Class 9 Mathematics — Original Notes
Ganita Manjari · Part 1 · Independent explanations and practice. Check the chapter title and exercise coverage against your school's current textbook edition.
How to study each chapter
- Read the important questions first and try answering from memory.
- Learn definitions, properties and formulae.
- Study the worked examples step by step.
- Attempt practice without looking at answers.
- Check solutions and record mistakes.
- Finish with a timed mixed test.
Chapter 1 — Orienting Yourself: The Use of Coordinates
Concept notesKey ideas
- A coordinate system describes the position of a point using ordered numbers.
- The Cartesian plane has a horizontal x-axis and a vertical y-axis.
- Their intersection is the origin, written (0, 0).
- A point is written (x, y): move x units horizontally, then y units vertically.
- Positive x is to the right; negative x is to the left. Positive y is upward; negative y is downward.
- The axes divide the plane into four quadrants. Points on an axis are not inside any quadrant.
Remember
In (−3, 2), the first number tells horizontal movement and the second tells vertical movement. Do not reverse the order.
Important questions with answers
1. What is the origin?
The point where the x-axis and y-axis intersect. Its coordinates are (0, 0).
2. What does an ordered pair (a, b) represent?
It represents a point whose horizontal coordinate is a and vertical coordinate is b.
3. In which quadrant does (−4, 5) lie?
Second quadrant: x is negative and y is positive.
4. Can a point on the x-axis belong to a quadrant?
No. Points on either axis are boundary points and do not belong to any quadrant.
Worked examples
Example 1: Locate (3, −2). Start at the origin, move 3 units right, then 2 units down. It lies in the fourth quadrant.
Example 2: Identify the coordinates of a point 4 units left and 1 unit up from the origin. Answer: (−4, 1).
Practice exercises
- Write the coordinates of the origin.
- Name the quadrant containing (−2, −7).
- Which axis contains (0, 6)?
- Describe how to plot (5, 3).
- State whether (−3, 0) lies in a quadrant.
MCQ
- The point (2, −5) lies in: (A) I (B) II (C) III (D) IV
- The y-coordinate of (−6, 4) is: (A) −6 (B) 4 (C) 6 (D) −4
Answer key
1. (0,0); 2. Third quadrant; 3. y-axis; 4. 5 right, 3 up; 5. No, it lies on x-axis. MCQ: 1-D, 2-B.
Long-answer / exam-style
Explain how to plot (−3, −4) on a Cartesian plane.
Begin at (0,0). Move 3 units left because x = −3. From there move 4 units down because y = −4. Mark the point and label it (−3,−4). Since both coordinates are negative, it lies in the third quadrant.
Chapter 2 — Introduction to Linear Polynomials
Key ideas
- A polynomial in one variable is an expression made from constants and non-negative integer powers of the variable, combined using addition, subtraction and multiplication.
- A term has a coefficient and a variable part. In 5x, the coefficient is 5.
- A linear polynomial in x has degree 1 and can be written ax + b, where a is not zero.
- The degree is the greatest exponent of the variable with a non-zero coefficient.
- To evaluate a polynomial, substitute the given value for the variable and simplify carefully.
- A zero of a polynomial is a value of the variable that makes the polynomial equal to zero.
Important questions with answers
1. What is a linear polynomial?
A polynomial of degree 1, such as 3x + 2.
2. What is the degree of 7x − 9?
1, because the highest power of x is 1.
3. Evaluate 2x + 5 when x = 3.
2(3) + 5 = 11.
4. Find the zero of x − 6.
Set x − 6 = 0, so x = 6.
Worked examples
Example 1: Evaluate 4x − 3 at x = −2: 4(−2) − 3 = −8 − 3 = −11.
Example 2: Find the zero of 3x + 12: 3x + 12 = 0 ⇒ 3x = −12 ⇒ x = −4.
Practice
- State the degree of 9x + 1.
- Evaluate 5x − 2 for x = 4.
- Find the zero of x + 8.
- Is 4x² + 1 a linear polynomial? Explain.
Answers: 1; 18; −8; No, its degree is 2.
MCQ
The degree of 6x − 11 is (A) 0 (B) 1 (C) 2 (D) 6. Answer: B.
Long-answer / exam-style
Find the zero of 5x − 20 and verify it.
Set 5x − 20 = 0. Then 5x = 20, so x = 4. Verification: 5(4) − 20 = 20 − 20 = 0. Hence 4 is the zero.
Chapter 3 — The World of Numbers
Key ideas
- Natural numbers are counting numbers: 1, 2, 3, … .
- Whole numbers include 0 and the natural numbers.
- Integers include positive numbers, negative numbers and zero.
- Rational numbers can be written as p/q, where p and q are integers and q is not zero.
- Terminating decimals and repeating decimals are rational numbers.
- Real numbers include rational and irrational numbers.
- An irrational number cannot be expressed as a ratio of two integers; its decimal expansion is non-terminating and non-repeating.
Important questions with answers
1. Is every integer rational?
Yes. Any integer n can be written as n/1.
2. Give an example of an irrational number.
√2 is irrational.
3. Is 0.75 rational?
Yes. 0.75 = 75/100 = 3/4.
4. What are real numbers?
The collection of all rational and irrational numbers.
Worked examples
Example 1: Convert 0.625 to a fraction: 0.625 = 625/1000 = 5/8.
Example 2: Classify −7: it is an integer, rational (−7/1), and real.
Practice
- Write 0.4 as a fraction in simplest form.
- Is √9 irrational? Explain.
- Give one integer that is not a natural number.
- Classify 2/5 as rational or irrational.
Answers: 2/5; No, √9 = 3; −1; rational.
Long-answer / exam-style
Explain the difference between rational and irrational numbers with examples.
A rational number can be expressed as p/q with integers p and q and q ≠ 0, for example 3/4 or −2. An irrational number cannot be expressed in this form, for example √2. Rational decimals terminate or repeat; irrational decimals neither terminate nor repeat.
Chapter 4 onward: More chapter-specific notes are being added in batches. Use the official NCERT book to confirm the exact topic sequence and your school's prescribed portions.