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 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

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

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:

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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:

ML Aggarwal Class 6 Solutions for ICSE Maths Model Question Paper 3

Section-A
Questions 1 to 8 is of 1 mark each.
Choose the correct answer from the given four options (1 to 8):
Question 1.
The difference between the place value and face value of 5 in the numeral 70542 is:
(a) 0
(b) 42
(c) 495
(d) 537
Solution:

Question 2.
he sum of the successor of 99 and the predecessor of 101 is
(a) 198
(b) 200
(c) 201
(d) 199
Solution:

Question 3.
If x and y are two co-prime numbers, then their LCM is

Solution:

Question 4.
Which of the following statement is true?
(a) |15 – 6| = |15| + |-6|
(b) Additive inverse of-3 is 3
(c) -1 lies on the right of 0 on the number line
(d) -4 is greater than -3
Solution:

Question 5.
The fraction equivalent to \(\frac{360}{540}\) is

Solution:

Question 6.
Which of the following numbers is divisible by 6?
(a) 5372
(b) 6495
(c) 7632
(d) 7568
Solution:

Question 7.
5 kg 5g is equal to
(a) 5.5 kg
(b) 5.05 kg
(c) 5.005 kg
(d) 5.0005 kg
Solution:

Question 8.
The ratio of a number of girls to the number of boys in a class is 7 : 5. If there are 15 boys in the class, then the number of girls in the class is
(a) 14
(b) 21
(c) 28
(d) 36
Solution:

Section-B
Questions 9 to 14 are of 2 marks each.
Question 9.
Write the greatest and the smallest 4-digit numbers using four different digits with the condition that the digit 4 occurs at tens place.
Solution:

Question 10.
Find the prime factorisation of 980.
Solution:

Question 11.
Find the value of-15 – (-2) – 71 – (-8) + 6.
Solution:

Question 12.
What fraction of the given figure is the shaded part?

Solution:

Question 13.
Write the mixed fraction \(7 \frac{3}{40}\) as a decimal number.
Solution:

Question 14.
The length of a pencil is 14 cm and its diameter is 7 mm. What is the ratio of the diameter of the pencil to that of its length?
Solution:

Section-C
Questions 15 to 24 are of 4 marks each.
Question 15.
Estimate the product 2459 x 653 by rounding off each factor to its
(i) greatest place
(ii) nearest hundreds.
Solution:

Question 16.
By using distributive laws, find 257 × 1007.
Solution:

Question 17.
Using number line, subtract (-3) from (-8).
Solution:

Question 18.
Find the greatest number that will divide 76, 113 and 186 leaving remainders 4, 5 and 6 respectively.
Solution:

Question 19.
Arrange the following integers in descending order:
-506, 2376, 2367, -311, -509, 245.
Solution:

Question 20.
Arrange the following fractions in ascending order \(\frac{5}{12}, \frac{1}{4}, \frac{7}{8}, \frac{5}{6}\).
Solution:

Question 21.
Simplify: \(3 \frac{5}{6}+4 \frac{3}{4}-5-1 \frac{3}{8}\).
Solution:

Question 22.
Write all possible natural numbers using the digits 3, 0, 8. Repetition of digits is not allowed. Also, find their sum.
Solution:

Question 23.
Munna and Munni are aged 14 years and 10 years. Their mother wants to divide ?180 between them in the ratio of their ages. How much does each get?
Solution:

Question 24.
Do the ratios 15 cm to 3 m and 15 seconds to 3 minutes form a proportion? Justify your answer.
Solution:

Section-D
Questions 25 to 29 are of 6 marks each.
Question 25.
To stitch a shirt, 2 m 25 cm cloth is needed. Out of 30 m cloth, how many shirts can be stitched and how much cloth will remain?
Solution:

Question 26.
Four bells are ringing at intervals of 40, 30, 36 and 45 minutes respectively. At what time will they ring together again if they start ringing simultaneously at 9 A.M.?
Solution:

Question 27.
I bought fruits worth ₹\(27 \frac{3}{4}\) and vegetables worth ₹\(10 \frac{1}{2}\). If I gave a fifty-rupee note to the shopkeeper, how much will I get back?
Solution:

Question 28.
Javed purchased vegetables weighing 10 kg. Out of this, 3 kg 450 g was potatoes, 2 kg 10 g was onions, 1 kg 750 g was tomatoes and the rest were green vegetables. What was the weight of green vegetables?
Solution:

Question 29.
To get a closer finish in a 100 m race, the runners are given different starting positions, depending upon how fast they can run. A starting position of +10 means that the runner starts 10 m in front of the starting line and -5 means that the runner starts 5 m behind the start line. The competitors starting positions are :
Abbas 0 Mohan -3 Rishi -20
Pete  + 10 Tarush + 15 Sahel + 5
(i) Who starts farthest from the finish line?
(ii) How far does Tarush have to run to reach the finish line?
(iii) How far Mohan has to run to reach the finish line?
(iv) How many metres are there between the starting positions of Rishi and Peter?
(v) What value is added if the students are asked to run 1 km daily?
Solution:

ML Aggarwal Class 6 Solutions for ICSE Maths Model Question Paper 2

Choose the correct answer from the given four options (1-2):
Question 1.
Every composite number has at least
(a) 1 factor
(b) 2 factors
(c) 3 factors
(d) 4 factors
Solution:

Question 2.
Which of the following collections form a set?
(a) Collection of 5 odd prime numbers
(b) Collection of 3 most intelligent students of your class
(c) Collection of 4 vowels in the English alphabet
(d) Collection of months of a year having less than 31 days.
Solution:

Question 3.
Find the prime factorisation of 1320.
Solution:

Question 4.
If A = {x : x is a positive multiple of 3 less than 20} and B = {x : x is an odd prime number less than 20}, then find n(A) + n(B).
Solution:

Question 5.
Reduce the fraction \(\frac{714}{1386}\) in its simplest form.
Solution:

Question 6.
Simplify the following:
\(2 \frac{3}{14}-3 \frac{5}{6}-\frac{2}{5}+2 \frac{1}{2}\)
Solution:

Question 7.
A number is divisible by 5 and 8 both. By what other numbers with that number be always divisible?
Solution:

Question 8.
Arrange the fractions \(\frac{2}{3}, \frac{7}{9}, \frac{5}{8}, \frac{3}{5}\) in ascending order.
Solution:

Question 9.
Find the smallest number of 5-digits which is divisible by 12, 15 and 18.
Solution:

Question 10.
Three bells are ringing continuously at intervals of 30, 40 and 45 minutes respectively. At what time will they ring together if they ring simultaneously at 5 A.M.?
Solution:

ML Aggarwal Class 6 Solutions for ICSE Maths Model Question Paper 1

Choose the correct answer from the given four options (1-2):
Question 1.
If the sum of two integers is -21 and one of them is -10 then the other is
(a) -32
(b) 32
(c) -11
(d) 11
Solution:

Question 2.
The number of natural numbers between the smallest natural number and the greatest 2-digit number is
(a) 90
(b) 97
(c) 98
(d) 99
Solution:

Question 3.
Find the value of 25 × 37 × 8 × 6 by suitable arrangement.
Solution:

Question 4.
Write four consecutive integers preceding -97.
Solution:

Question 5.
Write the greatest and the smallest 4-digit numbers using four different digits with the condition that 5 occurs at ten’s place.
Solution:

Question 6.
Write all possible natural numbers formed by the digits 7, 0 and 3. Repetition of digits is not allowed.
Solution:

Question 7.
Find the value of: -237 – (-328) + (-205) – 76 + 89.
Solution:

Question 8.
Abhijeet’s school is 3 km 520 m away from his home. One day while returning from his school, just after covering 1 km 370 m distance, he saw a woman who was bleeding, he took her to the nearest hospital which was 2 km 775 m away from that place and got her admitted. He came back to his home which was 4 km 565 m from the hospital.
(i) Find the distance covered by Abhijeet on that day.
(ii) What value of life is depicted by Abhijeet?
Solution:

Question 9.
Arrange the following integers in descending order:
-353, 207, -289, 702, -335, 0, -77.
Solution:

Question 10.
Find the smallest five-digit number which is exactly divisible by 254.
Solution:

ML Aggarwal Class 6 Solutions for ICSE Maths Chapter 15 Data Handling Check Your Progress

Question 1.
A die is thrown 25 times and the scores were as given below :
2, 1, 4, 6, 2, 3, 1, 5, 6, 3, 4, 5, 2, 1, 6, 6, 6, 3, 2, 2, 2, 4, 3, 2, 2
(i) Construct a data array.
(ii) Construct a tally chart and frequency distribution table.
Solution:

Question 2.
In primary school, the number of students in different classes is as follows:

Represent this data by a pictograph, using  ≃ 20 students.
Solution:

Question 3.
Observe the following bar graph, showing the marks scored by Gurmeet in the annual examination in different subjects: Answer the following questions:

(i) What is the scale of this bar graph?
(ii) In which subject Gurmeet obtained maximum marks?
(iii) In which subject she obtained minimum marks?
(iv) Name the subject(s) in which she got 80 or more marks.
Solution:

Question 4.
The following table shows the monthly expenditure of a family on various items:

Represent the data by a bar graph.
Solution:

Question 5.
Find the mean and the median of the following data:
5, 3, 12, 0, 7, 11, 4, 3, 9
Solution:

ML Aggarwal Class 6 Solutions for ICSE Maths Chapter 15 Data Handling Objective Type Questions

Mental Maths
Question 1.
Fill in the blanks:
(i) A ……. is a collection of numerical figures to give some information.
(ii) Each numerical figure in data is called an ………..
(iii) The number of times a particular observation occurs in a data is called … of the observation.
(iv) A data arranged in ascending or descending order is called an ………… data.
(v) A pictorial representation of a data is called a …………
(vi) If represent 5 students then  represent ………… students.
(vii) In a pictograph, if one bicycle represents 20 bicycles then 140 bicycles can be represented by ………… bicycles.
(viii)In a pictograph, if one electric bulb represents 10 bulbs then the picture of half a bulb will represent ………… bulbs.
(ix) In a pictograph, if the symbol represents 50 fruits baskets, then  represent ………… fruit baskets.
(x) The frequency of 8 is written symbolically as ………… using tally marks.
(xi) In a bar graph, the bars are of uniform …………
(xii) The mean of the first 8 natural numbers is …………
Solution:

Question 2.
State whether the following statements are true (T) or false (F):
(i) If the data is large, it is difficult to get information from raw data.
(ii) Pictographs and bar graphs help us in understanding and analyzing a data.
(iii) In a bar graph, bars can be drawn either vertically or horizontally.
(iv) In a bar graph, width of a bar has no significance. It is only for eye attraction.
(v) Usually, the tally marks are recorded in bunches of 5.
(vi) The frequency 9 is represented as filtlll using tally marks.
(vii) Mentioning of scale is necessary in pictographs.
(viii) Mean is always one of the number in a given data.
(ix) Median is always one of the numbers in a given data.
Solution:

Multiple Choice Questions
Observe the following pictograph which shows the number of ice cream cones sold by school canteen during a week. Chose the correct answer from the given options for questions 3 to 7:

Question 3.
The minimum number of ice cream cones were sold on:
(a) Monday
(b) Saturday
(c) Tuesday
(d) Thursday
Solution:
Thursday (d)

Question 4.
The maximum number of ice cream cones were sold on:
(a) Tuesday
(b) Friday
(c) Wednesday
(d) Thursday
Solution:
Tuesday (a)

Question 5.
Ratio of the number of ice cream cones sold on Saturday to the number of ice cream cones sold on Wednesday is.
(a) 3 : 2
(b) 2 : 3
(c) 4 : 5
(d) 4 : 7
Solution:

Question 6.
Total number of ice cream cones sold during the whole week was:
(a) 33
(b) 67
(c) 65
(d) 57
Solution:

Question 7.
If the cost of one ice cream cone is ₹20, then the sale value on Thursday was:
(a) ₹70
(b) ₹100
(c) ₹140
(d) ₹1340
Solution:

Observe the adjoining bar graph, showing the number of one-day international matches played by cricket teams of different countries. Choose the correct answer from the given four options for questions 8 to 11:

Question 8.
Which country played a maximum number of matches?
(a) India
(b) England
(c) Pakistan
(d) Australia
Solution:
Australia (d)

Question 9.
How many matches did South Africa play?
(a) 16
(b) 18
(c) 20
(d) 24
Solution:
18 (b)

Question 10.
How many more matches were played by India than Pakistan?
(a) 6
(b) 12
(c) 24
(d) 30
Solution:

Question 11.
Ratio of the number of matches played by India to the number of matches played by Sri Lanka is
(a) 4 : 5
(b) 5 : 4
(c) 4 : 3
(d) 7 : 6
Solution:

Choose the correct answer from the given four options for questions 12 and 13:
Question 12.
The mean of the first 6 odd natural numbers is
(a) 5
(b) 5.5
(c) 6
(d) 6.5
Solution:

Question 13.
The median of the numbers 4, 4, 7, 5, 7, 6, 1, 3, 11 is
(a) 7
(b) 6
(c) 5
(d) 4
Solution:

ML Aggarwal Class 6 Solutions for ICSE Maths

ML Aggarwal Class 6 Solutions for ICSE Maths Chapter 15 Data Handling Objective Type Questions

Mental Maths
Question 1.
Fill in the blanks:
(i) A ……. is a collection of numerical figures to give some information.
(ii) Each numerical figure in data is called an ………..
(iii) The number of times a particular observation occurs in a data is called … of the observation.
(iv) A data arranged in ascending or descending order is called an ………… data.
(v) A pictorial representation of a data is called a …………
(vi) If represent 5 students then represent ………… students.
(vii) In a pictograph, if one bicycle represents 20 bicycles then 140 bicycles can be represented by ………… bicycles.
(viii)In a pictograph, if one electric bulb represents 10 bulbs then the picture of half a bulb will represent ………… bulbs.
(ix) In a pictograph, if the symbol represents 50 fruits baskets, then represent ………… fruit baskets.
(x) The frequency of 8 is written symbolically as ………… using tally marks.
(xi) In a bar graph, the bars are of uniform …………
(xii) The mean of the first 8 natural numbers is …………
Solution:

Question 2.
State whether the following statements are true (T) or false (F):
(i) If the data is large, it is difficult to get information from raw data.
(ii) Pictographs and bar graphs help us in understanding and analyzing a data.
(iii) In a bar graph, bars can be drawn either vertically or horizontally.
(iv) In a bar graph, width of a bar has no significance. It is only for eye attraction.
(v) Usually, the tally marks are recorded in bunches of 5.
(vi) The frequency 9 is represented as filtlll using tally marks.
(vii) Mentioning of scale is necessary in pictographs.
(viii) Mean is always one of the number in a given data.
(ix) Median is always one of the numbers in a given data.
Solution:

Multiple Choice Questions
Observe the following pictograph which shows the number of ice cream cones sold by school canteen during a week. Chose the correct answer from the given options for questions 3 to 7:

Question 3.
The minimum number of ice cream cones were sold on:
(a) Monday
(b) Saturday
(c) Tuesday
(d) Thursday
Solution:
Thursday (d)

Question 4.
The maximum number of ice cream cones were sold on:
(a) Tuesday
(b) Friday
(c) Wednesday
(d) Thursday
Solution:
Tuesday (a)

Question 5.
Ratio of the number of ice cream cones sold on Saturday to the number of ice cream cones sold on Wednesday is.
(a) 3 : 2
(b) 2 : 3
(c) 4 : 5
(d) 4 : 7
Solution:

Question 6.
Total number of ice cream cones sold during the whole week was:
(a) 33
(b) 67
(c) 65
(d) 57
Solution:

Question 7.
If the cost of one ice cream cone is ₹20, then the sale value on Thursday was:
(a) ₹70
(b) ₹100
(c) ₹140
(d) ₹1340
Solution:

Observe the adjoining bar graph, showing the number of one-day international matches played by cricket teams of different countries. Choose the correct answer from the given four options for questions 8 to 11:

Question 8.
Which country played a maximum number of matches?
(a) India
(b) England
(c) Pakistan
(d) Australia
Solution:
Australia (d)

Question 9.
How many matches did South Africa play?
(a) 16
(b) 18
(c) 20
(d) 24
Solution:
18 (b)

Question 10.
How many more matches were played by India than Pakistan?
(a) 6
(b) 12
(c) 24
(d) 30
Solution:

Question 11.
Ratio of the number of matches played by India to the number of matches played by Sri Lanka is
(a) 4 : 5
(b) 5 : 4
(c) 4 : 3
(d) 7 : 6
Solution:

Choose the correct answer from the given four options for questions 12 and 13:
Question 12.
The mean of the first 6 odd natural numbers is
(a) 5
(b) 5.5
(c) 6
(d) 6.5
Solution:

Question 13.
The median of the numbers 4, 4, 7, 5, 7, 6, 1, 3, 11 is
(a) 7
(b) 6
(c) 5
(d) 4
Solution: