NCERT Class X Mathematics Chapter 1: Real Numbers

Chapter 1: Real Numbers

Comprehensive Study Notes, Day-by-Day Explanations, and Question Bank

Part 1: Day-to-Day Study Notes & Step-by-Step Explanations

Day 1: The Fundamental Theorem of Arithmetic

Objective: Understand prime factorization of composite numbers and its uniqueness.

Theorem 1.1 (Fundamental Theorem of Arithmetic): Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

Step-by-Step Explanation:

• Any natural number can be written as a product of its prime factors.
• We typically arrange prime factors in ascending order (e.g., $p_1 \le p_2 \le \dots \le p_n$).
• Example: Let's factorise 32760 using a factor tree. We get $32760 = 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 7 \times 13 = 2^3 \times 3^2 \times 5 \times 7 \times 13$.

Day 2: Finding HCF and LCM Using Prime Factorisation

Objective: Apply prime factorization to find Highest Common Factor (HCF) and Least Common Multiple (LCM), and understand their relationship.

Step-by-Step Explanation:

• HCF: Product of the smallest power of each common prime factor in the numbers.
• LCM: Product of the greatest power of each prime factor, involved in the numbers.
• Crucial Formula: For any two positive integers $a$ and $b$, $HCF(a, b) \times LCM(a, b) = a \times b$.
• Note for 3 numbers: $HCF(p, q, r) \times LCM(p, q, r) \ne p \times q \times r$.

Application Example: Check whether $4^n$ can end with the digit zero. For a number to end with zero, its prime factorisation must contain the prime 5. Since $4^n = (2)^{2n}$, the only prime factor is 2[cite: 102, 103]. Thus, it can never end with zero.

Day 3: Revisiting Irrational Numbers & Theorem 1.2

Objective: Prove the irrationality of numbers like $\sqrt{2}$ and $\sqrt{3}$ using proof by contradiction.

Theorem 1.2: Let $p$ be a prime number. If $p$ divides $a^2$, then $p$ divides $a$, where $a$ is a positive integer.

Step-by-Step Proof that $\sqrt{2}$ is irrational:

1. Assume the contrary: Let $\sqrt{2}$ be a rational number.
2. Write it in simplest form: $\sqrt{2} = \frac{a}{b}$ where $a$ and $b$ are coprime integers ($b \ne 0$)[cite: 191].
3. Rearrange and square: $b\sqrt{2} = a$ becomes $2b^2 = a^2$[cite: 192, 193].
4. This means 2 divides $a^2$, so by Theorem 1.2, 2 divides $a$. Let $a = 2c$.
5. Substitute back: $2b^2 = (2c)^2 = 4c^2 \implies b^2 = 2c^2$.
6. This means 2 divides $b^2$, so 2 divides $b$.
7. Contradiction: Both $a$ and $b$ are divisible by 2, contradicting that they are coprime[cite: 197, 199]. Thus, $\sqrt{2}$ is irrational.

Day 4: Combinations of Rational and Irrational Numbers

Objective: Prove expressions like $5 - \sqrt{3}$ are irrational.

Key Facts:

• The sum or difference of a rational and an irrational number is irrational.
• The product and quotient of a non-zero rational and irrational number is irrational.

Step-by-Step Proof for $5 - \sqrt{3}$:

1. Assume $5 - \sqrt{3}$ is rational, meaning $5 - \sqrt{3} = \frac{a}{b}$ (coprime integers, $b \ne 0$)[cite: 231, 232, 233].
2. Rearrange to isolate the root: $\sqrt{3} = 5 - \frac{a}{b} = \frac{5b - a}{b}$.
3. Since $a$ and $b$ are integers, $\frac{5b - a}{b}$ is rational, implying $\sqrt{3}$ is rational[cite: 237, 238].
4. This contradicts the known fact that $\sqrt{3}$ is irrational. Hence, $5 - \sqrt{3}$ is irrational[cite: 239, 240].

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Part 2: Multiple Choice Questions (MCQs)

Ideal for quick conceptual testing.

1. If $HCF(306, 657) = 9$, what is the $LCM(306, 657)$?

   • 22338
   • 23328
   • 28332
   • 32238

   Answer: (a) Use formula $LCM = \frac{a \times b}{HCF}$.

2. The prime factorisation of 140 is:

   • $2 \times 7 \times 10$
   • $2^2 \times 5 \times 7$
   • $2 \times 5^2 \times 7$
   • $2^2 \times 3 \times 7$

   Answer: (b)

3. According to Theorem 1.2, if a prime $p$ divides $a^2$, then:

   • $p$ divides $a$
   • $a$ divides $p$
   • $p^2$ divides $a$
   • None of the above

   Answer: (a)

4. The HCF of 96 and 404 is:

   • 2
   • 4
   • 8
   • 12

   Answer: (b)

5. The LCM of 24 and 36 is:

   • 48
   • 60
   • 72
   • 96

   Answer: (c)

6. The prime factorisation of 180 is:

   • $2^2 \times 3^2 \times 5$
   • $2 \times 3^2 \times 5$
   • $2^2 \times 3 \times 5$
   • $2^3 \times 3^2 \times 5$

   Answer: (a)

7. If $HCF(72, 120) = 24$, then $LCM(72,120)$ is:

   • 240
   • 360
   • 480
   • 600

   Answer: (b) Use $LCM = \frac{a \times b}{HCF}$.

8. Which of the following is irrational?

   • $\sqrt{16}$
   • $\sqrt{25}$
   • $\sqrt{7}$
   • $\frac{3}{4}$

   Answer: (c)

9. The decimal expansion of $\frac{7}{8}$ is:

   • Terminating
   • Non-terminating repeating
   • Non-terminating non-repeating
   • None of these

   Answer: (a)

10. The HCF of two consecutive integers is:

    • 0
    • 1
    • 2
    • The smaller number

    Answer: (b)

11. If $p$ is a prime number and $p$ divides the product $ab$, then:

    • $p$ divides $a$ or $p$ divides $b$
    • $p$ divides $a$ and $p$ divides $b$
    • $a$ divides $p$
    • $b$ divides $p$

    Answer: (a)

12. The LCM of two coprime numbers is equal to:

    • 1
    • Their difference
    • Their product
    • Their sum

    Answer: (c)

13. The HCF of 45 and 75 is:

    • 5
    • 10
    • 15
    • 25

    Answer: (c)

14. Which of the following has a terminating decimal expansion?

    • $\frac{13}{30}$
    • $\frac{7}{64}$
    • $\frac{11}{45}$
    • $\frac{17}{27}$

    Answer: (b)

15. The prime factorisation of 225 is:

    • $3^2 \times 5^2$
    • $3^3 \times 5$
    • $9 \times 25$
    • $15^2$

    Answer: (a)

16. Euclid’s Division Lemma states that for given positive integers $a$ and $b$, there exist unique integers $q$ and $r$ such that:

    • $a = bq + r$, $0 \le r < b$
    • $a = br + q$
    • $b = aq + r$
    • $a = b + q + r$

    Answer: (a)

17. The HCF of 17 and 23 is:

    • 1
    • 17
    • 23
    • 391

    Answer: (a)

18. If $LCM(a,b)=180$ and $HCF(a,b)=6$, and $a=30$, then $b$ is:

    • 18
    • 24
    • 36
    • 54

    Answer: (c)

19. The decimal expansion of $\frac{2}{11}$ is:

    • Terminating
    • Non-terminating repeating
    • Non-terminating non-repeating
    • Whole number

    Answer: (b)

20. Which of the following is a rational number?

    • $\sqrt{3}$
    • $\sqrt{2}$
    • $\frac{22}{7}$
    • $\pi$

    Answer: (c)

21. The HCF of 8, 12 and 16 is:

    • 2
    • 4
    • 8
    • 16

    Answer: (b)

22. The LCM of 8, 12 and 16 is:

    • 24
    • 32
    • 48
    • 96

    Answer: (c)

23. Which of the following fractions has a non-terminating repeating decimal?

    • $\frac{3}{25}$
    • $\frac{5}{16}$
    • $\frac{7}{15}$
    • $\frac{9}{40}$

    Answer: (c)

24. The prime factorisation of 84 is:

    • $2^2 \times 3 \times 7$
    • $2 \times 3 \times 14$
    • $4 \times 21$
    • $2^3 \times 3 \times 7$

    Answer: (a)

25. If $a$ and $b$ are coprime numbers, then $HCF(a,b)$ is:

    • 0
    • 1
    • $a$
    • $b$

    Answer: (b)

26. The LCM of 15 and 20 is:

    • 45
    • 60
    • 75
    • 90

    Answer: (b)

27. If $\sqrt{2}$ is irrational, then $5\sqrt{2}$ is:

    • Rational
    • Irrational
    • Integer
    • Whole number

    Answer: (b)

28. The HCF of 144 and 180 is:

    • 12
    • 18
    • 24
    • 36

    Answer: (d)

29. If a rational number has denominator of the form $2^m5^n$ (where $m,n$ are non-negative integers), then its decimal expansion is:

    • Terminating
    • Non-terminating repeating
    • Non-terminating non-repeating
    • Irrational

    Answer: (a)

30. The HCF of 27 and 36 using prime factorisation is:

    • 3
    • 6
    • 9
    • 12

    Answer: (c)

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Part 3: Graded Sample Question Bank

Very Short Answer Type (1 Mark)

1. Express 156 as a product of its prime factors. [1 Mark]
2. Express 84 as a product of its prime factors. [1 Mark]
3. Express 210 as a product of its prime factors. [1 Mark]
4. Express 360 as a product of its prime factors. [1 Mark]
5. Express 128 as a product of its prime factors. [1 Mark]
6. Express 945 as a product of its prime factors. [1 Mark]
7. Express 231 as a product of its prime factors. [1 Mark]
8. Express 144 as a product of its prime factors. [1 Mark]
9. Express 225 as a product of its prime factors. [1 Mark]
10. Express 396 as a product of its prime factors. [1 Mark]
11. Express 504 as a product of its prime factors. [1 Mark]
12. If $a = 2^2 \times 3^1$ and $b = 2 \times 2 \times 5$, find the $HCF(a, b)$. [1 Mark]
13. If $a = 2^3 \times 5$ and $b = 2^2 \times 3$, find the $HCF(a, b)$. [1 Mark]
14. If $a = 3^2 \times 7$ and $b = 3 \times 5^2$, find the $HCF(a, b)$. [1 Mark]
15. If $a = 2^4 \times 3$ and $b = 2^2 \times 3^2$, find the $HCF(a, b)$. [1 Mark]
16. If $a = 5^2 \times 11$ and $b = 5 \times 7$, find the $HCF(a, b)$. [1 Mark]
17. If $a = 2^2 \times 7^2$ and $b = 2 \times 7 \times 3$, find the $HCF(a, b)$. [1 Mark]
18. If $a = 3^3 \times 5$ and $b = 3^2 \times 2$, find the $HCF(a, b)$. [1 Mark]
19. If $a = 2^3 \times 11$ and $b = 2 \times 11^2$, find the $HCF(a, b)$. [1 Mark]
20. If $a = 7^2 \times 13$ and $b = 7 \times 13^2$, find the $HCF(a, b)$. [1 Mark]
21. If $a = 2^5 \times 3^2$ and $b = 2^3 \times 3$, find the $HCF(a, b)$. [1 Mark]
22. If $a = 3^2 \times 5^2$ and $b = 3 \times 5^3$, find the $HCF(a, b)$. [1 Mark]

Short Answer Type I (2 Marks)

23. Check whether $6^n$ can end with the digit 0 for any natural number $n$. [2 Marks]
24. Check whether $4^n$ can end with the digit 0 for any natural number $n$. [2 Marks]
25. Check whether $5^n$ can end with the digit 5 for any natural number $n$. [2 Marks]
26. Check whether $2^n$ can end with the digit 0 for any natural number $n$. [2 Marks]
27. Check whether $3^n$ can end with the digit 1 for any natural number $n$. [2 Marks]
28. Check whether $7^n$ can end with the digit 3 for any natural number $n$. [2 Marks]
29. Check whether $8^n$ can end with the digit 2 for any natural number $n$. [2 Marks]
30. Check whether $9^n$ can end with the digit 7 for any natural number $n$. [2 Marks]
31. Check whether $10^n$ can end with the digit 0 for any natural number $n$. [2 Marks]
32. Check whether $12^n$ can end with the digit 5 for any natural number $n$. [2 Marks]
33. Check whether $15^n$ can end with the digit 0 for any natural number $n$. [2 Marks]
34. Prove that $3\sqrt{2}$ is irrational, provided that $\sqrt{2}$ is irrational. [2 Marks]
35. Prove that $2\sqrt{3}$ is irrational, provided that $\sqrt{3}$ is irrational. [2 Marks]
36. Prove that $5\sqrt{7}$ is irrational, provided that $\sqrt{7}$ is irrational. [2 Marks]
37. Prove that $4\sqrt{5}$ is irrational, provided that $\sqrt{5}$ is irrational. [2 Marks]
38. Prove that $7\sqrt{2}$ is irrational, provided that $\sqrt{2}$ is irrational. [2 Marks]
39. Prove that $6\sqrt{3}$ is irrational, provided that $\sqrt{3}$ is irrational. [2 Marks]
40. Prove that $9\sqrt{11}$ is irrational, provided that $\sqrt{11}$ is irrational. [2 Marks]
41. Prove that $8\sqrt{13}$ is irrational, provided that $\sqrt{13}$ is irrational. [2 Marks]
42. Prove that $10\sqrt{5}$ is irrational, provided that $\sqrt{5}$ is irrational. [2 Marks]
43. Prove that $11\sqrt{3}$ is irrational, provided that $\sqrt{3}$ is irrational. [2 Marks]
44. Prove that $12\sqrt{7}$ is irrational, provided that $\sqrt{7}$ is irrational. [2 Marks]
45. Prove that $3 + 2\sqrt{5}$ is irrational. Provided that $\sqrt{5}$ is irrational.[2 Marks]
46. Prove that $4 + 3\sqrt{7}$ is irrational, provided that $\sqrt{7}$ is irrational. [2 Marks]
47. Prove that $5 - 2\sqrt{3}$ is irrational, provided that $\sqrt{3}$ is irrational. [2 Marks]
48. Prove that $7 + \sqrt{11}$ is irrational, provided that $\sqrt{11}$ is irrational. [2 Marks]
49. Prove that $6 - 4\sqrt{2}$ is irrational, provided that $\sqrt{2}$ is irrational. [2 Marks]
50. Prove that $8 + 5\sqrt{13}$ is irrational, provided that $\sqrt{13}$ is irrational. [2 Marks]
51. Prove that $9 - 3\sqrt{5}$ is irrational, provided that $\sqrt{5}$ is irrational. [2 Marks]
52. Prove that $2 + 7\sqrt{3}$ is irrational, provided that $\sqrt{3}$ is irrational. [2 Marks]
53. Prove that $10 - 6\sqrt{7}$ is irrational, provided that $\sqrt{7}$ is irrational. [2 Marks]
54. Prove that $1 + 4\sqrt{6}$ is irrational, provided that $\sqrt{6}$ is irrational. [2 Marks]
55. Prove that $12 - 5\sqrt{2}$ is irrational, provided that $\sqrt{2}$ is irrational. [2 Marks]
56. Prove that $\frac{5 + 3\sqrt{5}}{4}$ is irrational, provided that $\sqrt{5}$ is irrational. [2 Marks]
57. Prove that $\frac{7 + 2\sqrt{3}}{5}$ is irrational, provided that $\sqrt{3}$ is irrational. [2 Marks]
58. Prove that $\frac{9 - 4\sqrt{2}}{3}$ is irrational, provided that $\sqrt{2}$ is irrational. [2 Marks]
59. Prove that $\frac{6 + 5\sqrt{7}}{8}$ is irrational, provided that $\sqrt{7}$ is irrational. [2 Marks]
60. Prove that $\frac{11 - 3\sqrt{5}}{6}$ is irrational, provided that $\sqrt{5}$ is irrational. [2 Marks]
61. Prove that $\frac{4 + 7\sqrt{11}}{9}$ is irrational, provided that $\sqrt{11}$ is irrational. [2 Marks]
62. Prove that $\frac{8 - 2\sqrt{13}}{7}$ is irrational, provided that $\sqrt{13}$ is irrational. [2 Marks]
63. Prove that $\frac{3 + 4\sqrt{6}}{5}$ is irrational, provided that $\sqrt{6}$ is irrational. [2 Marks]
64. Prove that $\frac{10 - 5\sqrt{3}}{12}$ is irrational, provided that $\sqrt{3}$ is irrational. [2 Marks]
65. Prove that $\frac{2 + 9\sqrt{2}}{11}$ is irrational, provided that $\sqrt{2}$ is irrational. [2 Marks]
66. Prove that $\frac{13 - 6\sqrt{7}}{10}$ is irrational, provided that $\sqrt{7}$ is irrational. [2 Marks]
67. Explain why $7 \times 11 \times 13 + 13$ is a composite number. [2 Marks]
68. Explain why $5 \times 9 \times 11 + 11$ is a composite number. [2 Marks]
69. Explain why $4 \times 7 \times 15 + 15$ is a composite number. [2 Marks]
70. Explain why $6 \times 13 \times 17 + 17$ is a composite number. [2 Marks]
71. Explain why $8 \times 9 \times 19 + 19$ is a composite number. [2 Marks]
72. Explain why $3 \times 10 \times 21 + 21$ is a composite number. [2 Marks]
73. Explain why $2 \times 5 \times 23 + 23$ is a composite number. [2 Marks]
74. Explain why $7 \times 8 \times 25 + 25$ is a composite number. [2 Marks]
75. Explain why $9 \times 14 \times 29 + 29$ is a composite number. [2 Marks]
76. Explain why $11 \times 12 \times 31 + 31$ is a composite number. [2 Marks]
77. Explain why $4 \times 6 \times 27 + 27$ is a composite number. [2 Marks]

Short Answer Type II (3 Marks)

1. Prove that $\sqrt{5}$ is an irrational number. [3 Marks]
2. Prove that $\sqrt{7}$ is an irrational number. [3 Marks]
3. Prove that $\sqrt{11}$ is an irrational number. [3 Marks]
4. Prove that $\sqrt{13}$ is an irrational number. [3 Marks]
5. Prove that $\sqrt{17}$ is an irrational number. [3 Marks]
6. Prove that $\sqrt{19}$ is an irrational number. [3 Marks]
7. Prove that $\sqrt{23}$ is an irrational number. [3 Marks]
8. Prove that $\sqrt{29}$ is an irrational number. [3 Marks]
9. Prove that $\sqrt{31}$ is an irrational number. [3 Marks]
10. Prove that $\sqrt{37}$ is an irrational number. [3 Marks]
11. Prove that $\sqrt{41}$ is an irrational number. [3 Marks]
12. Find the LCM and HCF of 240 and 96 by the prime factorisation method and verify that $LCM \times HCF = \text{product of the two numbers}$. [3 Marks]
13. Find the LCM and HCF of 180 and 144 by the prime factorisation method and verify that $LCM \times HCF = \text{product of the two numbers}$. [3 Marks]
14. Find the LCM and HCF of 210 and 84 by the prime factorisation method and verify that $LCM \times HCF = \text{product of the two numbers}$. [3 Marks]
15. Find the LCM and HCF of 324 and 90 by the prime factorisation method and verify that $LCM \times HCF = \text{product of the two numbers}$. [3 Marks]
16. Find the LCM and HCF of 150 and 225 by the prime factorisation method and verify that $LCM \times HCF = \text{product of the two numbers}$. [3 Marks]
17. Find the LCM and HCF of 288 and 108 by the prime factorisation method and verify that $LCM \times HCF = \text{product of the two numbers}$. [3 Marks]
18. Find the LCM and HCF of 420 and 168 by the prime factorisation method and verify that $LCM \times HCF = \text{product of the two numbers}$. [3 Marks]
19. Find the LCM and HCF of 360 and 72 by the prime factorisation method and verify that $LCM \times HCF = \text{product of the two numbers}$. [3 Marks]
20. Find the LCM and HCF of 252 and 198 by the prime factorisation method and verify that $LCM \times HCF = \text{product of the two numbers}$. [3 Marks]
21. Find the LCM and HCF of 270 and 192 by the prime factorisation method and verify that $LCM \times HCF = \text{product of the two numbers}$. [3 Marks]
22. Find the LCM and HCF of 336 and 54 by the prime factorisation method and verify that $LCM \times HCF =\text{product of the two numbers}$. [3 Marks]

Long Answer / Application Based (4 Marks)

1. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point? Explain your mathematical reasoning. [4 Marks]
2. Two runners are running around a circular track. Aman completes one round in 15 minutes, while Rohan completes one round in 20 minutes. If they start together from the same point and run in the same direction, after how many minutes will they meet again at the starting point? Explain your reasoning. [4 Marks]
3. Neha and Priya walk around a circular park. Neha takes 24 minutes to complete one round, and Priya takes 16 minutes. If they start at the same time from the same point in the same direction, after how many minutes will they meet again at the starting point? Explain your reasoning. [4 Marks]
4. Two cyclists take 14 minutes and 21 minutes respectively to complete one round of a circular track. If they start together and move in the same direction, after how many minutes will they be together again at the starting point? Justify your answer. [4 Marks]
5. Rahul takes 28 minutes to complete one round of a circular ground, while Mohan takes 35 minutes. If both start at the same time from the same point and move in the same direction, after how many minutes will they meet again at the starting point? Explain clearly. [4 Marks]
6. Two athletes complete one round of a stadium in 18 minutes and 27 minutes respectively. If they begin together from the same point in the same direction, after how many minutes will they again reach the starting point together? Give mathematical reasoning. [4 Marks]
7. Anita and Kavya jog around a circular path. Anita takes 32 minutes for one round and Kavya takes 48 minutes. If they start simultaneously from the same point and move in the same direction, after how many minutes will they meet again at the starting point? Explain your reasoning. [4 Marks]
8. Two friends complete one round of a circular park in 9 minutes and 15 minutes respectively. If they start together and move in the same direction, after how many minutes will they be together again at the starting point? Explain your method. [4 Marks]
9. Arjun and Vivek take 20 minutes and 30 minutes respectively to complete one round of a circular field. If they start at the same time from the same point and go in the same direction, after how many minutes will they meet again at the starting point? Justify your answer. [4 Marks]
10. Two walkers take 25 minutes and 40 minutes respectively to go once around a circular garden. If they begin together and walk in the same direction, after how many minutes will they reach the starting point together again? Explain clearly. [4 Marks]
11. Seema and Ritu take 36 minutes and 45 minutes respectively to complete one round of a circular path. If they start together from the same point and move in the same direction, after how many minutes will they meet again at the starting point? Explain the mathematical reasoning. [4 Marks]
12. Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method. Show that the product of these three numbers is not equal to the product of their HCF and LCM. [4 Marks]
13. Find the HCF and LCM of 8, 48 and 160, using the prime factorisation method. Show that the product of these three numbers is not equal to the product of their HCF and LCM. [4 Marks]
14. Find the HCF and LCM of 9, 54 and 81, using the prime factorisation method. Show that the product of these three numbers is not equal to the product of their HCF and LCM. [4 Marks]
15. Find the HCF and LCM of 12, 90 and 150, using the prime factorisation method. Show that the product of these three numbers is not equal to the product of their HCF and LCM. [4 Marks]
16. Find the HCF and LCM of 15, 60 and 105, using the prime factorisation method. Show that the product of these three numbers is not equal to the product of their HCF and LCM. [4 Marks]
17. Find the HCF and LCM of 10, 45 and 75, using the prime factorisation method. Show that the product of these three numbers is not equal to the product of their HCF and LCM. [4 Marks]
18. Find the HCF and LCM of 14, 84 and 196, using the prime factorisation method. Show that the product of these three numbers is not equal to the product of their HCF and LCM. [4 Marks]
19. Find the HCF and LCM of 16, 64 and 96, using the prime factorisation method. Show that the product of these three numbers is not equal to the product of their HCF and LCM. [4 Marks]
20. Find the HCF and LCM of 18, 108 and 144, using the prime factorisation method. Show that the product of these three numbers is not equal to the product of their HCF and LCM. [4 Marks]
21. Find the HCF and LCM of 20, 100 and 140, using the prime factorisation method. Show that the product of these three numbers is not equal to the product of their HCF and LCM. [4 Marks]
22. Find the HCF and LCM of 21, 63 and 126, using the prime factorisation method. Show that the product of these three numbers is not equal to the product of their HCF and LCM. [4 Marks]

MCQs on Class 9 Maths: Chapter 3 Coordinate Geometry

Each question has 4 options. Choose the correct answer to improve your problem-solving skills.

1. The name of the horizontal line in the Cartesian plane which determines the position of a point is called:

• a. Origin
• b. $x$-axis
• c. $y$-axis
• d. Quadrants

Answer: b

5. If the coordinates of a point are $(0,-4)$, then it lies on:

• a. $x$-axis
• b. $y$-axis
• c. At origin
• d. Between $x$-axis and $y$-axis

Answer: b

Explanation: Since $x=0$ and $y=-4$, the point will lie on the negative $y$-axis $4$ units away from the origin.

11. Signs of the abscissa and ordinate of a point in the second quadrant are respectively:

• a. $+$, $+$
• b. $+$, $-$
• c. $-$, $+$
• d. $-$, $-$

Answer: c

Explanation: The signs of abscissa ($x$-value) and ordinate ($y$-value) in the second quadrant are $-$ and $+$ respectively.

14. Abscissa of all the points on the $x$-axis is:

• a. $0$
• b. $1$
• c. $2$
• d. Any number

Answer: d

Explanation: Abscissa of all the points on the $x$-axis can be any number. The coordinates of any point on the $x$-axis is $(x,0)$, where $x$ can take any value.

20. The point which lies on the $y$-axis at a distance of $5$ units in the negative direction of the $y$-axis is:

• a. $(5,0)$
• b. $(0,5)$
• c. $(-5,0)$
• d. $(0,-5)$

Answer: d

Explanation: The coordinates of any point on the $y$-axis is $(0,y)$. Given that the point lies in the negative direction, $y$ must be $-5$. Therefore, the point is $(0,-5)$.

Indian History MCQ GK Set-10

Indian History MCQ GK Set-10

Q.1 Which of the following conspiracy is related to the attack on Lord Hardinge?

A. Delhi Conspiracy

B. Gadhar Conspiracy

C. Hindu German Conspiracy

D. None of these

Answer: A. Delhi Conspiracy

Note: The Delhi Conspiracy case, also known as the Delhi-Lahore Conspiracy, refers to a conspiracy in 1912 to assassinate the then Viceroy of India, Lord Hardinge, on the occasion of transferring the capital of British India from Calcutta to Delhi.

Q.2 Rash Behari Bose is known for:

A. Delhi Conspiracy

B. Gadhar Conspiracy

C. Only A

D. Both A & B

Answer: D. Both A & B

Note: Following the attempt to assassinate Lord Hardinge, Rash Behari was forced to go into hiding. Later,  he became extensively involved as one of the leading figures of the Gadar Revolution that attempted to trigger a mutiny in India in February 1915 and fled to Japan after that incident.

Q.3 The principle of diarchy introduced by the Government of India Act, 1919 refers to which of the following?

A. Separation of Judiciary and Legislature

B. Parallel government at India and London

C. Division of subjects transferred to provinces into two categories

D. Rule by both British and Indian

Answer: C. Division of subjects transferred to provinces into two categories

Note: Under the diarchy system, the subjects of legislation were divided into central and provinces categories. The subjects delegated to the provinces were further divided into ‘reserved’ and ‘transferred’ categories.

Q.4 Mahatma Gandhi launched the Salt March from which of the following location?

A. Bombay

B. Champran

C. Sabaramati Ashram

D. Lahore

Answer: C. Sabaramati Ashram

Note:  The twenty five day march of Last March lasted from 12 March 1930 to 5 April 1930 as a direct action campaign of tax resistance and nonviolent protest against the British salt monopoly.

Q.5 Who among the following termed the Government of India Act, 1935 as the “Charter of Slavery”?

A. Mahatma Gandhi

B. Subhash Chandra Bose

C. Jawaharlal Nehru

D. Maulana Azad

Answer: C. Jawaharlal Nehru

Note: As per the Government of India Act of 1935 special powers were vested to the Governor-General and for that reason, Jawaharlal Nehru termed this situation as “Charter of Slavery”.

Q.6 Who among the following conceived the idea of Pakistan?

A. Muhammad Ali Jinnah

B. Muhammad Iqbal

C. Aga Khan

D. None of these

Answer: B. Muhammad Iqbal

Q.7 Swadeshi Movement started as a consequence of:

A. Partition of Bengal

B. Rowlatt Act

C. Champaran Satyagraha

D. None of these

Answer: A. Partition of Bengal

Q.8 Who among the following was the first Indian native ruler to accept the system of Subsidiary Alliance?

A. Nizam of Hyderabad

B. Scindia of Gwalior

C. Gaikwad of Baroda

D. None of these

Answer: A. Nizam of Hyderabad

Q.9 Lala Lajpat Rai died during the protest of:

A. Rowlatt Act

B. Simon Commission

C. Massacre of Jalliwanwala Bagh

D. None of these

Answer: B. Simon Commission

Q.10 Simon Commission was rejected by India because:

A. It was an all-white commission with Indian representation

B. It emphasized on sedition charges of protesters

C. It proposed partition of India

D. None of these

Answer: A. It was an all-white commission with Indian representation

Indian History MCQ GK Set-9

Indian History MCQ GK Set-9

Q.1 Who among the following was responsible for the first lawful widow remarriage among Hindu upper castes in India?

A. Raja Rammohan Roy

B. Mahadev Govind Ranade

C. Ishwar Chandra Vidyasagar

D. Dayananda Saraswati

Answer: C. Ishwar Chandra Vidyasagar

Q.2 Who among the following is known as the “Father of Bengali Prose”?

A. Rabindranath Tagore

B. Ishwar Chandra Vidyasagar

C. Chittaranjan Das

D. Sanjib Chattopadhay

Answer: B. Ishwar Chandra Vidyasagar

Q.3 Who among the following led the Theosophical society in India?

A. Dadabhai Naroji

B. Annie Besant

C. Dayananda Saraswati

D. Motilal Nehru

Answer: B. Annie Besant

Q.4 Which among the following idea was popularized through the Theosophical society?

I. Reincarnation, karma and drew inspiration from philosophy from Upanishads, Samkhya, Vedanta and Yoga school

II. Universal brotherhood among various religions

III. Abstaining from consumption of meat

A. Only I

B. Only II

C. Only II

D. Both I & II

Answer: A. Only I

Q.5 Which among the following was/were the primary philosophy of the Satyashodhak Samaj?

I. Spread of Service

II. Spread of education among women and lower castes

III. Complete abolition of caste system and inequalities 

A. Only I

B. Only II

C. Only III

D. I, II & III

Answer: D. I, II and III

Note: Satyashodhak Samaj (Truth-seekers’ Society) was a social reform society founded by Jyotirao Phule in Pune, Maharashtra, on 24 September 1873.

Q.6 Who among the following is regarded as the first female teacher of India?

A. Savitribai Phule

B. Satyavati Devi

C. Sarojini Naidu

D. Ela Bhatt

Answer: A. Savitribai Phule

Q.7 Brahmo Samaj was created for which of the following purpose?

I. Purify Hinduism and preach monotheism

II. Criticize social evils, oppose idolatry and emphasize human dignity

III. To make faith in doctrine of Karma

A. Only I

B. Only II

C. Only III

D. I, II and III

Answer: D. I, II and III

Q.8 Who among the following was the founder of the “Brahmo Samaj”?

A. Raja Ram Mohan Roy

B. Dwarkanath Tagore

C. Debendranath Tagore

D. None of these

Answer: A. Raja Ram Mohan Roy

Q.9 Who among the following was the founder of the “Young Bengals” a free-thinker group of Hindu College, Calcutta?

A. Henry Louis Vivian Derozio

B. Alexander Duff

C. Lal Behari Dey

D. Brajendra Nath Seal

Answer: A. Henry Louis Vivian Derozio

Note: 

Q.10 Which of the following reform is a legislative contribution of Raja Ram Mohan Roy?

A. Abolition of Sati

B. Establishment of Hindu Marriage Law

C. Demolishment of Caste System

D. None of these

Answer: A. Abolition of Sati

Note: Bengal Sati Regulation Act, 1829 legally abolished the sati system in India. Even after passing the law, in 1830, Ram Mohan Roy travelled to the United Kingdom as an ambassador of the Mughal Empire to ensure that Lord William Bentinck’s Bengal Sati Regulation, 1829 banning the practice of Sati was not overturned.

Indian History MCQ GK Set-8

Indian History MCQ GK Set-8

Q.1 Which of the following incident compelled Mahatma Gandhi to withdraw the nation’s cooperation from the British Government?

A. Jallianwala Bagh Massacre

B. Hanging of Bhagat Singh

C. Passing of Rowlatt Act

D. Both A and C

Answer: A. Both A & C

Q.2 Which of the following incident led to withdrawal of the Non-Cooperation Movement?

A. Chauri Chaura Incident

B. Khilafat Movement

C. Introduction of Civil Disobedience Movement

D. None of these

Answer: A. Chauri Chaura Incident

Note: The Chauri Chaura incident took place on 4 February 1922 at Chauri Chaura in the Gorakhpur district of the United Provinces (modern-day Uttar Pradesh), where protesters has clashed with policemen leading to death and Gandhi was against of violence.

Q.3 Which of the following is CORRECT about the Quit India Movement?

I. It was launched by Mahatma Gandhi on the Bombay Session of the All India Congress

II. It was launched on 9th August, 1942

III. It was started after the failure of Cripps’ Mission

A. Only I

B. Only II

C. Only III

D. I, II & III

Answer: D. I, II and III

Note: Cripps’ Mission was a mission to negotiate with the Indian National Congress a deal to obtain total co-operation during the World War II.

Q.4 Which of the following is the another name of the Quit India Movement?

A. August Movement

B. Purna Swaraj Movement

C. Cripps’ Movement

D. World War Movement

Answer: A. August Movement

Note: The Quit India Movement was also known as “August Kranti or August Movement” as it was started on 9th August, 1942.

Q.5 Who among the following is recognized as the youngest martyr of the Quit India Movement?

A. Kanaklata Barua

B. Tileswari Barua

C. Bhagat Singh

D. None of these

Answer: B. Tileswari Barua

Note: Tileswari Barua of Dhekiajuli, Assam died hoisting the national flag at the age of 12.

Q.6 Who among the following had given the slogan “Do or Die” at the Bombay session of the All India Congress and ratified the Quit India Movement?

A. Subhash Chandra Bose

B. Jawaharlal Nehru

C. Mahatma Gandhi

D. Vallabhbhai Patel

Answer: C. Mahatma Gandhi

Q.7 Who among the following was NOT a organizing member of the Champaran Satyagraha, the first Satyagraha led by Gandhi?

A. Jawaharlal Nehru

B. Rajendra Prasad

C. J.B. Kripalani

D. Brajkishore Prasad

Answer: A. Jawaharlal Nehru

Q.8 Who among the following has written the book “Poverty and Un-British Rule in India”, that brought the theory of wealth-drain to British?

A. Jawaharlal Nehru

B. Gopal Krishna Gokhale

C. Bal Gangadhar Tilak

D. Dadabhai Naroji

Answer: D. Dadabhai Naroji

Q.9 Who among the following is known as the “Beacon of light of Asia”?

A. Subhash Chandra Bose

B. Mahatma Gandhi

C. Dayananda Saraswati

D. Dadabhai Naroji

Answer: A. Subhash Chandra Bose

Q.10 The first Indian National Army was formed under who among the following?

A. Subhash Chandra Bose

B. Mohan Singh

C. Jaganath Rao Bhonsale

D. Prem Sahgal

Answer: B. Mohan Singh