NCERT Exemplar Problems Class 8 Mathematics Square-Square Root and Cube-Cube Root

NCERT Exemplar Problems Class 8 Mathematics Chapter 3 Square-Square Root and Cube-Cube Root

Multiple Choice Questions (MCQs)

Question1  196 is the square of
(a) 11                  (b) 12
(c) 14                  (d) 16
Solution.
(c) Square of 11 = 11 x 11 = 121
Square of 12 = 12 x 12 = 144
Square of 14 = 14 x 14 = 196
Clearly, 196 is the square of 14

Question 2 Which of the following is a square of an even number?
(a) 144 (b) 169
(c) 441 (d) 625
Solution.

Thus, 144 is a square of an even number.
Alternate Method
We know that, square of an even number is always an even number. Hence, 169, 441 and 625 are not even numbers. So, only 144 is an even number, which is the square of 12.

Question 3  A number ending in 9 will have the unit’s place of its square as
(a) 3                           (b) 9
(c) 1                            (d) 6
Solution.

Question 4 Which of the following will have 4 at the unit’s place?
(a) 14²                 (b) 62²                 (c) 27²             (d)35²
Solution.

Question 5  How many natural numbers lie between 5² and 6²?
(a) 9 (b) 10 (c)11 (d) 12
Solution. (b) The natural numbers lying between 5² and 6², i.e. between 25 and 36 are 26, 27, 28, 29, 30, 31,32, 33, 34 and 35.
Hence, 10 natural numbers lie between 5² and 6².

Question 6 Which of the following cannot be a perfect square?
(a) 841  (b) 529  (c) 198  (d) All of these
Solution.(c) We know that, a number ending with digits 2, 3, 7 or 8 can never be a perfect square. So, 198 cannot be written in the form of a perfect square.

Question 7 The one’s digit of the cube of 23 is
(a) 6 (b) 7 (c) 3 (d) 9
Solution. (b) We know that, the cubes of the numbers ending with digits 3 and 7, have 7 and 3 at one’s digit, respectively.
So, the one’s digit of the cube of 23 is 7.

Question 8 A square board has an area of 144 sq units. How long is each side of the board?
(a) 11 units  (b) 12 units  (c) 13 units  (d) 14 units
Solution.

Question 9

Solution.

Question 10  If one member of a Pythagorean triplet is 2m, then the other two members are

Solution.

Question 11 The sum of successive odd numbers 1, 3, 5, 7, 9, 11, 13 and 15 is
(a) 61   (b)   64   (c) 49   (d) 36
Solution. (b) We know that, the sum of first n odd natural numbers is n².
Given odd numbers are 1,3, 5, 7, 9,11,13 and 15.
So, number of odd numbers, n = 8
The sum of given odd numbers =n² = (8)² = 64

Question 12 The sum of first n odd natural numbers is
(a) 2n +1 (b)     n²  (c)     n² –1    (d) n² +1
Solution.

Question 13 Which of the following numbers is a perfect cube?
(a) 243   (b) 216   (c) 392 (d) 8640
Solution.(b) For option (a) We have, 243
Resolving 243 into prime factors, we have
243= 3 x 3 x 3 x 3 x 3
Grouping the factors in triplets of equal factors, we get
243 = (3 x 3 x 3) x 3 x 3
Clearly, in grouping, the factors in triplets of equal factors, we are left with two factors 3 x 3.
Therefore, 243 is not a perfect cube.
For option (b) We have, 216 Resolving 216 into prime factqrs, we have
216 = 2  x 2 x  2 x 3 x 3 x 3
Grouping the factors in triplets of equal factors, we get 216 = (2 x 2 x 2) x (3 x 3 x 3)
Clearly, in grouping, the factors of triplets of equal factors, no factor is left over.
So, 216 is a perfect cube.
For option (c) We have, 392
Resolving 392 into prime factors, we get
392 = 2 x 2 x 2 x 7 x 7
Grouping the factors in triplets of equal factors, we get
392 = (2 x 2 x 2) x 7 x 7
Clearly, in grouping, the factors in triplets of equal factors, we are left with two factors 7 x 7.
Therefore, 392 is not a perfect cube.
For option (d) We have, 8640
Resolving 8640 into prime factors, we get
8640=2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 5
Grouping the factors in triplets of equal factors, we get
8640 = (2 x 2 x 2) x (2 x 2 x 2) x (3 x 3 x 3) x 5
Clearly, in grouping, the factors in triplets of equal factors, we are left with one factor 5. Therefore, 8640 in not a perfect cube.
After solving, it is clear that option (b) is correct.

Question 14 The hypotenuse of a right angled triangle with its legs of lengths 3x  x  4x is
(a) 5X                                   (b )7x                                 (c) 16x                              (d) 25x
Solution.

Question 15  The next two numbers in the number pattern 1, 4, 9,16, 25,… are
(a) 35, 48 (b) 36, 49 (c) 36, 48 (d) 35, 49
Solution. (b) We have, 1,4, 9,16, 25, ….
The number pattern can be written as (1)², (2)², (3)², (4)², (5)²
Hence, the next two numbers are (6)² and (7)², i.e. 36 and 49.

Question 16  Which among 43²   , 67² , 52² , 59² would end with digit 1?
(a) 43²           (b)67²           (c)52²          (d)59²
Solution.

Question 17  A perfect square can never have the following digit in its one’s place.
(a) 1                                  (b) 8                             (c) 0                                    (d) 6
Solution.
(b) We know that, a number ending with digits 2, 3, 7 or 8 can never be a perfect square. Clearly, a perfect square can never have the digit 8 in its one’s place.

Question 18 Which of the following numbers is not a perfect cube?
(a) 216                   (b) 567                   (c) 125              (d) 343
Solution.
(b) 216=6 x 6 x 6, 567 = 3 x 3 x 3 x 3 x 7
125 = 5 x 5 x 5, 343 = 7 x 7 x 7
Clearly, 567 is not a perfect cube, because in grouping, the factors in triplets of equal factors, we are left with two factors 3 x 7.

Question 19

Solution.

Question 20

Solution.

Question 21  A perfect square number having n digits, where n is even, will have square root with

Solution. (b) A perfect square number having n digits, where n is even, will have square root with n/2 digit.

Question 22

solution.

Question 23

Solution.

Question 24

Solution.

Fill in the Blanks
In questions 25 to 48, fill in the blanks to make the statements true.

Question 25  There are________perfect squares between 1 and 100.
Solution.8
There are 8 perfect squares between 1 and 100, i.e. 4, 9,16, 25, 36, 49, 64 and 81.

Question 26  There are________ perfect cubes between 1 and 1000.
Solution.8
There are 8 perfect cubes between 1 and 1000, i.e. 8, 27,64,125, 216, 343 and 729.

Question 27 The unit’s digit in the square of 1294 is________
Solution. 6
We know that, the unit’s digit of the square of a number having digit .at unit’s place as 4 or 6 is 6.
Hence, the units digit in the square of 1294 is 6 as 4 x 4 = 16.

Question 28  The square of 500 will have zeroes.
Solution.  four
The square of 500 = (500)²
= 500 x 500 = 250000
Hence, the square of 500 will have four zeroes.

Question 29  There are natural numbers between n² and (n + l)²
Solution.

Question 30 The square root of 24025 will have________digits.
Solution.

Question 31 The square of 5.5 is________
Solution. 30.25
Square of 5.5= (5.5)² = 55 x 5.5= 30.25

Question 32  The square root of 5.3 x 5.3 is________
Solution.

Question 33  The cube of 100 will havfe________zeroes.
Solution. 6
Cubeof100 = 100³
= 100x100x100 = 1000000

Question 34  1m² =________ cm².
Solution.

Question 351m³ =________ cm³.
Solution.

Question 36 One’s digit in the cube of 38 is________
Solution.

Question 37 The square of 0.7 is________
Solution.0.49
Square of 0.7 = (0.7)² = 07 x 07 = 0.49

Question 38  The sum of first six odd natural numbers is________
Solution.

Question 39  The digit at the one’s place of 572 is________
Solution.

Question 40  The sides of a right angled triangle whose hypotenuse is 17cm, are________and________
Solution.

As, hypotenuse of right angled triangle is 17 cm.

Question 41

Solution.

Question 42 (1.2)³=________
Solution.

Question 43 The cube of an odd number is always an________number.
Solution.odd
We know that, the cubes of all odd natural numbers are odd.

Question 44  The cube root of a number x is denoted by________
Solution.

Question 45  The least number by which 125 be multiplied to make it a perfect square, is________
Solution.

Question 46 The least number by which 72 be multiplied to make it a perfect cube, is________
Solution. 3
Resolving 72 into prime factors, we get
72=2 x 2 x 2 x 3 x 3
Grouping the factors in triplets of equal factors, we get
72 = (2 x 2 x 2) x 3 x 3
We find that 2 occurs as a prime factor of 72 thrice, but 3 occurs as a prime factor only twice. Thus, if we multiply 72 by 3, 3 will also occurs as a prime factor thrice and the product will be 2 x 2 x 2 x 3 x 3 x 3, which is a perfect cube.
Hence, the least number, which should be multiplied with 72 to get perfect cube, is 3.

Question 47  The least number by which 72 be divided to make it a perfect cube, is________
Solution. 9
Resolving 72 into prime factors, we get
72=2 x 2 x 2 x 3 x 3
Grouping the factors in triplets of equal factors, we get
72 = (2 x 2 x 2) x 3 x 3
Clearly, if we divide 72 by 3 x 3, the quotient would be 2 x 2 x 2, which is a perfect cube. Hence, the least number by which 72 be divided to make it, a perfect cube, is 9.

Question 48 Cube of a number ending in 7 will end in the digit________
Solution 3
We know that, the cubes of the numbers ending in digits 3 or 7 ends in digits 7 or 3, respectively.
i.e 7 x 7 x 7 = 343
Hence, the cube of a number ending in 7 will end in the digit 3.

True/False
In questions 49 to 86, state whether the statements are True or False.

Question 49  The square of 86 will have 6 at the unit’s place.
Solution True
We know that, the unit’s digit of the square of a number having digit at unit’s place as 4 or 6 is 6.

Question 50 The sum of two perfect squares is a perfect square.
Solution False
e.g. 16 and 25 are the perfect squares, but 16 + 25 = 41 is not a perfect square.

Question 51 The product of twtfperfect squares is a perfect square.
Solution True
e.g. If 4 and 25 are the perfect square, then 4 x 25 = 100 is also a perfect square.
Clearly, the product of two perfect squares is a perfect square.

Question 52  There is no square number between 50 and 60.
Solution True
Numbers between 50 and 60 are 51,52, 53, 54, 55, 56, 57, 58 and 59.
We observed that there is no square number between 50 and 60.

Question 53 The square root of 1521 is 31.
Solution Falsie %
As, the square of 31 = (31)² = 31 x 31 = 961

Question 54 Each prime factor appears 3 times in its cube.
Solution True
If a3 is the cube and m is one of the prime factors of a. Then, m appears three times in a3.

Question 55 The square of 2.8 is 78.4.
Solution False
The square of 2.8 = (2.8)² = 2.8×2.8 = 7.84

Question 56 The cube of 0.4 is 0.064.
Solution True
Cube of 0.4 = (0.4)² = 0.4 x 0.4 x 0.4 = 0.064

Question 57 The square root of 0.9 is 0.3.
Solution False
As, the square of 0.3 = (0.3)² = 0.3 x 0.3 =0.09

Question 58  The square of every natural number is always greater than the number itself.
Solution False
1 is a natural number and square of 1 is not greater than 1.

Question 59  The cube root of 8000 is 200.
Solution

Question60 There are five perfect cubes between 1 and 100.
Solution False
There are eight perfect cubes between 1 and 100, i.e. 8,27,64,125,216,343,512 and 729.

Question 61 There are 200 natural numbers between 100² and 101².
Solution.

Question 62 The sum of first n odd natural numbers is n².
Solution.

Question 63  1000 is a perfect square.
Solution False
1000 = 2 x 2 x 2 x 5 x 5 x 5 = 2 ² x 5²x 2 x 5 Clearly, it is not a perfect square, because it has two unpaired factors 2 and 5.

Question 64  A perfect square can have 8 as its unit’s digit.
Solution False
A perfect square can never have 8 as its unit’s digit.

Question65

Solution

Question 66  All numbers of a Pythagorean triplet are odd.
Solution False
3, 4 and 5 are the numbers of Pythagorean triplet as 5² = 4 ² + 3 ² where, 4 is not an odd number.

Question 67  For an integer a, a³ is always greater than a².
Solution

Qustion68

Solution

Question 69  Let x and y be natural numbers. If x divides y, then x³ divides y³.
Solution.

Question 70 If a² ends in 5, then a³ ends in 25:
Solution.

Question 71 If a² ends in 9, thena³ ends in 7.
Solution

Question72

Solution.

Question 73  Square root of a number x is denoted by 4x.
Solution True
Square root of a number x is denoted by 4x.

Question 74 A number having 7 at its one’s place will have 3 at the unit’s place of its square.
Solution False
Square of 7 = 7 x 7 = 49
Square of 17 = 17 x 17 = 289
Square of 27 = 27 x 27 = 729
and so on.

Question 75 A number having 7 at its one’s place will have 3 at the one’s place of its cube.
Solution True
Cube of 7 = 7 x 7 x 7 =343
Cube of 17 = 17 x 17 x 17=4913
Cube of 27 = 27 x 27 x 27 = 19683
and so on.

Question 76 The cube of a one-digit number cannot be a two-digit number.
Solution

Question 77  Cube of an even number is odd.
Solution. False
We know that, the cube of an even number is always an even number,
e.g. 2 is an even number. Then, 2³ = 2 x 2 x 2 = 8
Clearly, 8 is also an even number.

Question 78 Cube of an odd number is even.
Solution. False
We know that, the cube of an odd number is always an odd number,
e.g. 3 is an odd number. Then, 33 = 3 x 3 x 3 = 27
Clearly, 27 is not an even number.

Question 79 Cube of an even number is even.
Solution. True
We know that, the cube of an even number is always an even number,
e.g. 4 is an even number. Then, 43 = 4 x 4 x 4 = 64
Clearly, 64 is also an even number.

Question 80 Cube of an odd number is odd.
Solution. True
We know that, the cube of an odd number is always an odd number,
e.g. 9 is an odd number.
Then,93 =9 x 9 x 9 = 729
Clearly, 729 is also an odd number.

Question 81  999 is a perfect cube.
Solution.

Question 82  363 x 81 is a perfect cube.
Solution.

Question 83 Cube roots of 8 are + 2 and – 2 .
Sol. False
Cube root of 8 is 2 only and cube root of – 8 is – 2.

Question 84

Solution.

Question 85 There is no cube root of a negative integer.
Solution.

Question 86 Square of a number is positive, so the cube of positive.
Solution.

Question 87  Write the first five square numbers.
Solution. First five square numbers are 1²,2², 3², 4² and 5², i.e. 1, 4, 9,16and 25.

Question 88  Write cubes of first three multiples of 3.
Solution. Since, the first three multiples of 3 are 3, 6 and 9.
Hence, the cubes of first three multiples of 3 are (3)³, (6)³ and (9)³, i.e. 27, 216 and 729.

Question 89  Show that 500 is not a perfect square.
Solution.

Question 90  Express 81 as the sum of first nine consecutive odd numbers.
Solution. 81= (9)² =1+3+ 5+ 7 + 9+ 11 + 13+ 15+ 17 = Sum of first nine consecutive odd numbers

Question 91 Using prime factorisation, find which of the following are perfect squares.
(a) 484                                     (b) 11250
(c) 841                                     (d) 729 .
Solution.

So, 484 is a perfect square.
(b) Prime factors of 11250 = 2 x (3 x 3) x (5 x 5) x (5 x 5)
As grouping, 2 has no pair.

So, 11250 is not a perfect square,
(c) Prime factors of 841 = (29 x 29)

As grouping, there is no unpaired factor left over. So, 841 is a perfect square.
(d) Prime factors of 729 = (3 x 3) x (3 x 3) x (3 x 3)
As grouping, there is no unpaired factor left over.

So, 729 is a perfect square.

Question 92  Using prime factorisatioji, find which of the following are perfect cubes,
(a) 128 (b) 343 (c) 729 (d) 1331
Solution.(a) We have, 128 = 2 x 2 x 2 x 2 x 2 x 2 x 2
Since, 2 remains after grouping in triplets.
So, 128 is not a perfect cube.
(b) We have, 343 = 7 x 7 x 7
Since, the prime factors appear in triplets.
So, 343 is a perfect cube.
(c) We have, 729 = 3 x 3 x 3 x 3 x 3 x3
Since, the prime factors appear in triplets.
So, 729 is a perfect cube.
(d) We have, 1331 =11x11x11
Since, the prime factors appear in triplets.
So, 1331 is a perfect cube.

Question 93 Using distributive law, find the squares of (a) 101 (b) 72
Sol. (a) We have, 101² = 101 x 101
= 101(100+ 1)= 10100+ 101 = 10201
(b) We have, 72² = 72 x 72 = 72 x (70 + 2)
= 5040+ 144= 5184

Question 94  Can a right angled triangle with sides 6cm, 10cm and 8cm be formed? Give reason.
Solution.

Question 95 Write the Pythagorean triplet whose one of the numbers is 4.
Solution.

Question 96  Using prime factorisation, find the square roots of (a) 11025 (b) 4761
Solution.

Question 97 Using prime factorisation, find the cube roots of
(a) 512
(b) 2197
Solution.

Question 98 Is 176 a perfect square? If not;- find the smallest number by which it should be multiplied to get a perfect square.
Solution.

Question 99 Is 9720 a perfect cube? If not, find the smallest number by which it should be divided to get a perfect cube.

Solution.Prime factors of 9720 = 2 x 2 x 2 x 3 x 3 x 3 x 3 x 3 x5
The prime factors 3 and 5 do not appear in group of triplets.
So, 9720 is not a perfect cube.
If we divide the number by 3 x 3 x 5, then the prime factorisation of the quotient will not contain 3 x 3 x 5 = 45.

Question 100 Write two Pythagorean triplets, each having one of the numbers as 5.
Sol.

Question 101  By what smallest number should 216 be divided, so that the quotient
is’ a perfect square? Also, find the square root of the quotient.
Solution.

Question 102 By what smallest number should 3600 be multiplied, so that the quotient is a perfect cube. Also, find the cube root of the quotient.
Solution. Prime factors of 3600 = 2x2x2x2x3x3x5x5
Grouping the factors into triplets of equal factors, we get

Question 103  Find the square root of the following by long division method.
(a) 1369 (b) 5625
Solution.

Question 104  Find the square root of the following by long division method. : (a) 27.04 (b) 1.44
Solution.

Question 105  What is the least number, that should be subtracted from 1385 to get a perfect square? Also, find the square root of the perfect square.
Solution.

Question 106  What is the least number that should be added to 6200 to make it a perfect square?
Solution.

Question 107  Find the least number of four digits that is a perfect square.
Solution.

Question 108  Find the greatest^umber of three digits that is a perfect square.
Solution.

Question 109  Find the least square, number, which is exactly divisible by 3, 4, 5, 6 and 8.
Solution.

Question 110 Find the length of the side of a square, if the length of its diagonal is 10 cm.
Solution.

Question 111  A decimal number is multiplied by itself. If the product is 51.84, then find the number.
Solution.

Question 112  Find the decimal fraction, which when multiplied by itself, gives 84.64.
Solution.

Question 113  A farmer wants to plough his square field of side 150m. How much area will he have to plough?
Solution.

Question 114  What will be the number of unit squares on each side of a square graph paper, if the total number of unit squares is 256?
Solution.

Question 115  If one side of a cube is 15m in Length, then find its volume.
Solution.

Question 116  The dimensions of a rectangular field are 80m and 18m. Find the length of its diagonal.
Solution.

Question 117 Find the area of a square field, if its perimeter is 96 m.
Solution.

Question 118 Find the length of each side of a cube, if its volume is 512 cm3.
Sol.

Question 119 Three numbers are in the ratio 1:2:3 and the sum of their cubes is 4500. Find the numbers.
Solution.

Question 120  How many square metres of carpet will be required for a square room of side 6.5m to be carpeted?
Solution.

Question 121 Find the side of a square, whose area is equal to the area of a rectangle with sides 6.4m and 2.5m.
Solution.

Question 122 Difference of two perfect cubes is” 189. If the cube root of the smaller of the two numbers is 3, then find the cube root of the larger number.
Solution.

Question 123  Find the number of plants in each row, if 1024 plants are arranged, so
that number of plants in a row is the same as the number of rows.
Solution.

Question 124  A hall has a capacity of 2704 seats. If the number of rows is equal to the number of seats in each row, then find the number of seats in each row.
Solution.

Question 125  A General wishes to draw up his 7500 soldiers in the form of a square. After arranging, he found out that some of them are left out. How many soldiers were left out?
Solution.

Question 126 8649 students were sitting in a lecture room in such a manner that there were as many students in the row as there were rows in the lecture room. How many students were there in each row of the lecture room?
Solution.

Question 127 Rahul walks 12m North from his house and turns West to walk 35m to reach his friend’s house. While returning, he walks diagonally from his friend’s house to reach back to his house. What distance did he walk, while returning?
Solution.

Question 128  A 5.5m long ladder is leaned against a wall. The ladder reaches the wall to a height of 4.4m. Find^the distance between the wall and the foot of the ladder.
Solution.

Question 129  A king wanted to reward his advisor, a wiseman of the kingdom. So, he asked the wiseman to name his own reward. The wiseman thanked the king, but said that he would ask only for some gold coins each day for a month. The coins were to be counted out in a pattern of one coin for the first day, 3 coins for the second day, 5 coins for the third day and so on for 30 days. Without making calculations, find how many coins will the advisor get in that month?
Solution.

Question 130  Find three numbers in the ratio 2 : 3 : 5, the sum of whose squares is 608.
Solution.

Question 131 Find the smallest square number divisible by each of the numbers 8, 9 and 10.
Solution.

Question 132

Solution.

Question 133  Find the square root of 324 by the method of repeated subtraction.
Solution.

Question 134 Three numbers are in the ratio 2 : 3 : 4. The sum of their cubes is 0.334125. Find the numbers.
Solution.

Question 135

Solution.

Question 136 

Solution.

Question137

Solution.

Question 138
A perfect square number has four digits, none of which is zero. The- digits from left to right have valfles, that are even, even, odd, even. Find the number.
Solution.
Suppose abed is a perfect square.
where, a = even number
b = even number
c = odd number
d = even number
Hence, 8836 is one of the number which satisfies the given condition.

Question 139  Put three different numbers in the circles, so that when you add the numbers at the end of each line, you always get a perfect square.

Solution.

Question 140 The perimeters of two squares are 40m and 96m, respectively. Find the perimeter of another square equal in area to the sum of the first two squares.
Solution.

Question 141  A three-digit perfect square is such that, if it is viewed upside down, the number seen is also a perfect square. What is the number?
[Hint The digits 1, 0 and 8 stay the same when viewed upside down, whereas 9 becomes 6 and 6 becomes 9]
Solution. Three-digit perfect squares are 196 and 961, which looks same when viewed upside down.

Question 142 13 and 31 is a strange pair of numbers, such that their squares 169 and 961 are also mirror of each other. Can yqu find two other such pairs?
Solution.

 

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