CBSE Class 8 Mathematics Square And Square Roots Worksheet Set 03

Question. Which of the following is a four digit perfect square number whose first two dig its and last two digits taken separately are also perfect squares ?
(A) 1681
(B) 1636
(C) 3664
(D) 6481
Answer : A

Question. If √16/49 = x/49 , find the value of 'x:
(A) 4
(B) 7
(C) 16
(D) 28
Answer : D

Question. Evaluate 36.1/102.4
(A) -32
(B) 72
(C) 19/32
(D) 29/62
Answer : C

Question. A group of students decided to collect as many rupees from each member of the group as is the number of members. If the total collection amounts to ~ 2209, what is the number of members in the group?
(A) 37
(B) 47
(C) 107
(D) 43
Answer : B

Question. Find the value of the expression √0.0009 + √0.01
(A) 3
(B) 0.3
(C) 1/3
(D) 0.9
Answer : B

Question. A general wishes to draw up his 36562 soldiers in the form of a solid square. After arranging them, he found that some of them are left over. How many are left?
(A) 36
(B) 65
(C) 81
(D) 97
Answer : C

Question. Find the value of √0.16/0.4
(A) 0.2
(B) 0.02
(C) 0.63
(D) 2√5/5
Answer : C

Question. The area of a square field is 80(244/729) sq. m. What is the length of each side of the field?
(A) 8.96 m
(B) 10.26 m
(C) 13.54 m
(D) 9.86 m
Answer : A

Question. If √2ⁿ = 64 , what is the value of 'n' ?
(A) 2
(B) 4
(C) 6
(D) 12
Answer : D

Question. If √2401 = √7^x ,find the value of ·x:
(A) 3
(B) 4
(C) 5
(D) 6
Answer : B

Question. Find the number, which when subtracted from its square results in 12.
(A) 6
(B) 5
(C) 4
(D) 8 
Answer : C

Question. Which of the following is the smallest number of 4 digits, which is not a perfect square?
(A) 1024
(B) 1089
(C) 1000
(D) 1156
Answer : C

Question. If √24 = 4.899, find the value of √8/3·
(A) 0.544
(B) 2.666
(C) 1.633
(D) 1.333
Answer : C

Question. Evaluate √4/3 - √3/4
(A) 1/2√3
(B) - 1/2√3
(C) 1
(D) 5√3/6
Answer : A

Question. Find the value of √0.064
(A) 0.8
(B) 0.08
(C) 0.008
(D) 0.253
Answer : D

Question. A gardener wants to plant 17956 trees and arranges them in such a way that there are as many rows as there are trees in a row. What is the number of trees in a row?
(A) 144
(B) 136
(C) 154
(D) 134
Answer : D

Question. Given x/√2.25 = 550 , find the value of 'x'.
(A) 825
(B) 82.5
(C) 3666.66
(D) 2
Answer : A

Question. Which is the smallest natural number which when added to the difference of squares of 17 and 13 gives a perfect square?
(A) 1
(B) 5
(C) 11
(D) 24
Answer : A

Question. Find the square of the following numbers.
(i) 32
(ii) 35
(iii) 86
(iv) 93
(v) 71
(vi) 46
Answer: (i) 1024
(ii) 1225
(iii) 7396
(iv) 8649
(v) 5041
(vi) 2116

Question. Write a pythagorean triplet whose one member is.
(i) 6
(ii) 14
(iii) 16
(iv) 18
Answer: (i) 6, 8, 10
(ii) 14, 48, 50
(iii) 16, 63, 65
(iv) 18, 80, 82

Question. What could be the possible ‘one’s’ digits of the square root of each of the following numbers?
(i) 9801
(ii) 99856
(iii) 998001
(iv) 657666025
Answer: (i) 1, 9
(ii) 4, 6
(iii) 1, 9
(iv) 5

Question. Without doing any calculation, find the numbers which are surely not perfect squares.
(i) 153
(ii) 257
(iii) 408
(iv) 441
Answer: (i), (ii), (iii)

Question. Find the square roots of 100 and 169 by the method of repeated subtraction.
Answer: 10, 13

Question. Find the square roots of the following numbers by the prime Factorisation Method.
(i) 729
(ii) 400
(iii) 1764
(iv) 4096
(v) 7744
(vi) 9604
(vii) 5929
(viii) 9216
(ix) 529
(x) 8100
Answer: (i) 27
(ii) 20
(iii) 42
(iv) 64
(v) 88
(vi) 98
(vii) 77
(viii) 96
(ix) 23
(x) 90

Question. For each of the following numbers, find the smallest whole number by which it should be multiplied so as to get a perfect square number. Also find the square root of the square number so obtained.
(i) 252
(ii) 180
(iii) 1008
(iv) 2028
(v) 1458
(vi) 768
Answer: (i) 7; 42
(ii) 5; 30
(iii) 7; 84
(iv) 3; 78
(v) 2; 54
(vi) 3; 48

Question. For each of the following numbers, find the smallest whole number by which it should be divided so as to get a perfect square. Also find the square root of the square number so obtained.
(i) 252
(ii) 2925
(iii) 396
(iv) 2645
(v) 2800
(vi) 1620
Answer: (i) 7; 6
(ii) 13; 15
(iii) 11; 6
(iv) 5; 23
(v) 7; 20
(vi) 5; 18

Question. The students of Class VIII of a school donated Rs 2401 in all, for Prime Minister's National Relief Fund. Each student donated as many rupees as the number of students in the class. Find the number of students in the class.
Answer: 49

Question. 2025 plants are to be planted in a garden in such a way that each row contains as many plants as the number of rows. Find the number of rows and the number of plants in each row.
Answer: 45 rows; 45 plants in each row

Question. Find the smallest square number that is divisible by each of the number 4,9 and 10.
Answer: 900

Question. Find the smallest square number that is divisible by each of the numbers 8,15 and 20.
Answer: 3600

Question. Find the least number that must be subtracted from 5607 so as to get a perfect square. Also find the square root of the perfect square.
Answer: 131; 74

Question. Find the greatest 5-digit number which is a perfect square.
Answer: 99856

Question. Find the least number that must be added to 1300 so as to get a perfect square. Also find the square root of the perfect square.
Answer: 69

Question. Find the square root of each of the following numbers by Division method.
(i) 2304
(ii) 4489
(iii) 3481
(iv) 529
(v) 3249
(vi) 1369
(vii) 5776
(viii) 7921
(ix) 576
(x) 1024
(xi) 3136
(xii) 900
Answer: (i) 48
(ii) 67
(iii) 59
(iv) 23
(v) 57
(vi) 37
(vii) 76
(viii) 89
(ix) 24
(x) 32
(xi) 56
(xii) 30

Question. Find the number of digits in the square root of each of the following numbers (without any calculation).
(i) 64
(ii) 144
(iii) 4489
(iv) 27225
(v) 390625
Answer: (i) 1
(ii) 2
(iii) 2
(iv) 3
(v) 3

Question. Find the square root of the following decimal numbers.
(i) 2.56
(ii) 7.29
(iii) 51.84
(iv) 42.25
(v) 31.36
Answer: (i) 1.6
(ii) 2.7
(iii) 7.2
(iv) 6.5
(v) 5.6

Question. Find the least number which must be subtracted from each of the following numbers so as to get a perfect square. Also find the square root of the perfect square so obtained.
(i) 402
(ii) 1989
(iii) 3250
(iv) 825
(v) 4000
Answer: (i) 2; 20
(ii) 53; 44
(iii) 1; 57
(iv) 41; 28
(v) 31; 63

Question. Find the least number which must be added to each of the following numbers so as to get a perfect square. Also find the square root of the perfect square so obtained.
(i) 525
(ii) 1750
(iii) 252
(iv) 1825
(v) 6412
Answer: (i) 4; 23
(ii) 14; 42
(iii) 4; 16
(iv) 24; 43
(v) 149; 81

Question. Find the length of the side of a square whose area is \( 441 \text{ m}^2 \).
Answer: 21 m

Question. In a right triangle ABC, \( \angle B = 90^\circ \).
(i) If AB = 6 cm, BC = 8 cm find AC
(ii) If AC = 13 cm, BC = 5 cm find AB
Answer: (i) 10 cm
(ii) 12 cm

Question. A gardener has 1000 plants. He wants to plant these in such a way that the number of rows and the number of columns remain same. Find the minimum number of plants he needs more for this.
Answer: 24 plants

Question. There are 500 children in a school. For a P.T. drill they have to stand in such a manner that the number of rows is equal to number of columns. How many children would be left out in this arrangement.
Answer: 16 children

Question 1. Find the square root of 6400
Answer: Prime factorization of 6400:
\(6400 = 64 \times 100\)
\(64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^6\)
\(100 = 10 \times 10 = 2^2 \times 5^2\)
\(6400 = 2^6 \times 2^2 \times 5^2 = 2^8 \times 5^2\)
\(\sqrt{6400} = \sqrt{2^8 \times 5^2} = 2^4 \times 5 = 16 \times 5 = 80\)
In simple words: The square root of 6400 is 80 because \(80 \times 80 = 6400\).

Exam Tip: For numbers ending in even zeros, take half the number of zeros and find the square root of the remaining leading number.

Question 2. Is 90 a perfect square
Answer: Prime factorization of 90:
\(90 = 2 \times 3 \times 3 \times 5 = 2^1 \times 3^2 \times 5^1\)
Since the prime factors 2 and 5 do not occur in pairs, 90 is not a perfect square.
In simple words: No, 90 is not a perfect square as its prime factors cannot be grouped into pairs.

Exam Tip: Any natural number ending with an odd number of zeros or ending in digits 2, 3, 7, or 8 is never a perfect square.

Question 3. Is 2352 a perfect square? If not find the smallest multiple of 2352 which is a perfect square. Find the square root of the new number.
Answer: Prime factorization of 2352:
\(2352 = 2 \times 2 \times 2 \times 2 \times 3 \times 7 \times 7 = 2^4 \times 3^1 \times 7^2\)
Since the prime factor 3 does not have a pair, 2352 is not a perfect square.
To make it a perfect square, we multiply 2352 by 3:
\(2352 \times 3 = 7056\)
Square root of the new number = \(\sqrt{7056} = \sqrt{2^4 \times 3^2 \times 7^2} = 2^2 \times 3 \times 7 = 4 \times 3 \times 7 = 84\)
In simple words: Multiplying 2352 by the unpaired factor 3 gives 7056, which has a square root of 84.

Exam Tip: To find the smallest multiplier to make a number a perfect square, find its prime factorization and multiply by the factors that lack a pair.

Question 4. Find the smallest number by which 9408 must be divided so that the quotient is a perfect square. Find the square root of the quotient.
Answer: Prime factorization of 9408:
\(9408 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 7 \times 7 = 2^6 \times 3^1 \times 7^2\)
The prime factor 3 does not have a pair. Therefore, 9408 must be divided by 3.
Quotient = \(\frac{9408}{3} = 3136\)
Square root of the quotient = \(\sqrt{3136} = \sqrt{2^6 \times 7^2} = 2^3 \times 7 = 8 \times 7 = 56\)
In simple words: Divide by the leftover unpaired prime factor 3 to get 3136, whose square root is 56.

Exam Tip: Division to form a perfect square follows the same rule as multiplication: divide out the unpaired prime factors.

Question 5. Without doing any calculation, find the numbers which are surely not perfect square.
i.) 153 ii.) 257 iii) 408 iv) 441
Answer:
i.) 153 ends in 3, so it is not a perfect square.
ii.) 257 ends in 7, so it is not a perfect square.
iii.) 408 ends in 8, so it is not a perfect square.
iv.) 441 ends in 1 and is equal to \(21^2\), so it may be a perfect square.
Numbers that are surely not perfect squares: 153, 257, and 408.
In simple words: Numbers ending in 2, 3, 7, or 8 can never be perfect squares.

Exam Tip: Memorize the ending digits of square numbers (0, 1, 4, 5, 6, 9) to quickly identify non-squares.

Question 6. Find the square root of the following numbers by the prime factorisation method.
i.) 400 ii.) 9604 iii.) 8100 iv.) 1764 v.) 5929 vi.) 9216
Answer:
i.) \(400 = 2^4 \times 5^2 \Rightarrow \sqrt{400} = 2^2 \times 5 = 20\)
ii.) \(9604 = 2^2 \times 7^4 \Rightarrow \sqrt{9604} = 2 \times 7^2 = 2 \times 49 = 98\)
iii.) \(8100 = 2^2 \times 3^4 \times 5^2 \Rightarrow \sqrt{8100} = 2 \times 3^2 \times 5 = 90\)
iv.) \(1764 = 2^2 \times 3^2 \times 7^2 \Rightarrow \sqrt{1764} = 2 \times 3 \times 7 = 42\)
v.) \(5929 = 7^2 \times 11^2 \Rightarrow \sqrt{5929} = 7 \times 11 = 77\)
vi.) \(9216 = 2^{10} \times 3^2 \Rightarrow \sqrt{9216} = 2^5 \times 3 = 32 \times 3 = 96\)
In simple words: Break each number into prime factors, group them in pairs, and take one factor from each pair.

Exam Tip: Always double-check your prime factorization by multiplying the extracted square root by itself.

Question 7. For each of the following numbers, find the smallest whole number by which it should be multiplied so as to get a perfect square. also find the square root of the square number so obtained.
i.) 252 ii.) 2925 iii.) 396 iv.) 2028 v.) 1458 vi.) 768
Answer:
i.) \(252 = 2^2 \times 3^2 \times 7\). Unpaired factor is 7. Multiply by 7. New number = 1764, square root = 42.
ii.) \(2925 = 3^2 \times 5^2 \times 13\). Unpaired factor is 13. Multiply by 13. New number = 38025, square root = 195.
iii.) \(396 = 2^2 \times 3^2 \times 11\). Unpaired factor is 11. Multiply by 11. New number = 4356, square root = 66.
iv.) \(2028 = 2^2 \times 3 \times 13^2\). Unpaired factor is 3. Multiply by 3. New number = 6084, square root = 78.
v.) \(1458 = 2 \times 3^6\). Unpaired factor is 2. Multiply by 2. New number = 2916, square root = 54.
vi.) \(768 = 2^8 \times 3\). Unpaired factor is 3. Multiply by 3. New number = 2304, square root = 48.
In simple words: Identify which prime factor lacks a pair and multiply the number by it.

Exam Tip: Write prime factorizations in exponential form to instantly spot which exponents are odd.

Question 8. For each of the following number, find the smallest whole number by which it should be divided so as to get a perfect square. Also find the square root of the square number so obtained.
i.) 252 ii.) 180 iii.) 1008 iv.) 2028 v.) 1458 vi.) 768
Answer:
i.) \(252 = 2^2 \times 3^2 \times 7\). Divide by 7. Quotient = 36, square root = 6.
ii.) \(180 = 2^2 \times 3^2 \times 5\). Divide by 5. Quotient = 36, square root = 6.
iii.) \(1008 = 2^4 \times 3^2 \times 7\). Divide by 7. Quotient = 144, square root = 12.
iv.) \(2028 = 2^2 \times 3 \times 13^2\). Divide by 3. Quotient = 676, square root = 26.
v.) \(1458 = 2 \times 3^6\). Divide by 2. Quotient = 729, square root = 27.
vi.) \(768 = 2^8 \times 3\). Divide by 3. Quotient = 256, square root = 16.
In simple words: Divide the number by the unpaired prime factor to eliminate the leftover and obtain a perfect square.

Exam Tip: The number to divide by is always the product of the prime factors that do not form pairs.

Question 9. The students of class viii of a school donated Rs 2401 in all, for prime minister's national relief fund. each student donated as many rupees as the no: of students in the class. Find the no: of students in the class.
Answer: Let the number of students in the class be \(x\).
Donation per student = Rs. \(x\)
Total collection = \(x \times x = x^2\)
Given total collection = Rs. 2401
\(x^2 = 2401\)
\(x = \sqrt{2401} = 49\)
In simple words: Since the number of students equals the donation per amount, finding the square root of 2401 gives 49 students.

Exam Tip: Word problems stating "each rows equals columns" or "each student gives as many rupees as students" always require finding a square root.

Question 10. 2116 plants are to be planted in a garden in such a way that each row contains as many plants as the no: of rows. Find the no: of rows and the no: of plants in each row.
Answer: Let the number of rows be \(x\).
Number of plants in each row = \(x\)
Total plants = \(x \times x = x^2\)
\(x^2 = 2116\)
\(x = \sqrt{2116} = 46\)
Number of rows = 46, and number of plants in each row = 46.
In simple words: The number of rows and plants per row is the square root of 2116, which is 46.

Exam Tip: Use the division method for larger numbers like 2116 to find square roots quickly and accurately.

Question 11. Find the smallest square no: that is divisible by each of the no: 4, 9 and 10.
Answer: First, find the LCM of 4, 9, and 10:
\(4 = 2^2\)
\(9 = 3^2\)
\(10 = 2 \times 5\)
LCM = \(2^2 \times 3^2 \times 5 = 4 \times 9 \times 5 = 180\)
Prime factorization of 180 = \(2^2 \times 3^2 \times 5\)
To make 180 a perfect square, multiply by the unpaired factor 5:
\(180 \times 5 = 900\)
In simple words: Find the LCM first, then multiply by any unpaired prime factors to get the smallest square number.

Exam Tip: The required number must be a multiple of the LCM and also a perfect square.

Question 12. Find the smallest square no: that is divisible by each of the no: 8, 15 and 20.
Answer: Find the LCM of 8, 15, and 20:
\(8 = 2^3\)
\(15 = 3 \times 5\)
\(20 = 2^2 \times 5\)
LCM = \(2^3 \times 3 \times 5 = 8 \times 3 \times 5 = 120\)
Prime factorization of 120 = \(2^3 \times 3 \times 5 = 2^2 \times 2 \times 3 \times 5\)
Unpaired factors are 2, 3, and 5. Multiply 120 by \(2 \times 3 \times 5 = 30\):
\(120 \times 30 = 3600\)
In simple words: The LCM is 120. Multiplying by 2, 3, and 5 gives 3600.

Exam Tip: Always verify that all prime factors in the final LCM product have even exponents.

Question 13. Find the least no: that must be subtracted from 5607 so as to get a perfect square. Also find the square root of the perfect square.
Answer: Using long division method for 5607:
\(74^2 = 5476\)
\(75^2 = 5625\)
Remainder when finding \(\sqrt{5607}\) by division method:
\(5607 - 5476 = 131\)
Therefore, 131 must be subtracted from 5607 to get a perfect square.
New perfect square = \(5607 - 131 = 5476\)
Square root = 74
In simple words: Subtracting the remainder from the number leaves a perfect square whose root is found via division.

Exam Tip: For subtraction problems, find the remainder using long division and subtract it directly.

Question 14. Find the greatest 4- digit no: which is a perfect square.
Answer: The greatest 4-digit number is 9999.
Using long division to find \(\sqrt{9999}\), the remainder is 198.
Greatest 4-digit perfect square = \(9999 - 198 = 9801\)
Square root = \(\sqrt{9801} = 99\)
In simple words: Subtract the remainder of 9999 from itself to get 9801, which is \(99^2\).

Exam Tip: The greatest 4-digit square is always \(99^2\).

Question 15. Find the least no: that must be added to 1300 so as to get a perfect square .Also finds the square root of the perfect square.
Answer: Note that \(36^2 = 1296\) and \(37^2 = 1369\).
Since \(36^2 < 1300 < 37^2\), the next perfect square is \(37^2 = 1369\).
Number to be added = \(1369 - 1300 = 69\)
Square root of the new number = 37
In simple words: Find the next higher square number (\(37^2 = 1369\)) and find the difference from 1300.

Exam Tip: For addition problems, always look at the next integer's square (\((n+1)^2\)).

Question 16. Find the square root of 12.25.
Answer: Convert to fraction: \(12.25 = \frac{1225}{100}\)
\(\sqrt{12.25} = \frac{\sqrt{1225}}{\sqrt{100}} = \frac{35}{10} = 3.5\)
In simple words: Removing the decimal gives 1225, whose square root is 35, placing the decimal back gives 3.5.

Exam Tip: Count decimal places; the square root will have half as many decimal places as the original number.

Question 17. Area of a square plot is \(2304\text{ m}^2\). find the side of the square plot. also find the square root of the perfect square.
Answer: Area of square = \((\text{side})^2\)
\((\text{side})^2 = 2304\)
Side = \(\sqrt{2304} = 48\text{ m}\)
In simple words: The side length is the square root of the area, which is 48 meters.

Exam Tip: Area has square units (\(\text{m}^2\)), while side length has linear units (\(\text{m}\)).

Question 18. find the square root of each of the following no: by division method.
i.) 2304 ii.) 4489 iii.) 3481 iv.) 529 v.) 3249 vi.) 1369 vii.) 5776 viii.) 7921 ix.) 576 x.) 1024 xi.) 3136 xii.) 900
Answer:
i.) \(\sqrt{2304} = 48\)
ii.) \(\sqrt{4489} = 67\)
iii.) \(\sqrt{3481} = 59\)
iv.) \(\sqrt{529} = 23\)
v.) \(\sqrt{3249} = 57\)
vi.) \(\sqrt{1369} = 37\)
vii.) \(\sqrt{5776} = 76\)
viii.) \(\sqrt{7921} = 89\)
ix.) \(\sqrt{576} = 24\)
x.) \(\sqrt{1024} = 32\)
xi.) \(\sqrt{3136} = 56\)
xii.) \(\sqrt{900} = 30\)
In simple words: Group digits in pairs from right to left and apply the long division method.

Exam Tip: The division method is much faster than prime factorization for 3-digit and 4-digit numbers.

Question 19. Find the square root of the following decimal no:
2.56 ii 7.29 iii. 51.84 iv 42.25 v 31.36
Answer:
i.) \(\sqrt{2.56} = 1.6\)
ii.) \(\sqrt{7.29} = 2.7\)
iii.) \(\sqrt{51.84} = 7.2\)
iv.) \(\sqrt{42.25} = 6.5\)
v.) \(\sqrt{31.36} = 5.6\)
In simple words: Ignore the decimal point, find the integer square root, and place the decimal point after one digit from the right.

Exam Tip: Ensure pairing of decimal digits starts from the decimal point moving outwards.

Question 20. Find at least no: which must be subtracted from each of the following no: so as to get a perfect square. Also find the square root of the perfect square so obtained.
i. 402 ii. 1989 iii. 3250 iv. 825 v. 4000
Answer:
i.) For 402: \(20^2 = 400\). Remainder = 2. Subtract 2. Perfect square = 400, square root = 20.
ii.) For 1989: \(44^2 = 1936\), \(45^2 = 2025\). Remainder = \(1989 - 1936 = 53\). Subtract 53. Perfect square = 1936, square root = 44.
iii.) For 3250: \(57^2 = 3249\). Remainder = 1. Subtract 1. Perfect square = 3249, square root = 57.
iv.) For 825: \(28^2 = 784\), \(29^2 = 841\). Remainder = \(825 - 784 = 41\). Subtract 41. Perfect square = 784, square root = 28.
v.) For 4000: \(63^2 = 3969\). Remainder = \(4000 - 3969 = 31\). Subtract 31. Perfect square = 3969, square root = 63.
In simple words: Find the remainder using long division and subtract it to reach the lower perfect square.

Exam Tip: The number to subtract is precisely the remainder obtained at the end of the long division process.

Question 21. Find the least no: which must be added to each of the following no: so as to get a perfect square. Also find the square root of the perfect square so obtained.
i. 525 ii. 1750 iii. 252 iv. 1825 v. 6412
Answer:
i.) For 525: \(22^2 = 484\), \(23^2 = 529\). Number to add = \(529 - 525 = 4\). Square root = 23.
ii.) For 1750: \(41^2 = 1681\), \(42^2 = 1764\). Number to add = \(1764 - 1750 = 14\). Square root = 42.
iii.) For 252: \(15^2 = 225\), \(16^2 = 256\). Number to add = \(256 - 252 = 4\). Square root = 16.
iv.) For 1825: \(42^2 = 1764\), \(43^2 = 1849\). Number to add = \(1849 - 1825 = 24\). Square root = 43.
v.) For 6412: \(80^2 = 6400\), \(81^2 = 6561\). Number to add = \(6561 - 6412 = 149\). Square root = 81.
In simple words: Find the next integer's square and calculate the difference from the given number.

Exam Tip: For addition, always exceed the current number to find the square of (\(\text{quotient} + 1\)).

Question 22. Find the length of the side of a square whose area is \(441\text{ m}^2\).
Answer: Area = \((\text{side})^2 = 441\)
Side = \(\sqrt{441} = 21\text{ m}\)
In simple words: The side length is 21 meters since \(21 \times 21 = 441\).

Exam Tip: Always include proper units in your final answer.

Question 23. In a right triangle ABC angle B = \(90^\circ\)
Answer: This is a setup statement for the subsequent sub-questions regarding right triangles using the Pythagorean theorem (\(AC^2 = AB^2 + BC^2\)).
In simple words: Sets up the right-angled triangle condition where B is the right angle.

Exam Tip: Identify the hypotenuse as the side opposite to the right angle \(B\).

Question 24. a. If AB=6 cm, BC=8cm, find AC
b. If AC=13cm,BC= 5cm, find AB
Answer:
a.) By Pythagoras theorem:
\(AC^2 = AB^2 + BC^2\)
\(AC^2 = 6^2 + 8^2 = 36 + 64 = 100\)
\(AC = \sqrt{100} = 10\text{ cm}\)
b.) By Pythagoras theorem:
\(AC^2 = AB^2 + BC^2\)
\(13^2 = AB^2 + 5^2\)
\(169 = AB^2 + 25\)
\(AB^2 = 169 - 25 = 144\)
\(AB = \sqrt{144} = 12\text{ cm}\)
In simple words: Apply Pythagoras theorem: square the legs, add or subtract, and take the square root.

Exam Tip: Remember the classic Pythagorean triples: (3, 4, 5), (6, 8, 10), and (5, 12, 13).

Question 25. A gardener has 1000 plants. He wants to plant this in such a way that the no: of rows and the no: of columns remain same. Find the minimum no: of plants he needs more for this.
Answer: We need to find the smallest number to be added to 1000 to make it a perfect square.
\(31^2 = 961\)
\(32^2 = 1024\)
Minimum plants needed = \(1024 - 1000 = 24\)
In simple words: The next square after 1000 is 1024, so he needs 24 more plants.

Exam Tip: Whenever a problem asks for the "minimum number needed more", find the next higher square.

Question 26. There are 500 children in a school. for a P.Ed. drill. They have to stand in such a manner that the no. of rows is equal to the no: of columns. How many children would be left out in this arrangement? [least no :]
Answer: We need to find the remainder when 500 is divided to find its lower square root.
\(22^2 = 484\)
\(23^2 = 529\)
Children left out = \(500 - 484 = 16\)
In simple words: Subtract the largest square less than 500 (which is 484) from 500 to get 16 children left out.

Exam Tip: "Left out" means finding the remainder or difference to the lower perfect square through subtraction.

Worksheet - 2 Chapter 2 - Squares and Square root

Question 1. A natural number is called a ____________ or ___________ if it is the square of some natural number.
Answer: perfect square or square number
In simple words: A number produced by squaring an integer is called a perfect square or square number.

Exam Tip: These are standard definitions; memorize key terminology.

Question 2. \(\sqrt{0.09} =\) _________ .
Answer: 0.3
In simple words: \(0.3 \times 0.3 = 0.09\), so the square root is 0.3.

Exam Tip: Be careful with decimal placement when taking square roots of decimals less than 1.

Question 3. \(\frac{1}{\sqrt{0.09}} \times \sqrt{5.76} =\) ________ .
Answer:
\(\sqrt{0.09} = 0.3\)
\(\sqrt{5.76} = 2.4\)
Expression = \(\frac{1}{0.3} \times 2.4 = \frac{2.4}{0.3} = 8\)
In simple words: Simplify each square root first: \(\frac{2.4}{0.3} = 8\).

Exam Tip: Simplify individual square roots before performing multiplication or division.

Question 4. \(121^2 - 120^2 =\) ___________ .
Answer: Using the algebraic identity \(a^2 - b^2 = (a-b)(a+b)\):
\(121^2 - 120^2 = (121 - 120)(121 + 120) = 1 \times 241 = 241\)
In simple words: Use the difference of squares formula to quickly calculate \(1 \times 241 = 241\).

Exam Tip: For consecutive numbers, \((n+1)^2 - n^2 = 2n + 1\), which here gives \(2(120) + 1 = 241\).

Question 5. Is ( 1, 2, 3 ) a Pythagorean triplet ?
Answer: Check if \(1^2 + 2^2 = 3^2\):
\(1 + 4 = 5\)
\(3^2 = 9\)
Since \(5 \neq 9\), (1, 2, 3) is not a Pythagorean triplet.
In simple words: No, because the sum of the squares of the two smaller numbers does not equal the square of the largest number.

Exam Tip: Always verify \(a^2 + b^2 = c^2\) where \(c\) is the largest number in the triplet.

Question 6. Is 2352 a perfect square? If not, find the smallest number by which 2352 must be multiplied so that the product is a perfect square. Find the square root of the perfect square obtained.
Answer: Prime factorization of 2352 = \(2^4 \times 3^1 \times 7^2\).
Since 3 is unpaired, 2352 is not a perfect square.
Multiply by 3 to get \(2352 \times 3 = 7056\).
Square root = \(\sqrt{7056} = 84\)
In simple words: Multiply by the unpaired factor 3 to get 7056, whose square root is 84.

Exam Tip: This is a repeated question reinforcing prime factorization pairing principles.

Question 7. Find the smallest number by which 9408 must be divided so that the quotient is a perfect square. Find the square root of the perfect square obtained.
Answer: Prime factorization of 9408 = \(2^6 \times 3^1 \times 7^2\).
Divide by 3 to remove the unpaired factor.
Quotient = \(\frac{9408}{3} = 3136\)
Square root = \(\sqrt{3136} = 56\)
In simple words: Divide by 3 to obtain 3136, with a square root of 56.

Exam Tip: Division removes excess unpaired factors just like multiplication supplies missing ones.

Question 8. The student of class VIII of a school donated Rs. 2401 for prime ministers National relief fund. Each student donated as many rupees as the number of students in the class. Find the number of students in the class.
Answer: Let the number of students be \(x\).
\(x^2 = 2401\)
\(x = \sqrt{2401} = 49\)
Number of students = 49.
In simple words: The number of students is the square root of 2401, which is 49.

Exam Tip: Set up equations of the form \(x^2 = \text{Total}\) for equal-distribution word problems.

Question 9. Find the least number which must be subtracted from 18265 so as to get a perfect square.
Answer: Using the division method for 18265:
\(135^2 = 18225\)
\(136^2 = 18496\)
Remainder when finding \(\sqrt{18265}\):
\(18265 - 18225 = 40\)
Therefore, 40 must be subtracted.
In simple words: Subtract the remainder 40 to get the perfect square 18225.

Exam Tip: Long division remainder gives the exact amount to subtract.

Question 10. Find the least number which must be added to 893304 to obtain a perfect square.
Answer: Using long division for 893304:
\(945^2 = 893025\)
\(946^2 = 894916\)
Since \(945^2 < 893304 < 946^2\), the next square is \(946^2 = 894916\).
Number to be added = \(894916 - 893304 = 1612\)
In simple words: Find the next integer square (\(946^2\)) and find the difference.

Exam Tip: For large numbers, carefully perform long division to determine the exact integer root.

Question 11. Find the square root of 2.9 correct to two places of decimal.
Answer: To find \(\sqrt{2.9}\) correct to two decimal places, calculate up to three decimal places (2.90 00 00):
Using long division method:
\(\sqrt{2.90} \approx 1.702\)
Rounding to two decimal places gives 1.70.
In simple words: By long division, \(\sqrt{2.9}\) is approximately 1.702, which rounds to 1.70.

Exam Tip: Always calculate one extra decimal place than required to round correctly.

Question 12. Find the square root of \(11\frac{2}{3}\) correct to two places of decimal.
Answer: Convert mixed fraction to improper fraction:
\(11\frac{2}{3} = \frac{35}{3} = 11.6667\)
\(\sqrt{11.6667} \approx 3.4156\)
Rounding to two decimal places gives 3.42.
In simple words: Convert to decimal 11.67 and find its square root, which rounds to 3.42.

Exam Tip: Convert mixed fractions into decimals or improper fractions before applying long division.

Question 13. Find the square root of \(21\frac{51}{169}\)
Answer: Convert mixed fraction to improper fraction:
\(21\frac{51}{169} = \frac{21 \times 169 + 51}{169} = \frac{3549 + 51}{169} = \frac{3600}{169}\)
\(\sqrt{\frac{3600}{169}} = \frac{\sqrt{3600}}{\sqrt{169}} = \frac{60}{13} = 4\frac{8}{13}\)
In simple words: Convert to an improper fraction (\(\frac{3600}{169}\)) and take square roots separately to get \(\frac{60}{13}\).

Exam Tip: Recognising perfect squares like 3600 and 169 saves time.

Question 14. Find the least perfect square which is divisible by 5, 6 and 8.
Answer: Find LCM of 5, 6, and 8:
\(5 = 5\)
\(6 = 2 \times 3\)
\(8 = 2^3\)
LCM = \(2^3 \times 3 \times 5 = 8 \times 3 \times 5 = 120\)
Prime factorization of 120 = \(2^3 \times 3 \times 5 = 2^2 \times 2 \times 3 \times 5\)
To make it a square, multiply by \(2 \times 3 \times 5 = 30\):
\(120 \times 30 = 3600\)
In simple words: The LCM is 120. Multiply by 30 to make all prime factor exponents even, yielding 3600.

Exam Tip: LCM gives the smallest divisible number; pairing all prime factors gives the smallest square divisible number.

Question 15. Find the smallest five digit number which is a perfect square.
Answer: The smallest 5-digit number is 10000.
Notice that \(10000 = 100^2\).
Therefore, 10000 is already the smallest 5-digit perfect square.
In simple words: 10000 is \(100^2\), making it the smallest 5-digit perfect square.

Exam Tip: Powers of 10 like 10000 are immediate perfect squares.

Question 16. Find the other two members of a Pythagorean triplet, one of the number of which is 16.
Answer: A Pythagorean triplet is given in the form \((2m, m^2 - 1, m^2 + 1)\).
Let \(2m = 16 \implies m = 8\).
Second member = \(m^2 - 1 = 8^2 - 1 = 64 - 1 = 63\)
Third member = \(m^2 + 1 = 8^2 + 1 = 64 + 1 = 65\)
The other two members are 63 and 65.
In simple words: Taking \(2m = 16\), we find \(m = 8\), giving the triplet members 63 and 65.

Exam Tip: Always use the standard formulas \(2m\), \(m^2 - 1\), and \(m^2 + 1\) for Pythagorean triplets.

Question 17. Check, is ( 12, 35, 37 ) a Pythagorean triplet?
Answer: Check if \(12^2 + 35^2 = 37^2\):
\(12^2 = 144\)
\(35^2 = 1225\)
\(144 + 1225 = 1369\)
\(37^2 = 1369\)
Since \(1369 = 1369\), (12, 35, 37) is a Pythagorean triplet.
In simple words: Yes, because the sum of the squares of 12 and 35 equals the square of 37.

Exam Tip: Verify equality of left-hand and right-hand sides to confirm Pythagorean triplets.