²). Don't mix them up!
🎯 CLASS 9-10 INTERMEDIATE MATHEMATICS
Advanced algebra, coordinate geometry, trigonometry for board exams
📊 Algebra - Polynomials & Equations (Class 9-10)
1. Algebraic Identities (Must Memorize!)
Basic Identities:
(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²
a² - b² = (a + b)(a - b)
(a + b)³ = a³ + 3a²b + 3ab² + b³
(a - b)³ = a³ - 3a²b + 3ab² - b³
a³ + b³ = (a + b)(a² - ab + b²)
a³ - b³ = (a - b)(a² + ab + b²)
(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
Advanced:
a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)
If a + b + c = 0, then a³ + b³ + c³ = 3abc
(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²
a² - b² = (a + b)(a - b)
(a + b)³ = a³ + 3a²b + 3ab² + b³
(a - b)³ = a³ - 3a²b + 3ab² - b³
a³ + b³ = (a + b)(a² - ab + b²)
a³ - b³ = (a - b)(a² + ab + b²)
(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
Advanced:
a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)
If a + b + c = 0, then a³ + b³ + c³ = 3abc
🔍 Example 7: Expand (2x + 3y)²
Using (a + b)² = a² + 2ab + b²
Here a = 2x, b = 3y
(2x + 3y)² = (2x)² + 2(2x)(3y) + (3y)²
= 4x² + 12xy + 9y²
Here a = 2x, b = 3y
(2x + 3y)² = (2x)² + 2(2x)(3y) + (3y)²
= 4x² + 12xy + 9y²
2. Factorization Methods
- Common Factor Method: Take out HCF (Example: 6x + 9 = 3(2x + 3))
- Grouping Method: Group terms and factor
- Identity Method: Use algebraic identities
- Middle Term Splitting: For quadratic expressions
🔍 Example 8: Factorize x² + 5x + 6
Find two numbers whose sum = 5 and product = 6
Numbers are 2 and 3 (2+3=5, 2×3=6)
x² + 5x + 6 = x² + 2x + 3x + 6
= x(x + 2) + 3(x + 2)
= (x + 2)(x + 3)
Numbers are 2 and 3 (2+3=5, 2×3=6)
x² + 5x + 6 = x² + 2x + 3x + 6
= x(x + 2) + 3(x + 2)
= (x + 2)(x + 3)
3. Linear Equations in Two Variables
Standard Form: ax + by + c = 0
Slope-Intercept Form: y = mx + c
Where m = slope, c = y-intercept
Methods to Solve Simultaneous Equations:
1. Substitution Method
2. Elimination Method
3. Cross Multiplication Method
Cross Multiplication:
For a&sub1;x + b&sub1;y + c&sub1; = 0 and a&sub2;x + b&sub2;y + c&sub2; = 0
x/(b&sub1;c&sub2; - b&sub2;c&sub1;) = y/(c&sub1;a&sub2; - c&sub2;a&sub1;) = 1/(a&sub1;b&sub2; - a&sub2;b&sub1;)
Slope-Intercept Form: y = mx + c
Where m = slope, c = y-intercept
Methods to Solve Simultaneous Equations:
1. Substitution Method
2. Elimination Method
3. Cross Multiplication Method
Cross Multiplication:
For a&sub1;x + b&sub1;y + c&sub1; = 0 and a&sub2;x + b&sub2;y + c&sub2; = 0
x/(b&sub1;c&sub2; - b&sub2;c&sub1;) = y/(c&sub1;a&sub2; - c&sub2;a&sub1;) = 1/(a&sub1;b&sub2; - a&sub2;b&sub1;)
🔍 Example 9: Solve 2x + 3y = 12 and 3x - 2y = 5
Using Elimination Method:
Multiply eq(1) by 2: 4x + 6y = 24 ...(3)
Multiply eq(2) by 3: 9x - 6y = 15 ...(4)
Add (3) and (4): 13x = 39
x = 3
Substitute x=3 in eq(1):
2(3) + 3y = 12
6 + 3y = 12
3y = 6
y = 2
Solution: x = 3, y = 2
Multiply eq(2) by 3: 9x - 6y = 15 ...(4)
Add (3) and (4): 13x = 39
x = 3
Substitute x=3 in eq(1):
2(3) + 3y = 12
6 + 3y = 12
3y = 6
y = 2
Solution: x = 3, y = 2
4. Quadratic Equations (High Weightage!)
Standard Form: ax² + bx + c = 0 (where a ≠ 0)
Quadratic Formula (Most Important!):
x = [-b ± √(b² - 4ac)] / 2a
Discriminant: D = b² - 4ac
• If D > 0: Two distinct real roots
• If D = 0: Two equal real roots (repeated root)
• If D < 0: No real roots (imaginary roots)
Sum and Product of Roots:
Sum of roots (α + β) = -b/a
Product of roots (αβ) = c/a
Formation of Equation from Roots:
x² - (sum of roots)x + (product of roots) = 0
x² - (α + β)x + αβ = 0
Quadratic Formula (Most Important!):
x = [-b ± √(b² - 4ac)] / 2a
Discriminant: D = b² - 4ac
• If D > 0: Two distinct real roots
• If D = 0: Two equal real roots (repeated root)
• If D < 0: No real roots (imaginary roots)
Sum and Product of Roots:
Sum of roots (α + β) = -b/a
Product of roots (αβ) = c/a
Formation of Equation from Roots:
x² - (sum of roots)x + (product of roots) = 0
x² - (α + β)x + αβ = 0
🔍 Example 10: Solve 2x² - 7x + 3 = 0
Here a = 2, b = -7, c = 3
Using quadratic formula:
x = [-(-7) ± √((-7)² - 4(2)(3))] / 2(2)
x = [7 ± √(49 - 24)] / 4
x = [7 ± √25] / 4
x = [7 ± 5] / 4
x&sub1; = (7 + 5)/4 = 12/4 = 3
x&sub2; = (7 - 5)/4 = 2/4 = 1/2
Roots are: x = 3 or x = 1/2
Using quadratic formula:
x = [-(-7) ± √((-7)² - 4(2)(3))] / 2(2)
x = [7 ± √(49 - 24)] / 4
x = [7 ± √25] / 4
x = [7 ± 5] / 4
x&sub1; = (7 + 5)/4 = 12/4 = 3
x&sub2; = (7 - 5)/4 = 2/4 = 1/2
Roots are: x = 3 or x = 1/2
5. Arithmetic Progressions (AP)
Definition: Sequence where difference between consecutive terms is constant
Key Formulas:
nth term: an = a + (n-1)d
Where: a = first term, d = common difference, n = number of terms
Sum of n terms:
Sn = n/2 [2a + (n-1)d]
OR
Sn = n/2 [a + l]
Where l = last term
Common Difference: d = an - an-1
Key Formulas:
nth term: an = a + (n-1)d
Where: a = first term, d = common difference, n = number of terms
Sum of n terms:
Sn = n/2 [2a + (n-1)d]
OR
Sn = n/2 [a + l]
Where l = last term
Common Difference: d = an - an-1
🔍 Example 11: Find 20th term and sum of first 20 terms of AP: 3, 7, 11, 15...
a = 3, d = 7-3 = 4, n = 20
20th term: a20 = a + (n-1)d
a20 = 3 + (20-1)×4
a20 = 3 + 76 = 79
Sum: S20 = n/2 [2a + (n-1)d]
S20 = 20/2 [2(3) + (20-1)×4]
S20 = 10 [6 + 76]
S20 = 10 × 82 = 820
20th term: a20 = a + (n-1)d
a20 = 3 + (20-1)×4
a20 = 3 + 76 = 79
Sum: S20 = n/2 [2a + (n-1)d]
S20 = 20/2 [2(3) + (20-1)×4]
S20 = 10 [6 + 76]
S20 = 10 × 82 = 820
💡 Quick Trick for AP:
If you know first and last term, use Sn = n/2(a + l). It's much faster than the other formula!
📐 Coordinate Geometry & Trigonometry (Class 9-10)
1. Coordinate Geometry Formulas
Distance Formula:
Distance between (x&sub1;, y&sub1;) and (x&sub2;, y&sub2;):
d = √[(x&sub2; - x&sub1;)² + (y&sub2; - y&sub1;)²]
Section Formula (Internal Division):
Point dividing line joining (x&sub1;, y&sub1;) and (x&sub2;, y&sub2;) in ratio m:n:
x = (mx&sub2; + nx&sub1;)/(m + n)
y = (my&sub2; + ny&sub1;)/(m + n)
Midpoint Formula:
Midpoint = [(x&sub1; + x&sub2;)/2, (y&sub1; + y&sub2;)/2]
Area of Triangle:
With vertices (x&sub1;, y&sub1;), (x&sub2;, y&sub2;), (x&sub3;, y&sub3;):
Area = (1/2)|x&sub1;(y&sub2; - y&sub3;) + x&sub2;(y&sub3; - y&sub1;) + x&sub3;(y&sub1; - y&sub2;)|
Distance between (x&sub1;, y&sub1;) and (x&sub2;, y&sub2;):
d = √[(x&sub2; - x&sub1;)² + (y&sub2; - y&sub1;)²]
Section Formula (Internal Division):
Point dividing line joining (x&sub1;, y&sub1;) and (x&sub2;, y&sub2;) in ratio m:n:
x = (mx&sub2; + nx&sub1;)/(m + n)
y = (my&sub2; + ny&sub1;)/(m + n)
Midpoint Formula:
Midpoint = [(x&sub1; + x&sub2;)/2, (y&sub1; + y&sub2;)/2]
Area of Triangle:
With vertices (x&sub1;, y&sub1;), (x&sub2;, y&sub2;), (x&sub3;, y&sub3;):
Area = (1/2)|x&sub1;(y&sub2; - y&sub3;) + x&sub2;(y&sub3; - y&sub1;) + x&sub3;(y&sub1; - y&sub2;)|
🔍 Example 12: Find distance between points A(3, 4) and B(6, 8)
Using distance formula:
d = √[(x&sub2; - x&sub1;)² + (y&sub2; - y&sub1;)²]
d = √[(6 - 3)² + (8 - 4)²]
d = √[3² + 4²]
d = √[9 + 16]
d = √25 = 5 units
d = √[(x&sub2; - x&sub1;)² + (y&sub2; - y&sub1;)²]
d = √[(6 - 3)² + (8 - 4)²]
d = √[3² + 4²]
d = √[9 + 16]
d = √25 = 5 units
2. Introduction to Trigonometry
Basic Trigonometric Ratios:
In right triangle with angle θ:
sin θ = Perpendicular/Hypotenuse = P/H
cos θ = Base/Hypotenuse = B/H
tan θ = Perpendicular/Base = P/B
cosec θ = 1/sin θ = H/P
sec θ = 1/cos θ = H/B
cot θ = 1/tan θ = B/P
Key Relations:
tan θ = sin θ/cos θ
cot θ = cos θ/sin θ
In right triangle with angle θ:
sin θ = Perpendicular/Hypotenuse = P/H
cos θ = Base/Hypotenuse = B/H
tan θ = Perpendicular/Base = P/B
cosec θ = 1/sin θ = H/P
sec θ = 1/cos θ = H/B
cot θ = 1/tan θ = B/P
Key Relations:
tan θ = sin θ/cos θ
cot θ = cos θ/sin θ
3. Trigonometric Values (Memorize!)
| Angle θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan θ | 0 | 1/√3 | 1 | √3 | ∞ |
| cosec θ | ∞ | 2 | √2 | 2/√3 | 1 |
| sec θ | 1 | 2/√3 | √2 | 2 | ∞ |
| cot θ | ∞ | √3 | 1 | 1/√3 | 0 |
💡 Trick to Remember Trig Values:
For sin: √0/2, √1/2, √2/2, √3/2, √4/2 (0° to 90°). For cos: Just reverse sin values! For tan: sin/cos.
4. Trigonometric Identities (Very Important!)
Fundamental Identities:
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ
Derived Forms:
sin²θ = 1 - cos²θ
cos²θ = 1 - sin²θ
tan²θ = sec²θ - 1
cot²θ = cosec²θ - 1
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ
Derived Forms:
sin²θ = 1 - cos²θ
cos²θ = 1 - sin²θ
tan²θ = sec²θ - 1
cot²θ = cosec²θ - 1
🔍 Example 13: If sin θ = 3/5, find cos θ and tan θ
Using sin²θ + cos²θ = 1
(3/5)² + cos²θ = 1
9/25 + cos²θ = 1
cos²θ = 1 - 9/25 = 16/25
cos θ = 4/5
tan θ = sin θ/cos θ
tan θ = (3/5)/(4/5) = 3/4
(3/5)² + cos²θ = 1
9/25 + cos²θ = 1
cos²θ = 1 - 9/25 = 16/25
cos θ = 4/5
tan θ = sin θ/cos θ
tan θ = (3/5)/(4/5) = 3/4
5. Heights and Distances (Application)
- Angle of Elevation: Angle above horizontal when looking up
- Angle of Depression: Angle below horizontal when looking down
- Line of Sight: Straight line from observer's eye to object
🔍 Example 14: A tower is 50m high. Find distance from base if angle of elevation is 45°
Let distance = x
tan 45° = Height/Distance
1 = 50/x
x = 50 m
Distance from base = 50 meters
tan 45° = Height/Distance
1 = 50/x
x = 50 m
Distance from base = 50 meters
⚠️ Common Mistake in Trigonometry:
Students confuse sin-1, cos-1 (inverse functions) with 1/sin, 1/cos (reciprocals). sin-1x means "angle whose sine is x", NOT 1/sin! Be careful!
📊 Statistics & Probability (Class 9-10)
1. Measures of Central Tendency
Mean (Average):
For ungrouped data: Mean = (Sum of observations)/(Number of observations)
For grouped data:
Mean = Σ(fixi)/Σfi
Where fi = frequency, xi = class mark
Median:
Middle value when data arranged in order
For n observations:
• If n is odd: Median = (n+1)/2 th term
• If n is even: Median = average of (n/2)th and (n/2 + 1)th terms
For grouped data:
Median = l + [(n/2 - cf)/f] × h
Where: l = lower limit of median class,
cf = cumulative frequency before median class,
f = frequency of median class, h = class width
Mode:
Most frequently occurring value
For grouped data:
Mode = l + [(f&sub1; - f&sub0;)/(2f&sub1; - f&sub0; - f&sub2;)] × h
For ungrouped data: Mean = (Sum of observations)/(Number of observations)
For grouped data:
Mean = Σ(fixi)/Σfi
Where fi = frequency, xi = class mark
Median:
Middle value when data arranged in order
For n observations:
• If n is odd: Median = (n+1)/2 th term
• If n is even: Median = average of (n/2)th and (n/2 + 1)th terms
For grouped data:
Median = l + [(n/2 - cf)/f] × h
Where: l = lower limit of median class,
cf = cumulative frequency before median class,
f = frequency of median class, h = class width
Mode:
Most frequently occurring value
For grouped data:
Mode = l + [(f&sub1; - f&sub0;)/(2f&sub1; - f&sub0; - f&sub2;)] × h
🔍 Example 15: Find mean of 5, 8, 12, 15, 20
Mean = Sum/Count
Mean = (5 + 8 + 12 + 15 + 20)/5
Mean = 60/5 = 12
Mean = (5 + 8 + 12 + 15 + 20)/5
Mean = 60/5 = 12
2. Basic Probability
Probability Formula:
P(Event) = (Number of favorable outcomes)/(Total number of outcomes)
Range: 0 ≤ P(E) ≤ 1
• P(Impossible event) = 0
• P(Sure event) = 1
Complementary Events:
P(E) + P(not E) = 1
P(not E) = 1 - P(E)
P(Event) = (Number of favorable outcomes)/(Total number of outcomes)
Range: 0 ≤ P(E) ≤ 1
• P(Impossible event) = 0
• P(Sure event) = 1
Complementary Events:
P(E) + P(not E) = 1
P(not E) = 1 - P(E)
🔍 Example 16: A die is rolled. Find probability of getting an even number
Total outcomes = 6 (1, 2, 3, 4, 5, 6)
Favorable outcomes (even) = 3 (2, 4, 6)
P(even) = 3/6 = 1/2 = 0.5
Favorable outcomes (even) = 3 (2, 4, 6)
P(even) = 3/6 = 1/2 = 0.5
💡 Statistics Quick Tip:
For symmetric distributions: Mean = Median = Mode. If Mean > Median, distribution is positively skewed. If Mean < Median, negatively skewed.
🎓 CLASS 11-12 ADVANCED MATHEMATICS
Calculus, vectors, 3D geometry, matrices for boards and competitive exams
∞ Calculus - Limits, Derivatives & Integrals
1. Limits (Class 11)
Standard Limits (Memorize!):
lim(x→0) (sin x)/x = 1
lim(x→0) (tan x)/x = 1
lim(x→0) (1 - cos x)/x = 0
lim(x→∞) (1 + 1/x)x = e
lim(x→0) (ax - 1)/x = log a
lim(x→0) (ex - 1)/x = 1
Properties:
lim(x→a) [f(x) + g(x)] = lim(x→a) f(x) + lim(x→a) g(x)
lim(x→a) [f(x) × g(x)] = lim(x→a) f(x) × lim(x→a) g(x)
lim(x→a) [f(x)/g(x)] = [lim(x→a) f(x)]/[lim(x→a) g(x)]
lim(x→0) (sin x)/x = 1
lim(x→0) (tan x)/x = 1
lim(x→0) (1 - cos x)/x = 0
lim(x→∞) (1 + 1/x)x = e
lim(x→0) (ax - 1)/x = log a
lim(x→0) (ex - 1)/x = 1
Properties:
lim(x→a) [f(x) + g(x)] = lim(x→a) f(x) + lim(x→a) g(x)
lim(x→a) [f(x) × g(x)] = lim(x→a) f(x) × lim(x→a) g(x)
lim(x→a) [f(x)/g(x)] = [lim(x→a) f(x)]/[lim(x→a) g(x)]
🔍 Example 17: Find lim(x→0) (sin 3x)/x
Multiply and divide by 3:
= lim(x→0) 3 × (sin 3x)/(3x)
= 3 × lim(x→0) (sin 3x)/(3x)
= 3 × 1 [Using standard limit]
= 3
= lim(x→0) 3 × (sin 3x)/(3x)
= 3 × lim(x→0) (sin 3x)/(3x)
= 3 × 1 [Using standard limit]
= 3
2. Differentiation (Class 11 & 12)
Basic Differentiation Formulas:
d/dx (xn) = nxn-1
d/dx (constant) = 0
d/dx (ex) = ex
d/dx (ax) = ax log a
d/dx (log x) = 1/x
d/dx (sin x) = cos x
d/dx (cos x) = -sin x
d/dx (tan x) = sec²x
d/dx (cot x) = -cosec²x
d/dx (sec x) = sec x tan x
d/dx (cosec x) = -cosec x cot x
Rules:
Product Rule: d/dx [u·v] = u(dv/dx) + v(du/dx)
Quotient Rule: d/dx [u/v] = [v(du/dx) - u(dv/dx)]/v²
Chain Rule: dy/dx = (dy/du) × (du/dx)
d/dx (xn) = nxn-1
d/dx (constant) = 0
d/dx (ex) = ex
d/dx (ax) = ax log a
d/dx (log x) = 1/x
d/dx (sin x) = cos x
d/dx (cos x) = -sin x
d/dx (tan x) = sec²x
d/dx (cot x) = -cosec²x
d/dx (sec x) = sec x tan x
d/dx (cosec x) = -cosec x cot x
Rules:
Product Rule: d/dx [u·v] = u(dv/dx) + v(du/dx)
Quotient Rule: d/dx [u/v] = [v(du/dx) - u(dv/dx)]/v²
Chain Rule: dy/dx = (dy/du) × (du/dx)
🔍 Example 18: Differentiate y = x³ sin x
Using Product Rule: d/dx(uv) = u(dv/dx) + v(du/dx)
Here u = x³, v = sin x
dy/dx = x³ · d/dx(sin x) + sin x · d/dx(x³)
= x³ · cos x + sin x · 3x²
= x³ cos x + 3x² sin x
= x²(x cos x + 3 sin x)
Here u = x³, v = sin x
dy/dx = x³ · d/dx(sin x) + sin x · d/dx(x³)
= x³ · cos x + sin x · 3x²
= x³ cos x + 3x² sin x
= x²(x cos x + 3 sin x)
3. Applications of Derivatives
- Equation of Tangent: y - y&sub1; = (dy/dx)x=x1 · (x - x&sub1;)
- Equation of Normal: y - y&sub1; = -1/(dy/dx)x=x1 · (x - x&sub1;)
- Increasing Function: dy/dx > 0
- Decreasing Function: dy/dx < 0
- Maxima/Minima: dy/dx = 0 and check d²y/dx²
- If d²y/dx² < 0 at that point → Maxima
- If d²y/dx² > 0 at that point → Minima
4. Integration (Class 12 - Highest Weightage!)
Basic Integration Formulas:
∫ xn dx = xn+1/(n+1) + C (n ≠ -1)
∫ 1/x dx = log|x| + C
∫ ex dx = ex + C
∫ ax dx = ax/log a + C
∫ sin x dx = -cos x + C
∫ cos x dx = sin x + C
∫ sec²x dx = tan x + C
∫ cosec²x dx = -cot x + C
∫ sec x tan x dx = sec x + C
∫ cosec x cot x dx = -cosec x + C
∫ 1/√(1-x²) dx = sin-1x + C
∫ 1/(1+x²) dx = tan-1x + C
Properties:
∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx
∫ k·f(x) dx = k ∫ f(x) dx
Definite Integral:
∫ab f(x) dx = F(b) - F(a)
Where F(x) is antiderivative of f(x)
∫ xn dx = xn+1/(n+1) + C (n ≠ -1)
∫ 1/x dx = log|x| + C
∫ ex dx = ex + C
∫ ax dx = ax/log a + C
∫ sin x dx = -cos x + C
∫ cos x dx = sin x + C
∫ sec²x dx = tan x + C
∫ cosec²x dx = -cot x + C
∫ sec x tan x dx = sec x + C
∫ cosec x cot x dx = -cosec x + C
∫ 1/√(1-x²) dx = sin-1x + C
∫ 1/(1+x²) dx = tan-1x + C
Properties:
∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx
∫ k·f(x) dx = k ∫ f(x) dx
Definite Integral:
∫ab f(x) dx = F(b) - F(a)
Where F(x) is antiderivative of f(x)
🔍 Example 19: Integrate ∫ (3x² + 2x + 1) dx
∫ (3x² + 2x + 1) dx
= 3∫ x² dx + 2∫ x dx + ∫ 1 dx
= 3(x³/3) + 2(x²/2) + x + C
= x³ + x² + x + C
= 3∫ x² dx + 2∫ x dx + ∫ 1 dx
= 3(x³/3) + 2(x²/2) + x + C
= x³ + x² + x + C
5. Integration by Parts
Formula: ∫ u·v dx = u∫v dx - ∫[(du/dx)·∫v dx] dx
ILATE Rule (Choose u in this order):
I - Inverse trigonometric
L - Logarithmic
A - Algebraic
T - Trigonometric
E - Exponential
ILATE Rule (Choose u in this order):
I - Inverse trigonometric
L - Logarithmic
A - Algebraic
T - Trigonometric
E - Exponential
🔍 Example 20: Integrate ∫ x·ex dx
Using ILATE: u = x (Algebraic), v = ex (Exponential)
∫ x·ex dx = x∫ex dx - ∫[d/dx(x)·∫ex dx] dx
= x·ex - ∫1·ex dx
= x·ex - ex + C
= ex(x - 1) + C
∫ x·ex dx = x∫ex dx - ∫[d/dx(x)·∫ex dx] dx
= x·ex - ∫1·ex dx
= x·ex - ex + C
= ex(x - 1) + C
🔥 Calculus = 30+ marks in Class 12!
Master differentiation (all formulas) and integration (all methods). Practice 50+ problems. Calculus is the highest scoring chapter if you practice enough!
📐 Vectors & 3D Geometry (Class 12)
1. Vector Basics
Vector Representation:
a⃗ = axi + ayj + azk
Where i, j, k are unit vectors along x, y, z axes
Magnitude:
|a⃗| = √(ax² + ay² + az²)
Unit Vector:
â = a⃗/|a⃗|
Position Vector:
Vector from origin to point (x, y, z)
r⃗ = xi + yj + zk
a⃗ = axi + ayj + azk
Where i, j, k are unit vectors along x, y, z axes
Magnitude:
|a⃗| = √(ax² + ay² + az²)
Unit Vector:
â = a⃗/|a⃗|
Position Vector:
Vector from origin to point (x, y, z)
r⃗ = xi + yj + zk
2. Vector Operations
Addition: a⃗ + b⃗ = (ax+bx)i + (ay+by)j + (az+bz)k
Scalar Multiplication: k·a⃗ = kaxi + kayj + kazk
Dot Product (Scalar Product):
a⃗ · b⃗ = |a⃗||b⃗|cos θ
= axbx + ayby + azbz
Properties:
• If a⃗ · b⃗ = 0, then a⃗ ⊥ b⃗ (perpendicular)
• a⃗ · a⃗ = |a⃗|²
Cross Product (Vector Product):
a⃗ × b⃗ = |a⃗||b⃗|sin θ n̂
Where n̂ is unit vector perpendicular to both
Using determinant:
a⃗ × b⃗ = |i j k|
|ax ay az|
|bx by bz|
Properties:
• If a⃗ × b⃗ = 0⃗, then a⃗ ∥ b⃗ (parallel)
• |a⃗ × b⃗| = Area of parallelogram
Scalar Multiplication: k·a⃗ = kaxi + kayj + kazk
Dot Product (Scalar Product):
a⃗ · b⃗ = |a⃗||b⃗|cos θ
= axbx + ayby + azbz
Properties:
• If a⃗ · b⃗ = 0, then a⃗ ⊥ b⃗ (perpendicular)
• a⃗ · a⃗ = |a⃗|²
Cross Product (Vector Product):
a⃗ × b⃗ = |a⃗||b⃗|sin θ n̂
Where n̂ is unit vector perpendicular to both
Using determinant:
a⃗ × b⃗ = |i j k|
|ax ay az|
|bx by bz|
Properties:
• If a⃗ × b⃗ = 0⃗, then a⃗ ∥ b⃗ (parallel)
• |a⃗ × b⃗| = Area of parallelogram
🔍 Example 21: Find a⃗ · b⃗ if a⃗ = 2i + 3j + k and b⃗ = i - 2j + 3k
a⃗ · b⃗ = axbx + ayby + azbz
= (2)(1) + (3)(-2) + (1)(3)
= 2 - 6 + 3
= -1
= (2)(1) + (3)(-2) + (1)(3)
= 2 - 6 + 3
= -1
3. Three Dimensional Geometry
Direction Cosines:
If line makes angles α, β, γ with x, y, z axes:
Direction cosines: l = cos α, m = cos β, n = cos γ
Property: l² + m² + n² = 1
Equation of Line:
Vector form: r⃗ = a⃗ + λb⃗
Cartesian form: (x-x&sub1;)/a = (y-y&sub1;)/b = (z-z&sub1;)/c
Equation of Plane:
Vector form: r⃗ · n⃗ = d
Cartesian form: ax + by + cz + d = 0
Distance from Point to Plane:
Distance from (x&sub1;, y&sub1;, z&sub1;) to ax + by + cz + d = 0:
D = |ax&sub1; + by&sub1; + cz&sub1; + d| / √(a² + b² + c²)
If line makes angles α, β, γ with x, y, z axes:
Direction cosines: l = cos α, m = cos β, n = cos γ
Property: l² + m² + n² = 1
Equation of Line:
Vector form: r⃗ = a⃗ + λb⃗
Cartesian form: (x-x&sub1;)/a = (y-y&sub1;)/b = (z-z&sub1;)/c
Equation of Plane:
Vector form: r⃗ · n⃗ = d
Cartesian form: ax + by + cz + d = 0
Distance from Point to Plane:
Distance from (x&sub1;, y&sub1;, z&sub1;) to ax + by + cz + d = 0:
D = |ax&sub1; + by&sub1; + cz&sub1; + d| / √(a² + b² + c²)
💡 Vectors Quick Tip:
For perpendicular vectors: Use dot product = 0. For parallel vectors: Use cross product = 0. Remember these conditions!
📊 Matrices & Determinants (Class 12)
1. Matrix Basics
Types of Matrices:
• Row Matrix: 1 row, n columns [1×n]
• Column Matrix: m rows, 1 column [m×1]
• Square Matrix: m = n [m×m]
• Diagonal Matrix: aij = 0 for i ≠ j
• Identity Matrix: I (diagonal elements = 1, others = 0)
• Zero Matrix: All elements = 0
• Symmetric: A = AT (aij = aji)
• Skew-Symmetric: A = -AT (aij = -aji)
Properties:
(A + B)T = AT + BT
(AB)T = BTAT (Note: Order reverses!)
(AT)T = A
• Row Matrix: 1 row, n columns [1×n]
• Column Matrix: m rows, 1 column [m×1]
• Square Matrix: m = n [m×m]
• Diagonal Matrix: aij = 0 for i ≠ j
• Identity Matrix: I (diagonal elements = 1, others = 0)
• Zero Matrix: All elements = 0
• Symmetric: A = AT (aij = aji)
• Skew-Symmetric: A = -AT (aij = -aji)
Properties:
(A + B)T = AT + BT
(AB)T = BTAT (Note: Order reverses!)
(AT)T = A
2. Matrix Operations
- Addition/Subtraction: Only if same order
- Multiplication: If A is [m×n] and B is [n×p], then AB is [m×p]
- Properties: AB ≠ BA (not commutative), but (AB)C = A(BC) (associative)
3. Determinants
2×2 Matrix:
|a b|
|c d| = ad - bc
3×3 Matrix (Expansion along first row):
|a&sub1; b&sub1; c&sub1;|
|a&sub2; b&sub2; c&sub2;| = a&sub1;(b&sub2;c&sub3;-b&sub3;c&sub2;) - b&sub1;(a&sub2;c&sub3;-a&sub3;c&sub2;) + c&sub1;(a&sub2;b&sub3;-a&sub3;b&sub2;)
|a&sub3; b&sub3; c&sub3;|
Properties:
• |A| = |AT|
• |AB| = |A| × |B|
• |kA| = kn|A| (for n×n matrix)
• If two rows/columns identical, |A| = 0
• Interchanging rows/columns changes sign
|a b|
|c d| = ad - bc
3×3 Matrix (Expansion along first row):
|a&sub1; b&sub1; c&sub1;|
|a&sub2; b&sub2; c&sub2;| = a&sub1;(b&sub2;c&sub3;-b&sub3;c&sub2;) - b&sub1;(a&sub2;c&sub3;-a&sub3;c&sub2;) + c&sub1;(a&sub2;b&sub3;-a&sub3;b&sub2;)
|a&sub3; b&sub3; c&sub3;|
Properties:
• |A| = |AT|
• |AB| = |A| × |B|
• |kA| = kn|A| (for n×n matrix)
• If two rows/columns identical, |A| = 0
• Interchanging rows/columns changes sign
4. Inverse of Matrix
For 2×2 Matrix:
If A = [a b], then A-1 = 1/(ad-bc) × [d -b]
[c d] [-c a]
General Formula:
A-1 = adj(A) / |A|
(Only if |A| ≠ 0)
Properties:
• A·A-1 = A-1·A = I
• (AB)-1 = B-1A-1
• (A-1)-1 = A
• (AT)-1 = (A-1)T
If A = [a b], then A-1 = 1/(ad-bc) × [d -b]
[c d] [-c a]
General Formula:
A-1 = adj(A) / |A|
(Only if |A| ≠ 0)
Properties:
• A·A-1 = A-1·A = I
• (AB)-1 = B-1A-1
• (A-1)-1 = A
• (AT)-1 = (A-1)T
5. System of Linear Equations
Matrix Method:
For AX = B:
X = A-1B
Cramer's Rule:
For 2 equations in 2 variables:
x = Dx/D, y = Dy/D
Where D = coefficient determinant
Dx = replace x-column with constants
Dy = replace y-column with constants
For AX = B:
X = A-1B
Cramer's Rule:
For 2 equations in 2 variables:
x = Dx/D, y = Dy/D
Where D = coefficient determinant
Dx = replace x-column with constants
Dy = replace y-column with constants
🔍 Example 22: Find determinant of [2 3; 1 4]
|2 3|
|1 4| = (2×4) - (3×1)
= 8 - 3
= 5
|1 4| = (2×4) - (3×1)
= 8 - 3
= 5
⚠️ Matrix Multiplication Mistake:
AB ≠ BA in matrices! Matrix multiplication is NOT commutative. Also, (AB)T = BTAT (order reverses). Students often make these mistakes in exams!
🎯 SCORING STRATEGIES & EXAM TIPS
How to score 100/100, chapter weightage, common mistakes, book comparison
💯 How to Score 100/100 in Mathematics
1. Preparation Strategy (Before Exam)
- Master NCERT First: Solve every single exercise problem, every example. NCERT = 60% board paper!
- Formula Sheet: Make your own formula sheet for each chapter. Revise daily in morning.
- Practice, Practice, Practice: Solve minimum 15-20 sample papers before exam.
- Previous Year Papers: Solve last 10 years papers. Patterns repeat!
- Time Management: Practice full 3-hour papers with timer. Speed matters!
- Weak Areas: Identify your weak chapters. Practice them MORE, not avoid them.
- Daily Practice: 2-3 hours daily for math. Skip a day = skills decrease.
2. Chapter-wise Weightage (Class 12)
| Chapter | Marks | Difficulty | Priority |
|---|---|---|---|
| Integrals | 12 | Medium | Very High |
| Relations & Functions | 10 | Easy | High |
| Matrices & Determinants | 10 | Medium | High |
| Continuity & Differentiability | 10 | Medium | High |
| Applications of Derivatives | 8 | Medium | High |
| Applications of Integrals | 8 | Easy | High |
| Probability | 10 | Easy-Medium | High |
| Vectors | 6 | Medium | Medium |
| 3D Geometry | 6 | Medium-Hard | Medium |
| Differential Equations | 6 | Medium | Medium |
| Linear Programming | 6 | Easy | High |
3. Exam Day Strategy
- First 15 Minutes: Read entire paper carefully. Choose questions wisely.
- Time Allocation: 1.5 minutes per mark (12 mark question = 18 minutes max)
- Question Selection: Start with easiest for confidence boost
- Show All Steps: Even if you know shortcut, show working for partial marks
- Circle Final Answer: Helps examiner spot answer quickly
- Graphs/Diagrams: Draw neatly with pencil, label all parts
- Check Units: Always mention units in answers (cm², degrees, etc.)
- Avoid Overwriting: Cut once with single line if mistake, don't make it messy
- Last 20 Minutes: Recheck all calculations, verify formulas used
4. Presentation Tips
- Neat Handwriting: Worth 3-5 marks! Examiner should read easily.
- Margins: Leave proper margins (2cm left, 1.5cm right)
- Spacing: Don't cramp answers. Leave space between questions.
- Underlining: Underline formulas, theorems, final answers
- Step Numbering: Number your steps (Step 1, Step 2...) for clarity
- Rough Work: Do on left page, final answer on right page
- Blue/Black Pen: Use only blue or black pen. Avoid other colors.
🎯 100/100 Formula:
NCERT (Complete) + Reference Book (Practice) + 15 Sample Papers + 10 Previous Years + Good Presentation = 100/100! It's achievable with consistent effort!
⚠️ Common Mistakes & How to Avoid
Algebra Mistakes
- Mistake: (a+b)² = a² + b² ❌
Correct: (a+b)² = a² + 2ab + b² ✓
Tip: Never forget middle term! - Mistake: √(a² + b²) = a + b ❌
Correct: Cannot simplify further ✓
Tip: Square root doesn't distribute over addition! - Mistake: (x+2)(x+3) = x² + 6 ❌
Correct: x² + 5x + 6 ✓
Tip: Use FOIL method properly
Trigonometry Mistakes
- Mistake: sin 2θ = 2 sin θ ❌
Correct: sin 2θ = 2 sin θ cos θ ✓ - Mistake: cos²θ + sin²θ = 2 ❌
Correct: cos²θ + sin²θ = 1 ✓ - Mistake: tan θ = sin θ + cos θ ❌
Correct: tan θ = sin θ / cos θ ✓
Calculus Mistakes
- Mistake: d/dx(x²) = x ❌
Correct: d/dx(x²) = 2x ✓
Tip: Bring power down, reduce power by 1 - Mistake: ∫ 1/x dx = x²/2 ❌
Correct: ∫ 1/x dx = log|x| + C ✓
Tip: Special case for power -1 - Mistake: Forgetting "+ C" in indefinite integration ❌
Correct: Always write "+ C" ✓
Tip: C is integration constant, mandatory!
Coordinate Geometry Mistakes
- Mistake: Distance = √[(x&sub2; - x&sub1;) + (y&sub2; - y&sub1;)] ❌
Correct: Distance = √[(x&sub2; - x&sub1;)² + (y&sub2; - y&sub1;)²] ✓
Tip: Square the differences! - Mistake: Slope = (x&sub2; - x&sub1;)/(y&sub2; - y&sub1;) ❌
Correct: Slope = (y&sub2; - y&sub1;)/(x&sub2; - x&sub1;) ✓
Tip: Rise over run!
Sign and Calculation Mistakes
- Sign errors: (-2)^2 = -4 (WRONG) Correct: 4
- Fraction mistakes: 1/2 + 1/3 = 2/5 (WRONG) Correct: 5/6
- Decimal errors: 0.5 x 0.5 = 0.25 (not 0.05)
- BODMAS violation: Always follow order!
💡 Avoiding Silly Mistakes:
Practice writing neatly. Double-check calculations. Don't rush. Silly mistakes cost 10-15 marks! Take 2 minutes to verify your answer before moving on.
📚 RD Sharma vs RS Aggarwal - Complete Comparison
RD Sharma
Best For: In-depth understanding, JEE preparation alongside boards
Pros:
- ✓ Extremely detailed explanations for every concept
- ✓ Large variety of questions (2000+ per class)
- ✓ Questions arranged difficulty-wise (Easy → Hard)
- ✓ Excellent for JEE Mains preparation
- ✓ Conceptual clarity through theory sections
- ✓ Multiple methods shown for same problem
- ✓ Previous year board questions included
Cons:
- ✗ Very lengthy (intimidating for weak students)
- ✗ Time-consuming to complete fully
- ✗ Some questions too advanced for boards
- ✗ Can overwhelm average students
Recommended For:
- Students targeting 95%+ in boards
- JEE/NEET aspirants (Math/Bio students)
- Those with 6+ months preparation time
- Students comfortable with math
RS Aggarwal
Best For: Quick revision, board exam focus, average students
Pros:
- ✓ Concise and to-the-point
- ✓ Strictly follows CBSE pattern
- ✓ Perfect for last-minute preparation
- ✓ Formula summaries at chapter start
- ✓ Sufficient questions for boards (not overwhelming)
- ✓ Easier to complete in limited time
- ✓ Good for scoring 85-90% quickly
Cons:
- ✗ Less variety compared to RD Sharma
- ✗ Not ideal for competitive exam prep
- ✗ Theory explanations are brief
- ✗ May not cover some advanced concepts
Recommended For:
- Students targeting 85-90% in boards
- Those with limited preparation time
- Commerce/Arts students with math
- Quick revision before exams
Head-to-Head Comparison
| Feature | RD Sharma | RS Aggarwal |
|---|---|---|
| Content Depth | Very Deep | Moderate |
| Question Count | 2000+ | 800-1000 |
| Difficulty Level | Easy to Very Hard | Easy to Medium |
| Time Required | 6-8 months | 3-4 months |
| JEE Preparation | Excellent | Basic |
| Board Focus | High | Very High |
| Price Range | ₹600-800 | ₹400-600 |
| Page Count | 1200+ | 600-700 |
Expert Recommendation
- For 100/100 Target: NCERT + RD Sharma (Selective important questions)
- For 90-95% Target: NCERT + RS Aggarwal (Complete)
- For 80-85% Target: NCERT (Thoroughly) + RS Aggarwal (Selective)
- For Last 2 Months: NCERT + Sample Papers only
💡 Smart Strategy:
Use both! NCERT first (complete), then RS Aggarwal (complete), then RD Sharma (only high-weightage chapters for extra practice). Don't compare with peers - follow your own pace!
⚡ Quick Calculation Tricks & Shortcuts
Mental Math Shortcuts
- Squaring numbers ending in 5:
Example: 35² = (3×4) hundred + 25 = 1225
Formula: (n5)² = n(n+1) hundred + 25 - Multiply by 11:
Example: 23 x 11 = 253 (Add digits: 2_3, middle = 2+3 = 5)
For 67x11 = 737 (6_7, middle = 6+7 = 13, carry 1) - Multiply by 5:
Divide by 2, then multiply by 10
Example: 86 x 5 = (86/2) x 10 = 43 x 10 = 430 - Percentage to Fraction:
25% = 1/4, 50% = 1/2, 75% = 3/4
20% = 1/5, 33.33% = 1/3, 66.67% = 2/3
Trigonometry Shortcuts
- Allied Angles:
sin(90° - A) = cos A
cos(90° - A) = sin A
tan(90° - A) = cot A - Negative Angles:
sin(-A) = -sin A
cos(-A) = cos A
tan(-A) = -tan A
Algebra Speed Tips
- Quick Factorization:
x² - y² = (x+y)(x-y) [Difference of squares]
Use this before trying other methods! - Quadratic Sum/Product:
Instead of formula, sometimes sum = -b/a, product = c/a is faster - Completing the Square:
x² + bx = (x + b/2)² - (b/2)²
Calculus Shortcuts
- Standard Limits: Memorize all 10 standard limits
- L'Hospital Rule: For 0/0 or ∞/∞ forms
lim[f(x)/g(x)] = lim[f'(x)/g'(x)] - Integration: If you see f'(x)/f(x), answer is log|f(x)|
⚠️ Don't Overuse Shortcuts:
Shortcuts are for speed, not replacement of concepts! In board exams, show full working. Use shortcuts only for verification or when specifically asked for quick method.