Worksheet Solutions-Real numbers-10th class-Mathematics-English Medium-2026-2027

WORKSHEET - 1 Class X — Mathematics

Chapter: 1. Real Numbers Topic: Division Algorithm & Euclid's Lemma

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Q1. In a = bq + r (Division algorithm):
a is _______, q is _______, r is _______, b is _______ and condition for "r" is _______.

a = Dividend,    b = Divisor

q = Quotient,    r = Remainder

Condition for r: 0 ≤ r < b

Q2. Find the HCF of 56 and 98, using Euclid's Division algorithm.

Applying Euclid's Division Lemma (a = bq + r):

98 = 56 × 1 + 42    (r = 42 ≠ 0)

56 = 42 × 1 + 14    (r = 14 ≠ 0)

42 = 14 × 3 + 0     (r = 0)

Since remainder is 0, HCF(56, 98) = 14.

Q3. Use Euclid's Division Algorithm to find the HCF of 135 and 225.

Here 225 > 135. Using a = bq + r:

225 = 135 × 1 + 90    (r = 90 ≠ 0)

135 = 90 × 1 + 45      (r = 45 ≠ 0)

90 = 45 × 2 + 0        (r = 0)

Therefore, HCF(135, 225) = 45.

Q4. If the HCF of 12 and 84 is expressed as 7m - 2, then find the value of "m".

First, find HCF of 12 and 84:

84 = 12 × 7 + 0  ⇒  HCF(12, 84) = 12

Given HCF = 7m - 2

⇒ 7m - 2 = 12 ⇒ 7m = 14 ⇒ m = 2

Q5. Find the greatest number that divides 245 and 1029 leaving the remainder 5 in each case.

Subtract remainder 5 from given numbers:

245 - 5 = 240  and  1029 - 5 = 1024

Now, find HCF(240, 1024) using Euclid's Algorithm:

1024 = 240 × 4 + 64  |  240 = 64 × 3 + 48

64 = 48 × 1 + 16       |  48 = 16 × 3 + 0

The required greatest number is 16.

WORKSHEET - 2 Class X — Mathematics

Chapter: 1. Real Numbers Topic: Prime Factorisation

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Q1. Find the prime factorisation of 60.

Dividing 60 by prime factors step-by-step:

60 = 2 × 30 = 2 × 2 × 15 = 2 × 2 × 3 × 5

Prime factorisation of 60 = 2² × 3 × 5

Q2. Find the prime factorisation of the following:

(a) 84
84 = 2 × 42
= 2 × 2 × 21
= 2² × 3 × 7

(b) 180
180 = 2 × 90
= 2 × 2 × 45 = 2 × 2 × 3 × 15
= 2² × 3² × 5

(c) 360
360 = 2 × 180
= 2 × 2 × 2 × 3² × 5
= 2³ × 3² × 5

(d) 945
945 = 3 × 315
= 3 × 3 × 105 = 3 × 3 × 3 × 35
= 3³ × 5 × 7

Q3. Find the smallest number that must be multiplied to 540 to make it a perfect square.

Prime factorisation of 540:

540 = 2 × 2 × 3 × 3 × 3 × 5 = (2²) × (3²) × 3 × 5

Here, prime factors 3 and 5 do not have pairs.

To make it a perfect square, we must multiply by 3 × 5 = 15.

Required smallest number = 15.

Q4. Check whether 8ⁿ can end with digit '0' for any natural number n. Justify your answer.

If a number ends with digit 0, its prime factorisation must contain both 2 and 5.

Now, 8ⁿ = (2³)ⁿ = 2³ⁿ

The only prime factor of 8ⁿ is 2. It does not contain 5 as a prime factor.

By Fundamental Theorem of Arithmetic, 8ⁿ can NEVER end with digit 0 for any n ∈ N.

WORKSHEET - 3 Class X — Mathematics

Chapter: 1. Real Numbers Topic: Fundamental Theorem of Arithmetic - HCF & LCM

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Q1. Two numbers have prime factorizations: A = 2² × 3 × 5 and B = 2 × 3² × 7.
(a) HCF = _______      (b) LCM = _______

(a) HCF = Product of smallest powers of common prime factors

         = 2¹ × 3¹ = 6

(b) LCM = Product of greatest powers of all prime factors

         = 2² × 3² × 5 × 7 = 4 × 9 × 5 × 7 = 1260

Q2. Find HCF of 24 and 36 by prime factorisation.

24 = 2 × 2 × 2 × 3 = 2³ × 3¹

36 = 2 × 2 × 3 × 3 = 2² × 3²

HCF(24, 36) = 2² × 3¹ = 4 × 3 = 12

Q3. Using prime factorisation, find HCF and LCM of 84 and 180. Also verify HCF × LCM = product of numbers.

84 = 2² × 3 × 7  |  180 = 2² × 3² × 5

HCF = 2² × 3 = 12  |  LCM = 2² × 3² × 5 × 7 = 1260

Verification: HCF × LCM = 12 × 1260 = 15,120

Product of numbers = 84 × 180 = 15,120. (Verified!)

Q4. Determine the least number that is divisible by 15, 20 and 24.

The required least number is the LCM(15, 20, 24):

15 = 3 × 5  |  20 = 2² × 5  |  24 = 2³ × 3

LCM = 2³ × 3 × 5 = 8 × 3 × 5 = 120

The least divisible number is 120.

Q5. Find the greatest number dividing 306 and 657 exactly.

The required greatest number is HCF(306, 657):

306 = 2 × 3² × 17  |  657 = 3² × 73

HCF(306, 657) = 3² = 9.

WORKSHEET - 4 Class X — Mathematics

Chapter: 1. Real Numbers Topic: Rational Numbers & Decimal Expansions

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Q1. (a) When does a rational number have a terminating decimal expansion?
(b) What type of decimal expansion does a rational number have if its denominator (in lowest form) contains primes other than 2 or 5?

(a) A rational number p/q has a terminating decimal expansion when prime factorisation of denominator q is of the form 2ⁿ × 5^m (n, m ≥ 0).

(b) It will have a Non-terminating recurring (repeating) decimal expansion.

Q2. Without actual division, state whether the decimal expansion of the following is terminating or non-terminating recurring?
(a) 7/8          (b) 9/15

(a) 7/8: Denominator = 8 = 2³ = 2³ × 5⁰. Form 2ⁿ × 5^m ⇒ Terminating.

(b) 9/15: In simplest form = 3/5. Denominator = 5 = 2⁰ × 5¹. Form 2ⁿ × 5^m ⇒ Terminating.

Q3. Write the decimal expansion of the following without actual division:
(a) 1/8          (b) 13/30

(a) 1/8 = 1 / 2³ = (1 × 5³) / (2³ × 5³) = 125 / 1000 = 0.125

(b) 13/30 = 13 / (3 × 10) = 0.4333... = 0.43̄ (Non-terminating recurring)

Q4. Express 0.375 as a rational form (p/q).

0.375 = 375 / 1000

Dividing numerator and denominator by 125:

0.375 = 3 / 8

Q5. Express 0.27 (0.2727...) as a rational number.

Let x = 0.2727... — (1)

Multiplying by 100: 100x = 27.2727... — (2)

Subtract (1) from (2): 99x = 27

x = 27 / 99 = 3 / 11

WORKSHEET - 5 Class X — Mathematics

Chapter: 1. Real Numbers Topic: Proofs of Irrational Numbers

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Q1. Prove that √3 is irrational.

Assume to the contrary that √3 is rational. Then √3 = a/b (where a, b are co-prime integers, b ≠ 0).

⇒ a = b√3 ⇒ Squaring both sides: a² = 3b² — (1)

Thus, 3 divides a², which means 3 divides a. Let a = 3c.

Substitute in (1): (3c)² = 3b² ⇒ 9c² = 3b² ⇒ b² = 3c²

Thus, 3 divides b², so 3 divides b.

Hence, 3 is a common factor of both a and b, contradicting that a, b are co-prime. Therefore, √3 is irrational.

Q2. Prove that 2 + √3 is irrational.

Let us assume 2 + √3 is rational.

Then 2 + √3 = a/b (where a, b are integers, b ≠ 0).

⇒ √3 = (a/b) - 2 ⇒ √3 = (a - 2b) / b

Since a, b are integers, (a - 2b)/b is rational.

This implies √3 is rational, which contradicts the known fact that √3 is irrational.

Hence, 2 + √3 is irrational.

Q3. Prove that √3 + 5√2 is irrational.

Assume x = √3 + 5√2 is rational.

⇒ x - 5√2 = √3 ⇒ Squaring both sides: (x - 5√2)² = (√3)²

⇒ x² - 10√2x + 50 = 3 ⇒ x² + 47 = 10√2x

⇒ √2 = (x² + 47) / (10x)

Since x is rational, (x² + 47)/(10x) is rational, implying √2 is rational.

This is a contradiction. Thus, √3 + 5√2 is irrational.

WORKSHEET - 6 (Part 1) Class X — Mathematics

Chapter: 1. Real Numbers Topic: Logarithms — Forms, Evaluations & Laws

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Q1. Convert into logarithmic form:
(i) 10³ = 1000      (ii) 2⁴ = 16      (iii) 3² = 9

(i) 10³ = 1000 ⇒ log₁₀ 1000 = 3

(ii) 2⁴ = 16 ⇒ log₂ 16 = 4

(iii) 3² = 9 ⇒ log₃ 9 = 2

Q2. Convert into exponential form:
(i) log₂ 8 = 3      (ii) log₅ 25 = 2      (iii) log₁₀ 100 = 2

(i) log₂ 8 = 3 ⇒ 2³ = 8

(ii) log₅ 25 = 2 ⇒ 5² = 25

(iii) log₁₀ 100 = 2 ⇒ 10² = 100

Q3. Evaluate:
(i) log₃ 3    (ii) log₃ 81    (iii) log₁₀ 1    (iv) log₅ 125    (v) log₁₀ 1000    (vi) log₁₀ 0.01

(i) log₃ 3 = 1

(ii) log₃ 3⁴ = 4

(iii) log₁₀ 1 = 0

(iv) log₅ 5³ = 3

(v) log₁₀ 10³ = 3

(vi) log₁₀ 10^-2 = -2

Q4. Write the laws of logarithms:

a) Product Law: log_a (mn) = log_a m + log_a n

b) Quotient Law: log_a (m/n) = log_a m - log_a n

c) Power Law: log_a (m^k) = k · log_a m

Q5. Expand using laws of logarithms:

a) log(5x) = log 5 + log x

b) log(3y/2) = log 3 + log y - log 2

c) log(7a³) = log 7 + 3 log a

d) log(4x²/y) = log 4 + 2 log x - log y

WORKSHEET - 6 (Part 2) Class X — Mathematics

Chapter: 1. Real Numbers Topic: Logarithms — Simplification & Proofs

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Q6. Simplify:

a) log 8 + log 5 = log(40)

b) log 12 - log 3 = log(4)

c) 3 log 2 = log(2³) = log 8

d) 2 log x - log y = log(x²/y)

e) log₁₀ 8 - log₁₀ 2 = log₁₀ 4

f) log₁₀ 50 + log₁₀ 2 = log₁₀ 100 = 2

Q7. Evaluate (Given: log 2 = 0.3010, log 3 = 0.4771):
(a) log 6     (b) log 9     (c) log(8/3)     (d) log₁₀ 20

a) log 6 = log(2 × 3) = log 2 + log 3 = 0.3010 + 0.4771 = 0.7781

b) log 9 = log(3²) = 2 log 3 = 2 × 0.4771 = 0.9542

c) log(8/3) = 3 log 2 - log 3 = 3(0.3010) - 0.4771 = 0.9030 - 0.4771 = 0.4259

d) log₁₀ 20 = log₁₀ (2 × 10) = log₁₀ 2 + 1 = 0.3010 + 1 = 1.3010

Q8. If log a = 2 and log b = 3, find the values of:
(a) log(ab)          (b) log(a²/b)

a) log(ab) = log a + log b = 2 + 3 = 5

b) log(a²/b) = 2 log a - log b = 2(2) - 3 = 4 - 3 = 1

Q9. If x² + y² = 18xy, then prove that 2 log(x - y) = 4 log 2 + log x + log y.

Given: x² + y² = 18xy

Subtract 2xy from both sides:

x² + y² - 2xy = 18xy - 2xy  ⇒  (x - y)² = 16xy

Taking log on both sides:

log(x - y)² = log(16xy)

⇒ 2 log(x - y) = log 16 + log x + log y

⇒ 2 log(x - y) = log(2⁴) + log x + log y

⇒ 2 log(x - y) = 4 log 2 + log x + log y. (Hence Proved!)

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Real Numbers - 10th Class - 20 MCQ's - Worksheet4 - Complete Chapter - English Medium

Chapter 1: Real Numbers Practice

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1. If two positive integers \(a\) and \(b\) are written as \(a = x^3 y^2\) and \(b = x y^3\), where \(x\) and \(y\) are prime numbers, then \(\text{HCF}(a, b)\) is:

• A) \(x y\)
• B) \(x y^2\)
• C) \(x^3 y^3\)
• D) \(x^2 y^2\)

2. Use Euclid's division algorithm to find the HCF of 196 and 38220.

• A) 196
• B) 98
• C) 195
• D) 382

3. Show that any positive even integer is of the form \(6q\), \(6q + 2\), or \(6q + 4\). What are the possible values for remainder \(r\) when an integer is divided by 6?

• A) 0, 1, 2, 3
• B) 1, 3, 5
• C) 0, 1, 2, 3, 4, 5
• D) 0, 2, 4

4. Find the prime factorisation of 3825.

• A) \(3 \times 5^2 \times 17\)
• B) \(3^2 \times 5^2 \times 17\)
• C) \(3^3 \times 5 \times 17\)
• D) \(3^2 \times 5 \times 85\)

5. The LCM of two numbers is 1820 and their HCF is 26. If one of the numbers is 130, find the other number.

• A) 364
• B) 260
• C) 455
• D) 520

6. Three bells toll together at intervals of 9, 12, and 15 minutes respectively. If they toll together now, after how many hours will they toll together next?

• A) 2 hours
• B) 3 hours
• C) 5 hours
• D) 6 hours

7. For what natural number \(n\) will \(4^n\) end with the digit 0?

• A) \(n = 5\)
• B) \(n = 10\)
• C) Any even number \(n\)
• D) No natural number \(n\)

8. Which of the following non-zero numbers will have a non-terminating repeating decimal expansion?

• A) \(\frac{13}{3125}\)
• B) \(\frac{17}{8}\)
• C) \(\frac{64}{455}\)
• D) \(\frac{6}{15}\)

9. After how many places of decimals will the rational number \(\frac{14587}{1250}\) terminate?

• A) 2 decimal places
• B) 3 decimal places
• C) 4 decimal places
• D) 5 decimal places

10. Write the decimal expansion of \(\frac{7}{80}\) without performing long division.

• A) 0.0875
• B) 0.875
• C) 0.00875
• D) 0.0785

11. If \(a\) and \(b\) are coprime positive integers, then \(a^2\) and \(b^2\) are:

• A) Not coprime
• B) Coprime
• C) Even integers
• D) Prime numbers

12. Which of the following is an irrational number?

• A) \(2.141414...\)
• B) \(\sqrt{0.04}\)
• C) \(3 + \sqrt{7}\)
• D) \(\frac{\sqrt{12}}{\sqrt{3}}\)

13. The product of a non-zero rational number and an irrational number is always:

• A) Rational
• B) Irrational
• C) An integer
• D) Zero

14. Convert the logarithmic equation \(\log_3 81 = 4\) into exponential form.

• A) \(4^3 = 81\)
• B) \(3^4 = 81\)
• C) \(81^{1/4} = 3\)
• D) \(3^{81} = 4\)

15. Evaluate the expression: \(\log_{10} 0.001\).

• A) 3
• B) \(-3\)
• C) \(-2\)
• D) \(\frac{1}{3}\)

16. Write \(\log\left(\sqrt{\frac{x^3}{y^2}}\right)\) in expanded form using logarithmic laws.

• A) \(\frac{3}{2}\log x - \log y\)
• B) \(3\log x - 2\log y\)
• C) \(\frac{3}{2}\log x + \log y\)
• D) \(\frac{1}{2}(3\log x - 2\log y)\)

17. Simplify into a single logarithm: \(3\log 2 + 2\log 5 - \log 20\).

• A) \(\log 10\)
• B) \(\log 5\)
• C) \(\log 2\)
• D) \(\log 100\)

18. Solve for \(x\): \(\log_2(x + 3) + \log_2(x - 3) = 4\).

• A) \(x = 5\)
• B) \(x = \pm 5\)
• C) \(x = 25\)
• D) \(x = 4\)

19. If \(\log_a b = x\), what is the value of \(\log_{\frac{1}{a}} b\)?

• A) \(x\)
• B) \(-x\)
• C) \(\frac{1}{x}\)
• D) \(a^x\)

20. If \(a^2 + b^2 = 7ab\), show that \(\log\left(\frac{a+b}{3}\right)\) is equal to:

• A) \(\frac{1}{2}(\log a + \log b)\)
• B) \(\log a + \log b\)
• C) \(2(\log a + \log b)\)
• D) \(\frac{1}{3}(\log a + \log b)\)

Real Numbers - 10th Class - 20 MCQ's - Worksheet3 - Complete Chapter - English Medium-Textbook

Chapter 1: Real Numbers

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1. According to Euclid's Division Algorithm, for positive integers \(a\) and \(b\), there exist unique integers \(q\) and \(r\) satisfying \(a = bq + r\). Which condition must the remainder \(r\) satisfy?

• A) \(0 < r \le b\)
• B) \(0 \le r < b\)
• C) \(0 \le r \le b\)
• D) \(r > b\)

2. Use Euclid's division algorithm to find the Highest Common Factor (HCF) of 135 and 225.

• A) 15
• B) 30
• C) 45
• D) 90

3. Any positive odd integer can be expressed in which of the following forms (where \(q\) is some integer)?

• A) \(4q\) or \(4q + 2\)
• B) \(4q + 1\) or \(4q + 3\)
• C) \(2q\) only
• D) \(6q\) or \(6q + 4\)

4. Expressing 5005 as a product of prime factors yields:

• A) \(5 \times 7 \times 11 \times 13\)
• B) \(5 \times 11 \times 13 \times 17\)
• C) \(3 \times 5 \times 7 \times 13\)
• D) \(5^2 \times 7 \times 11\)

5. For any two positive integers \(a\) and \(b\), which relation between their HCF and LCM is always true?

• A) \(\text{HCF}(a, b) + \text{LCM}(a, b) = a \times b\)
• B) \(\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b\)
• C) \(\frac{\text{HCF}(a, b)}{\text{LCM}(a, b)} = a \times b\)
• D) \(\text{HCF}(a, b) \times a = \text{LCM}(a, b) \times b\)

6. Given that \(\text{HCF}(306, 657) = 9\), what is the \(\text{LCM}(306, 657)\)?

• A) 22338
• B) 12338
• C) 22448
• D) 30600

7. Why can't the number \(6^n\) end with the digit zero for any natural number \(n\)?

• A) Because 6 is not a prime number.
• B) Because its prime factorisation \(2^n \times 3^n\) does not contain 5.
• C) Because \(6^n\) is always odd.
• D) Because prime factorisation of 6 contains only odd numbers.

8. A rational number \(\frac{p}{q}\) (in simplest form) has a terminating decimal expansion if and only if the prime factorization of \(q\) is of the form:

• A) \(2^n 3^m\)
• B) \(2^n 5^m\)
• C) \(3^n 5^m\)
• D) \(7^n 5^m\)

9. Which of the following rational numbers has a terminating decimal expansion?

• A) \(\frac{77}{210}\)
• B) \(\frac{129}{2^2 \cdot 5^7 \cdot 7^5}\)
• C) \(\frac{15}{1600}\)
• D) \(\frac{29}{343}\)

10. Convert the rational number \(\frac{23}{2^3 \cdot 5^2}\) into decimal form without actual division.

• A) 0.23
• B) 0.115
• C) 0.0115
• D) 1.15

11. Let \(p\) be a prime number. If \(p\) divides \(a^2\) (where \(a\) is a positive integer), then:

• A) \(p\) must be equal to \(a\)
• B) \(p\) divides \(a\)
• C) \(a\) divides \(p\)
• D) \(p^2\) divides \(a\)

12. If \(p\) and \(q\) are distinct prime numbers, then \(\sqrt{p} + \sqrt{q}\) is always:

• A) Rational
• B) An integer
• C) Irrational
• D) A whole number

13. Evaluating the expression \((2\sqrt{3} + \sqrt{5})(2\sqrt{3} - \sqrt{5})\) yields:

• A) 7 (which is rational)
• B) 7 (which is irrational)
• C) 13 (which is rational)
• D) \(\sqrt{7}\) (which is irrational)

14. Convert the exponential statement \(5^3 = 125\) into its logarithmic equivalent.

• A) \(\log_3 125 = 5\)
• B) \(\log_5 125 = 3\)
• C) \(\log_{125} 5 = 3\)
• D) \(\log_5 3 = 125\)

15. What is the value of \(\log_2 \left(\frac{1}{16}\right)\)?

• A) 4
• B) \(-4\)
• C) \(\frac{1}{4}\)
• D) \(-2\)

16. Express \(\log\left(\frac{343}{125}\right)\) in expanded logarithmic form.

• A) \(3\log 7 + 3\log 5\)
• B) \(3(\log 7 - \log 5)\)
• C) \(7\log 3 - 5\log 3\)
• D) \(\frac{3\log 7}{3\log 5}\)

17. Express \(2\log 3 + 3\log 5 - 5\log 2\) as a single logarithm.

• A) \(\log\left(\frac{1125}{32}\right)\)
• B) \(\log\left(\frac{225}{32}\right)\)
• C) \(\log\left(\frac{1125}{10}\right)\)
• D) \(\log(1125 - 32)\)

18. Solve for \(x\) if \(2\log 5 + \frac{1}{2}\log 9 - \log 3 = \log x\).

• A) \(x = 5\)
• B) \(x = 15\)
• C) \(x = 25\)
• D) \(x = 75\)

19. What is the value of \(2^{2 + \log_2 3}\)?

• A) 7
• B) 12
• C) 6
• D) 14

20. If \(x^2 + y^2 = 25xy\), which of the following logarithmic identities holds true?

• A) \(2\log(x+y) = 3\log 3 + \log x + \log y\)
• B) \(2\log(x+y) = \log x + \log y + 3\log 2\)
• C) \(\log(x+y) = \log 25 + \log x + \log y\)
• D) \(2\log(x-y) = 3\log 3 + \log x + \log y\)