Important Questions for Class 9 Maths Chapter Wise
CBSE Class 9 Maths | NCERT Important Questions with Key Concepts and Formulas | MyClass24
Important Questions for Class 9 Maths lays the groundwork for the concepts students will meet in Class 10 board exams and beyond, which is why practising important questions for the Class 9 Maths chapter-wise is one of the most effective ways to prepare. This guide, prepared by the academic team at MyClass24, walks through every chapter of the CBSE Class 9 Maths (NCERT) syllabus with a quick concept refresher, a mobile-friendly formula table, and a curated list of important questions that students commonly get asked in school tests, unit exams, and the final assessment. Whether you are revising Important questions for Number Systems, brushing up on circle theorems, or preparing for a surprise test on Statistics, use this page as a single chapter-wise revision companion.
How to use this page: read the short concept summary for each chapter, memorise the formulas in the table below it, and then attempt the important questions on your own before checking the textbook solution. This active-recall approach, chapter by chapter, is far more effective than passive reading and will help you retain formulas for a longer time.
Find the chapter-wise Important Questions for Class 9 Maths
Chapter-wise Important Questions for Class 9 Maths: Introduction
Chapter 1: Number Systems
This chapter builds the foundation for algebra by introducing rational and irrational numbers, their decimal expansions, and operations on real numbers. A strong grip here makes later chapters like polynomials and coordinate geometry much easier.
Key Concepts & Formulas
Concept | Rule / Formula |
Rational number | Can be written as p/q, where q ≠ 0 |
Irrational number | Cannot be written as p/q; non-terminating, non-repeating decimal |
Law of exponents | aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ |
Rationalising factor of √a | √a, since √a × √a = a |
Identity | (a + b)(a − b) = a² − b² |
Important Questions
Represent √2 and √3 on the number line using the spiral method.
Insert three rational numbers between 3/5 and 4/5.
Simplify expressions involving surds by rationalising the denominator.
Express 0.363636… (repeating) in the form p/q.
Find the value of x if 2ˣ⁻¹ = 32, using laws of exponents.
Prove that the sum or product of a rational and an irrational number is irrational.
Chapter 2: Polynomials
Polynomials introduce students to expressions with one or more terms and their degrees. Questions from this chapter frequently test factorisation, the Remainder Theorem, and algebraic identities, all of which carry good weightage in exams.
Key Concepts & Formulas
Identity / Term | Formula |
Zero of a polynomial | Value of x for which p(x) = 0 |
Remainder Theorem | Remainder of p(x) ÷ (x − a) is p(a) |
Factor Theorem | (x − a) is a factor of p(x) if p(a) = 0 |
(a + b)² | a² + 2ab + b² |
(a − b)³ | a³ − 3a²b + 3ab² − b³ |
a³ + b³ | (a + b)(a² − ab + b²) |
Important Questions
Find the zero of the polynomial p(x) = 2x − 5.
Use the Factor Theorem to check whether (x − 1) is a factor of a given cubic polynomial.
Factorise x³ − 23x² + 142x − 120 using the Factor Theorem.
Expand (2x + 3y − 4z)² using the appropriate identity.
Evaluate 105 × 106 using suitable identities without direct multiplication.
If x + y + z = 0, prove that x³ + y³ + z³ = 3xyz.
Chapter 3: Coordinate Geometry
This short but scoring chapter deals with plotting points on the Cartesian plane. Students should be comfortable identifying quadrants, signs of coordinates, and reading ordered pairs correctly, since diagram-based questions are common.
Key Concepts & Formulas
Term | Details |
Cartesian plane | Formed by x-axis and y-axis intersecting at origin O(0, 0) |
Quadrant I | (+, +) |
Quadrant II | (−, +) |
Quadrant III | (−, −) |
Quadrant IV | (+, −) |
Abscissa / Ordinate | x-coordinate is abscissa; y-coordinate is ordinate |
Important Questions
Plot the points A(2, 3), B(−2, 3), C(−2, −3) and D(2, −3) and name the figure formed.
Write the coordinates of a point that lies on the y-axis at a distance of 4 units above the origin.
In which quadrant does the point (−5, 7) lie? Justify your answer.
Find the mirror image of the point (3, −4) about the x-axis and y-axis.
State whether a point of the form (a, 0) can lie inside any quadrant.
Chapter 4: Linear Equations in Two Variables
This chapter connects algebra with geometry, since every linear equation in two variables represents a straight line. Expect questions on writing equations, finding solutions, and graphing lines correctly on graph paper.
Key Concepts & Formulas
Concept | Formula / Note |
General form | ax + by + c = 0, where a, b are not both zero |
Solution of equation | Infinite solutions exist for one linear equation in two variables |
Equation of x-axis | y = 0 |
Equation of y-axis | x = 0 |
Line parallel to x-axis | y = k (a constant) |
Line parallel to y-axis | x = k (a constant) |
Important Questions
Write 2x + 3y = 12 in the form y = mx + c and find two solutions.
Draw the graph of the linear equation x − 2y = 4 and read off the intercepts.
If (2, k) is a solution of the equation 3x + y = 8, find the value of k.
Express the statement 'the cost of a notebook is twice the cost of a pen' as a linear equation in two variables.
Find the point where the graph of 3x + 4y = 12 cuts the x-axis and the y-axis.
Chapter 5: Introduction to Euclid's Geometry
This chapter introduces the axiomatic approach to geometry through Euclid's definitions, axioms, and postulates. Questions usually test conceptual understanding rather than calculation, so reasoning-based answers are important.
Key Concepts & Formulas
Term | Meaning |
Axiom | A statement accepted as true without proof, common to all branches of mathematics |
Postulate | An assumption specific to geometry |
Euclid's 1st postulate | A straight line may be drawn from any one point to any other point |
Euclid's 5th postulate | Basis of the concept of parallel lines |
Important Questions
State any two of Euclid's axioms with a suitable example.
Explain the difference between an axiom and a postulate with examples.
If A, B, C are three points on a line such that AB = BC, then show that B is the mid-point of AC using Euclid's axioms.
Why is Euclid's fifth postulate considered different from the other four?
Explain how 'things which coincide with one another are equal to one another' applies to two congruent triangles.
Chapter 6: Lines and Angles
Lines and angles is a theorem-heavy, high-scoring chapter. Students should memorise all the standard results and practise proving them step by step, since long-answer proofs are frequently asked in exams.
Key Concepts & Formulas
Result | Statement |
Linear pair axiom | Sum of adjacent angles on a straight line is 180° |
Vertically opposite angles | Always equal |
Corresponding angles | Equal, when a transversal cuts two parallel lines |
Alternate interior angles | Equal, for parallel lines |
Co-interior angles | Supplementary (sum = 180°), for parallel lines |
Angle sum property of a triangle | Sum of all interior angles = 180° |
Exterior angle theorem | Exterior angle = sum of the two interior opposite angles |
Important Questions
Prove that the sum of the three angles of a triangle is 180°.
If two parallel lines are cut by a transversal, prove that the bisectors of the interior angles on the same side form a right angle.
In a figure, if ray OC stands on line AB such that ∠AOC = ∠BOC, prove that ∠AOC = 90°.
Find the value of x if two angles of a linear pair are (3x − 5)° and (2x + 15)°.
Prove that the exterior angle of a triangle is equal to the sum of the two interior opposite angles.
Chapter 7: Triangles
Congruence criteria and inequalities in triangles form the core of this chapter. Since proofs from this chapter are almost always asked in board-pattern exams, practising each congruence rule with different figures is essential.
Key Concepts & Formulas
Congruence Criterion | Full Form |
SAS | Side–Angle–Side |
ASA | Angle–Side–Angle |
AAS | Angle–Angle–Side |
SSS | Side–Side–Side |
RHS | Right angle–Hypotenuse–Side (only for right triangles) |
Angle opposite longer side | Is always the larger angle |
Triangle inequality | Sum of any two sides > the third side |
Important Questions
Prove that angles opposite to equal sides of a triangle are equal (Isosceles Triangle Theorem).
Two sides AB and BC and median AM of triangle ABC are respectively equal to sides PQ, QR and median PN of triangle PQR; prove that triangle ABC is congruent to triangle PQR.
In a triangle, prove that the side opposite to the larger angle is longer.
Show that the angles of an equilateral triangle are 60° each.
AD is an altitude of an isosceles triangle ABC with AB = AC. Prove that AD bisects BC as well as angle A.
Chapter 8: Quadrilaterals
This chapter extends triangle congruence to properties of parallelograms, rectangles, rhombuses, and the mid-point theorem. Diagram-based proofs are common, so labelling figures accurately before solving is a good habit to build.
Key Concepts & Formulas
Property | Statement |
Angle sum of quadrilateral | Sum of all interior angles = 360° |
Parallelogram | Opposite sides equal and parallel; diagonals bisect each other |
Rectangle | Parallelogram with all angles 90° and equal diagonals |
Rhombus | Parallelogram with all sides equal and diagonals bisecting at 90° |
Square | Rectangle and rhombus combined |
Mid-point theorem | Line joining mid-points of two sides is parallel to the third side and half its length |
Important Questions
Prove that the diagonals of a parallelogram bisect each other.
Prove that a quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel.
ABCD is a rhombus; show that the diagonal AC bisects angle A as well as angle C.
Using the mid-point theorem, prove that the quadrilateral formed by joining the mid-points of the sides of a quadrilateral is a parallelogram.
In a parallelogram, prove that the bisectors of any two consecutive angles intersect at a right angle.
Chapter 9: Circles
Circles is a concept-heavy chapter that tests properties of chords, arcs, and cyclic quadrilaterals. Understanding the reasoning behind each theorem, rather than rote memorisation, helps in solving unfamiliar figure-based questions.
Key Concepts & Formulas
Theorem | Statement |
Equal chords | Subtend equal angles at the centre |
Perpendicular from centre | Bisects the chord |
Equal chords and distance | Equal chords are equidistant from the centre |
Angle in a semicircle | Is always a right angle |
Angles in the same segment | Are equal |
Cyclic quadrilateral | Sum of opposite angles = 180° |
Important Questions
Prove that equal chords of a circle subtend equal angles at the centre.
Prove that the perpendicular from the centre of a circle to a chord bisects the chord.
Prove that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
Prove that the sum of either pair of opposite angles of a cyclic quadrilateral is 180°.
AB and CD are two chords of a circle such that AB = CD; prove that arc AB = arc CD.
Chapter 10: Heron's Formula
This compact chapter allows students to find the area of a triangle using only its side lengths, which is especially useful for scalene and irregular triangles. It is a scoring, application-based chapter that also connects with mensuration in real-life problems.
Key Concepts & Formulas
Term | Formula |
Semi-perimeter (s) | s = (a + b + c) / 2 |
Heron's formula | Area = √[s(s − a)(s − b)(s − c)] |
Area of equilateral triangle | (√3 / 4) × side² |
Area of isosceles triangle (Heron's) | Use a = b, apply Heron's formula directly |
Important Questions
Find the area of a triangle whose sides are 18 cm, 24 cm, and 30 cm using Heron's formula.
A triangular park has sides 40 m, 24 m, and 32 m; find the cost of levelling it at a given rate per square metre.
Find the area of an isosceles triangle whose base is 12 cm and equal sides are 10 cm each.
A quadrilateral field's area is to be found by dividing it into two triangles using a diagonal; apply Heron's formula to each part.
The sides of a triangle are in the ratio 12:17:25 and its perimeter is 540 cm; find its area.
Chapter 11: Surface Areas and Volumes
This chapter is entirely formula-based and covers solids such as cubes, cuboids, cylinders, cones, and spheres. Since problems often combine two or more solids, memorising every formula in a table and practising unit conversions is highly recommended.
Key Concepts & Formulas
Solid | Curved/Lateral Surface Area | Total Surface Area | Volume |
Cube (side a) | 4a² | 6a² | a³ |
Cuboid (l, b, h) | 2h(l + b) | 2(lb + bh + hl) | l × b × h |
Cylinder (r, h) | 2πrh | 2πr(r + h) | πr²h |
Cone (r, h, slant l) | πrl | πr(l + r) | (1/3)πr²h |
Sphere (r) | 4πr² | 4πr² | (4/3)πr³ |
Hemisphere (r) | 2πr² | 3πr² | (2/3)πr³ |
Important Questions
Find the volume and total surface area of a cylinder with radius 7 cm and height 10 cm.
A cone and a hemisphere have equal bases and equal heights; find the ratio of their volumes.
The radius of a sphere is doubled; find the ratio of the volumes of the two spheres.
How many litres of water can a hemispherical tank of radius 1.75 m hold?
A cylindrical pillar has diameter 50 cm and height 3.5 m; find the cost of painting its curved surface at a given rate.
Chapter 12: Statistics
Statistics deals with the collection, organisation, and graphical representation of data using bar graphs, histograms, and frequency polygons, along with finding the mean, median, and mode. This chapter is highly scoring since most questions are direct applications of formulas.
Key Concepts & Formulas
Measure | Formula |
Mean (ungrouped) | Sum of observations ÷ Number of observations |
Median (n odd) | Value of the [(n + 1)/2]th observation |
Median (n even) | Average of (n/2)th and [(n/2) + 1]th observations |
Mode | Observation occurring most frequently |
Class mark | (Upper limit + Lower limit) / 2 |
Important Questions
Find the mean, median, and mode of a given set of ungrouped data.
Draw a histogram for a grouped frequency distribution with unequal class widths.
Construct a frequency polygon for a given data set without drawing a histogram.
The mean of 20 observations was found to be 15; later it was found that an observation was misread. Find the correct mean.
Represent the given data using a bar graph and interpret the results in one line.
Quick Revision Tips for Class 9 Maths
Make a one-page formula sheet for each chapter and revise it a day before every test.
Attempt NCERT in-text and exercise questions first before moving to reference books.
For geometry chapters (Lines and Angles, Triangles, Quadrilaterals, Circles), always draw a neat, labelled figure before starting the proof.
Practise at least five previous years' important questions from every chapter to understand the exam pattern.
Time yourself while solving Statistics and Surface Areas and Volumes questions, since these are calculation-heavy and speed matters.
Why Practise Chapter-Wise Important Questions?
Solving important questions for Class 9 Maths chapter wise helps students identify high-weightage topics, spot recurring question patterns, and build the problem-solving speed needed for board-pattern exams in later years. Instead of revising randomly, a chapter-wise approach ensures that no topic — from Number Systems to Statistics — is left untouched. Keep this page bookmarked and revisit it before every unit test, and pair it with regular NCERT textbook practice for the best results.
MyClass24 will keep updating this Class 9 Maths important questions page with more practice problems, sample papers, and chapter notes, so students preparing for CBSE and state board exams always have a reliable, exam-focused resource in one place.