CBSE Class 9 Maths Ganita Manjari Part 1 Ch 03 The World of Numbers MCQs with Answers Set 01

Answer: (c) Riya is incorrect because on the number line \( -\frac{3}{4} \) lies to the left of −0.6.

Explanation:
1. \( -\frac{3}{4} = -0.75 \).
2. For negative numbers, the one with the larger absolute value is further left on the number line, so it is smaller.
3. Since −0.75 lies to the left of −0.6, \( -\frac{3}{4} \) comes first. Correct order: \( -\frac{3}{4}, -0.6, 0, \frac{2}{3} \).

Answer: (c) Infinitely many rational numbers exist between two numbers.

Explanation:
1. The average \( \frac{2 + 3}{2} = 2.5 \) does give one number between 2 and 3.
2. But we can repeat the process: between 2 and 2.5 lies 2.25, between 2 and 2.25 lies 2.125, and so on.
3. So there are infinitely many rational numbers between any two numbers. This is called the density property.

Answer: (b) Q

Explanation:
1. Closure means the operation must give a natural number for every pair, without exception.
2. \( 3 - 5 = -2 \), which is not a natural number. One such example is enough to disprove closure.
3. R's single example cannot prove closure for all pairs, and P's statement is false.

Answer: (a) \( x^2 = 5 \)

Explanation:
1. \( x = \sqrt{5} \); 5 is not a perfect square, so \( \sqrt{5} \) is irrational.
2. \( y = 3 \), \( z = 0.2 \) and \( u = \sqrt{4} = 2 \) are all rational.

Answer: (c) Terminating

Explanation:
1. \( 16 = 2^4 \). The denominator has no prime factor other than 2 (and 5).
2. So the decimal terminates: \( \frac{11}{16} = 0.6875 \).

Answer: (a) \( \frac{1}{2} \)

Explanation:
1. \( \frac{2}{5} = 0.4 \) and \( \frac{3}{5} = 0.6 \).
2. \( \frac{1}{2} = 0.5 \), which lies between 0.4 and 0.6.
3. \( \frac{5}{6} \approx 0.83 \), \( \frac{4}{5} = 0.8 \) and \( \frac{1}{5} = 0.2 \) are all outside the range.

Answer: (c) \( \sqrt{2} \)

Explanation:
1. The square root of a natural number that is not a perfect square is irrational.
2. Since 2 is not a perfect square, \( \sqrt{2} \) is irrational.
3. 2 + 3 = 5, 2 × 3 = 6 and \( 2^3 = 8 \) are all integers, so they are rational.

Answer: (d) \( \frac{2}{11} \)

Explanation:
1. Let \( x = 0.181818… \). Two digits repeat, so multiply by 100: \( 100x = 18.1818… \).
2. Subtract: \( 99x = 18 \Rightarrow x = \frac{18}{99} = \frac{2}{11} \).

Answer: (b) The decimal digits repeat in a fixed pattern.

Explanation:
1. Every non-terminating decimal whose digits repeat in a fixed block is rational.
2. Here the block '27' repeats, so the number is rational.
3. Having infinitely many digits alone is not enough, because irrational numbers also have infinitely many digits.

Answer: (c) 8 m

Explanation:
1. Below sea level is written as −6 and above as +2.
2. Change = final − initial = \( 2 - (-6) = 8 \) m (upwards).

Answer: (d) 0.123451234512345…

Explanation:
1. In (a), (b) and (c) the digits never settle into a repeating block, so they are non-terminating non-repeating (irrational).
2. In (d), the block 12345 repeats again and again, so it is a recurring decimal, which is rational.
3. So (d) is the odd one out.

Answer: (c) Since \( 1.4 < \sqrt{2} < 1.5 \), it should lie between 1.4 and 1.5.

Explanation:
1. \( 1.4^2 = 1.96 \) and \( 1.5^2 = 2.25 \). Since 1.96 < 2 < 2.25, we get \( 1.4 < \sqrt{2} < 1.5 \).
2. So \( \sqrt{2} \approx 1.414 \) lies between 1.4 and 1.5, just left of 1.5.
3. Every irrational number has an exact position on the number line, so (d) is wrong.

Answer: (b) \( \frac{5}{9} \)

Explanation:
1. Let \( x = 0.555… \). Then \( 10x = 5.555… \).
2. Subtract: \( 9x = 5 \Rightarrow x = \frac{5}{9} \).

Answer: (c) \( \frac{2}{3} \)

Explanation:
1. \( \frac{2}{3} = 0.666… \), which never terminates but is still rational.
2. The other options (0.75, 0.55, 0.9) all terminate, so they do not challenge her claim.

Answer: (c) Chitra

Explanation:
1. Writing \( \frac{2}{5} = \frac{20}{50} \) and \( \frac{3}{5} = \frac{30}{50} \), we already find \( \frac{21}{50}, \frac{22}{50}, … \) between them.
2. Using larger denominators gives even more, without end.
3. So there are infinitely many rational numbers between them, as Chitra says.

Answer: (b) Counting (natural) numbers begin from 1

Explanation:
1. Natural numbers are the counting numbers 1, 2, 3, …; counting starts from 1.
2. Zero is included in the whole numbers 0, 1, 2, 3, …, but it is not a natural number.
3. 0 is neither negative nor irrational.

Answer: (c) 4 − 9 = −5

Explanation:
1. To show natural numbers are not closed under subtraction, we need two natural numbers whose difference is not natural.
2. \( 4 - 9 = -5 \), and −5 is not a natural number.
3. The other options all give natural numbers.

Answer: (b) Identity and zero property

Explanation:
1. \( x + 0 = x \) shows that 0 is the additive identity.
2. \( x \times 0 = 0 \) is the zero property of multiplication.

Answer: (d) A is false, but R is true.

Explanation:
1. Closure would need \( a \div b \) to be rational for every pair of rational numbers, including b = 0.
2. Since \( a \div 0 \) is not defined, rational numbers are not closed under division. The Assertion is false.
3. The Reason is a true statement, and in fact it is the reason the Assertion fails.

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. Natural numbers are 1, 2, 3, …, so 0 is not natural; integers are …, −2, −1, 0, 1, 2, …, so 0 is an integer. The Assertion is true.
2. The Reason is true and states exactly these two definitions, so it correctly explains the Assertion.