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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.
Answer:
(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.
Answer:
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:

Answer:
(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 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

ML Aggarwal Class 6 Solutions for ICSE Maths Model Question Paper 6 acts as the best resource during your learning and helps you score well in your exams.

ML Aggarwal Class 6 Solutions for ICSE Maths Model Question Paper 6

Questions 1 to 8 are of 1 mark each.
Choose the correct answer from the given four options (1 to 8):
Question 1.
The value of the expression \(\frac{5}{3}\)x² – 1 when x = -2 is

Solution.

Question 2.
By joining any two points of a circle, we obtain its
(a) radius
(b) circumference
(c) diameter
(d) chord
Solution:
chord (d)

Question 3.
Which of the following statement is true?
(a) Every closed curve is a polygon
(b) Every closed simple curve is a polygon
(c) Every simple curve made up entirely of line segment is a polygon
(d) Every simple closed curve made up entirely of line segments is a polygon.
Solution:

Question 4.
The median of the numbers 3, 1,0, 6, 5, 3, 4, 1, 2, 2 is
(a) 2
(b) 2.5
(c) 3
(d) none of these
Solution:

Question 5.
If the perimeter of a regular octagon is 72 cm, then its side is
(a) 6 cm
(b) 8 cm
(c) 9 cm
(d) 12 cm
Solution:

Question 6.
If Anandi’s present age is x years and her father’s age is 3 years less than 4 times her age, then her father’s present age is
(a) (4x – 3) years
(b) (3x – 4) years
(c) 4(x – 3) years
(d) (4x + 3) years
Solution:

Question 7.
The number of lines of symmetry which a quadrilateral cannot have is
(a) 1
(b) 2
(c) 3
(d) 4
Solution:

Question 8.
The number of bisectors that can be drawn of a given angle is
(a) 1
(b) 2
(c) 4
(d) infinitely many
Solution:
1 (a)

Section-B
Questions 9 to 14 are of 2 marks each.
Question 9.
A cuboidal box has height h cm. Its length is 4 times the height and the breadth is 7 cm less than the length. Express the length and the breadth of the box in terms of its height.
Solution:

Question 10.
In the given figure, name the point(s)
(i) in the interior of ∠EOD.
(ii) in the exterior of ∠FOE.

Solution:

Question 11.
Write the following statement in mathematical form using literals, numbers and the signs of basic operations:
“Three times a number x is equal to 12 less than twice the number y.”
Solution:

Question 12.
If the area of a rectangular plot is 240 sq. m and its breadth is 12 m, then find the perimeter of the plot.
Solution:

Question 13.
On a squared paper, sketch a hexagon with exactly one line of symmetry.
Solution:

Question 14.
Find the area of the region enclosed by the given polygon.

Solution:

Section-C
Questions 15 to 24 are of 4 marks each.
Question 15.
In the given figure, count the number of segments and name them.

Solution:

Question 16.
In the given figure, state which of the angles marked with small letters are acute, obtuse, reflex or right angle (you may judge the nature of angle by observation).

Solution:

Question 17.
There are 40 employees in a Government Office. They were asked how many children they have. The result was:
1, 2, 3, 1, 0, 2, 0, 1, 2, 2, 1, 3, 5, 2, 0, 0, 2. 4, 1, 1
2, 2, 0, 3, 0, 0, 2, 1, 3, 6, 0, 2, 1, 0, 3, 2, 2, 2, 1, 4
(i) Arrange the above data in ascending order.
(ii) Construct frequency distribution table for the given data.
Solution:

Question 18.
A survey was carried out on 32 students of class VI in a school. Data about different modes of transport used by them to travel to school was displayed in a pictograph as under:

Observe the pictograph and answer the following questions:
Observe the pictograph and answer the following questions:
(i) Which is the most popular mode of transport?
(ii) What is the number of students who travel either by cycle or walking?
(iii) What are the advantages of using a school bus as a mode of transport?
(iv) What mode of transport would you suggest and why?
Solution:

Question 19.
If p = 4, q = 3 and r = -2, then find the value of the algebraic expression \(\frac{p^{2}+q^{2}-r^{2}}{p q+q r-p r}\).
Solution:

Question 20.
A room is 5 m long and 3 m 50 cm wide. How many square metres of carpet is needed to cover the floor of the room completely?
Solution:

Question 21.
Solve the linear equation
3(2x – 1) = 5 – (3x – 2).
Solution:

Question 22.
Copy the given figure on a squared paper and complete the figure such that the resultant figure is symmetrical about the dotted line.

Solution:

Question 23.
Draw a net of a square pyramid.
Solution:

Question 24.
Draw a line segment of length 7.5 cm and construct its axis of symmetry.
Solution:

Section-D
Questions 25 to 29 are of 6 marks each.
Question 25.
A survey was carried out on 150 families of a colony about the consumption of milk per day. The result was recorded as:

Represent the above data by a vertical bar graph, choosing scale: 1 unit height = 6 families. What are the advantages of taking milk every day?
Solution:

Question 26.
Draw a rough sketch of a regular hexagon. Connecting three of its vertices, draw
(i) an isosceles triangle
(ii) an equilateral triangle
(iii) a right-angled triangle.
Solution:

Question 27.
The cost of cultivating a rectangular field at the rate of ₹5 per square metre is ₹2880. If the length of the field is 32 m, find the cost of fencing the field at the rate of ₹11.25 per metre.
Solution:

Question 28.
By using a ruler and compass, construct an angle of 45° and bisect it. Measure any one part.
Solution:

Question 29.
Look at the following matchstick pattern of polygons. Complete the table. Also, write the general rule that gives the number of matchsticks.

Solution:

ML Aggarwal Class 6 Solutions for ICSE Maths Model Question Paper 5 acts as the best resource during your learning and helps you score well in your exams.

ML Aggarwal Class 6 Solutions for ICSE Maths Model Question Paper 5

Choose the correct answer from the given four options (1-2):
Question 1.
The number of lines of symmetry of a protractor is
(a) 0
(b) 1
(c) 2
(d) unlimited
Solution:
1 (b)

Question 2.
If the perimeter of a regular pentagon is 60 cm, then its every side is
(a) 10 cm
(b) 12 cm
(c) 15 cm
(d) 20 cm
Solution:

Question 3.
If the perimeter of a square is 42 cm, then find its area.
Solution:

Question 4.
Using a ruler and compass, construct an angle of 90°.
Solution:
Steps of Construction:

Question 5.
On a squared paper, sketch a quadrilateral with exactly two lines of symmetry. Also, sketch the lines of symmetry.
Solution:

Question 6.
If the area of a rectangular plot is 396 sq. m and its breadth is 18 m. Find the length of the plot and the cost of fencing it at the rate of ₹7.50 per metre.
Solution:

Question 7.
Draw a line segment AB of length 6.4 cm. Take a point P on AB such that AP = 4.5 cm. Draw a perpendicular to AB at P. (use ruler and compass).
Solution:

Question 8.
Copy the given figure. How many lines of symmetry it has? Draw its all lines of symmetry.

Solution:

Question 9.
In the given figure, all adjacent sides are at right angles. Find:
(i) the perimeter of the figure.
(ii) the area enclosed by the figure.

Solution:

Question 10.
Copy the given figure on a squared paper and complete the figure such that the resultant figure is symmetrical about both the dotted lines.

Solution:

https://ncertsolutionsguru.dreamwidth.org/
https://www.racked.com/users/NCERTSolutionsGuru
https://github.com/NCERTSolutionsGuru

ML Aggarwal Class 6 Solutions for ICSE Maths Model Question Paper 4 acts as the best resource during your learning and helps you score well in your exams.

ML Aggarwal Class 6 Solutions for ICSE Maths Model Question Paper 4

Choose the correct answer from the given four options (1-2):
Question 1.
Avanti’s present age is y years and her mother’s age is 4 years less than 3 times her age, then her mother’s present age is
(a) (3y + 4) years
(b) (4y – 3) years
(c) (3y – 4) years
(d) 3(y – 4) years
Solution:

Question 2.
Which of the following statements is false?
(a) Every square is a rhombus.
(b) An equilateral triangle is a regular polygon.
(c) A triangle having all acute angles is scalene.
(d) Every square is a regular polygon.
Solution:

Question 3.
If = 3, q = -2 and r = -1, find the value of: 2p² + 3q – r² + 2pr – 5pqr.
Solution:

Question 4.
Fill in the following blanks:
(i) A polygon is a closed simple curve made up of entirely ………….
(ii) A cuboid has 6 rectangular faces, edges and ………. vertices.
Solution:

Question 5.
Give reason(s) of using triangular shapes and not polygonal shapes consisting of four or more sides in constructing structures like electric towers and bridges. What value is added in using triangular shapes?
Solution:

Question 6.
Look at the following pattern of squares fonned by matchsticks:

Find the rule that gives the number of matchsticks required in terms of the number of squares formed.
Solution:

Question 7.
Name each of the following triangles in two ways (you may judge by observation or use ruler and protractor):

Solution:

Question 8.
Draw a net of a regular tetrahedron.
Solution:

Question 9.
Solve the linear equation:
4 – 3(5x + 2) = 4(7 – 3x).
Also, verify the solution.
Solution:

Question 10.
Use the given figure to name:
(i) parallel lines
(ii) concurrent lines
(ii) collinear points
(iv) two opposite rays.

Solution: