Chapter 1 : Number Systems
Review of representation of natural numbers, integers and rational numbers on the number line. Rational numbers as recurring/ terminating decimals. Operations on real numbers. 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. Rationalization (with precise meaning) of real numbers of the type (and their combinations) where x and y are natural number and a and b are integers. 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.)
Class 9 Maths Chapter 15 Probability MCQ Questions
MCQ Questions Class 9 Maths Chapter 13 Surface Areas and Volumes
Class 9 Maths Chapter 11 Constructions MCQ Questions
Circles MCQ Questions Class 9 Maths Chapter 10
Quadrilaterals MCQ Questions Class 9 Maths Chapter 8
Class 9 Maths Chapter 2 Polynomials MCQ Questions
Class 9 Maths Chapter 14 Statistics MCQ Questions
Class 9 Maths Chapter 12 Heron's Formula MCQ Questions
Class 9 Maths Chapter 7 Triangles MCQ Questions
Chapter 3: Coordinate Geometry
The Cartesian plane, coordinates of a point, names and terms associated with the coordinate plane, notations, plotting points in the plane.
Chapter 4: Linear Equations in Two Variables
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. Explain 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 with algebraic and graphical solutions being done simultaneously.
Chapter 6: Lines and Angles
1. (Motivate) If a ray stands on a line, then the sum of the two adjacent angles so formed is 1800 and 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 1800
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.
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Class 9 Maths MCQ Chapter 4 Linear Equations in Two Variables
Class 9 Maths Chapter 3 Coordinate Geometry MCQs
MCQ Questions Class 9 Maths Chapter 1 Number Systems
Class 9 Maths Question Paper PT2 (2021) - Amity Vasundhara
CBSE Class 9 Maths Question Paper Annual Exam 2020 - DPS Ranchi
CBSE Class 9 Maths Question Paper Annual Exam 2019 - DPS Ranchi
CBSE Class 9 Maths Question Paper Annual Exam 2018 - DPS Ranchi
Test Paper on Polynomials, Quadrilaterals and Probability - Class 9 Maths
Herons Formula Extra Questions Class 9 Maths
Chapter 7: Triangles
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. (Motivate) 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.
Chapter 12: Heron’s Formula
Area of a triangle using Heron's formula (without proof).
Chapter 14: Statistics Introduction to Statistics:
Collection of data, presentation of data — tabular form, ungrouped / grouped, bar graphs, histograms. Mental Maths Revision from Support Material
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Areas of Parallelogram and Triangles Question Bank Class 9 Maths
Areas of Parallelograms Maths Assignment Worksheet Class 9
Class 9 Maths Question Paper First Term Exam (2021- 22)- Homerton Grammar School
CBSE Class 9 Maths Assignment Worksheet (3) - Brilliant College Of Commerce
CBSE Class 9 Maths Assignment Worksheet (2) - Brilliant College Of Commerce
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Class 9 Trigonometry & Pythagoras Test Paper - 2021
CBSE Class 9 Maths Solved Sample Paper (10) - 2020
CBSE Class 9 Maths Solved Sample Paper (9) - 2020
CBSE Class 9 Maths Solved Sample Paper (8) - 2020
TERM- 2
Chapter 2 : Polynomials
Polynomials 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. Zeroes of a polynomial. 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) x 3 ± y3 = (x ± y)(x2 ∓ xy + y2 ) and their use in factorization of polynomials.
Chapter 8: Quadrilaterals
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 is half of it and (motivate) its converse.
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Chapter 10: Circles
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 centre and (motivate) its converse.
2.(Motivate) The perpendicular from the centre of a circle to a chord bisects the chord and conversely, the line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.
3. (Motivate) Equal chords of a circle (or of congruent circles) are equidistant from the centre (or their respective centre) and conversely.
4. (Motivate) 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.
5. (Motivate) Angles in the same segment of a circle are equal.
6. (Motivate) The sum of either of the pair of the opposite angles of a cyclic quadrilateral is 180° and its converse.
Chapter 11: Constructions
Construction of bisectors of line segments and angles of measure 60o , 90o , 45o etc., equilateral triangles. Construction of a triangle given its base, sum/difference of the other two sides and one base angle.
Chapter 13: Surface Areas and Volumes
Surface areas and volumes of cubes, cuboids, spheres (including hemispheres) and right circular cylinders/cones.
Chapter 15: Probability
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)
Mental Maths
Revision from Support Material
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Short Notes
Polynomials
Constants : A symbol having a fixed numerical value is called a constant.
Example : 7, 3, -2, 3/7, etc. are all constants.
Variables : A symbol which may be assigned different numerical values is known avariable.
Example : cumference of circle
r - radius of circle Where 2 & are constants. while C and r are variable
Algebraic expressions : A combination of constants and variables. Connected by some or all of the operations +, -, X and is known as algebraic expression.
Terms : The several parts of an algebraic expression separated by '+' or '-' operations are called the terms of the expression.
Polynomials : An algebraic expression in which the variables involved have only nonnegative integral powers is called a polynomial.
Coefficients : In the polynomial coefficient of respectively and we also say that +1 is the constant term in it. Degree of a polynomial in one variable : In case of a polynomial in one variable the highest power of the variable is called the degree of the polynomial. Classification of polynomials on the basis of degree.
Classification of polynomials on the basis of no. of terms
Constant polynomial : A polynomial containing one term only, consisting a constant term is called a constant polynomial the degree of non-zero constant polynomial is zero.
Zero polynomial : A polynomial consisting of one term, namely zero only is called a zero polynomial.The degree of zero polynomial is not defined.
Zeroes of a polynomial : Let p(x).be a polynomial. If then we say that is zero of the polynomial of p(x).
Remark : Finding the zeroes of polynomial p(x) means solving the equation p(x)=0.
Remainder theorem : Let be a polynomial of degree and let a be any real number. When is divided by then the remainder is
Factor theorem : Let be a polynomial of degree and let a be any real number.
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Number Systems
* Natural numbers are - 1, 2, 3, ................. denoted by N.
* Whole numbers are - 0, 1, 2, 3, .................. denoted by W.
* Integers - ....... -3, -2, -1, 0, 1, 2, 3, .................. denoted by Z.
* Rational numbers - All the numbers which can be written in the form p/q, are called rational numbers where p and q are integers.
* Irrational numbers - A number s is called irrational, if it cannot be written in the form p/q where p and q are integers and
* The decimal expansion of a rational number is either terminating or non terminating recurring. Thus we say that a number whose decimal expansion is either terminating or non terminating recurring is a rational number.
* The decimal expansion of a irrational number is non terminating non recurring.
* All the rational numbers and irrational numbers taken together.
* Make a collection of real number.
* A real no is either rational or irrational.
* If r is rational and s is irrational then r+s, r-s, r.s are always irrational numbers but r/s may be rational or irrational.
* Every irrational number can be represented on a number line using Pythagoras theorem.
* Rationalization means to remove square root from the denominator.
Linear Equations in Two Variables
An equation of the form ax + by + c = 0 where a, b, c are real numbers and x, y are variables, is called a linear equation in two variables.
Here ‘a’ is called coefficient of x, ‘b’ is called coefficients of y and c is called constant term.
Eg. 6x + 2y + 5 = 0, 5x – 2y + 3 = 0 etc.
SOLUTIONS OF LINEAR EQUATION
The value of the variable which when substituted for the variable in the equation satisfies the equation i.e. L.H.S. and R.H.S. of the equation becomes equal, is called the solution or root of the equation.
RULES FOR SOLVING AN EQUATION
(i) Same quantity can be added to both sides of an equation without changing the equality.
(ii) Same quantity can be subtracted from both sides of an equation without changing the equality.
(iii) Both sides of an equation may be multiplied by a same non-zero number without changing the equality.
(iv) Both sides of an equation may be divided by a same non-zero number without changing the equality.
A LINEAR EQUATION EN TWO VARIABLES
HAS INFINITELY MANY SOLUTIONS
We can get many many solutions in the following way.
Pick a value of your choice for x (say x = 2) in 2x + 3y = 12. then the equation reduces to 4+3y = 12, which is a linear equation in one variable. On solving this, you get y = 8/3. So (2, 8/3) is another solution of 2x + 3y = 12. Similarly, choosing x = —5, you find that the equation becomes — 10 + 3y = 12. This gives y =22/3. So, (-5, 22/3) is another solution of 2x + 3y = 12. So there is no end to different solutions of a linear equation in two variables.
Note : An easy way of getting a solution is to take x = 0 and get the corresponding value of y. Similarly, we can put y =0 and obtain the corresponding value of x.
GRAPH OF LINEAR EQUATIONS
The graph of an equation in x and y is the set of all points whose coordinates satisfy the equation :
In order to draw the graph of a linear equation ax + by + c = 0 may follow the following algorithm.
Step 1: Obtain the linear equation ax + by+ c = 0
Step 2 : Express yin terms of x i.e. y = -((ax + b)/c))
Step 3 : Put any two or three values for x and calculate the corresponding values of y from the expression values of y from the expression obtained in step 2. Let we get points as (α1, β1), (α2, β2), (α3, β3)
Step 4 : Plot points (α1, β1), (α2, β2), (α3, β3) on graph paper.
Step 5 : Join the points marked in step 4 to obtain. The line obtained is the graph of the equation ax+by+c= 0
Note : (i) The reason that a degree one polynomial equation ax + by + c = 0 is called a linear equation is that its geometrical representation is a straight line.
(ii) The graph of the equation of the form y = kx is a line which always passes through the origin.
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Surface Areas and Volumes Class 9 Questions & Answers
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Areas of Parallelogram and Triangles Class 9 Questions
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Introduction to Euclid's Geometry Class 9 Maths Extra Questions with Answers
Introduction to Euclid's Geometry
The Greeks developed geometry is a systematic manner Euclid (300 B.C.) a greek mathematician, father of geometry introduced the method of proving mathematical results by using deductive logical reasoning and the previously proved result. The Geometry of plane figure is know as "Euclidean Geometry".
Axioms :
The basic facts which are taken for granted without proof are called axioms some
Euclid's axioms are
(i) Things which are equal to the same thing are equal to one another. i.e.
(ii) If equals are added to equals, the wholes are equal i.e.
(iii) If equals are subtracted from equals, the remainders are equal i.e.
(iv) Things which coincide with one another are equal to one another.
(v) The whole is greater than the part.
Postulates :
Axioms are the general statements, postulates are the axioms relating to a particular field.
Educlid's five postulates are.
(i) A straight line may be drawn from any one point to any other point.
(ii) A terminated line can be produced indefinitely.
(iii) A circle can be drawn with any centre and any radius.
(iv) All right angles are equal to one another.(v) If a straight line falling on two straight lines makes the interior angles on the same
side of it taken together less than two right angles, then the two straight lines, if produced indefinitely meet on that side on which the angles are less than two right angles.
Statements : A sentence which is either true or false but not both, is called a statement. eg. (i) 4+9=6 If is a false sentence, so it is a statement.
(ii) Sajnay is tall. This is not a statement because he may be tall for certain persons and may not be taller for others.
Theorems : A statement that requires a proof is called a theorem.
eg. (i) The sum of the angles of triangle is 1800.
(ii) The angles opposite to equal sides of a triangles are equal. Corollary - Result deduced from a theorem is called its corollary.
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Chapter Lines and Angles NCERT Solutions For Class 9 Maths
Chapter Triangles NCERT Solutions For Class 9 Maths
Chapter Circles NCERT Solutions for Class 9 Maths
Quadrilaterals NCERT Solutions for Class 9
Areas of Parallelograms and Triangles NCERT Solutions For Class 9
Surface Areas and Volumes NCERT Solutions for Class 9
Statistics NCERT Solutions for Class 9 Maths
Heron's Formula NCERT Solutions For Class 9
Lines and Angles
Types of Angles -
-
(1) Acute angle - An acute angle measure between 00 and 900.
-
(2) Right angle - A right angle is exactly equal to 900.
-
(3) Obtuse angle - An angle greater than 900 but less than 1800.
-
(4) Straight angle - A straight angle is equal to 1800.
-
(5) Reflex angle - An angle which is greater than 1800 but less than 3600 is called a reflex angle.
-
(6) Complementary angles - Two angles whose sum is 900 are called complementary angles.
-
(7) Supplementary angle - Two angles whose sum is 1800 are called supplementary angles.
-
(8) Adjacent angles - Two angles are adjacent, if they have a common vertex, a common arm and their non common arms are on different sides of common arm.
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(9) Linear pair - Two angles form a linear pair, if their non-common arms form a line.
-
(10) Vertically opposite angles - Vertically opposite angles are formed when two lines intersect each other at a point.
TRANSVERSAL
(a) Corresponding angles
(b) Alternate interior angles
(c) Alternate exterior angles
(d) Interior angles on the same side of the transversal.
-
* If a transversal intersects two parallel lines, then
(i) each pair of corresponding angles is equal.
(ii) each pair of alternate interior angles is equal.
(iii) each pair of interior angle on the same side of the transversal is supplementary. -
* If a transversal interacts two lines such that, either
(i) any one pair of corresponding angles is equal, or
(ii) any one pair of alternate interior angles is equal or
(iii) any one pair of interior angles on the same side of the transversal is supplementary then the lines are parallel. -
* Lines which are parallel to a given line are parallel to each other.
-
* The sum of the three angles of a triangle is 1800.
-
* If a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles.
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Probability NCERT Solutions for Class 9
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Statistics
GRAPHICAL REPRESENTATION OF DATA
The representation of data through diagrams helps us to visualize the Whole meaning of a numerical distribution at a single glance. There are various methods of representing the data by means of graphs.
(A) Bar graphs
(B) Histograms of uniform width, and of varying widths
(C) Frequency polygons
(1) Bar graph : It is the simplest and most widely used graph in which the numerical data is represented by bars (rectangles) of equal width. . In a bar graph :
(i) The width of each bar can be any, but widths of all the bars should be the same.
(ii) The space between consecutive bars should also be same.
The height (or length) of a bar is proportional to the numerical data it represents.
(2) Histogram : A histogram is a graphical representation of a frequency distribution in the form of rectangles one after the other.
The bases of these rectangles represent the magnitudes of the variables of the class boundaries. So these are taken along the x-axis.
The heights of these represent the frequencies of the corresponding magnitudes of the variable of the class-boundaries.
(3) Frequency polygon : After drawing the histogram of a frequency distribution, when the mid-points of the respective tops of the rectangles are joined by the line segment, the figure so obtained is known as frequency polygon.
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When number of terms is large and variate are descrete, variate can accept some
particular values only under finite limits and is repeated then it is called descrete frequency
distribution.
(iii) Continuous frequency distribution :
When number of terms is large and variate is continuous, i.e., variate can accept all values under finite limits and they are repeated then it is called continuous frequency distribution.
The mid-point of the top of the first rectangle is joined to the mid-point of the earlier interval (imagined) on the x-axis. Similarly the mid-point of the top of the last rectangle is joined to the mid-point of the next interval (imagined) on the x-axis.
Note : If both histogram and frequency polygon are to be drawn, then it is advisable first to draw histogram and then join the mid-points of the tops of the rectangles of the histogram to get frequency polygon.
But if only frequency polygon is to be drawn then first represent the class marks along x-axis and frequencies along y-axis and then plot the point corresponding to the frequency at each class mark.
Note : Frequency polygons can also be drawn independently without drawing histograms. For this, we require the mid-points of the class-intervals used in the data.
These mid-points of the class-intervals are called class-marks.
Probability
MEANING OF PROBABILITY
In everyday life, we come across statements such as :
1. It will probably rain today.
2. I doubt that he will pass the test.
3. Most probably, Kavita will stand first in the annual examination.
4. Chances are high that the prices of diesel will go Up.
5. There is a 50-50 chance of India winning a toss in today’s match.
The words ‘probably’, ‘doubt’, ‘most probably’, ‘chances’, etc. used in the statements
above involve an element of uncertainty. For example, in (1), ‘probably rain’ will mean it may
rain or may not rain today. We are predicting rain today based on our past experience when it
rained under similar conditions. Similar predictions are
also made in other cases listed in (2) to (5).
The uncertainty of ‘probably’ etc. can be measured numerically be means of `probability’ in many cases.
SOME IMPORTANT OBJECTS
(i) Coin : We know that a coin has two faces : They are called Head and Tail.
(ii) Die : (Dice) : Die is a solid in the form of a cube, having six faces.
Each face has some marking of dots as shown above.
One face has one dot, second face two dots, third face has three dots and ... so on. We take them as 1, 2, 3, 4, 5, 6. Plural of die is dice.
(iii) Cards : A pack of cards has 52 cards out of which 26 are red cards and 26 are black cards.
(a) 26 red cards contain 13 cards of diamond (♦) and 13 cards of heart (♥).
(b) 26 black cards contain 13 cards spade (♠) and 13 cards of club (♣)
(c) 13 cards are 1, 2, 3, ..., 10, Jack, Queen and King.
(d) Card having 1 is also called an ace.
PROBABILITY — AN EXPERIMENTAL (EMPIRICAL) APPROACH.
A trial is an action which results in one or several outcomes. An event for an experiment is the collection of some outcomes of the experiment.
Let n be the total number of trials. The empirical probability P(E) of an event E happening, is given by
Number of trials in which the event happened
Note : The empirical probability depends on the number of trials.
SOME BASIC TERMS AND CONCEPTS:
1. An Experiment : An action or operation resulting in two or more outcomes is called an experiment
EX. (i) Tossing of a coin is an experiment.
There are two possible outcomes head or tail
(ii) Drawing a card from a pack of 52 cards is an experiment There are 52 possible outcomes.
2. Sample space : The set of all possible outcomes of an experiment is called the sample space, denoted by S. An element of S is called a sample point.
Ex. (i) In the experiment of tossing of a coin, the sample space has two points corresponding to head (H) and Tail (T) i.e S {H,T}.
Ex. (ii) When we throw a die then any one of the numbers 1, 2, 3, 4, 5 and 6 will come up. So the sample space, S = {1, 2, 3, 4, 5, 6}
3. Event: Any subset of sample space is an event.
Ex. (i) If the experiment is done throwing a die which has faces numbered 1 to 6, then S= {1,2,3,4, 5,6},A= {1,3,5},B {2, 4, 6}, the null set Φ and S itself are some events with respect to S.The null set Φ is called the impossible event or null event.
(ii) Getting 7 when a die is thrown is called a null event.
The entire sample space is called the certain event.
Some Notes :
(a) The probability of an event lies between 0 and 1, i.e., It can be any fraction from 0 to 1.
(b) The sum of the probabilities of all the possible outcomes of a trial is 1.
(c) Probability of the occurance of an event + Probability of the non-occurrence of that event = 1
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