## Online Education ML Aggarwal Class 8 Solutions for ICSE Maths Chapter 18 Mensuration Ex 18.1

Question 1.
The length and breadth of a rectangular field are in the ratio 9 : 5. If the area of the field is 14580 square metres, find the cost of surrounding the field with a fence at the rate of ₹3·25 per metre.
Solution:

Question 2.
A rectangle is 16 m by 9 m. Find a side of the square whose area equals the area of the rectangle. By how much does the perimeter of the rectangle exceed the perimeter of the square?
Solution:

Question 3.
Two adjacent sides of a parallelogram are 24 cm and 18 cm. If the distance between longer sides is 12 cm, find the distance between shorter sides.
Solution:

Question 4.
Rajesh has a square plot with the measurement as shown in the given figure. He wants to construct a house in the middle of the plot. A garden is developed around the house. Find the total cost of developing a garden around the house at the rate of ₹50 per m2.

Solution:

Question 5.
A flooring tile has a shape of a parallelogram whose base is 18 cm and the corresponding height is 6 cm. How many such tiles are required to cover a floor of area 540 m2? (If required you can split the tiles in whatever way you want to fill up the comers).
Solution:

Question 6.
An ant is moving around a few food pieces of different shapes scattered on the floor. For which food piece would the ant have to take a longer round?

Solution:

Question 7.
In the adjoining figure, the area enclosed between the concentric circles is 770 cm2. If the radius of the outer circle is 21 cm, calculate the radius of the inner circle.

Solution:

Question 8.
A copper wire when bent in the form of a square encloses an area of 121 cm2. If the same wire is bent into the form of a circle, find the area of the circle.
Solution:

Question 9.
From the given figure, find
(i) the area of ∆ ABC
(ii) length of BC
(iii) the length of altitude from A to BC

Solution:

Question 10.
A rectangular garden 80 m by 40 m is divided into four equal parts by two cross-paths 2.5 m wide. Find
(i) the area of the cross-paths.
(ii) the area of the unshaded portion.

Solution:

Question 11.
In the given figure, ABCD is a rectangle. Find the area of the shaded region.

Solution:

Question 12.
In the adjoining figure, ABCD is a square grassy lawn of area 729 m2. A path of uniform width runs all around it. If the area of the path is 295 m2, find
(i) the length of the boundary of the square field enclosing the lawn and the path.
(ii) the width of the path.

Solution:

## Online Education for ML Aggarwal Class 7 Solutions for ICSE Maths Chapter 9 Linear Equations and Inequalities Ex 9.2

ML Aggarwal Class 7 Solutions Online Education Chapter 9 Linear Equations and Inequalities Ex 9.2 for ICSE Understanding Mathematics acts as the best resource during your learning and helps you score well in your exams.

## Online Education ML Aggarwal Class 7 Solutions for ICSE Maths Chapter 9 Linear Equations and Inequalities Ex 9.2

Question 1.
If 7 is added to five times a number, the result is 57. Find the number.
Solution:

Question 2.
Find a number, such that one-fourth of the number is 3 more than 7.
Solution:

Question 3.
A number is as much greater than 15 as it is less than 51. Find the number.
Solution:

Question 4.
If $$\frac { 1 }{ 2 }$$ is subtracted from a number and the difference is multiplied by 4, the result is 5. What is the number?
Solution:

Question 5.
The sum of two numbers is 80 and the greater number exceeds twice the smaller by 11. Find the numbers.
Solution:

Question 6.
Find three consecutive odd natural numbers whose sum is 87.
Solution:

Question 7.
In a class of 35 students, the number of girls is two-fifths of the number of boys. Find the number of girls in the class.
Solution:

Question 8.
A chair costs ₹ 250 and the table costs ₹ 400. If a housewife purchased a certain number of chairs and two tables for ₹ 2800, find the number of chairs she purchased.
Solution:

Question 9.
Aparna got ₹ 27840 as her monthly salary and over-time. Her salary exceeds the overtime by ₹ 16560. What is her monthly salary?
Solution:

Question 10.
Heena has only ₹ 2 and ₹ 5 coins in her purse. If in all she has 80 coins in her purse amounting to ₹ 232, find the number of ₹ 5 coins.
Solution:

Question 11.
A purse contains ₹ 550 in notes of denominations of ₹ 10 and ₹ 50. If the number of ₹ 50 notes is one less than that of ₹ 10 notes, then find the number of ₹ 50 notes.
Solution:

Question 12.
After 12 years, 1 shall be 3 times as old as I was 4 years ago. Find my present age.
Solution:

Question 13.
Two equal sides of an isosceles triangle are 3x – 1 and 2x + 2. The third side is 2x units. Find x and the perimeter of the triangle.
Solution:

Question 14.
The length of a rectangle plot is 6 m less than thrice its breadth. Find the dimensions of the plot if its perimeter is 148 m.
Solution:

Question 15.
Two complementary angles differ by 20°. Find the measure of each angle.
Solution:

## Online Education for Selina Concise Mathematics Class 8 ICSE Solutions Chapter 20 Area of Trapezium and a Polygon

Selina Publishers Concise Mathematics Class 8 ICSE Solutions Chapter 20 Area of Trapezium and a Polygon

### Area of Trapezium and a Polygon Exercise 20A – Selina Concise Mathematics Class 8 ICSE Solutions

Question 1.
Find the area of a triangle, whose sides are :
(i) 10 cm, 24 cm and 26 cm
(ii) 18 mm, 24 mm and 30 mm
(iii) 21 m, 28 m and 35 m
Solution:

Question 2.
Two sides of a triangle are 6 cm and 8 cm. If height of the triangle corresponding to 6 cm side is 4 cm ; find :
(i) area of the triangle
(ii) height of the triangle corresponding to 8 cm side.
Solution:

Question 3.
The sides of a triangle are 16 cm, 12 cm and 20 cm. Find :
(i) area of the triangle ;
(ii) height of the triangle, corresponding to the largest side ;
(iii) height of the triangle, corresponding to the smallest side.
Solution:

Question 4.
Two sides of a triangle are 6.4 m and 4.8 m. If height of the triangle corresponding to 4.8 m side is 6 m; find :
(i) area of the triangle ;
(ii) height of the triangle corresponding to 6.4 m side.
Solution:

Question 5.
The base and the height of a triangle are in the ratio 4 : 5. If the area of the triangle is 40 m2; find its base and height.
Solution:

Question 6.
The base and the height of a triangle are in the ratio 5 : 3. If the area of the triangle is 67.5 m2; find its base and height.
Solution:

Question 7.
The area of an equilateral triangle is 144√3 cm2; find its perimeter.
Solution:

Question 8.
The area of an equilateral triangle is numerically equal to its perimeter. Find its perimeter correct to 2 decimal places.
Solution:

Question 9.
A field is in the shape of a quadrilateral ABCD in which side AB = 18 m, side AD = 24 m, side BC = 40m, DC = 50 m and angle A = 90°. Find the area of the field.
Solution:

Question 10.
The lengths of the sides of a triangle are in the ratio 4 : 5 : 3 and its perimeter is 96 cm. Find its area.
Solution:

Question 11.
One of the equal sides of an isosceles triangle is 13 cm and its perimeter is 50 cm. Find the area of the triangle.
Solution:

Question 12.
The altitude and the base of a triangular field are in the ratio 6 : 5. If its cost is ₹ 49,57,200 at the rate of ₹ 36,720 per hectare and 1 hectare = 10,000 sq. m, find (in metre) dimensions of the field,
Solution:

Question 13.
Find the area of the right-angled triangle with hypotenuse 40 cm and one of the other two sides 24 cm.
Solution:

Question 14.
Use the information given in the adjoining figure to find :
(i) the length of AC.
(ii) the area of a ∆ABC
(iii) the length of BD, correct to one decimal place.

Solution:

### Area of Trapezium and a Polygon Exercise 20B – Selina Concise Mathematics Class 8 ICSE Solutions

Question 1.
Find the length and perimeter of a rectangle, whose area = 120 cm2 and breadth = 8 cm
Solution:

Question 2.
The perimeter of a rectangle is 46 m and its length is 15 m. Find its :
(ii) area
(iii) diagonal.
Solution:

Question 3.
The diagonal of a rectangle is 34 cm. If its breadth is 16 cm; find its :
(i) length
(ii) area
Solution:

Question 4.
The area of a small rectangular plot is 84 m2. If the difference between its length and the breadth is 5 m; find its perimeter.
Solution:

Question 5.
The perimeter of a square is 36 cm; find its area
Solution:

Question 6.
Find the perimeter of a square; whose area is : 1.69 m2
Solution:

Question 7.
The diagonal of a square is 12 cm long; find its area and length of one side.
Solution:

Question 8.
The diagonal of a square is 15 m; find the length of its one side and perimeter.
Solution:

Question 9.
The area of a square is 169 cm2. Find its:
(i) one side
(ii) perimeter
Solution:

Question 10.
The length of a rectangle is 16 cm and its perimeter is equal to the perimeter of a square with side 12.5 cm. Find the area of the rectangle.
Solution:

Question 11.
The perimeter of a square is numerically equal to its area. Find its area.
Solution:

Question 12.
Each side of a rectangle is doubled. Find the ratio between :
(i) perimeters of the original rectangle and the resulting rectangle.
(ii) areas of the original rectangle and the resulting rectangle.
Solution:

Question 13.
In each of the following cases ABCD is a square and PQRS is a rectangle. Find, in each case, the area of the shaded portion.
(All measurements are in metre).

Solution:

Question 14.
A path of uniform width, 3 m, runs around the outside of a square field of side 21 m. Find the area of the path.
Solution:

Question 15.
A path of uniform width, 2.5 m, runs around the inside of a rectangular field 30 m by 27 m. Find the area of the path.
Solution:

Question 16.
The length of a hall is 18 m and its width is 13.5 m. Find the least number of square tiles, each of side 25 cm, required to cover the floor of the hall,
(i) without leaving any margin.
(ii) leaving a margin of width 1.5 m all around. In each case, find the cost of the tiles required at the rate of Rs. 6 per tile
Solution:

Question 17.
A rectangular field is 30 m in length and 22m in width. Two mutually perpendicular roads, each 2.5 m wide, are drawn inside the field so that one road is parallel to the length of the field and the other road is parallel to its width. Calculate the area of the crossroads.
Solution:

Question 18.
The length and the breadth of a rectangular field are in the ratio 5 : 4 and its area is 3380 m2. Find the cost of fencing it at the rate of ₹75 per m.
Solution:

Question 19.
The length and the breadth of a conference hall are in the ratio 7 : 4 and its perimeter is 110 m. Find:
(i) area of the floor of the hall.
(ii) number of tiles, each a rectangle of size 25 cm x 20 cm, required for flooring of the hall.
(iii) the cost of the tiles at the rate of ₹ 1,400 per hundred tiles.
Solution:

### Area of Trapezium and a Polygon Exercise 20C – Selina Concise Mathematics Class 8 ICSE Solutions

Question 1.
The following figure shows the cross-section ABCD of a swimming pool which is trapezium in shape.
If the width DC, of the swimming pool is 6.4cm, depth (AD) at the shallow end is 80 cm and depth (BC) at deepest end is 2.4m, find Its area of the cross-section.

Solution:

Question 2.
The parallel sides of a trapezium are in the ratio 3 : 4. If the distance between the parallel sides is 9 dm and its area is 126 dm2 ; find the lengths of its parallel sides.
Solution:

Question 3.
The two parallel sides and the distance between them are in the ratio 3 : 4 : 2. If the area of the trapezium is 175 cm2, find its height.
Solution:

Question 4.
A parallelogram has sides of 15 cm and 12 cm; if the distance between the 15 cm sides is 6 cm; find the distance between 12 cm sides.
Solution:

Question 5.
A parallelogram has sides of 20 cm and 30 cm. If the distance between its shorter sides is 15 cm; find the distance between the longer sides.
Solution:

Question 6.
The adjacent sides of a parallelogram are 21 cm and 28 cm. If its one diagonal is 35 cm; find the area of the parallelogram.
Solution:

Question 7.
The diagonals of a rhombus are 18 cm and 24 cm. Find:
(i) its area ;
(ii) length of its sides.
(iii) its perimeter;
Solution:

Question 8.
The perimeter of a rhombus is 40 cm. If one diagonal is 16 cm; find :
(i) its another diagonal
(ii) area
Solution:

Question 9.
Each side of a rhombus is 18 cm. If the distance between two parallel sides is 12 cm, find its area.
Solution:

Question 10.
The length of the diagonals of a rhombus is in the ratio 4 : 3. If its area is 384 cm2, find its side.
Solution:

Question 11.
A thin metal iron-sheet is rhombus in shape, with each side 10 m. If one of its diagonals is 16 m, find the cost of painting its both sides at the rate of ₹ 6 per m2.
Also, find the distance between the opposite sides of this rhombus.
Solution:

Question 12.
The area of a trapezium is 279 sq.cm and the distance between its two parallel sides is 18 cm. If one of its parallel sides is longer than the other side by 5 cm, find the lengths of its parallel sides.
Solution:

Question 13.
The area of a rhombus is equal to the area of a triangle. If base of ∆ is 24 cm, its corresponding altitude is 16 cm and one of the diagonals of the rhombus is 19.2 cm. Find its other diagonal.
Solution:

Question 14.
Find the area of the trapezium ABCD in which AB//DC, AB = 18 cm, ∠B = ∠C = 90°, CD = 12 cm and AD = 10 cm.
Solution:

### Area of Trapezium and a Polygon Exercise 20D – Selina Concise Mathematics Class 8 ICSE Solutions

Question 1.
Find the radius and area of a circle, whose circumference is :
(i) 132 cm
(ii) 22 m
Solution:

Question 2.
Find the radius and circumference of a circle, whose area is :
(i) 154 cm2
(ii) 6.16 m2
Solution:

Question 3.
The circumference of a circular table is 88 m. Find its area.
Solution:

Question 4.
The area of a circle is 1386 sq.cm ; find its circumference.
Solution:

Question 5.
Find the area of a flat circular ring formed by two concentric circles (circles with same centre) whose radii are 9 cm and 5 cm.
Solution:

Question 6.
Find the area of the shaded portion in each of the following diagrams :

Solution:

Question 7.
The radii of the inner and outer circumferences of a circular running track are 63 m and 70 m respectively. Find :
(i) the area of the track ;
(it) the difference between the lengths of the two circumferences of the track.
Solution:

Question 8.
A circular field cf radius 105 m has a circular path of uniform width of 5 m along and inside its boundary. Find the area of the path.
Solution:

Question 9.
There is a path of uniform width 7 m round and outside a circular garden of diameter 210 m. Find the area of the path.
Solution:

Question 10.
A wire, when bent in the form of a square encloses an area of 484 cm2. Find :
(i) one side of the square ;
(ii) length of the wire ;
(iii) the largest area enclosed; if the same wire is bent to form a circle.
Solution:

Question 11.
A wire, when bent in the form of a square; encloses an area of 196 cm2. If the same wire is bent to form a circle; find the area of the circle.
Solution:

Question 12.
The radius of a circular wheel is 42 cm. Find the distance travelled by it in :
(i) 1 revolution ;
(ii) 50 revolutions ;
(iii) 200 revolutions ;
Solution:

Question 13.
The diameter of a wheel is 0.70 m. Find the distance covered by it in 500 revolutions. If the wheel takes 5 minutes to make 500 revolutions; find its speed in :
(i) m/s
(ii) km/hr.
Solution:

Question 14.
A bicycle wheel, diameter 56 cm, is making 45 revolutions in every 10 seconds. At what speed in kilometre per hour is the bicycle travelling ?
Solution:

Question 15.
A roller has a diameter of 1.4 m. Find :
(i) its circumference ;
(ii) the number of revolutions it makes while travelling 61.6 m.
Solution:

Question 16.
Find the area of the circle, length of whose circumference is equal to the sum of the lengths of the circumferences with radii 15 cm and 13 cm.
Solution:

Question 17.
A piece of wire of length 108 cm is bent to form a semicircular arc bounded by its diameter. Find its radius and area enclosed.
Solution:

Question 18.
In the following figure, a rectangle ABCD enclosed three circles. If BC = 14 cm, find the area of the shaded portion (Take π = 22/7)

Solution:

## Online Education for Understanding ICSE Mathematics Class 9 ML Aggarwal Solved Solutions

Get Latest Edition of ML Aggarwal Class 9 Solutions PDF Download on LearnInsta.com. It provides step by step solutions for ML Aggarwal Maths for Class 9 ICSE Solutions Pdf Download. You can download the Understanding ICSE Mathematics Class 9 ML Aggarwal Solved Solutions with Free PDF download option, which contains chapter wise solutions. APC Maths Class 9 Solutions ICSE all questions are solved and explained by expert Mathematic teachers as per ICSE board guidelines.

APC Understanding ICSE Mathematics Class 9 ML Aggarwal Solutions 2019 Edition for 2020 Examinations

ML Aggarwal Class 9 Maths Chapter 1 Rational and Irrational Numbers

ML Aggarwal Class 9 Maths Chapter 2 Compound Interest

• Chapter 2 Compound Interest Ex 2.1
• Chapter 2 Compound Interest Ex 2.2
• Chapter 2 Compound Interest Ex 2.3
• Chapter 2 Compound Interest Multiple Choice Questions
• Chapter 2 Compound Interest Chapter Test

ML Aggarwal Class 9 Maths Chapter 3 Expansions

• Chapter 3 Expansions Ex 3.1
• Chapter 3 Expansions Ex 3.2
• Chapter 3 Expansions Multiple Choice Questions
• Chapter 3 Expansions Chapter Test

ML Aggarwal Class 9 Maths Chapter 4 Factorisation

• Chapter 4 Factorisation Ex 4.1
• Chapter 4 Factorisation Ex 4.2
• Chapter 4 Factorisation Ex 4.3
• Chapter 4 Factorisation Ex 4.4
• Chapter 4 Factorisation Ex 4.5
• Chapter 4 Factorisation Multiple Choice Questions
• Chapter 4 Factorisation Chapter Test

ML Aggarwal Class 9 Maths Chapter 5 Simultaneous Linear Equations

• Chapter 5 Simultaneous Linear Equations Ex 5.1
• Chapter 5 Simultaneous Linear Equations Ex 5.2
• Chapter 5 Simultaneous Linear Equations Ex 5.3
• Chapter 5 Simultaneous Linear Equations Ex 5.4
• Chapter 5 Simultaneous Linear Equations Multiple Choice Questions
• Chapter 5 Simultaneous Linear Equations Chapter Test

ML Aggarwal Class 9 Maths Chapter 6 Problems on Simultaneous Linear Equations

• Chapter 6 Problems on Simultaneous Linear Equations Ex 6
• Chapter 6 Problems on Simultaneous Linear Equations Multiple Choice Questions
• Chapter 6 Problems on Simultaneous Linear Equations Chapter Test

ML Aggarwal Class 9 Maths Chapter 7 Quadratic Equations

• Chapter 7 Quadratic Equations Ex 7
• Chapter 7 Quadratic Equations Multiple Choice Questions
• Chapter 7 Quadratic Equations Chapter Test

ML Aggarwal Class 9 Maths Chapter 8 Indices

• Chapter 8 Indices Ex 8
• Chapter 8 Indices Multiple Choice Questions
• Chapter 8 Indices Chapter Test

ML Aggarwal Class 9 Maths Chapter 9 Logarithms

• Chapter 9 Logarithms Ex 9.1
• Chapter 9 Logarithms Ex 9.2
• Chapter 9 Logarithms Multiple Choice Questions
• Chapter 9 Logarithms Chapter Test

ML Aggarwal Class 9 Maths Chapter 10 Triangles

• Chapter 10 Triangles Ex 10.1
• Chapter 10 Triangles Ex 10.2
• Chapter 10 Triangles Ex 10.3
• Chapter 10 Triangles Ex 10.4
• Chapter 10 Triangles Multiple Choice Questions
• Chapter 10 Triangles Chapter Test

ML Aggarwal Class 9 Maths Chapter 11 Mid Point Theorem

• Mid Point Theorem Ex 11
• Mid Point Theorem Multiple Choice Questions
• Mid Point Theorem Chapter Test

ML Aggarwal Class 9 Maths Chapter 12 Pythagoras Theorem

• Chapter 12 Pythagoras Theorem Ex 12
• Chapter 12 Pythagoras Theorem Multiple Choice Questions
• Chapter 12 Pythagoras Theorem Chapter Test

ML Aggarwal Class 9 Maths Chapter 13 Rectilinear Figures

• Chapter 13 Rectilinear Figures Ex 13.1
• Chapter 13 Rectilinear Figures Ex 13.2
• Chapter 13 Rectilinear Figures Multiple Choice Questions
• Chapter 13 Rectilinear Figures Chapter Test

ML Aggarwal Class 9 Maths Chapter 14 Theorems on Area

• Chapter 14 Theorems on Area Ex 14
• Chapter 14 Theorems on Area Multiple Choice Questions
• Chapter 14 Theorems on Area Chapter Test

ML Aggarwal Class 9 Maths Chapter 15 Circle

• Chapter 15 Circle Ex 15.1
• Chapter 15 Circle Ex 15.2
• Chapter 15 Circle Multiple Choice Questions
• Chapter 15 Circle Chapter Test

ML Aggarwal Class 9 Maths Chapter 16 Mensuration

• Chapter 16 Mensuration Ex 16.1
• Chapter 16 Mensuration Ex 16.2
• Chapter 16 Mensuration Ex 16.3
• Chapter 16 Mensuration Ex 16.4
• Chapter 16 Mensuration Multiple Choice Questions
• Chapter 16 Mensuration Chapter Test

ML Aggarwal Class 9 Maths Chapter 17 Trigonometric Ratios

• Chapter 17 Trigonometric Ratios Ex 17
• Chapter 17 Trigonometric Ratios Multiple Choice Questions
• Chapter 17 Trigonometric Ratios Chapter Test

ML Aggarwal Class 9 Maths Chapter 18 Trigonometric Ratios and Standard Angles

• Chapter 18 Trigonometric Ratios and Standard Angles Ex 18.1
• Chapter 18 Trigonometric Ratios and Standard Angles Ex 18.2
• Chapter 18 Trigonometric Ratios and Standard Angles Multiple Choice Questions
• Chapter 18 Trigonometric Ratios and Standard Angles Chapter Test

ML Aggarwal Class 9 Maths Chapter 19 Coordinate Geometry

• Chapter 19 Coordinate Geometry Ex 19.1
• Chapter 19 Coordinate Geometry Ex 19.2
• Chapter 19 Coordinate Geometry Ex 19.3
• Chapter 19 Coordinate Geometry Ex 19.4
• Chapter 19 Coordinate Geometry Multiple Choice Questions
• Chapter 19 Coordinate Geometry Chapter Test

ML Aggarwal Class 9 Maths Chapter 20 Statistics

• Chapter 20 Statistics Ex 20.1
• Chapter 20 Statistics Ex 20.2
• Chapter 20 Statistics Ex 20.3
• Chapter 20 Statistics Multiple Choice Questions
• Chapter 20 Statistics Chapter Test

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## ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 1 Rational Numbers Ex 1.2

ML Aggarwal Class 9 Solutions Chapter 1 Rational Numbers Ex 1.2 for ICSE Understanding Mathematics acts as the best resource during your learning and helps you score well in your exams.

## ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 1 Rational Numbers Ex 1.2

Question 1.
Prove that $$\sqrt{5}$$ is an irrational number. Hence show that $$\frac {2}{3}$$$$\sqrt{5}$$ is an irrational number.
Solution:
Let $$\sqrt{5}$$ is a rational number
Let $$\sqrt{5}$$ = $$\frac {p}{q}$$ where p and q are integer and q > 0, p and q have no common factor except 1
Squaring both sides

⇒ p2 = 5q2
∴ 5q2 is divisible by 5
∴ p2 is also divisible by 5
⇒ p is divisible by 5
Let p = 5k where k is an integer
squaring both sides
p2 = 25 k2
⇒ 5q2 = 25k2
⇒ q2 = 5k2
∴ 5k2 is divisible by 5
∴ q2 is also divisible by 5
⇒ q is divisible by 5
∴ p and q are both divisible by 5
our supposition is wrong as p and q have no common factor
∴ $$\sqrt{5}$$ is an irrational number
Now in $$\frac {2}{3}$$$$\sqrt{5}$$ , $$\frac {2}{3}$$ is a rational number and $$\sqrt{5}$$ is an irrational number.

But product of a rational number and an irrational number is also an irrational number
∴ $$\frac {2}{3}$$$$\sqrt{5}$$ is an irrational number.
Hence proved.

Question 2.
Prove that $$\sqrt{7}$$ is an irrational number.
Solution:
Let $$\sqrt{7}$$ is a rational number
Let $$\sqrt{7}$$ = $$\frac {p}{q}$$
Where p and q are integers, q ≠ 0 and p and q have no common factor
Squaring both sides,

⇒ p2 = 7q2
∴ p2 is a multiple of 7
⇒ p is multiple of 7
Let p = 7 m
Where m is an integer
∴ Then (7 m)2 = 7q2 ⇒ 49 m2 = 7q2
⇒ q2 = 7 m2
∴ q2 is multiple of 7
⇒ q is multiple of 7
p and q both are multiple of 7
Which is not possible
Hence $$\sqrt{7}$$ is not a reational number
∴ $$\sqrt{7}$$ is an irrational number

Question 3.
Prove that $$\sqrt{6}$$ is an irrational number.
Solution:
Let $$\sqrt{6}$$ is a rational number
and $$\sqrt{6}$$ = $$\frac {p}{q}$$ where p and q are integers and q ≠ 0 and have no common factor

= p2 = 6q2 ………(i)
∴ p2 is divisible by 2 which is a prime
p is also divisible by 2
Let p = 2k where k is an integer
∴ Substituting the value of p in (i)
(2k)2 = 6q2 ⇒ 4k2 = 6q2
⇒ 2k2 = 3q2
∴ q2 is divisible by 2
⇒ q is divisible
p and q both are divisible by 2
Which is not possible as p and q both have
no common factor
Hence $$\sqrt{6}$$ is an irrational number

Question 4.
Prove that $$\frac{1}{\sqrt{11}}$$ is an irrational number.
Solution:
Let $$\frac{1}{\sqrt{11}}$$ is a rational number
Let $$\frac{1}{\sqrt{11}}$$ = $$\frac {p}{q}$$ where p and q are integers
and q ≠ 0 and have no common factor Squaring both sides

∴ q2 is divisible by 11
⇒ q is divisible by 11
Let q = 11k where k is an integer squaring
q2 = 121k2
Substituting the value of q in (i)
∴ 121k2 = 11p2
⇒ 11k2 = p2
∴ p2 is divisible by 11
⇒ p is divisible by 11
∴ p and q both are divisible by 11
But it is not possible
∴ $$\frac{1}{\sqrt{11}}$$ is an irrational number

Question 5.
Prove that $$\sqrt{2}$$ is an irrational number. Hence show that 3 – $$\sqrt{2}$$ is an irrational number.
(i) Let $$\sqrt{2}$$ be a rational number, then by definition
$$\sqrt{2}$$ = $$\frac {p}{ q}$$ where p, q are integers ,q>0, p and q have no common factor.
Since, 12 – 1, 22 = 4 and 1 < 0 < 4, It follows that

In particular, if q = 1, then we get 1 < p < 2 But, there is no integer between 1 and 2. ∴ q ≠ 1 so q > 1

As 2 and q are both integers, 2q is an integer. On the other hand, q > 1 and p,q have no common factor. So p2 and q have no common factor. It follows that $$\frac {p}{q}$$ is not an integer. Thus, we arrive at a contradiction. Hence $$\sqrt{2}$$ is not a rational number.

If possible, let 3 – $$\sqrt{2}$$ is an rational number say r (r ≠ 0), then
3 – $$\sqrt{2}$$ = r ⇒ – $$\sqrt{2}$$ = r – 3 ⇒ $$\sqrt{2}$$ = 3 – r
As r is a rational number and r ≠ 0, Then 3 – r is rational
⇒ $$\sqrt{2}$$ is rational, which is wrong, Hence 3 – $$\sqrt{2}$$ is irrational number.

Question 6.
Prove that $$\sqrt{3}$$ is an irrational number. Hence, show that $$\frac{2}{5}$$$$\sqrt{3}$$ is an irrational number.
Solution:
Let $$\sqrt{3}$$ is a rational number
and let $$\sqrt{3}$$ = $$\frac{p}{q}$$ where p and q are integers,
q ≠ 0 and have no common factors both sides
Squaring both sides

p2 is divisible by 3
⇒ p is divisible by 3
Let p = 3k where k is an integer
Squaring both sides
p2 = 9k2
Substituting the value of p2 in (i)
9k2 = 3q2 ⇒ q2 = 3k2
∴ q2 is divisible by 3
⇒ q is divisible by 3
∴ p and q both are divisible by 3
But it is not pissible
∴ $$\sqrt{3}$$ is an irrational number
Now in $$\frac{2}{5}$$$$\sqrt{3}$$
2 and 5 both are rational numbers.
∴ $$\frac{2}{5}$$$$\sqrt{3}$$ is irrational number as product of rational and irrational is irrational
Hence $$\frac{2 \sqrt{3}}{5}$$ is an irrational number.

Question 7.
Prove that √5 is an irrational number.
Hence, show that -3 + 2√5 is an irrational number.
Let $$\sqrt{5}$$ is a rational number
and let $$\sqrt{5}$$ = $$\frac {p}{q}$$ where p and q are integers,
q ≠ 0 and have no common factors both sides
Squaring both sides

p2 is divisible by 5
⇒ p is divisible by 5
Let p = 5k where k is an integer
Squaring both sides
p2 = 25k2
Substituting the value of p2 in (i)
25k2 = 5q2 => q2 = 5k2
q2 is divisible by 5
⇒ is divisible by 5
∴ p and q both are divisible by 5
But it is not possible
$$\sqrt{5}$$ is an irrational number
Now in – 3 + 2$$\sqrt{5}$$
– 3 and 2 both are rational numbers
∴ 2$$\sqrt{5}$$ is irrational number as product of a rational and irrational is irrational
Hence – 3 + 2$$\sqrt{5}$$ is an irrational number

Question 8.
Prove that the following numbers are irrational:

(i) Suppose that 5 + $$\sqrt{2}$$ is rational number Say r (r ≠ 0) then
5 + $$\sqrt{2}$$ = r $$\sqrt{2}$$ = r – 5
As r is rational number, then r – 5 is also rational number.
⇒ $$\sqrt{2}$$ is rational number, which is wrong,
∴ our supposition is wrong.
Hence, 5 + $$\sqrt{2}$$ is irrational number.

(ii) 3 – 5$$\sqrt{3}$$
Suppose 3 – 5$$\sqrt{3}$$ is a rational
and let 3 – 5$$\sqrt{3}$$ = r
⇒ 5$$\sqrt{3}$$ = 3 – r = > 73 = $$\sqrt{3}=\frac{3-r}{5}$$
∵ r is a rational number 3-r
∴ $$\frac{3-r}{5}$$ is also a rational number
But $$\sqrt{3}$$ is an irrational number
∴It is not possible
∴ 3 – 5$$\sqrt{3}$$ is an irrational number

(iii) 2$$\sqrt{3}$$ – 7
Let 2$$\sqrt{3}$$ – 7 is a rational number
and let 2$$\sqrt{3}$$ – 7 = r
= > 2$$\sqrt{3}$$ = r + 7 ⇒ $$\sqrt{3}=\frac{r+7}{2}$$
∴ r is a rational number
∴ $$\frac{r+7}{2}$$ is also a rational number
But $$\sqrt{3}$$ is an irrational number
∴ It is not possible
2$$\sqrt{3}$$ – 7 is an irrational number

(iv) $$\sqrt{2}$$ + $$\sqrt{5}$$
Suppose $$\sqrt{2}$$ + $$\sqrt{5}$$ isa rational number and
let x = $$\sqrt{2}$$ + $$\sqrt{5}$$
Squaring both sides,

$$\sqrt{10}$$ is a rational number
But it is not true as $$\sqrt{10}$$ is an irrational number
∴ Our supposition is wrong
∴ $$\sqrt{2}$$ + $$\sqrt{5}$$ is an irrational number.

## ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 1 Rational Numbers Ex 1.1

ML Aggarwal Class 9 Solutions Chapter 1 Rational Numbers Ex 1.1 for ICSE Understanding Mathematics acts as the best resource during your learning and helps you score well in your exams.

## ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 1 Rational Numbers Ex 1.1

Question 1.
Insert a rational number between $$\frac {2}{9}$$ and $$\frac {3}{8}$$, and arrange in descending order.
Solution:
A rational number between $$\frac {2}{9}$$ and $$\frac {3}{8}$$

Question 2.
Insert two rational numbers between, $$\frac {1}{3}$$ and $$\frac {1}{4}$$, and arrange in ascending order.
Solution:
A rational number between and $$\frac {1}{3}$$ and $$\frac {1}{4}$$

A rational number between and $$\frac {1}{4}$$ and $$\frac {7}{24}$$

Question 3.
Insert two rational numbers between – $$\frac {1}{3}$$ and – $$\frac {1}{2}$$ and arrange in ascending order.
Solution:
L.C.M. of 3 and 2 is 6

∴ Two rational numbers between $$\frac {- 2}{6}$$ and $$\frac {- 3}{6}$$

Question 4.
Insert 3 rational numbers between $$\frac {1}{3}$$ and $$\frac {4}{5}$$ and arrange in descending order.
Solution:
A rational number between $$\frac {1}{3}$$ and $$\frac {4}{5}$$

Question 5.
Insert three rational numbers between 4 and 4.5.
Solution:

∵ 4 < 4.0625 < 4.125 < 4.25 < 4.5
∴ Three rational numbers between 4 and 4.5 are 4.0625, 4.125, 4.25

Question 6.
Find six rational numbers between 3 and 4.
Solution:
Six rational numbers between 3 and 4
First rational number between 3 and 4

Question 7.
Find five rational numbers between $$\frac {3}{5}$$ and $$\frac {4}{5}$$.
Solution:
Five rational numbers between $$\frac {3}{5}$$ and $$\frac {4}{5}$$
Multiplying and dividing by 5 + 1 = 6

Question 8.
Find ten rational numbers between $$\frac {- 2}{5}$$ and $$\frac {1}{7}$$
Solution:
Ten rational numbers between $$\frac {- 2}{5}$$ and $$\frac {1}{7}$$
LCM of 5 and 7 = 35

Question 9.
Find six rational numbers between $$\frac {1}{2}$$ and $$\frac {2}{3}$$.
Solution:
Six rational number between $$\frac {1}{2}$$ and $$\frac {2}{3}$$
LCM of 2, 3 = 6