CBSE Class 9 Maths Ganita Manjari Part 1 Ch 03 The World of Numbers Long Answer and Case Study Questions with Answers

5 Mark Questions

Question 1. A square park has side 5 m. A diagonal path is constructed across it.
(a) Find the length of the diagonal.
(b) Simplify it.
(c) Identify the number type.
(d) Explain its representation on the number line.
(e) Compare it with a rational value.
Answer:
(a) Diagonal = \( \sqrt{5^2 + 5^2} = \sqrt{50} \) m
(b) \( \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2} \) m
(c) \( \sqrt{2} \) is irrational, and a non-zero rational (5) times an irrational is irrational, so \( 5\sqrt{2} \) is irrational.
(d) Draw a right triangle on the number line with legs 5 and 5 units, starting at 0. The hypotenuse has length \( \sqrt{50} \). With the compass at 0 and radius equal to the hypotenuse, draw an arc to cut the number line. That point is \( 5\sqrt{2} \).
(e) \( 7^2 = 49 < 50 < 50.41 = 7.1^2 \), so \( 7 < 5\sqrt{2} < 7.1 \). In fact \( 5\sqrt{2} \approx 7.07 \) m, just a little more than 7 m.
In simple words: The diagonal is 5√2 m, about 7.07 m, a length that can be drawn exactly but never written as a fraction.

Exam Tip: Always simplify square roots by taking out perfect-square factors, such as √50 = 5√2.

Question 2. A city planner is designing a smart park. The park includes walking tracks, water pipelines and digital sensors. The measurements involve numbers like \( \frac{3}{4} \), \( \sqrt{2} \), 0.666… and −5 (representing underground depth). The planner must classify numbers correctly to ensure accurate calculations and avoid errors in design.
(i) Which number represents an irrational value? (a) \( \frac{3}{4} \) (b) 0.666… (c) \( \sqrt{2} \) (d) −5
(ii) Which number has a recurring decimal form? (a) \( \frac{3}{4} \) (b) 0.666… (c) \( \sqrt{2} \) (d) −5
(iii) Which number represents underground depth? (a) \( \frac{3}{4} \) (b) 0.666… (c) \( \sqrt{2} \) (d) −5
(iv) Which number is a terminating decimal? (a) \( \frac{3}{4} \) (b) 0.666… (c) \( \sqrt{2} \) (d) −5
(v) Write 0.666… in \( \frac{p}{q} \) form.
Answer:
(i) (c) \( \sqrt{2} \): 2 is not a perfect square.
(ii) (b) 0.666…: the digit 6 repeats.
(iii) (d) −5: a negative number shows a level below the ground.
(iv) (a) \( \frac{3}{4} = 0.75 \), which ends.
(v) Let \( x = 0.666… \). Then \( 10x = 6.666… \), so \( 9x = 6 \Rightarrow x = \frac{6}{9} = \frac{2}{3} \).
In simple words: √2 is irrational, 0.666… repeats, −5 is below ground, and 3/4 ends as 0.75.

Exam Tip: In (v), subtract the two equations so the repeating part cancels.

Question 3. A delivery vehicle moves along a straight road, starting from −5 km, then travelling to 3 km, moving back to −2 km and finally reaching 4 km. Determine the total distance travelled and the net displacement of the vehicle. Also interpret the result by comparing the two values.
Answer:
Leg 1: −5 to 3 = \( |3 - (-5)| = 8 \) km
Leg 2: 3 to −2 = \( |-2 - 3| = 5 \) km
Leg 3: −2 to 4 = \( |4 - (-2)| = 6 \) km
Total distance = 8 + 5 + 6 = 19 km
Net displacement = final − initial = \( 4 - (-5) = \) 9 km (in the positive direction)
Interpretation: The distance (19 km) is much greater than the displacement (9 km) because the vehicle changed direction and went back over part of the road. They would be equal only if the vehicle moved in one direction without turning back.
In simple words: The vehicle drove 19 km in total but finished only 9 km from where it started.

Exam Tip: Use absolute values for each leg of distance, but not for displacement.

Question 4. A student considers the numbers \( \sqrt{5}, \sqrt{25}, \sqrt{3}, \sqrt{36} \).
(a) Evaluate the expressions wherever possible.
(b) Classify each number as rational or irrational.
(c) How many irrational numbers are there in the group?
(d) Provide a clear explanation for your classification (rational / irrational).
(e) Out of \( \sqrt{5}, \sqrt{25}, \sqrt{3} \) and \( \sqrt{36} \), whose decimal expansion is non-terminating non-repeating?
Answer:
(a) \( \sqrt{25} = 5 \) and \( \sqrt{36} = 6 \). \( \sqrt{5} \approx 2.236… \) and \( \sqrt{3} \approx 1.732… \) cannot be written exactly as decimals.
(b) Rational: \( \sqrt{25}, \sqrt{36} \). Irrational: \( \sqrt{5}, \sqrt{3} \).
(c) There are two irrational numbers.
(d) 25 and 36 are perfect squares, so their roots are integers, which can be written as \( \frac{p}{q} \). 5 and 3 are not perfect squares, so their roots cannot be written as \( \frac{p}{q} \).
(e) \( \sqrt{5} \) and \( \sqrt{3} \).
In simple words: Square roots of perfect squares are whole numbers; the others go on forever without repeating.

Exam Tip: Irrational numbers are exactly those with non-terminating, non-repeating decimals; link (b) and (e) in your answer.

Case Study Based Questions

Question 5. A school is constructing a square playground of side 2 m for a model. The diagonal is calculated for placing a rope across it.
(i) Find the length of the diagonal.
(ii) State whether the diagonal is rational or irrational.
(iii) Justify your answer using appropriate reasoning.
OR
Write the decimal expansion of the diagonal correct up to 3 decimal places. State whether the decimal expansion is terminating or non-terminating.
Answer:
(i) Diagonal = \( \sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2} \) m
(ii) The diagonal is irrational.
(iii) \( \sqrt{2} \) is irrational because 2 is not a perfect square. Multiplying an irrational number by a non-zero rational number (2) gives an irrational number, so \( 2\sqrt{2} \) is irrational.
OR \( 2\sqrt{2} = 2 \times 1.41421… = 2.82842… \approx \) 2.828 m. The decimal expansion is non-terminating and non-repeating.
In simple words: The rope must be 2√2 m, about 2.828 m, and its decimal never ends or repeats.

Exam Tip: When rounding to 3 decimal places, look at the 4th decimal digit (here 4, so 2.828 stays).

Question 6. A construction engineer uses different types of numbers while designing a structure. He uses \( \sqrt{5} \) for diagonal measurements, 0.75 for material ratio and −3 to represent underground depth.
(i) Which of the following is irrational? (a) 0.75 (b) −3 (c) \( \sqrt{5} \) (d) 1
(ii) The decimal 0.75 is: (a) Irrational (b) Terminating (c) Recurring (d) Undefined
(iii) The number −3 represents a: (a) Natural number (b) Whole number (c) Integer (d) Irrational number
(iv) Which of the following statements is correct? (a) All integers are irrational (b) All rational numbers are integers (c) Some rational numbers are integers (d) Irrational numbers are finite
Answer:
(i) (c) \( \sqrt{5} \): 5 is not a perfect square.
(ii) (b) Terminating: 0.75 ends after two digits (\( = \frac{3}{4} \)).
(iii) (c) Integer: −3 is negative, so it is not natural or whole, but it is an integer.
(iv) (c) Some rational numbers are integers, for example \( 4 = \frac{4}{1} \). Others, such as \( \frac{1}{2} \), are not.
In simple words: √5 is irrational, 0.75 terminates, −3 is an integer, and only some rational numbers are integers.

Exam Tip: Remember the chain: natural ⊂ whole ⊂ integers ⊂ rational numbers.

Question 7. A weather station records temperature changes throughout the day. In the morning, the temperature is −3°C. By afternoon, it rises to \( \frac{5}{2} \)°C. The system tracks the change using number line concepts and also calculates differences to predict weather patterns.
(a) Find the change in temperature.
(b) Identify the type of number \( \frac{5}{2} \) is.
(c) Convert \( \frac{5}{2} \) into decimal form and classify the decimal.
OR
(d) Represent both temperatures on the number line and determine which temperature is closer to 0°C.
Answer:
(a) Change = \( \frac{5}{2} - (-3) = \frac{5}{2} + 3 = \frac{11}{2} = 5.5 \)°C (a rise).
(b) \( \frac{5}{2} \) is a rational number (in the form \( \frac{p}{q} \), q ≠ 0).
(c) \( \frac{5}{2} = 2.5 \), a terminating decimal.
OR (d) −3 is 3 units to the left of 0, and 2.5 is 2.5 units to the right of 0. Since 2.5 < 3, \( \frac{5}{2} \)°C is closer to 0°C.
In simple words: The temperature rose by 5.5°C, and the afternoon reading of 2.5°C is closer to zero than −3°C.

Exam Tip: 'Closer to 0' means smaller distance from 0, so compare the numbers without their signs.