Squares and Square Roots Test

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Squares and Square Roots Test - 5

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Question 1
1 / -0

3136 students are sitting in an examination hall in such a manner that there are as many students in a row as there are rows in the examination hall. How many rows are there in the examination hall?

A
56

B
64

C
49

D
36

E
42

Solution

This is the correct answer.
Let the number of rows in the examination hall be a.
Since, the number of students in a row is same as the number of rows in the examination hall,
∴ Number of students in a row = a
Number of students in 'a' rows = a × a = a²
It is given that the total number of students in the examination hall = 3136
∴ a² = 3136
or, a =
or, a = (By prime factorisation)
or, a = 2 × 2 × 2 × 7
or, a = 56
Hence, there are 56 rows in the examination hall.
Question 2
1 / -0

A principal wished to arrange his students, who were 1943 in number, in the form of a square and found that there were 7 students left over. How many students were there in each row?

A
32

B
36

C
44

D
40

E
48

Solution

Let the number of students in each row be a.
Then, number of rows = a
∴ Number of students in 'a' rows = a × a = a²
It is given that after arranging the students in a square, 7 students were left over.
∴ Total number of students = a² + 7
But the total number of students is 1943.
Or a² = 1943 - 7
Or a² = 1936
Or a = Or a =
Or a = 11 × 2 × 2
Or a = 44
Thus, there were 44 students in each row.
Question 3
1 / -0

The product of two numbers is 1008 and their quotient is . Find the numbers.

A
36 and 28

B
36 and 31

C
36 and 26

D
36 and 30

E
36 and 25

Solution

This is the correct answer.
Let one of the two numbers be a.
Since, the product of numbers is 1008, the other number is .
It is given that the quotient of the numbers is .
∴ =
or, =
or, a² =
or, a² = 1296
or, a =
or, a = 36
Hence, the numbers are 36 and = 28.
Question 4
1 / -0

Find the smallest square number which is divisible by each of the numbers 8, 9 and 10.

A
3240

B
3400

C
3600

D
3960

E
4320

Solution

This is the correct answer.
The smallest number divisible by each of the numbers 8, 9 and 10 is their LCM.
The LCM of 8, 9 and 10 is 2 × 4 × 9 × 5 = 360
Resolving 360 into prime factors,
360 = 2 × 2 × 2 × 3 × 3 × 5
Grouping the factors into pairs of equal factors, we have
360 = (2 × 2) × (3 × 3) × 2 × 5
Hence, there is no factor to form pairs with 2 and 5.
Thus, to make 360 a perfect square, it must be multiplied by 2 × 5 = 10.
Hence, the smallest square number divisible by 8, 9 and 10 is 3600.
Question 5
1 / -0

Find the smallest number by which 176 must be multiplied so that the product becomes a perfect square.

A
6

B
3

C
2

D
11

E
22

Solution

This is the correct answer.
Resolving 176 into prime factors, we get
176 = 2 × 2 × 2 × 2 × 11
Grouping the factors into pairs of equal factors, we get
176 = (2 × 2) × (2 × 2) × 11
Hence, the prime factor 2 occurs in pairs, but there is no prime factor to form a pair with 11.
Therefore, the number must be multiplied by 11 so that it becomes a perfect square.
Question 6
1 / -0

What will be the square root of 12,100 by using the prime factorisation method?

A
110

B
99

C
88

D
96

E
108

Solution

This is the correct answer.
Resolving 12100 into prime factors, we get
12100 = 2 × 2 × 5 × 5 × 11 × 11
Grouping the factors into pairs of equal factors, we get
12100 = (2 × 2) × (5 × 5) × (11 × 11)
Now, taking one factor from each pair, we get
= 2 × 5 × 11 = 110
Question 7
1 / -0

What will be the square of 575 by using the identity (a - b)² = a² - 2ab + b²?

A
3,29,375

B
3,30,025

C
3,90,625

D
3,57,625

E
3,30,625

Solution

This is the correct answer.
The given number is 575.
Using the identity (a - b)² = a² - 2ab + b², we get
575² = (600 - 25)²
= 600² - 2 × 600 × 25 + 25²
= 3,60,000 - 30,000 + 625
= 3,30,625
Question 8
1 / -0

What will be the square of 189 by using the identity (a - b)² = a² - 2ab + b²?

A
35,721

B
39,681

C
35,479

D
35,611

E
44,521

Solution

This is the correct answer.
Using the identity (a - b)² = a² - 2ab + b², we get
189² = (200 - 11)²
= 40,000 - 2 × 200 × 11 + 11²
= 40,000 - 4400 + 121
= 35,721
Question 9
1 / -0

What will be the square of 211 by using the identity (a + b)² = a² + 2ab + b²?

A
44,411

B
44,279

C
44,521

D
40,561

E
35,479

Solution

This is the correct answer.
Using the identity (a + b)² = a² + 2ab + b², we get
211² = (200 + 11)²
= 200² + 2 × 200 × 11 + 11²
= 40,000 + 4400 + 121
= 44,521
Question 10
1 / -0

What will be the square of 509 by using the identity (a + b)² = a² + 2ab + b²?

A
2,59,009

B
2,58,919

C
2,50,981

D
2,59,081

E
2,41,081

Solution

This is the correct oanswer.
Using the identity (a + b)² = a² + 2ab + b², we get
509² = (500 + 9)²
= (500)² + 2 × 500 × 9 + (9)²
= 2,50,000 + 9000 + 81
= 2,59,081
Question 11
1 / -0

Which of the following are Pythagorean triplets?

A
12, 34, 35

B
10, 26, 28

C
18, 79, 81

D
8, 14, 15

E
14, 48, 50

Solution

This is the correct answer.
For any natural number m, we know that 2 m, m² - 1, m² + 1 is a Pythagorean triplet.
Here, 2 m = 14
Or, m = 7
∴ m² - 1
= 49 - 1
= 48
And, m² + 1
= 49 + 1
= 50
Thus, 14, 48 and 50 are Pythagorean triplets.
Question 12
1 / -0

Which of the following are the other two Pythagorean triplets of 16?

A
63 and 65

B
65 and 67

C
67 and 69

D
61 and 63

E
62 and 64

Solution

This is the correct answer.
For any natural number m>1, we know that 2 m, m² - 1 and m² + 1 is a Pythagorean triplet.
Here, 2 m = 16
Or, m = 8
∴ m² - 1
= 64 - 1
= 63
And, m² + 1
= 64 + 1
= 65
Thus, 63 and 65 are the Pythagorean triplets of 16.
Question 13
1 / -0

Find the smallest number by which 1,27,008 must be multiplied so that the product is a perfect square.

A
7

B
2

C
5

D
3

E
11

Solution

This is the correct answer.
Resolving 1,27,008 into prime factors, we get
1,27,008 = 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 × 7 × 7
Grouping the factors into pairs of equal factors, we get
1,27,008 = (2 × 2) × (2 × 2) × (3 × 3) × (3 × 3) × (7 × 7) × 2
For a number to be a perfect square, it should be possible to pair all its prime factors. In this case, 2 is without a pair.
Thus, the number must be multiplied by 2 so that the product is a perfect square.
Hence, the smallest number by which 1,27,008 should be multiplied is 2.
Question 14
1 / -0

A warrior arranged his men, who were 11,032 in number, in the form of a square and found that there were 7 men left over. How many men were there in each row?

A
105

B
90

C
70

D
120

E
110

Solution

This is the correct answer.
Let the number of men in each row be 'a'.
Then, number of rows = a
∴ Total number of men in 'a' rows = a × a = a²
It is given that after arranging the men in the form of a square, 7 men were left over.
∴ Total number of men = a² + 7
But, the total number of men were 11,032.
or, a² + 7 = 11,032
or, a² = 11,032 - 7
or, a² = 11,025
or, a =
or, a =
or, a = 3 × 5 × 7
or, a = 105
Thus, there were 105 men in each row.
Question 15
1 / -0

Find the smallest square number which is divisible by each of the numbers 7, 8 and 9.

A
7560

B
6552

C
7056

D
7006

E
6048

Solution

This is the correct answer.
The smallest number which is divisible by each of the numbers 7, 8 and 9 is their LCM.
The LCM of 7, 8 and 9 is 4 × 2 × 9 × 7 = 504
Resolving 504 into prime factors, we get
504 = 2 × 2 × 2 × 3 × 3 × 7
Grouping the factors into pairs of equal factors, we get
504 = (2 × 2) × 2 × (3 × 3) × 7
Hence, there is no factor to form a pair with 2 and 7.
Thus, to make 504 a perfect square, it must be multiplied it by 2 × 7 = 14.
Hence, the smallest square number which is divisible by each of the numbers 7, 8 and 9 is 504 × 14 = 7056.