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Maths Master Notes (Ch 1-14 Deep Dive)

Detailed Analysis of Complete Syllabus

CHAPTER 1: REAL NUMBERS

1. Proving \( \sqrt{p} \) is irrational (Guarantee 3 Marks).
2. Proving \( a + b\sqrt{p} \) is irrational (Guarantee 2 Marks).
3. HCF/LCM Word Problems (Bells, Circular Track).

1. Types of Numbers

• Prime Number: Has exactly two factors: 1 and itself (e.g., 2, 3, 5, 7).
• Composite Number: Has more than two factors (e.g., 4, 6, 8, 9).
Note: 1 is neither Prime nor Composite.

2. Fundamental Theorem of Arithmetic

            "Every composite number can be written as a product of primes in a unique way."

            Application: Finding HCF & LCM

            • HCF: Product of smallest power of each common prime factor.

            • LCM: Product of greatest power of each prime factor involved.

Relation Formula:
\( \text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b \)
(Valid only for 2 numbers).

3. Irrationality Proofs

Proof: \( \sqrt{2} \) is irrational.
1. Let \( \sqrt{2} = a/b \) (where a, b are coprime).
2. \( 2 = a^2/b^2 \Rightarrow 2b^2 = a^2 \).
3. So, 2 divides \( a^2 \), hence 2 divides \( a \). Let \( a=2c \).
4. \( 2b^2 = (2c)^2 = 4c^2 \Rightarrow b^2 = 2c^2 \).
5. So, 2 divides \( b^2 \), hence 2 divides \( b \).
6. Contradiction: \( a, b \) share factor 2. Hence assumption wrong.

CHAPTER 2: POLYNOMIALS

1. Degree & Types

Type	Degree	Form	Max Zeroes
Linear	1	\( ax + b \)	1
Quadratic	2	\( ax^2 + bx + c \)	2
Cubic	3	\( ax^3 + bx^2 + cx + d \)	3

2. Geometrical Meaning

Zeroes are the x-coordinates of the points where graph \( y = p(x) \) intersects the X-axis.

Graph Shapes (Quadratic):
• \( a > 0 \): Parabola opens UP \( \cup \)
• \( a < 0 \): Parabola opens DOWN \( \cap \)
• Touch X-axis: Repeated Roots.
• Cut X-axis: Distinct Roots.

3. Relationship of Zeroes

            If \( \alpha, \beta \) are zeroes of \( ax^2 + bx + c \):

            • \( \alpha + \beta = -\frac{b}{a} \)

            • \( \alpha \beta = \frac{c}{a} \)

To Form a Polynomial:
If Sum (S) and Product (P) are given:
\( p(x) = k[x^2 - Sx + P] \)

CHAPTER 3: PAIR OF LINEAR EQUATIONS

1. Forms & Consistency

System: \( a_1x + b_1y + c_1 = 0 \), \( a_2x + b_2y + c_2 = 0 \).

Condition	Graph	Solution	Type
\( a_1/a_2 \neq b_1/b_2 \)	Intersecting	Unique	Consistent
\( a_1/a_2 = b_1/b_2 = c_1/c_2 \)	Coincident	Infinite	Dependent
\( a_1/a_2 = b_1/b_2 \neq c_1/c_2 \)	Parallel	None	Inconsistent

2. Algebraic Methods

            Substitution Method:

            1. Find value of 'y' from eq 1.

            2. Put it in eq 2 -> Solve for 'x'.

            3. Put 'x' back in step 1 -> Find 'y'.

            Elimination Method (Recommended):

            1. Multiply equations to equate coefficients of x or y.

            2. Add or Subtract to eliminate that variable.

            3. Solve for remaining variable.

3. Word Problems Categorization

1. Fraction: Let fraction be \( x/y \). condition 1 -> eq 1. condition 2 -> eq 2.
2. Age: Present age \( x \). 5 years ago \( x-5 \). 5 years later \( x+5 \).
3. Digits: Unit digit \( y \), Ten's digit \( x \). Number = \( 10x + y \). Reversing = \( 10y + x \).
4. Upstream: Speed \( (x-y) \). Downstream: \( (x+y) \). Time = Dist/Speed.

CHAPTER 4: QUADRATIC EQUATIONS

1. Standard Form & Factorization

\( ax^2 + bx + c = 0 \). (a \( \neq \) 0).
Factorization: Split middle term \( b \) into \( p+q \) such that \( p \times q = a \times c \).

2. Quadratic Formula

            \[ x = \frac{-b \pm \sqrt{D}}{2a} \]
            here Discriminant \( D = b^2 - 4ac \).
        

3. Nature of Roots

Value of D	Nature
\( D > 0 \)	2 Real Distinct Roots
\( D = 0 \)	2 Real Equal Roots
\( D < 0 \)	No Real Roots

Problem Solving Strategy:
If question says "Find k for equal roots", immediately set \( D = 0 \) i.e., \( b^2 - 4ac = 0 \).

CHAPTER 5: ARITHMETIC PROGRESSIONS

1. Basics

Sequence where difference between consecutive terms is constant.
\( a, a+d, a+2d \dots \)

2. General Term (nth Term)

            \[ a_n = a + (n-1)d \]
            • \( a_n \): Value of term at position n.

            • \( n \): Position number (always natural number).
        

3. Sum of First n Terms

            \[ S_n = \frac{n}{2}[2a + (n-1)d] \]
            OR (if last term \( l \) is known):

            \[ S_n = \frac{n}{2}(a + l) \]
        

Selection of Terms:
• 3 terms: \( a-d, a, a+d \) (Sum = 3a).
• 4 terms: \( a-3d, a-d, a+d, a+3d \) (Sum = 4a).

CHAPTER 6: TRIANGLES

1. BPT Theorem Proof (4/5 Marks - Very Important).
2. Similarity Problems (Shadow/Ladder type).
3. Ratio of Areas (Concept often asked).

1. Basic Proportionality Theorem (BPT)

Statement: If line \( DE \parallel BC \), then \( \frac{AD}{DB} = \frac{AE}{EC} \).
Proof Steps (Memorize Diagram):
1. Area(\( \Delta ADE \))/\( \Delta BDE \) = \( AD/DB \) (Common height EN).
2. Area(\( \Delta ADE \))/\( \Delta DEC \) = \( AE/EC \) (Common height DM).
3. \( \Delta BDE \) and \( \Delta DEC \) are equal in area (Same base DE, between same parallels).
4. Hence Ratio 1 = Ratio 2. \( \therefore AD/DB = AE/EC \).

2. Similarity Criteria

            • AAA / AA: Most used. Angles equal = Similar.

            • Ratio of Areas: \( \frac{\text{Ar}(\Delta_1)}{\text{Ar}(\Delta_2)} = (\frac{S_1}{S_2})^2
            \).
        

CHAPTER 7: COORDINATE GEOMETRY

1. Distance Formula (Equidistant points - Find x/y).
2. Section Formula (Find ratio k:1).
3. Mid-point formula (Parallelogram vertices).

Parallelogram Trick:
If A, B, C, D are vertices of a parallelogram, diagonals bisect each other.
\( \text{Mid}(AC) = \text{Mid}(BD) \)
\( \frac{x_1+x_3}{2} = \frac{x_2+x_4}{2} \) and \( \frac{y_1+y_3}{2} = \frac{y_2+y_4}{2} \).

CHAPTER 8: TRIGONOMETRY

1. Prove Identity: \( \frac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = \frac{1}{\sec\theta - \tan\theta} \) type.
2. Values: Evaluate \( 5\cos^2 60 + 4\sec^2 30 - \tan^2 45 \).

1. Trig Table (Hand Trick)

            Sin \(\theta\) Rule: \( \sqrt{0,1,2,3,4} / 2 \).

            • 0°: 0

            • 30°: 1/2

            • 45°: \( 1/\sqrt{2} \)

            • 60°: \( \sqrt{3}/2 \)

            • 90°: 1

2. Identities Strategy

Golden Rule for Proofs:
1. Convert Tan/Sec/Cosec/Cot to Sin and Cos.
2. Take LCM.
3. Apply \( \sin^2 A + \cos^2 A = 1 \).

CHAPTER 9: APPLICATIONS OF TRIG

            1. Broken Tree: Tree breaks, top touches ground. Forms right triangle.

            2. Two Ships: Angles of depression. Use \( \tan \theta \) twice.

CHAPTER 10: CIRCLES

1. Theorem 10.2: Tangents from external point are equal (Must Do Proof).
2. Quadrilateral circumscribing circle: AB + CD = AD + BC.
3. Angle between tangents + Angle at center = 180°.

Key Property (Angle Inclination):
The angle between two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the center.
\( \angle APB + \angle AOB = 180^\circ \).

1. Tangent Proof (Theorem 10.2)

Proof (RHS Method):
In \( \Delta OAP \) and \( \Delta OBP \):
• \( \angle OAP = \angle OBP = 90^\circ \) (Radius \( \perp \) Tangent).
• \( OP = OP \) (Common).
• \( OA = OB \) (Radii).
• By RHS, \( \Delta OAP \cong \Delta OBP \). Hence \( PA = PB \).

CHAPTER 11: AREAS RELATED TO CIRCLES

1. Area of Minor Segment (Use \( \theta/360 \times \pi r^2 - \text{Area of } \Delta \)).
2. Clock hand problems.
3. Horse tied to a peg (Quadrant Area).

1. Important Formulas

            • Area of Sector: \( \frac{\theta}{360} \times \pi r^2 \)

            • Length of Arc: \( \frac{\theta}{360} \times 2\pi r \)

            • Area of Quadrant: \( \frac{1}{4} \pi r^2 \) (Since \( \theta=90^\circ \)).

Clock Logic:
• Minute Hand: Moves \( 6^\circ \) per minute.
(e.g., 20 mins = \( 20 \times 6 = 120^\circ \)).
• Hour Hand: Moves \( 30^\circ \) per hour.

Cost Problems:
\( \text{Total Cost} = \text{Rate} \times \text{Quantity (Area/Length)} \).

CHAPTER 12: SURFACE AREAS & VOLUMES

1. Combination: Capsule (Cyl + 2 Hemi).
2. Melting: \( V_{big} = n \times V_{small} \).
3. Embankment (Hollow Cylinder).

1. Formula Cheat Sheet

Solid	CSA	TSA	Volume
Cylinder	\( 2\pi rh \)	\( 2\pi r(r+h) \)	\( \pi r^2h \)
Cone	\( \pi rl \)	\( \pi r(l+r) \)	\( \frac{1}{3}\pi r^2h \)
Sphere	\( 4\pi r^2 \)	\( 4\pi r^2 \)	\( \frac{4}{3}\pi r^3 \)
Hemisphere	\( 2\pi r^2 \)	\( 3\pi r^2 \)	\( \frac{2}{3}\pi r^3 \)

Embankment (Hollow Cylinder):
Earth dug out (Cylinder \( \pi r^2 h \)) = Volume of Embankment.
Vol of Embankment = \( \pi (R^2 - r^2) \times H \).
(Where \( R \) = outer radius, \( r \) = inner radius).

CHAPTER 13: STATISTICS

1. Missing Frequency (f1, f2) in Median.
2. Calculate Mean (Assumed Mean).

1. Mean (\( \bar{x} \))

Assumed Mean Method:
1. Choose middle \( x_i \) as 'a'.
2. Find deviation \( d_i = x_i - a \).
3. \( \bar{x} = a + \frac{\sum f_i d_i}{\sum f_i} \).

2. Median (Most Important)

            \[ \text{Median} = l + \left( \frac{\frac{n}{2} - cf}{f} \right) \times h \]
            Correctly Finding Median Class:

            Step 1: Calculate N (Total Freq).

            Step 2: Find N/2.

            Step 3: Check CF column. The first value > N/2 indicates the Median Class.

Empirical Formula (1 Mark):
\( 3 \times \text{Median} = \text{Mode} + 2 \times \text{Mean} \)

CHAPTER 14: PROBABILITY

1. Cards (Face cards, Red cards prob).
2. Dice (Doublet, Sum > 10).
3. Leap Year (53 Sundays).

1. Sample Spaces

Coins:
2 Coins (4): {HH, HT, TH, TT}
3 Coins (8): {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

Dice (Doublets):
(1,1), (2,2), (3,3), (4,4), (5,5), (6,6) -> Total 6.

Leap Year Logic:
Leap Year = 366 days = 52 Weeks + 2 Extra Days.
Extra days can be: (Sun,Mon), (Mon,Tue)...(Sat,Sun).
P(53 Sundays) = 2/7.
Non-Leap Year = 365 days = 52 Weeks + 1 Extra Day.
P(53 Sundays) = 1/7.

🎯 FINAL EXAM CHECKLIST

Real Numbers: Proof of \( \sqrt{3} \) irrational.
Triangles: BPT Proof.
Circles: Tangent Proof.
Stats: Median Formula.