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NCERT Solutions for Class 9 Maths
Read Notes and Videos,Practice,Mock Tests for Class 9 Maths
Maths is one of the vital subjects in Maths stream and it is also of very much importance while preparing for various competitive exams and hence it is essential to prepare it from the very beginning. Understanding the concepts and applying them in the provided situation can improvise your learning as well as knowledge ability.
Maths is a very important resource for students of CBSE Class 9.Maths is an interesting subject for the enhancement of analytical and problem solving skills.
Unfortunately, many students find it difficult because of no interest or lack of better knowledge to comprehend. Maths is interesting when comprehended well and practice with easy to hard problems and students find it difficult because most of them do not have their basics clear for the subject. Lack of sufficient practice tool is a major reason.
School Connect Online students are not only solving and understanding questions from Maths from NCERT or Syllabus but also they are solving questions to practice Maths Olympiad this helps students to become expert
To help you with this, we have learning notes and videos, Practice questions for NCERT, CBSE, with Solutions for Class 9 Maths.
NCERT solutions of Class 9 Maths includes solutions of all questions and more to help and support practice with more relevant and important questions. These solutions have been provided and prepared by the most experienced teachers and important resources. A very simple approach has been followed while solving the questions and designing the NCERT Solutions for Class 9 Maths. Students will find it extremely easy to understand the problems and how to go about solving them.
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Unlike other subjects, rote learning cannot help you to excel in Maths,as it requires more Learning approach and critical ability to think and solve. Even for other subjects, rote learning is not recommended as it won’t be of much use after a point of time. Maths, from the very beginning, needs one to think logically to enhance analytical ability.Maths needs practice and with practice, anything can be perfected, even the way you think and analyze.
So, it is of utmost importance to take Maths seriously right from your early school life. Getting your basics clear at this point will make it easier for you to handle difficult concepts as you grow older.
Maths of Class 9 standard plays a vital role in strengthening the fundamentals of the subject with comprehensive knowledge and understanding.So, one must prepare for this subject in a serious manner and get all their doubts clear,with maximum practice more and more to master the problem solving skill. Otherwise, they won’t be able to cope with the subject later, especially if they are planning to take up Maths in their Higher Secondary level.
CBSE Class 9 Maths Unit-Wise Weightage
Statistics & Probability
UNIT I: NUMBER SYSTEMS
1. Real Numbers (18 Periods)
1. Review of representation of natural numbers, integers, rational numbers on the number line. Representation of terminating / non-terminating recurring decimals on the number line through successive magnification. Rational numbers as recurring/ terminating decimals. Operations on real numbers.
2. Examples of non-recurring/non-terminating decimals. Existence of non-rational numbers (irrational numbers) such as √2, √3 and their representation on the number line. Explaining that every real number is represented by a unique point on the number line and conversely, viz. every point on the number line represents a unique real number.
3. Definition of nth root of a real number.
4. Existence of √x for a given positive real number x and its representation on the number line with geometric proof.
5. Rationalization (with precise meaning) of real numbers of the type 1/(a+b √x) and 1/(√x + √y) (and their combinations) where x and y are naturalnumber and a and b are integers.
6. Recall of laws of exponents with integral powers. Rational exponents with positive real bases (to be done by particular cases, allowing learner to arrive at the general laws.)
UNIT II: ALGEBRA
1. Polynomials (23 Periods)
Definition of a polynomial in one variable, with examples and counter examples. Coefficients of a polynomial, terms of a polynomial and zero polynomial. Degree of a polynomial. Constant, linear, quadratic and cubic polynomials. Monomials, binomials, trinomials. Factors and multiples. Zeros of a polynomial. Motivate and State the Remainder Theorem with examples. Statement and proof of the Factor Theorem. Factorization of ax2 + bx + c, a ≠ 0 where a, b and c are real numbers, and of cubic polynomials using the Factor Theorem.
Recall of algebraic expressions and identities. Verification of identities:
(x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx
(x ± y)3 = x3 ± y3 ± 3xy (x ± y)
x³ ± y³ = (x ± y) (x² ± xy + y²)
x3 + y3 + z3 – 3xyz = (x + y + z) (x2 + y2 + z2 – xy – yz – zx) and their use in factorization of polynomials.
2. Linear Equations in Two Variables (14 Periods)
Recall of linear equations in one variable. Introduction to the equation in two variables.
Focus on linear equations of the type ax + by + c = 0. Prove that a linear equation in two variables has infinitely many solutions and justify their being written as ordered pairs of real numbers, plotting them and showing that they lie on a line. Graph of linear equations in two variables. Examples, problems from real life, including problems on Ratio and Proportion and with algebraic and graphical solutions being done simultaneously.
UNIT III: COORDINATE GEOMETRY
1. Coordinate Geometry (6 Periods)
The Cartesian plane, coordinates of a point, names and terms associated with the coordinate plane, notations, plotting points in the plane.
UNIT IV: GEOMETRY
1. Introduction to Euclid’s Geometry (6 Periods)
History – Geometry in India and Euclid’s geometry. Euclid’s method of formalizing observed phenomenon into rigorous Maths with definitions, common/obvious notions, axioms/postulates and theorems. The five postulates of Euclid. Equivalent versions of the fifth postulate. Showing the relationship between axiom and theorem, for example:
(Axiom) 1. Given two distinct points, there exists one and only one line through them.
(Theorem) 2. (Prove) Two distinct lines cannot have more than one point in common.
2. Lines and Angles (13 Periods)
1. (Motivate) If a ray stands on a line, then the sum of the two adjacent angles so formed is 180oand the converse.
2. (Prove) If two lines intersect, vertically opposite angles are equal.
3. (Motivate) Results on corresponding angles, alternate angles, interior angles when a transversal intersects two parallel lines.
4. (Motivate) Lines which are parallel to a given line are parallel.
5. (Prove) The sum of the angles of a triangle is 180o.
6. (Motivate) If a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles.
3. Triangles (20 Periods)
1. (Motivate) Two triangles are congruent if any two sides and the included angle of one triangle is equal to any two sides and the included angle of the other triangle (SAS Congruence).
2. (Prove) Two triangles are congruent if any two angles and the included side of one triangle is equal to any two angles and the included side of the other triangle (ASA Congruence).
3. (Motivate) Two triangles are congruent if the three sides of one triangle are equal to three sides of the other triangle (SSS Congruence).
4. (Motivate) Two right triangles are congruent if the hypotenuse and a side of one triangle are equal (respectively) to the hypotenuse and a side of the other triangle (RHS Congruence).
5. (Prove) The angles opposite to equal sides of a triangle are equal.
6. (Motivate) The sides opposite to equal angles of a triangle are equal.
7. (Motivate) Triangle inequalities and relation between ‘angle and facing side’ inequalities in triangles.
4. Quadrilaterals (10 Periods)
1. (Prove) The diagonal divides a parallelogram into two congruent triangles.
2. (Motivate) In a parallelogram opposite sides are equal, and conversely.
3. (Motivate) In a parallelogram opposite angles are equal, and conversely.
4. (Motivate) A quadrilateral is a parallelogram if a pair of its opposite sides is parallel and equal.
5. (Motivate) In a parallelogram, the diagonals bisect each other and conversely.
6. (Motivate) In a triangle, the line segment joining the mid points of any two sides is parallel to the third side and in half of it and (motivate) its converse.
5. Area (7 Periods)
Review concept of area, recall area of a rectangle.
1. (Prove) Parallelograms on the same base and between the same parallels have the same area.
2. (Motivate) Triangles on the same (or equal base) base and between the same parallels are equal in area.
6. Circles (15 Periods)
Through examples, arrive at definition of circle and related concepts-radius, circumference, diameter, chord, arc, secant, sector, segment, subtended angle.
1. (Prove) Equal chords of a circle subtend equal angles at the center and (motivate) its converse.
2. (Motivate) The perpendicular from the center of a circle to a chord bisects the chord and conversely, the line drawn through the center of a circle to bisect a chord is perpendicular to the chord.
3. (Motivate) There is one and only one circle passing through three given non-collinear points.
4. (Motivate) Equal chords of a circle (or of congruent circles) are equidistant from the center (or their respective centers) and conversely.
5. (Prove) The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.
6. (Motivate) Angles in the same segment of a circle are equal.
7. (Motivate) If a line segment joining two points subtends equal angle at two other points lying on the same side of the line containing the segment, the four points lie on a circle.
8. (Motivate) The sum of either of the pair of the opposite angles of a cyclic quadrilateral is 180oand its converse.
7. Constructions (10 Periods)
1. Construction of bisectors of line segments and angles of measure 60o, 90o, 45o etc., equilateral triangles.
2. Construction of a triangle given its base, sum/difference of the other two sides and one base angle.
3. Construction of a triangle of given perimeter and base angles.
UNIT V: MENSURATION
1. Areas (4 Periods)
Area of a triangle using Heron’s formula (without proof) and its application in finding the area of a quadrilateral.
2. Surface Areas and Volumes (12 Periods)
Surface areas and volumes of cubes, cuboids, spheres (including hemispheres) and right circular cylinders/cones.
UNIT VI: STATISTICS & PROBABILITY
1. Statistics (13 Periods)
Introduction to Statistics: Collection of data, presentation of data – tabular form, ungrouped / grouped, bar graphs, histograms (with varying base lengths), frequency polygons. Mean, median and mode of ungrouped data.
2. Probability (9 Periods)
History, repeated experiments and observed frequency approach to probability. Focus is on empirical probability. (A large amount of time to be devoted to group and to individual activities to motivate the concept; the experiments to be drawn from real-life situations, and from examples used in the chapter on statistics)
1. Maths – Textbook for class IX – NCERT Publication
2. Guidelines for Maths Laboratory in Schools, class IX – CBSE Publication
3. Laboratory Manual – Maths, secondary stage – NCERT Publication
4. Maths exemplar problems for class IX, NCERT publication.
Tips and Tricks to learn Maths
- Attempt as many conceptual questions on those derivations as you can. This will give you a better understanding of the subject.
- Look for new questions on similar context from previous year’s question papers, sample papers and other model test papers.
- Always time yourself while solving sample papers as it will help you better manage time during the exam.
- Revision of notes and memorizing the formulas and the definition of difficult terms helps a lot while last minute preparation. Keep a few lists of topics separately for quick revision and it must contain the important formulae, derivations and important definitions.
- Solve as many Class 9 sample papers you can, but try to solve at least 3-4 completely. Practice past year question papers & model test papers.
- Time management is a must; take out time for re-creation as well along side preparation.
- Try and finish all the topics and questions in the NCERT books.CBSE board questions are majorly based on the question in this book only.
Reference Links –
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