These Chapter Wise Solutions for class 9 Maths are neat with an easy method in order that class 9 students can easily understand each sum’s solutions. The Class 9 Maths textbook features a total of 15 chapters which are divided into seven units.
Class 9 Maths Solutions develop reasoning skills in order that students are able to unravel all the sums once the concept is obvious. Class 9 Maths Solutions are easier and therefore the students also are advised to practice Examples of Class 9 Maths Solutions to urge a thought of question pattern within the final examination .
Solving these Class 9th Maths solutions in each chapter will assure positive results. Chapters covered within the 9th Math textbook are Sets,numeration system , Polynomials, Introduction to Euclid’s Geometry, Quadrilaterals, Surface Areas, Heron’s Formula, Constructions and Volumes, Statistics, Probability etc.
Class 9 Maths Chapter 1: SETS
This is the first chapter of class 9 maths. Read it properly to learn the detail concept.
- A well-defined collection of objects is called a set.The objects in a set are called its members or elements.
- Methods of writing sets:
Roster method or tabular form:In this method the elements of the sets are written in a curly bracket. We can write the elements only once and separate them by commas.In this method order of an element is not important but all the elements should be written.
Set builder form for rule method or description method:In set builder form,list of elements are not written but we write the general element using a variable followed by a vertical line or colon and write the property of the variable.
- Below examples of each set is written in both the methods:
Roster method or tabular form | Set builder form for rule method |
A = { 2, 4, 6, 8, 10} | A = {x | x is an even natural number less than 11} |
B = {1,4,9,16,25,36,49} | B = { x | x is a perfect square number between 1 to 50} |
C = { red, orange, yellow, green, blue, indigo,violet} | C = { y | y is a colour in the rainbow} |
X = {-4,-3,-2,-1,0,1,2} | X = {x | x is an integer and , -5< x < 3} |
- Set having a finite number of elements is termed as a finite set, otherwise it is termed as an infinite set.
- Number of distinct elements contained in a finite set is termed as the Cardinal number of the set.We denote the Cardinal number of a set B by n(B).
- A set which does not contain any element is called the empty set.It is denoted by .
- Set which consists of only one element is called a singleton set.
- If every element of A is in B and every element of B is in A then that two sets A and B are equal.
- If n(A) = n(B) Then two finite sets A and B are said to be equivalent.
- When every element of set A is in Set B then A is called subset of B is denoted by A⊆B.
- Here are some examples for singleton set,finite set, infinite set and null set.
Name of set | Example |
Singleton Set | P = {2} P is the set of even prime numbers. |
Empty Set orNull Set | B = {x | x is natural number between 6 and 7.}∴ B = { } or φ |
Finite Set | C = {p | p is a number from1 to 30 divisible by 5}C = {5,10,15,20,25} |
Infinite Set | W = {0,1, 2, 3, 4, . . . } |
- The set of all possible subsets of a set A is called the power set of A denoted by P(A).
- If A contains n elements, then there are 2^{n} subsets of A.
- The union of two sets A and B denoted by A U B, is the set of all the elements each one of which is either in A or in B or in both A and B.
- The intersection of two sets A and B, denoted by A ∩ B, is the set of all elements common to both A and B.
- Two sets A and B are said to be disjoint, if A ∩ B = φ
- For any two sets A and B, the set of all those elements of A which are not in B is called their difference,is denoted by A-B.
- Let be ζ the universal set and let A⊆ ζ, then the set of all those elements of ζ which are not in A is termed as complement of A, denoted by A’.
- Important points :
(i) Every set is a subset of itself. i.e. A ⊆ A
(ii) Empty set is a subset of every set i.e. φ ⊆ A
(iii) If A = B then A ⊆ B and B⊆ A
(iv) If A ⊆ B and B ⊆ A then A = B
- Basically what is an Universal set
Universal set: Let’s assume a bigger set which will accommodate all the given sets under consideration which in general is known as Universal set.
Let’s take an example:
A football team of 11 students is to be selected from a college. Here all the students from college who play football is the Universal set. A team of 11 football players is a subset of that Universal set. Generally, the universal set is denoted by ‘U’ and in Venn diagram it is represented by a rectangle.
- Properties of complement of a set.
(i) No elements are common in A and A’
(ii) A⊆ U and A’ ⊆ U
(iii) Complement of set U is empty set. U’ = φ
(iv) Complement of an empty set is U. φ’= U
- Properties of Intersection of sets
(1) A ∩ B = B A
(2) If A⊆ B then A ∩ B = A
(3) If A ∩ B = B then B ⊆ A
(4) A ∩ B ⊆ A and A ∩ B ⊆ B
(5) A ∩ A’ = φ
(6) A ∩ A = A
(7) A ∩ φ = φ
- Properties of Union of sets
(1) A U B = B U A
(2) If A⊆B then A U B = B
(3) A⊆A U B, B⊆A U B
(4) A U A’= U
(5) A U A= A
(6) A U φ = A
- Distributive Law for Sets:
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
- De Morgan’s law:
(A ∪ B)^{’} = A^{’} ∩ B^{’}
(A ∩ B)^{’} = A^{’} ∪ B^{’}
- For solving the sums based on Venn diagram we use the formula:
n (A U B ) = n (A) + n (B) – n (A∩B)
means n (A) + n (B) = n (A U B ) + n (A∩B)
Want to learn class 9 science? Click Here
Chapter 2: BASIC CONCEPTS IN GEOMETRY
Around 2000 to 3000 years ago there are some pyramids which are made in Egypt, and you can see the shape of this pyramid, the shape are like they are made by some engineer but at that time there were no engineers, so how this shapes are so perfect, so this is all because of the word Geometry where geo means ‘earth’ and metry means ‘measuring’.
Now we will see what are the basic concepts of geometry in class 9 maths.
Coordinates of points and distance:
Now from the above number line you can see all the integers are denoted by some alphabets.On the number line you can see point A denotes -5, point B denotes -3 and same as for the points C,O,D and E.Hence we can say that the coordinates of point D is 1 and co-ordinate of point E is 3 show the distance between D and E will be 2.
d (E, D) = 3 – 1 = 2
∴l(ED) = 2
d (E, D) = l(ED) = 2
Similarly d (D, E) = 2
Thus, we can find the distance between two points on a number line by counting the number of units.
The co-ordinates of points C and O are -2 and 0 respectively. We know that 0 > -2.
Therefore, distance between points C and O = 0 – (-2) = +2
- To get a distance between two points we have to subtract smaller co-ordinates from the larger co-ordinate.
- The distance between any two points is a non-negative real number.
Betweenness:
If A, B, C are three distinct collinear points, there are three possibilities.
If d (A, B) + d (B, C) = d (A, C) then it is said that point B is between A and C. The betweenness is shown as A – B – C.
- Line segment :
The union set of point P, point Q and points between P and Q is called segment PQ. Segment PQ is written as seg PQ. Seg PQ means seg QP.
Point P and point Q are called the end points of seg PQ.
The distance between the endpoints of a segment is termed as length of the segment. That is l(PQ) = d (P,Q)
l(PQ) = 10 then PQ = 10
- Ray XY :
Suppose, X and Y are two distinct points. The union set of all points on seg XY and the points P such that X – Y – P, is called ray XY.
Here point X is called the end point of ray XY.
- Line XY :
The union set of points on ray XY and opposite ray of the XY is called line XY.
The set of points of seg XY is a subset of points of line XY.
- Congruent segments :
If the length of two segments is equal then the two segments are congruent.
If l(PQ) = l(XY) then seg PQ ≅ seg XY.
- Properties of congruent segments :
Reflexivity : seg PQ≅ seg PQ
Symmetry : If seg PQ ≅ seg RS then seg RS ≅ seg PQ
Transitivity : If seg PQ ≅ seg RS and seg RS ≅ seg XY then seg PQ ≅ seg XY
- Midpoint of a segment :
If P-M-Q and seg PM ≅ seg MQ, then M is called the midpoint of seg PQ.
As we can see that every segment has one and only one midpoint.
- Comparison of segments :
If length of segment XY is less than the length of segment PQ, it is written as seg XY < seg PQ or seg PQ > seg XY.
The comparison of segments depends upon their Lengths.
- Perpendicularity of segments or rays :
If the lines containing two segments, two rays or a ray and a segment are perpendicular to each other then the two segments, two rays or the segment and the ray are said to be perpendicular to each other.
- Distance of a point from a line :
If seg CD ⏊ line AB and the point D lies on line AB then the length of seg CD is called the distance of point C from line AB.
The point D is called the foot of the perpendicular.
If l(CD) = a, then the point C is at a distance of ‘a’ from the line AB.
Conditional statements and converse
- The statements which are written in the ‘If-then’ form are called conditional statements.
- The part of the statement following ‘If’ is called the antecedent.
- The part following ‘then’ is called the consequent.
- If the antecedent and consequent in a given conditional statement are interchanged,the resulting statement is called the converse of the given statement.
Euclid’s postulates
- Infinite lines can be drawn passing through a single point.
- There is one and only one line passing through two points.
- A radius of a circle can be drawn taking any point as its centre.
- All right angles are congruent with each other.
- If two interior angles formed on one side of a transversal of two lines add up to less than two right angles then the lines produced in that direction intersect each other.
- Though Euclid defined a point, a line, and a plane, the definitions are not accepted by mathematicians. Therefore, these terms are now taken as undefined.
- Axioms or postulates are the presumption which are obvious universal truths. They are not proved.
- Theorems are statements which are proved, using definitions, axioms, previously proved statements and deductive reasoning.
Theorems and some properties:
Linear pair axiom:If there is a ray on a line, then the sum of the two adjacent angles so formed by the ray is 180°.
Vertically opposite angles are equal when two lines intersect each other.
If a transversal intersects two parallel lines, then
- every pair of corresponding angles is equal,
- each pair of alternate interior angles is equal,
- each pair of interior angles on the same side of the transversal is supplementary.
4. If a transversal intersects two lines such that, either
- any one pair of corresponding angles is equal, or
- any one pair of alternate interior angles is equal, or
- any one pair of interior angles on the same side of the transversal is supplementary,then the lines are parallel.
5. Lines which are parallel to a given line are mostly parallel to each other.
6. The sum of the three angles of a triangle is 180°.
7. If a side of a triangle is produced, the exterior angle of a triangle so formed is equal to the sum of the two interior opposite angles.
Class 9 Maths Chapter 3: LINES AND ANGLES
1. If a ray stands on a line, then the sum of the 2 adjacent angles so formed is 180° and vice versa. This property is named because of the Linear pair axiom.
2. If two lines intersect one another , then the vertically opposite angles are equal.
3. 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 angles on an equivalent side of the transversal is supplementary.
4. If a transversal intersects two lines such , either
(i) anybody pair of corresponding angles is equal, or
(ii) anybody pair of alternate interior angles is equal, or
(iii) anybody pair of interior angles on an equivalent side of the transversal is supplementary, then the lines are parallel.
5. Lines which are parallel to a given line are parallel to every other.
6. The sum of the three angles of a triangle is 180°.
7. If a side of a triangle is produced, the outside angle so formed is adequate to the sum of the 2 interior opposite angles.
Chapter 4: TRIANGLES
1. Two figures are congruent, if they’re of an equivalent shape and of an equivalent size.
2. Two circles of an equivalent radii are congruent.
3. Two squares of equivalent sides are congruent.
4. If two triangles ABC and PQR are congruent under the correspondence A ↔ P,
B ↔ Q and C ↔ R, then symbolically, it’s expressed as Δ ABC ≅ Δ PQR.
5. If two sides and therefore therefore the included angle of 1 triangle are adequate to two sides and the included angle of the opposite triangle, then the 2 triangles are congruent (SAS Congruence Rule).
6. If two angles and therefore therefore the included side of 1 triangle are adequate to two angles and the included side of the opposite triangle, then the 2 triangles are congruent (ASA Congruence Rule).
7. If two angles and one side of 1 triangle are adequate to two angles and therefore the corresponding side of the opposite triangle, then the 2 triangles are congruent (AAS Congruence Rule).
8. Angles opposite to equal sides of a triangle are equal.
9. Sides opposite to equal angles of a triangle are equal.
10. Each angle of an equiangular triangle is 60°.
11. If three sides of 1 triangle are adequate to three sides of the opposite triangle, then the 2
triangles are congruent (SSS Congruence Rule).
12. If in two right triangles, hypotenuse and one side of a triangle are adequate to the hypotenuse and one side of other triangle, then the 2 triangles are congruent (RHS Congruence
Rule).
13. During a triangle, the angle opposite to the longer side is larger (greater).
14. During a triangle, the side opposite to the larger (greater) angle is longer.
15. Sum of any two sides of a triangle is bigger than the third side.
Class 9 Maths Chapter5: QUADRILATERALS
1. Sum of the angles of a quadrilateral is 360°.
2. A diagonal of a parallelogram divides it into two congruent triangles.
3. During a parallelogram,
(i) opposite sides are equal (ii) opposite angles are equal
(iii) diagonals bisect one another.
4. A quadrilateral may be a parallelogram, if
(i) opposite sides are equal or (ii) opposite angles are equal
or (iii) diagonals bisect one another
or (iv)a pair of opposite sides is equal and parallel.
5. Diagonals of a rectangle bisect one another and are equal and vice-versa.
6. Diagonals of a rhombus bisect one another at right angles and vice-versa.
7. Diagonals of a square bisect one another at right angles and are equal, and vice-versa.
8. The line-segment joining the mid-points of any two sides of a triangle is parallel to the third side and is half it.
9. A line through the midpoint of a side of a triangle parallel to a different side bisects the third side.
10. The quadrilateral formed by joining the mid-points of the edges of a quadrilateral, in order, is a parallelogram.
Chapter 6: AREAS OF PARALLELOGRAMS AND TRIANGLES
1. Area of a figure may be a number (in some unit) related to the a part of the plane enclosed by that figure.
2. Two congruent figures have equal areas but the converse needn’t be true.
3. If a planar region formed by a figure T is formed from two non-overlapping planar regions formed by figures P and Q, then ar (T) = ar (P) + ar (Q), where ar (X) denotes the world of figure X.
4. Two figures are said to get on an equivalent base and between equivalent parallels, if they need a common base (side) and therefore the vertices, (or the vertex) opposite to the common base of each figure lie on a line parallel to the bottom.
5. Parallelograms on an equivalent base (or equal bases) and between equivalent parallels are equal in area.
6. Area of a parallelogram is the product of its base and therefore the corresponding altitude.
7. Parallelograms on an equivalent base (or equal bases) and having equal areas lie between the same parallels.
8. If a parallelogram and a triangle are on an equivalent base and between equivalent parallels, then the area of the Triangulum is half the world of the parallelogram.
9. Triangles on an equivalent base (or equal bases) and between equivalent parallels are equal in area.
10. Area of a triangle is half the merchandise of its base and therefore the corresponding altitude.
11. Triangles on an equivalent base (or equal bases) and having equal areas lie between equivalent parallels.
12. A median of a triangle divides it into two triangles of equal areas.
Class 9 Maths Chapter 7: CIRCLES
1. A circle is that the collection of all points during a plane, which are equidistant from a hard and fast point in the plane.
2. Equal chords of a circle (or of congruent circles) subtend equal angles at the centre.
3. If the angles subtended by two chords of a circle (or of congruent circles) at the centre (corresponding centres) are equal, the chords are equal.
4. The perpendicular from the centre of a circle to a chord bisects the chord.
5. The road drawn through the centre of a circle to bisect a chord is perpendicular to the chord.
6. There’s one and just one circle passing through three non-collinear points.
7. Equal chords of a circle (or of congruent circles) are equidistant from the centre (or corresponding centres).
8. Chords equidistant from the centre (or corresponding centres) of a circle (or of congruent circles) are equal.
9. If two arcs of a circle are congruent, then their corresponding chords are equal and conversely if two chords of a circle are equal, then their corresponding arcs (minor, major) are congruent.
10. Congruent arcs of a circle subtend equal angles at the centre.
11. The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining a part of the circle.
12. Angles within the same segment of a circle are equal.
13. Angle during a semicircle may be a right angle.
14. If a line segment joining two points subtends equal angles at two other points lying on
the same side of the road containing the road segment, the four points lie on a circle.
15. The sum of either pair of opposite angles of a cyclic quadrilateral is 180°.
16. If sum of a pair of opposite angles of a quadrilateral is 180°, then it is a quadrilateral.
Chapter 8: SURFACE AREAS AND VOLUMES
1. area of a cuboid = 2 (lb + bh + hl)
2. Area of a cube = 6a^{2}
3. Curved area of a cylinder = 2πrh
4. Total area of a cylinder = 2πr(r + h)
5. Curved area of a cone = πrl
6. Total area of a right circular cone = πrl + πr^{2}, i.e., πr (l + r)
7. Area of a sphere of radius r = 4 π r^{2}
8. Curved area of a hemisphere = 2π r^{2}
9. Total area of a hemisphere = 3π r^{2}
10. Volume of a cuboid = l × b × h
11. Volume of a cube = a^{3}
12. Volume of a cylinder = π r^{2}h
13. Volume of a cone =1/3 x r^{2}h
14. Volume of a sphere of radius r = 4/3 x r3
15. Volume of a hemisphere = 2/3 x r^{3}
Class 9 Maths Chapter 9: STATISTICS
1. Facts or figures, collected with a particular purpose, are called data.
2. Statistics is that the area of study handling the presentation, analysis and interpretation of data.
3. How data are often presented graphically within the sort of bar graphs, histograms and frequency polygons.
4. The three measures of central tendency for ungrouped data are:
(i) Mean : it’s found by adding all the values of the observations and dividing it by the total number of observations. it’s denoted by x.
(ii) Median : it’s the worth of the middle-most observation (s).
(iii) Mode : The mode that is the most often occurring observation.
Chapter 10: PROBABILITY
1. An occasion for an experiment is that the collection of some outcomes of the experiment.
2. The empirical (or experimental) probability P(E) of an occasion E is given by
P(E) =(Number of trials in which E has happened)/(Total number of trials)
3. The Probability of an occasion lies between 0 and 1 (0 and 1 inclusive).
Let us consider an easy experiment. A bag contains 4 balls of an equivalent size.
Three of them are white and therefore the fourth is black. you’re alleged to pick one ball at random without seeing it. Then obviously, the possibility of getting a white ball is more.
In Mathematical language, when possibility of an expected event is expressed
in number, it’s called ‘Probability’ . it’s expressed as a fraction or percentage using the subsequent formula.
For a random experiment, if sample space is ‘S’and ‘A’ is an expected event then probability of ‘A’ is P(A). It’s given by the following formula.
P(A) = (Number of sample points in event A)/(Number of sample points in sample spaces)= n(A)/n(S)
This was briefly about class 9 maths. I hope you have understood each concept thoroughly. I suggest you learn each chapter properly and practice problems to master solving any problem of class 9 maths.
You can also ask your questions in the comment section below. I would try to answer each of your questions as soon as possible.