COMPREHENSIVE LECTURE NOTES FOR CLASS 10 MATHEMATICS (EXTENDED EDITION)

COMPREHENSIVE LECTURE NOTES FOR CLASS 10 MATHEMATICS (EXTENDED EDITION)

INTRODUCTION

Mathematics is a fundamental subject that serves as the foundation for various fields, including engineering, physics, economics, and many others. This module addresses the essential topics outlined in the Class 10 Mathematics syllabus, enabling students to comprehend foundational concepts and excel in their upcoming board examinations. The primary objective is to ensure that students not only understand the material but can also effectively apply their knowledge to solve complex problems.

This extended edition builds on the core notes with additional theorems, derivations, formula sheets, extra worked examples, common mistakes, and practice questions for each chapter.

TABLE OF CONTENTS

1. Number Systems

2. Polynomials

3. Coordinate Geometry

4. Linear Equations in Two Variables

5. Quadratic Equations

6. Arithmetic Progressions

7. Triangles

8. Trigonometry

9. Statistics

10. Probability

11. Conclusion

12. Quick Formula Sheet

13. NUMBER SYSTEMS

Definition

A number system is a method for expressing numbers; it serves as a mathematical notation for representing numbers from a specific set using digits or other symbols in a consistent manner. The two most prevalent types are rational and irrational numbers, which together make up the set of Real Numbers.

Rational Numbers

A rational number can be expressed in the form p/q, where p and q are integers, and q ≠ 0. Examples include:

• 1/2

• -3/4

• 5 (which can be represented as 5/1)

• 0.333... = 1/3 (repeating decimals are rational)

Irrational Numbers

Irrational numbers cannot be expressed as a fraction of two integers. They possess non-repeating and non-terminating decimal expansions. Examples include:

• √2 = 1.41421356...

• π = 3.14159265...

• e = 2.71828182...

• √3, √5, √7 (square roots of non-perfect squares)

Euclid's Division Lemma

For any two positive integers a and b, there exist unique integers q and r such that:

a = bq + r, where 0 ≤ r < b

This lemma is the foundation of the Euclidean algorithm for finding the HCF (Highest Common Factor) of two numbers.

Example: Find HCF of 455 and 42 using Euclid's Division Algorithm

455 = 42 × 10 + 35

42 = 35 × 1 + 7

35 = 7 × 5 + 0

Since the remainder is now 0, the HCF is 7.

Fundamental Theorem of Arithmetic

Every composite number can be expressed (factorized) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.

Example: Express 156 as a product of its prime factors

156 = 2 × 78 = 2 × 2 × 39 = 2 × 2 × 3 × 13 = 2^2 × 3 × 13

Finding HCF and LCM using Prime Factorization

• HCF = product of the smallest power of each common prime factor

• LCM = product of the greatest power of each prime factor involved

• Important relation: HCF(a, b) × LCM(a, b) = a × b

Example: Find HCF and LCM of 6 and 20

6 = 2 × 3

20 = 2^2 × 5

HCF = 2 (smallest power of common factor 2)

LCM = 2^2 × 3 × 5 = 60

Check: HCF × LCM = 2 × 60 = 120 = 6 × 20 ✓

Proving Irrationality (Proof by Contradiction)

To prove √2 is irrational:

1. Assume √2 is rational, so √2 = p/q where p, q are co-prime integers, q ≠ 0.

2. Then 2 = p^2/q^2, so p^2 = 2q^2, meaning p^2 is even, so p is even.

3. Let p = 2m. Then (2m)^2 = 2q^2 → 4m^2 = 2q^2 → q^2 = 2m^2, so q is also even.

4. This contradicts the assumption that p and q are co-prime (both cannot be even).

5. Therefore, √2 is irrational.

Example: Converting a Decimal to a Fraction

To convert the decimal 0.75 into a fraction:

1. Write it as 75/100.

2. Simplify by dividing both the numerator and denominator by their greatest common divisor (25):

   75 ÷ 25 / 100 ÷ 25 = 3/4

Terminating and Non-Terminating Decimal Expansions

A rational number p/q (in lowest terms) has a terminating decimal expansion if and only if q is of the form 2^m × 5^n (where m, n are non-negative integers). Otherwise, the decimal expansion is non-terminating and repeating.

Example: Without actual division, state whether 13/3125 has a terminating decimal expansion.

3125 = 5^5, which is of the form 2^m × 5^n (m = 0, n = 5). So the decimal expansion terminates.

Finding Square Roots of Non-Perfect Squares

To find √7, one method is to use successive averages (the Babylonian/Heron method):

1. Start with two numbers, say 2 and 3, as they are close to √7.

2. Calculate the average: (2 + 3) / 2 = 2.5.

3. Divide 7 by 2.5 to get 2.8, then average 2.5 and 2.8 to get 2.65.

4. Repeat until the estimate stabilizes (√7 ≈ 2.6458).

Common Mistakes to Avoid

• Confusing "not a fraction" with "irrational" — all integers are rational, even though they aren't written as fractions initially.

• Forgetting that 0 is a rational number (0/1).

• Assuming a decimal that "looks like it doesn't repeat" within a few digits is automatically irrational — check for a repeating pattern further out.

Practice Questions

1. Use Euclid's algorithm to find the HCF of 867 and 255.

2. Prove that √5 is irrational.

3. Find the LCM and HCF of 26 and 91, and verify the product relationship.

4. Without actual division, determine whether 17/8 has a terminating decimal expansion.

5. POLYNOMIALS

Definition

A polynomial is an expression of the form:

P(x) = a_n·x^n + a_(n-1)·x^(n-1) + ... + a_1·x + a_0

where a_i are constants, x is a variable, and n is a non-negative integer.

Types of Polynomials by Number of Terms

1. Monomial: A polynomial with one term, e.g., 5x^3.

2. Binomial: A polynomial with two terms, e.g., 2x + 3.

3. Trinomial: A polynomial with three terms, e.g., x^2 + 2x + 1.

Types of Polynomials by Degree

1. Linear polynomial: degree 1, e.g., 3x + 2 (at most 1 zero)

2. Quadratic polynomial: degree 2, e.g., x^2 - 5x + 6 (at most 2 zeros)

3. Cubic polynomial: degree 3, e.g., x^3 - 6x^2 + 11x - 6 (at most 3 zeros)

Degree of a Polynomial

The degree of a polynomial is defined as the highest power of the variable in the expression. For example, the polynomial 2x^3 - 3x^2 + 4x - 5 has a degree of 3.

Geometrical Meaning of Zeros of a Polynomial

The zeros of a polynomial P(x) are the x-values where the graph y = P(x) intersects the x-axis.

• A linear polynomial's graph is a straight line and has exactly one zero.

• A quadratic polynomial's graph is a parabola and can have 0, 1, or 2 zeros (depending on whether it touches or crosses the x-axis).

• A cubic polynomial's graph can have up to 3 zeros.

Relationship Between Zeros and Coefficients

For a Quadratic Polynomial ax^2 + bx + c:

If α and β are the zeros:

Sum of zeros: α + β = -b/a

Product of zeros: α·β = c/a

For a Cubic Polynomial ax^3 + bx^2 + cx + d:

If α, β, γ are the zeros:

α + β + γ = -b/a

αβ + βγ + γα = c/a

αβγ = -d/a

Example: Find a quadratic polynomial whose zeros are 3 and -2.

Sum of zeros = 3 + (-2) = 1

Product of zeros = 3 × (-2) = -6

Polynomial: x^2 - (sum)x + (product) = x^2 - x - 6

Division Algorithm for Polynomials

If P(x) and g(x) are polynomials with g(x) ≠ 0, then there exist polynomials q(x) and r(x) such that:

P(x) = g(x)·q(x) + r(x), where r(x) = 0 or degree of r(x) < degree of g(x)

Example: Divide P(x) = x^3 - 3x^2 + 5x - 3 by g(x) = x^2 - 2

Step 1: x^3 ÷ x^2 = x, multiply g(x) by x: x^3 - 2x, subtract: -3x^2 + 7x - 3

Step 2: -3x^2 ÷ x^2 = -3, multiply g(x) by -3: -3x^2 + 6, subtract: 7x - 9

Quotient q(x) = x - 3, Remainder r(x) = 7x - 9

Remainder Theorem

The Remainder Theorem states that if a polynomial P(x) is divided by (x - a), the remainder is P(a).

Example

Let P(x) = 2x^3 - 3x^2 + 4x - 5. To find the remainder when divided by (x - 2):

P(2) = 2(2)^3 - 3(2)^2 + 4(2) - 5 = 16 - 12 + 8 - 5 = 7

Thus, the remainder is 7.

Factor Theorem

(x - a) is a factor of polynomial P(x) if and only if P(a) = 0.

Example: Check whether (x - 1) is a factor of P(x) = x^3 - 3x^2 + 3x - 1.

P(1) = 1 - 3 + 3 - 1 = 0

Since P(1) = 0, (x - 1) is a factor.

Factorizing Quadratic Polynomials (Splitting the Middle Term)

Example: Factorize 6x^2 + 5x - 6

Find two numbers whose product = 6 × (-6) = -36 and sum = 5 → these are 9 and -4

6x^2 + 9x - 4x - 6 = 3x(2x + 3) - 2(2x + 3) = (3x - 2)(2x + 3)

Common Mistakes to Avoid

• Forgetting the sign convention: sum of zeros is -b/a, not b/a.

• Mixing up quotient and remainder degree rules during long division.

Practice Questions

1. Find the zeros of the polynomial 4x^2 - 4x + 1 and verify the relationship between zeros and coefficients.

2. Divide 3x^3 + x^2 + 2x + 5 by x^2 + 2x + 1 and find the quotient and remainder.

3. Find a cubic polynomial with the sum of zeros 2, sum of products of zeros taken two at a time -7, and product of zeros -14.

4. COORDINATE GEOMETRY

Definition

Coordinate Geometry (or analytic geometry) is the study of geometric figures using a coordinate system. The most common system is the Cartesian coordinate system, where points are defined using ordered pairs (x, y).

Distance Formula

The distance formula calculates the distance d between two points A(x1, y1) and B(x2, y2):

d = √[(x2 - x1)^2 + (y2 - y1)^2]

Example

To find the distance between points A(1, 2) and B(4, 6):

d = √[(4-1)^2 + (6-2)^2] = √[3^2 + 4^2] = √[9 + 16] = √25 = 5

Distance of a Point from the Origin

For point P(x, y), distance from origin O(0,0):

d = √(x^2 + y^2)

Section Formula

The coordinates of a point P(x, y) that divides the line segment joining A(x1, y1) and B(x2, y2) internally in the ratio m1:m2 are:

x = (m1·x2 + m2·x1) / (m1 + m2)

y = (m1·y2 + m2·y1) / (m1 + m2)

Example: Find the point which divides the line segment joining (4, -3) and (8, 5) in the ratio 3:1.

x = (3×8 + 1×4) / (3+1) = (24+4)/4 = 7

y = (3×5 + 1×(-3)) / (3+1) = (15-3)/4 = 3

Point = (7, 3)

Midpoint Formula (special case of section formula where ratio = 1:1)

Midpoint = ((x1+x2)/2, (y1+y2)/2)

Example: Find the midpoint of A(2, 3) and B(6, 7).

Midpoint = ((2+6)/2, (3+7)/2) = (4, 5)

Area of a Triangle Using Coordinates

For vertices A(x1,y1), B(x2,y2), C(x3,y3):

Area = (1/2)|x1(y2-y3) + x2(y3-y1) + x3(y1-y2)|

Example: Find the area of the triangle with vertices (1,1), (4,1), and (4,5).

Area = (1/2)|1(1-5) + 4(5-1) + 4(1-1)|

 = (1/2)|-4 + 16 + 0| = (1/2)(12) = 6 square units

Condition for Collinearity of Three Points

Three points are collinear if the area of the triangle formed by them is 0.

Common Mistakes to Avoid

• Mixing up the order of subtraction in the distance formula (order does not matter since terms are squared, but consistency avoids sign errors).

• Forgetting the negative sign or absolute value in the area formula.

Practice Questions

1. Find the distance between P(-5, 7) and Q(-1, 3).

2. Determine if points (1,5), (2,3), and (-2,-11) are collinear.

3. Find the ratio in which the point (-4, 6) divides the line segment joining A(-6,10) and B(3,-8).

4. LINEAR EQUATIONS IN TWO VARIABLES

Definition

A linear equation in two variables is an equation of the form:

ax + by + c = 0

where a, b, and c are constants (a and b not both zero).

Graphing Linear Equations

Every linear equation can be graphically represented as a straight line. The solutions to the equation correspond to the points that lie on this line. A pair of linear equations in two variables can be represented and solved using both graphical and algebraic methods.

Graphical Method — Nature of Solutions

For two equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0:

• If a1/a2 ≠ b1/b2 → lines intersect at exactly one point → unique solution (consistent)

• If a1/a2 = b1/b2 = c1/c2 → lines coincide → infinitely many solutions (consistent, dependent)

• If a1/a2 = b1/b2 ≠ c1/c2 → lines are parallel → no solution (inconsistent)

Example: Check the nature of solutions for 2x + 3y = 6 and 4x + 6y = 18

a1/a2 = 2/4 = 1/2, b1/b2 = 3/6 = 1/2, c1/c2 = 6/18 = 1/3

Since a1/a2 = b1/b2 ≠ c1/c2, the lines are parallel → no solution.

Algebraic Methods of Solving

Substitution Method

Solve the equations:

1. 2x + 3y = 6

2. x - y = 1

From the second equation, express x:

x = y + 1

Substituting into the first equation:

2(y + 1) + 3y = 6

2y + 2 + 3y = 6

5y + 2 = 6

5y = 4

y = 4/5

Now substitute y back into x = y + 1:

x = 4/5 + 1 = 9/5

Thus, the solution is (9/5, 4/5).

Elimination Method

Solve: 3x + 2y = 11 and 2x + 3y = 4

Multiply first equation by 3 and second by 2:

9x + 6y = 33

4x + 6y = 8

Subtract: 5x = 25 → x = 5

Substitute back: 3(5) + 2y = 11 → 2y = -4 → y = -2

Solution: (5, -2)

Cross-Multiplication Method

For a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0:

x / (b1c2 - b2c1) = y / (c1a2 - c2a1) = 1 / (a1b2 - a2b1)

Word Problem Example

The sum of two numbers is 8, and the sum of their reciprocals is 8/15. Find the numbers.

Let numbers be x and y.

x + y = 8

1/x + 1/y = 8/15 → (x+y)/xy = 8/15 → 8/xy = 8/15 → xy = 15

x and y are roots of t^2 - 8t + 15 = 0 → (t-3)(t-5) = 0 → t = 3 or 5

Numbers are 3 and 5.

Common Mistakes to Avoid

• Sign errors when moving terms across the equals sign.

• Forgetting to check the consistency condition before assuming a unique solution exists.

Practice Questions

1. Solve using elimination: 4x + 3y = 24 and 3x + 4y = 25.

2. For what value of k does the system kx + 3y = k-3 and 12x + ky = k have no solution?

3. Solve graphically: x + y = 5 and 2x - y = 4.

4. QUADRATIC EQUATIONS

Definition

A quadratic equation is an equation of the form:

ax^2 + bx + c = 0

where a ≠ 0.

Methods of Solving Quadratic Equations

Method 1: Factorization

Example: Solve x^2 - 5x + 6 = 0

Split middle term: x^2 - 2x - 3x + 6 = 0

x(x-2) - 3(x-2) = 0

(x-2)(x-3) = 0

x = 2 or x = 3

Method 2: Completing the Square

Example: Solve x^2 + 6x + 5 = 0

x^2 + 6x = -5

x^2 + 6x + 9 = -5 + 9

(x+3)^2 = 4

x + 3 = ±2

x = -1 or x = -5

Method 3: Quadratic Formula

The solutions of a quadratic equation can be obtained using the quadratic formula:

x = [-b ± √(b^2 - 4ac)] / 2a

Example

Solve 2x^2 - 4x - 6 = 0:

1. Identify a = 2, b = -4, c = -6.

2. Calculate the discriminant D:

   D = b^2 - 4ac = (-4)^2 - 4(2)(-6) = 16 + 48 = 64

3. Apply the quadratic formula:

   x = [-(-4) ± √64] / (2·2) = (4 ± 8) / 4

This results in two solutions:

• x = 3

• x = -1

Nature of Roots (Discriminant Analysis)

The discriminant D = b^2 - 4ac determines the nature of roots:

• D > 0: two distinct real roots

• D = 0: two equal real roots (repeated root)

• D < 0: no real roots (roots are imaginary/complex)

Example: Find the nature of roots of 2x^2 - 3x + 5 = 0

D = (-3)^2 - 4(2)(5) = 9 - 40 = -31

Since D < 0, there are no real roots.

Word Problem Example

A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.

Let speed = x km/h. Time = 360/x hours.

360/x - 360/(x+5) = 1

360(x+5) - 360x = x(x+5)

1800 = x^2 + 5x

x^2 + 5x - 1800 = 0

Using the quadratic formula: x = [-5 ± √(25+7200)]/2 = [-5 ± 85]/2

x = 40 (rejecting the negative root)

Speed of the train is 40 km/h.

Common Mistakes to Avoid

• Forgetting to check that a ≠ 0 before calling an equation quadratic.

• Sign errors when computing the discriminant, especially with negative b or c.

• Not rejecting extraneous/negative roots in real-world word problems (e.g., negative speed, negative length).

Practice Questions

1. Solve by factorization: 3x^2 - 2x - 1 = 0.

2. Find the value of k for which the equation kx^2 + 6x + 1 = 0 has equal roots.

3. The sum of the squares of two consecutive positive integers is 365. Find the integers.

4. ARITHMETIC PROGRESSIONS

Definition

An arithmetic progression (A.P.) is a sequence of numbers in which the difference between consecutive terms is constant. The n-th term can be expressed as:

a_n = a + (n-1)d

where a is the first term and d is the common difference.

Example

Find the 10th term of the A.P. where a = 2 and d = 3:

a_10 = 2 + (10-1) · 3 = 2 + 27 = 29

Sum of First n Terms of an A.P.

S_n = n/2 [2a + (n-1)d]

Alternatively, if the last term (l = a_n) is known:

S_n = n/2 (a + l)

Example: Find the sum of the first 15 terms of the A.P. 3, 7, 11, 15, ...

a = 3, d = 4, n = 15

S_15 = 15/2 [2(3) + (15-1)(4)] = 15/2 [6 + 56] = 15/2 (62) = 465

Finding the Number of Terms

Example: How many terms of the A.P. 24, 21, 18, ... must be taken so that their sum is 78?

a = 24, d = -3

S_n = n/2 [2(24) + (n-1)(-3)] = 78

n/2 [48 - 3n + 3] = 78

n(51 - 3n) = 156

51n - 3n^2 = 156

3n^2 - 51n + 156 = 0

n^2 - 17n + 52 = 0

(n-4)(n-13) = 0

n = 4 or n = 13 (both valid, since terms become negative after a point but sum can repeat)

Applications and Word Problems

Example: A contract on construction job specifies a penalty for delay: Rs 200 for the first day, Rs 250 for the second day, Rs 300 for the third day, and so on, each delay costing Rs 50 more than the preceding day. How much money does the contractor have to pay as penalty if he has delayed the work by 30 days?

This is an A.P. with a = 200, d = 50, n = 30

S_30 = 30/2 [2(200) + 29(50)] = 15[400 + 1450] = 15(1850) = 27,750

The contractor has to pay Rs 27,750 as penalty.

Common Mistakes to Avoid

• Confusing a_n (the n-th term) with S_n (the sum of n terms) — a very common board-exam error.

• Sign errors when d is negative.

Practice Questions

1. Which term of the A.P. 3, 8, 13, 18, ... is 78?

2. Find the sum of first 40 positive integers divisible by 6.

3. If the 3rd and 9th terms of an A.P. are 4 and -8 respectively, find which term is zero.

4. TRIANGLES

Definition

A triangle is a polygon with three edges and three vertices. The fundamental properties include congruence criteria:

1. SSS (Side-Side-Side): If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.

2. SAS (Side-Angle-Side): If two sides and the included angle of one triangle are equal to the corresponding parts of another triangle, they are congruent.

3. ASA (Angle-Side-Angle): If two angles and the included side of one triangle are equal to the corresponding parts of another triangle, they are congruent.

Example

If triangle ABC has sides AB = 5, AC = 7, and BC = 5, it can be demonstrated that triangle ABC is isosceles because two sides are equal.

Similarity of Triangles

Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional (in the same ratio).

Criteria for Similarity

1. AA (Angle-Angle): If two angles of one triangle are equal to two angles of another, the triangles are similar.

2. SSS (Side-Side-Side) Similarity: If corresponding sides of two triangles are proportional, they are similar.

3. SAS (Side-Angle-Side) Similarity: If one angle of a triangle equals one angle of another and the sides including these angles are proportional, the triangles are similar.

Basic Proportionality Theorem (Thales' Theorem)

If a line is drawn parallel to one side of a triangle intersecting the other two sides at distinct points, the other two sides are divided in the same ratio.

If DE ∥ BC in triangle ABC, with D on AB and E on AC, then:

AD/DB = AE/EC

Example: In triangle ABC, DE ∥ BC. If AD = 4, DB = 6, and AE = 5, find EC.

AD/DB = AE/EC

4/6 = 5/EC

EC = (5×6)/4 = 7.5

Pythagoras Theorem

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

(Hypotenuse)^2 = (Base)^2 + (Perpendicular)^2

Example: In a right triangle, if the two legs are 6 cm and 8 cm, find the hypotenuse.

Hypotenuse^2 = 6^2 + 8^2 = 36 + 64 = 100

Hypotenuse = 10 cm

Converse of Pythagoras Theorem

If the square of one side of a triangle equals the sum of squares of the other two sides, then the angle opposite the first side is a right angle.

Areas of Similar Triangles

The ratio of areas of two similar triangles equals the square of the ratio of their corresponding sides.

Area(△1)/Area(△2) = (side1/side2)^2

Example: Two similar triangles have corresponding sides in the ratio 4:9. Find the ratio of their areas.

Ratio of areas = 4^2 : 9^2 = 16:81

Common Mistakes to Avoid

• Assuming triangles are similar just because they "look" similar without checking a valid similarity criterion.

• Mixing up the ratio of sides with the ratio of areas (remember to square the side ratio for area).

Practice Questions

1. In triangle ABC, D and E are points on AB and AC such that DE ∥ BC. If AD/DB = 3/5 and AC = 5.6 cm, find AE.

2. Prove that in a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides (Pythagoras Theorem).

3. The areas of two similar triangles are 81 cm² and 49 cm². If a side of the larger triangle is 18 cm, find the corresponding side of the smaller triangle.

4. TRIGONOMETRY

Definition

Trigonometry explores the relationships between the angles and sides of triangles. The primary trigonometric ratios are:

• Sine (sin)

• Cosine (cos)

• Tangent (tan)

• Cosecant (cosec), Secant (sec), Cotangent (cot) — the reciprocal ratios

Trigonometric Ratios

sin θ = Opposite / Hypotenuse

cos θ = Adjacent / Hypotenuse

tan θ = Opposite / Adjacent

cosec θ = 1/sin θ = Hypotenuse/Opposite

sec θ = 1/cos θ = Hypotenuse/Adjacent

cot θ = 1/tan θ = Adjacent/Opposite

Example

In a right triangle where the opposite side is 3 units and the hypotenuse is 5 units, we can determine sin θ:

sin θ = 3/5

Standard Angle Values Table

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 undefined

Trigonometric Identities

1. sin^2 θ + cos^2 θ = 1

2. 1 + tan^2 θ = sec^2 θ

3. 1 + cot^2 θ = cosec^2 θ

Example: Prove that (1 - cos θ)(1 + cos θ) = sin^2 θ

LHS = 1 - cos^2 θ = sin^2 θ (using identity 1) = RHS ✓

Example: If sin θ = 3/5, find cos θ and tan θ.

Using sin^2 θ + cos^2 θ = 1:

cos^2 θ = 1 - 9/25 = 16/25

cos θ = 4/5

tan θ = sin θ / cos θ = (3/5)/(4/5) = 3/4

Heights and Distances (Applications)

Angle of Elevation: the angle formed by the line of sight with the horizontal when looking upward at an object.

Angle of Depression: the angle formed by the line of sight with the horizontal when looking downward at an object.

Example: The angle of elevation of the top of a tower from a point on the ground, 30 m away from the foot of the tower, is 30°. Find the height of the tower.

tan 30° = height/30

1/√3 = height/30

height = 30/√3 = 10√3 ≈ 17.32 m

Common Mistakes to Avoid

• Using degree values for tan 90° (it is undefined, not 0).

• Forgetting to rationalize the denominator in final answers (e.g., 30/√3 should become 10√3).

• Confusing angle of elevation and depression in word problems — draw a diagram first.

Practice Questions

1. Prove that (sec θ - tan θ)(sec θ + tan θ) = 1.

2. From the top of a 60 m high building, the angle of depression of the top and bottom of a tower are observed to be 30° and 60° respectively. Find the height of the tower.

3. If cot θ = 7/8, evaluate (1 + sin θ)(1 - sin θ) / (1 + cos θ)(1 - cos θ).

4. STATISTICS

Definition

Statistics involves the collection, analysis, interpretation, presentation, and organization of data. Key measures include:

• Mean: The average of a set of numbers.

• Median: The middle value when data is arranged in order.

• Mode: The most frequently occurring value.

Example

Given the data set: {2, 3, 5, 3, 7}:

• Mean: (2 + 3 + 5 + 3 + 7) / 5 = 4

• Median: Arranged data: {2, 3, 3, 5, 7} → Median = 3

• Mode: 3 (appears most frequently)

Mean of Grouped Data

Direct Method

Mean (x̄) = Σ(fi·xi) / Σfi

where fi is the frequency and xi is the class mark (midpoint) of each class interval.

Assumed Mean Method

x̄ = a + Σ(fi·di) / Σfi, where di = xi - a (a is the assumed mean)

Step-Deviation Method

x̄ = a + h × [Σ(fi·ui) / Σfi], where ui = (xi - a)/h and h is the class width

Example: Find the mean of the following data using the direct method.

Class Interval: 0-10, 10-20, 20-30, 30-40, 40-50

Frequency: 7, 10, 15, 8, 10

Class marks (xi): 5, 15, 25, 35, 45

Σfi = 50

Σ(fi·xi) = 7(5) + 10(15) + 15(25) + 8(35) + 10(45) = 35+150+375+280+450 = 1290

Mean = 1290/50 = 25.8

Median of Grouped Data

Median = l + [(n/2 - cf) / f] × h

where l = lower boundary of median class, n = total frequency, cf = cumulative frequency before median class, f = frequency of median class, h = class width

Mode of Grouped Data

Mode = l + [(f1 - f0) / (2f1 - f0 - f2)] × h

where l = lower boundary of modal class, f1 = frequency of modal class, f0 = frequency of class before modal class, f2 = frequency of class after modal class, h = class width

Empirical Relationship

Mode = 3(Median) - 2(Mean)

This relationship is useful when only two of the three measures are known and a quick estimate of the third is required.

Cumulative Frequency and Ogives

A cumulative frequency curve (ogive) is used to graphically determine the median of grouped data. There are two types: "less than" ogive and "more than" ogive; their intersection point gives the median.

Common Mistakes to Avoid

• Using the wrong class boundaries (inclusive vs exclusive series) when computing the median/mode class.

• Forgetting that the modal class is the one with the highest frequency, not the class containing the mode value from the empirical formula.

Practice Questions

1. Find the mean of the given grouped data using the step-deviation method (construct your own frequency table with 5 classes).

2. The median of the following data is 28.5. Find the missing frequencies.

3. Using the empirical formula, find the mode if mean = 24 and median = 26.

4. PROBABILITY

Definition

Probability quantifies the likelihood that an event will occur. It is defined as:

P(E) = Number of favorable outcomes / Total number of outcomes

Example

If a die is rolled, what is the probability of rolling an even number? The even numbers are 2, 4, and 6, yielding 3 favorable outcomes out of 6 total outcomes:

P(Even) = 3/6 = 1/2

Key Terms

• Experiment: An action with a well-defined set of outcomes (e.g., tossing a coin).

• Sample Space: The set of all possible outcomes.

• Event: A subset of the sample space.

• Equally Likely Outcomes: Outcomes that have the same chance of occurring.

Complementary Events

P(E) + P(not E) = 1, so P(not E) = 1 - P(E)

Example: The probability of it raining tomorrow is 0.3. What is the probability that it will not rain?

P(not rain) = 1 - 0.3 = 0.7

Range of Probability

0 ≤ P(E) ≤ 1

• P(E) = 0 means the event is impossible.

• P(E) = 1 means the event is certain.

Example: A bag contains 5 red, 8 blue, and 7 green balls. A ball is drawn at random. Find the probability that it is:

(a) red: P(red) = 5/20 = 1/4

(b) not blue: P(not blue) = 1 - 8/20 = 12/20 = 3/5

(c) either red or green: P(red or green) = (5+7)/20 = 12/20 = 3/5

Cards Problems (Standard Deck of 52 Cards)

A standard deck has 52 cards: 4 suits (hearts, diamonds, clubs, spades) of 13 cards each; hearts and diamonds are red, clubs and spades are black; each suit has cards numbered 2-10, plus Jack, Queen, King, and Ace.

Example: One card is drawn from a well-shuffled deck. Find the probability of getting a king of red color.

There are 2 red kings (king of hearts, king of diamonds) out of 52 cards.

P(red king) = 2/52 = 1/26

Common Mistakes to Avoid

• Forgetting to reduce the fraction to simplest form.

• Miscounting the total number of outcomes in card or dice problems (always double check: dice = 6, coin = 2, deck of cards = 52).

Practice Questions

1. Two dice are thrown simultaneously. Find the probability of getting a sum of 8.

2. A box contains 90 discs numbered 1 to 90. Find the probability that a disc drawn bears a two-digit number.

3. A game consists of tossing a coin 3 times and noting the outcome each time. Find the probability of getting exactly two heads.

4. CONCLUSION

In conclusion, mastering the concepts discussed in this module is essential for achieving excellence in Class 10 Mathematics. Continuous practice, a thorough understanding of formulas, and the application of these concepts in problem-solving will prepare you for your board examinations. Remember, the key to success lies in consistent study and practice. Best of luck with your preparations for the 2026 Board Exams!

12. QUICK FORMULA SHEET

Number Systems

• Euclid's Division Lemma: a = bq + r, 0 ≤ r < b

• HCF × LCM = product of the two numbers

Polynomials

• Sum of zeros (quadratic) = -b/a; Product of zeros = c/a

• Cubic: sum = -b/a, sum of products two at a time = c/a, product = -d/a

Coordinate Geometry

• Distance: d = √[(x2-x1)^2 + (y2-y1)^2]

• Section formula: x = (m1x2+m2x1)/(m1+m2), y = (m1y2+m2y1)/(m1+m2)

• Area of triangle = (1/2)|x1(y2-y3)+x2(y3-y1)+x3(y1-y2)|

Quadratic Equations

• x = [-b ± √(b²-4ac)] / 2a

• Discriminant D = b² - 4ac

Arithmetic Progressions

• a_n = a + (n-1)d

• S_n = n/2[2a + (n-1)d] = n/2(a + l)

Trigonometry

• sin²θ + cos²θ = 1

• 1 + tan²θ = sec²θ

• 1 + cot²θ = cosec²θ

Statistics

• Mean (direct) = Σfixi / Σfi

• Mode = 3(Median) - 2(Mean)

• Mode (grouped) = l + [(f1-f0)/(2f1-f0-f2)] × h

• Median (grouped) = l + [(n/2 - cf)/f] × h

Probability

• P(E) = favorable outcomes / total outcomes

• P(E) + P(not E) = 1