Case Study / Scenario-Based MCQs for Sub-Topics of Topic 1: Numbers & Numeriacal Applications

Question 1. Case Study: Sorting Numbers for a Project

A group of students collected various numbers for a project on number systems. The numbers collected are: $5, -3, 0, \frac{1}{2}, \sqrt{7}, 2.\overline{3}, -8.1, \sqrt{16}, \pi, -4/5$. They need to classify these numbers into different categories.

Based on the collected numbers, identify the numbers that are integers:

(A) $5, -3, 0, \sqrt{16}$

(B) $5, -3, 0, -8.1$

(C) $5, 0, \frac{1}{2}, -4/5$

(D) $\sqrt{7}, \pi$

Question 2. Referring to the Case Study above, which of the collected numbers are rational numbers?

(A) $5, -3, 0, \frac{1}{2}, 2.\overline{3}, -8.1, \sqrt{16}, -4/5$

(B) $5, -3, 0, \frac{1}{2}, \sqrt{7}, \sqrt{16}, -4/5$

(C) $5, -3, 0, 2.\overline{3}, -8.1, \pi, -4/5$

(D) $\sqrt{7}, \pi, 2.\overline{3}, -8.1$

Question 3. Referring to the Case Study above, which of the collected numbers are irrational numbers?

(A) $\sqrt{16}, \pi$

(B) $\sqrt{7}, \pi$

(C) $\frac{1}{2}, 2.\overline{3}$

(D) $-3, -8.1$

Question 4. Referring to the Case Study above, which statements are correct about the collected numbers?

(A) All integers are also whole numbers.

(B) 0 is an integer but not a natural number.

(C) $\sqrt{16}$ is a real number.

(D) $-8.1$ is an integer.

Question 5. Referring to the Case Study above, if the students also considered prime and composite numbers from the positive integers in their list, which would be the smallest prime and smallest composite number respectively?

(A) 5 and $\sqrt{16}$

(B) 5 and None (as $\sqrt{16}$ is not in original list)

(C) $\sqrt{16}$ and 5

(D) 5 and -3

Question 1. Case Study: Understanding Large Numbers in India

Mr. Sharma is reading a report about government finances in India. He encounters numbers like $\textsf{₹} 1,75,86,93,124$ (one hundred seventy-five crore eighty-six lakh ninety-three thousand one hundred twenty-four) and wants to compare it with figures presented in the International System. He also sees figures written out in words and needs to convert them to digits.

Based on the number $\textsf{₹} 1,75,86,93,124$, what is the place value of the digit 8 in the Indian System?

(A) Ten Lakhs

(B) Lakhs

(C) Ten Crores

(D) Crores

Question 2. Referring to the Case Study above, how would the number $\textsf{₹} 1,75,86,93,124$ be read in the International System?

(A) One billion seventy-five million eight hundred sixty-nine thousand three hundred twenty-four

(B) One hundred seventy-five million eight hundred sixty-nine thousand three hundred twenty-four

(C) One billion seven hundred fifty-eight million six hundred ninety-three thousand one hundred twenty-four

(D) One hundred seventy-five crore eighty-six million nine hundred thirty-one thousand two hundred four

Question 3. Referring to the Case Study above, the report mentions a figure of 'fifty crore fifty lakh'. How is this written in figures in the Indian System?

(A) 50,00,50,000

(B) 50,50,00,000

(C) 50,50,000

(D) 5,05,00,000

Question 4. Referring to the Case Study above, Mr. Sharma sees that the budget allocated for a specific project is $\textsf{₹} 500,000$. How would this amount be referred to in the Indian System?

(A) 50 Lakh

(B) 5 Lakh

(C) 5 Crore

(D) 50 Thousand

Question 5. Referring to the Case Study above, Mr. Sharma also needs to record some dates using Roman numerals. He needs to write the year 2023. How would this be represented?

(A) MMXXIII

(B) MMCXXIII

(C) CCXXIII

(D) MXMXIII

Question 1. Case Study: Planning a Walking Route

Ria is planning a walking route and marking key points on a map that resembles a number line. Her starting point is at 0. She needs to mark a shop at -2 km, a park at $1.5$ km, a library at $\frac{5}{2}$ km, and a viewpoint at approximately $\sqrt{5}$ km. She is using different methods to locate these points accurately.

Where should Ria mark the shop and the park on the number line relative to her starting point?

(A) Shop at 2 km to the right, park at 1.5 km to the left.

(B) Shop at 2 km to the left, park at 1.5 km to the right.

(C) Shop at -2 km, park at 1.5 km. Both are positive distances but directions are opposite from 0.

(D) Shop at 2 km distance, park at 1.5 km distance. Direction is not indicated by the sign.

Question 2. Referring to the Case Study above, to mark the library at $\frac{5}{2}$ km, Ria should locate the point that is equivalent to:

(A) $2.5$ km

(B) $2 \frac{1}{2}$ km

(C) Halfway between 2 km and 3 km.

(D) All of the above.

Question 3. Referring to the Case Study above, to locate the viewpoint at approximately $\sqrt{5}$ km, Ria uses a geometric construction. Which method involving a right-angled triangle starting from 0 could she use?

(A) A triangle with legs of length 1 and 1.

(B) A triangle with legs of length 1 and 2.

(C) A triangle with legs of length $\sqrt{2}$ and $\sqrt{3}$.

(D) A triangle with hypotenuse 5.

Question 4. Referring to the Case Study above, if Ria needed to mark a point at $1.732$ km (approximately $\sqrt{3}$), and she wanted to show its location between $1.7$ and $1.8$, which technique would be most helpful?

(A) Finding a common denominator.

(B) Using Roman numerals.

(C) Successive magnification.

(D) Prime factorization.

Question 5. Referring to the Case Study above, the set of all possible points Ria could mark on her continuous map represents which set of numbers?

(A) Rational numbers

(B) Irrational numbers

(C) Integers

(D) Real numbers

Question 1. Case Study: Comparing Performance Scores

In a competition, four participants received scores represented by different types of numbers: Amit scored -5 points, Priya scored $3 \frac{1}{4}$ points, Rahul scored $3.5$ points, and Sneha scored $\sqrt{10}$ points. They need to order the participants based on their scores from lowest to highest.

Before ordering, which of the following is a correct comparison between Amit and Priya's scores?

(A) Amit's score $>$ Priya's score

(B) Amit's score $<$ Priya's score

(C) Amit's score $=$ Priya's score

(D) Cannot compare scores of different types.

Question 2. Referring to the Case Study above, what are Priya's and Rahul's scores in decimal form, respectively?

(A) $3.25$ and $3.5$

(B) $3.14$ and $3.5$

(C) $3.4$ and $3.5$

(D) $3.5$ and $3.25$

Question 3. Referring to the Case Study above, what is the approximate value of Sneha's score, $\sqrt{10}$?

(A) Between 3 and 4

(B) Exactly 3.16

(C) Less than 3

(D) Greater than 4

Question 4. Referring to the Case Study above, which of the following is the correct order of participants from lowest score to highest score?

(A) Amit, Priya, Sneha, Rahul

(B) Amit, Sneha, Priya, Rahul

(C) Amit, Priya, Rahul, Sneha

(D) Priya, Rahul, Sneha, Amit

Question 5. Referring to the Case Study above, the absolute value of Amit's score is:

(A) -5

(B) 5

(C) 0

(D) $|-5|$

Question 1. Case Study: Shopping and Finance

Mrs. Gupta goes shopping. She buys groceries for $\textsf{₹} 567.50$, clothes for $\textsf{₹} 1250.75$, and medicines for $\textsf{₹} 345.25$. She gives the shopkeeper a $\textsf{₹} 2000$ note and a $\textsf{₹} 500$ note. She also had a previous balance of $\textsf{₹} -150.50$ in her digital wallet.

What is the total cost of Mrs. Gupta's shopping?

(A) $\textsf{₹} (567.50 + 1250.75 + 345.25)$

(B) $\textsf{₹} 2163.50$

(C) $\textsf{₹} 2163.00$

(D) $\textsf{₹} 2164.00$

Question 2. Referring to the Case Study above, how much total money did Mrs. Gupta give to the shopkeeper?

(A) $\textsf{₹} 2000 + \textsf{₹} 500 = \textsf{₹} 2500$

(B) $\textsf{₹} 2050$

(C) $\textsf{₹} 2500.50$

(D) $\textsf{₹} (2000, 500)$

Question 3. Referring to the Case Study above, how much change should Mrs. Gupta receive from the shopkeeper?

(A) Total money given - Total cost

(B) $\textsf{₹} 2500 - \textsf{₹} 2163.50$

(C) $\textsf{₹} 336.50$

(D) $\textsf{₹} 337.50$

Question 4. Referring to the Case Study above, what is Mrs. Gupta's new balance in her digital wallet after adding the change received to her previous balance?

(A) $\textsf{₹} -150.50 + \textsf{₹} 336.50$

(B) $\textsf{₹} 186.00$

(C) $\textsf{₹} (336.50 - 150.50)$

(D) $\textsf{₹} 185.00$

Question 5. Referring to the Case Study above, suppose the shopkeeper had a special offer: buy clothes worth over $\textsf{₹} 1000$ and get groceries worth $\textsf{₹} 100$ free. If Mrs. Gupta's original grocery bill was $\textsf{₹} 567.50$, what would her adjusted grocery bill be?

(A) $\textsf{₹} 567.50 - \textsf{₹} 100 = \textsf{₹} 467.50$

(B) $\textsf{₹} 567.50$ (The offer is for clothes, not groceries bill reduction)

(C) $\textsf{₹} 567.50 - 100 = \textsf{₹} 467.50$

(D) Cannot be determined from the given information.

Question 1. Case Study: Grouping and Arranging Items

In a warehouse, items are being packed. There are 15 boxes of item A and 20 boxes of item B. Workers are asked to group them. One worker adds 15 boxes of A to 20 boxes of B and gets 35 boxes. Another worker adds 20 boxes of B to 15 boxes of A and gets 35 boxes. A supervisor notices that whether they group $(15+20)+5$ or $15+(20+5)$ more boxes, the total is the same. They also need to arrange 108 items in groups, checking if they can make groups of 3 or 9.

The observation that adding 15 boxes of A to 20 boxes of B or vice versa gives the same total illustrates which property?

(A) Associative property of addition

(B) Commutative property of addition

(C) Distributive property

(D) Closure property of addition

Question 2. Referring to the Case Study above, the supervisor's observation about grouping $(15+20)+5$ and $15+(20+5)$ illustrates which property?

(A) Commutative property of addition

(B) Associative property of addition

(C) Identity property of addition

(D) Closure property of addition

Question 3. Referring to the Case Study above, to check if 108 items can be arranged in groups of 3, which divisibility test can be applied?

(A) Check if the last digit is 0, 3, 6, or 9.

(B) Check if the sum of the digits is divisible by 3.

(C) Check if the number formed by the last two digits is divisible by 3.

(D) Check if the number is even.

Question 4. Referring to the Case Study above, can 108 items be arranged in groups of 9? Use a divisibility test.

(A) Yes, because the last digit is 8.

(B) Yes, because the sum of digits ($1+0+8=9$) is divisible by 9.

(C) No, because 108 is not a prime number.

(D) Yes, because 108 is divisible by 3.

Question 5. Referring to the Case Study above, if the warehouse manager multiplies the number of boxes of item A (15) by the number of boxes of item B (20), he gets 300. If he considers the set of whole numbers, is this operation closed within that set?

(A) Yes, because 15 and 20 are whole numbers, and 300 is also a whole number.

(B) No, because the numbers are large.

(C) Yes, but only for positive numbers.

(D) No, multiplication is not closed in whole numbers.

Question 1. Case Study: Baking a Cake

Rina is baking a cake. The recipe requires $1 \frac{1}{2}$ cups of flour, $\frac{3}{4}$ cup of sugar, and $0.5$ cup of milk. She has measuring cups marked with fractions and a measuring jug marked with decimals. She needs to convert between fractions and decimals to use her tools effectively and also scale the recipe.

What are the decimal equivalents of the required amounts of flour and sugar?

(A) Flour: $1.5$ cups, Sugar: $0.75$ cups

(B) Flour: $1.12$ cups, Sugar: $0.75$ cups

(C) Flour: $1.5$ cups, Sugar: $0.34$ cups

(D) Flour: $1.12$ cups, Sugar: $0.34$ cups

Question 2. Referring to the Case Study above, Rina needs $0.5$ cup of milk. How can this amount be expressed as a fraction in simplest form?

(A) $\frac{5}{10}$

(B) $\frac{1}{2}$

(C) $\frac{2.5}{5}$

(D) Both (A) and (B)

Question 3. Referring to the Case Study above, suppose Rina decides to make half the recipe. How much flour will she need in fraction form?

(A) $\frac{1}{2} \times 1 \frac{1}{2}$

(B) $\frac{1}{2} \times \frac{3}{2} = \frac{3}{4}$ cups

(C) $1 \frac{1}{4}$ cups

(D) Half of $1.5$ cups = $0.75$ cups

Question 4. Referring to the Case Study above, Rina finds another recipe that uses $\frac{6}{8}$ cup of butter. How can this fraction be reduced to its simplest form?

(A) Divide numerator and denominator by 2 to get $\frac{3}{4}$.

(B) Divide numerator and denominator by their HCF, which is 2.

(C) Divide numerator and denominator by 2 to get $\frac{3}{4}$.

(D) Multiply numerator and denominator by the same number.

Question 5. Referring to the Case Study above, Rina sees a different recipe uses $1.8$ cups of sugar. How can she express this as a mixed number in simplest form?

(A) $1 \frac{8}{10} = 1 \frac{4}{5}$

(B) $1 + \frac{8}{10} = 1 + \frac{4}{5} = 1 \frac{4}{5}$

(C) $\frac{18}{10} = \frac{9}{5} = 1 \frac{4}{5}$

(D) All of the above methods result in $1 \frac{4}{5}$.

Question 1. Case Study: Analyzing Measurements and Ratios

A scientist is analyzing data from different experiments. Some measurements are given as fractions, some as decimals. They encounter values like $\frac{1}{3}$, $0.75$, $\sqrt{2}$, $1.414$, and $0.121221222...$. They need to understand the nature of these numbers based on their decimal expansions and classify them as rational or irrational.

Based on its decimal expansion $0.333...$, the number $\frac{1}{3}$ is classified as:

(A) Terminating decimal, Rational

(B) Non-terminating recurring decimal, Rational

(C) Non-terminating non-recurring decimal, Irrational

(D) Terminating decimal, Irrational

Question 2. Referring to the Case Study above, the measurement $0.75$ is classified as:

(A) Non-terminating recurring decimal, Rational

(B) Terminating decimal, Irrational

(C) Terminating decimal, Rational

(D) Non-terminating non-recurring decimal, Irrational

Question 3. Referring to the Case Study above, the number $\sqrt{2}$ has a decimal expansion that starts as $1.414...$. This number is classified as:

(A) Rational, because it's a square root.

(B) Irrational, because its decimal expansion is non-terminating and non-recurring.

(C) Rational, because $1.414$ is a terminating decimal.

(D) Irrational, because it cannot be written as a fraction of two integers.

Question 4. Referring to the Case Study above, the number $0.121221222...$ is classified as:

(A) Non-terminating recurring decimal, Rational

(B) Terminating decimal, Irrational

(C) Non-terminating non-recurring decimal, Irrational

(D) Terminating decimal, Rational

Question 5. Referring to the Case Study above, if the scientist needs to work with an expression like $\frac{1}{\sqrt{3}}$, they might rationalize the denominator. What is the rationalized form?

(A) $\frac{\sqrt{3}}{3}$

(B) $\frac{1}{3}$

(C) $\sqrt{3}$

(D) $\frac{1.732}{3}$

Question 1. Case Study: Arranging Sweets for Distribution

A sweet shop prepared 120 laddoos and 144 barfis for a festival. The owner wants to pack them into boxes such that each box contains only one type of sweet, and the number of sweets in each box is the same for both types of sweets. They want to find the largest possible number of sweets in each box. They also discuss the properties of the numbers 120 and 144, including their factors and multiples.

Which of the following are factors of 120?

(A) 10

(B) 12

(C) 15

(D) All of the above

Question 2. Referring to the Case Study above, which of the following are multiples of 12?

(A) 6

(B) 24

(C) 120

(D) Both (B) and (C)

Question 3. Referring to the Case Study above, the owner checks if 120 is divisible by 5. Which divisibility test is used here?

(A) Check if the sum of digits is divisible by 5.

(B) Check if the last digit is 0 or 5.

(C) Check if the number is even.

(D) Check if the number is a multiple of 10.

Question 4. Referring to the Case Study above, the owner wants to find the largest possible number of sweets in each box. This problem requires finding the:

(A) LCM of 120 and 144

(B) HCF of 120 and 144

(C) Sum of 120 and 144

(D) Product of 120 and 144

Question 5. Referring to the Case Study above, which statement about 120 and 144 is correct based on prime factorization?

Prime factorization of $120 = 2^3 \times 3 \times 5$

Prime factorization of $144 = 2^4 \times 3^2$

(A) Both 120 and 144 are prime numbers.

(B) The common prime factors are 2, 3, and 5.

(C) The HCF will involve powers of 2 and 3.

(D) The LCM will involve only powers of 2 and 3.

Question 1. Case Study: School Activities Schedule

A school is planning three repeating activities: a music club meeting every 12 days, a science club meeting every 15 days, and a sports practice every 20 days. All three activities had a combined session today. The principal wants to know when all three activities will coincide again. They also need to decide on the largest number of students to put in equal groups for two different tasks, one requiring 48 students and another 72 students.

To find when all three activities will coincide again, the school needs to calculate the:

(A) HCF of 12, 15, and 20

(B) LCM of 12, 15, and 20

(C) Product of 12, 15, and 20

(D) Sum of 12, 15, and 20

Question 2. Referring to the Case Study above, what is the LCM of 12, 15, and 20?

(A) 3

(B) 60

(C) 180

(D) 120

Question 3. Referring to the Case Study above, if the activities started together today, when will they coincide again?

(A) In 3 days

(B) In 60 days

(C) In 180 days

(D) In 120 days

Question 4. Referring to the Case Study above, to find the largest number of students to put in equal groups for tasks requiring 48 and 72 students, the school needs to calculate the:

(A) LCM of 48 and 72

(B) HCF of 48 and 72

(C) Sum of 48 and 72

(D) Product of 48 and 72

Question 5. Referring to the Case Study above, what is the HCF of 48 and 72?

(A) 12

(B) 24

(C) 48

(D) 72

Question 1. Case Study: Understanding Number Theory Concepts

A mathematics teacher is explaining key concepts to students. They use the division of 26 by 4 as an example: $26 = 4 \times 6 + 2$. They then introduce the idea that for any two positive integers, a unique quotient and remainder exist. They also discuss how to find the HCF of large numbers using a step-by-step process based on this idea. Finally, they emphasize that every composite number can be broken down into a unique set of prime building blocks.

In the example $26 = 4 \times 6 + 2$, which number represents the divisor?

(A) 26

(B) 4

(C) 6

(D) 2

Question 2. Referring to the Case Study above, the statement that for any two positive integers $a$ and $b$, there exist unique integers $q$ and $r$ such that $a = bq + r$, where $0 \leq r < b$, is known as:

(A) The Fundamental Theorem of Arithmetic

(B) Euclid’s Division Algorithm

(C) Euclid’s Division Lemma

(D) The Remainder Theorem

Question 3. Referring to the Case Study above, the step-by-step process for finding the HCF of large numbers is called:

(A) Prime Factorization

(B) Long Division

(C) Euclid’s Division Algorithm

(D) The Fundamental Theorem of Arithmetic

Question 4. Referring to the Case Study above, the principle that every composite number can be uniquely expressed as a product of prime numbers is stated by the:

(A) Euclid’s Division Lemma

(B) Euclid’s Division Algorithm

(C) Fundamental Theorem of Arithmetic

(D) Unique Remainder Theorem

Question 5. Referring to the Case Study above, which concept is used to find the HCF of two numbers by repeatedly applying the division lemma?

(A) $HCF(a, b) = HCF(a+b, b)$

(B) $HCF(a, b) = HCF(a-b, b)$

(C) $HCF(a, b) = HCF(b, r)$ where $a=bq+r$

(D) $HCF(a, b) = \frac{a \times b}{LCM(a, b)}$

Question 1. Case Study: Working with Large and Small Quantities

A scientist is working with very large and very small numbers. They measure the distance between two stars as $4 \times 10^{16}$ meters and the size of a bacterium as $2 \times 10^{-6}$ meters. They also need to calculate the volume of a cube with side length $10^3$ cm and determine how many times larger the star distance is compared to the bacterium size. They use exponents and standard form in their calculations.

What is the value of the side length of the cube, $10^3$ cm, in standard form?

(A) $1 \times 10^3$ cm

(B) 1000 cm

(C) $1 \times 10^0$ cm

(D) $1.0 \times 10^3$ cm

Question 2. Referring to the Case Study above, how many times larger is the distance between the stars compared to the size of the bacterium?

(A) $\frac{4 \times 10^{16}}{2 \times 10^{-6}}$

(B) $2 \times 10^{16 - (-6)} = 2 \times 10^{22}$

(C) $2 \times 10^{10}$

(D) $2 \times 10^{16/6}$

Question 3. Referring to the Case Study above, what is the volume of the cube with side length $10^3$ cm?

(A) $(10^3)^3$ cubic cm

(B) $10^{3+3+3} = 10^9$ cubic cm

(C) $10^{3 \times 3} = 10^9$ cubic cm

(D) $10^6$ cubic cm

Question 4. Referring to the Case Study above, the scientist also considers a very thin wire with diameter $5 \times 10^{-4}$ meters. Which is smaller, the bacterium size ($2 \times 10^{-6}$ m) or the wire diameter ($5 \times 10^{-4}$ m)?

(A) Bacterium size

(B) Wire diameter

(C) They are equal

(D) Cannot compare numbers with different exponents.

Question 5. Referring to the Case Study above, the scientist needs to calculate $(10^2)^0$. What is the value?

(A) 100

(B) 10

(C) 1

(D) 0

Question 1. Case Study: Designing and Measuring Spaces

An architect is designing a square-shaped courtyard with an area of 225 square meters. They need to find the length of each side of the courtyard. They also need to calculate the diagonal of a rectangular room with length 8 meters and width 6 meters, which involves using the Pythagorean theorem. They recall the properties of square numbers and methods for finding square roots.

What is the length of each side of the square courtyard with an area of 225 square meters?

(A) $\sqrt{225}$ meters

(B) 15 meters

(C) 25 meters

(D) Both (A) and (B)

Question 2. Referring to the Case Study above, the architect needs to calculate the diagonal of the rectangular room. Which set of numbers forms a Pythagorean triplet corresponding to the room's dimensions and diagonal?

(A) (6, 8, 10)

(B) (6, 8, $\sqrt{100}$)

(C) (6, 8, $\sqrt{6^2+8^2}$)

(D) All of the above.

Question 3. Referring to the Case Study above, the number 225 ends in 5. Which statement is true about perfect squares ending in 5?

(A) They end in 05.

(B) They end in 25.

(C) Their square root ends in 5.

(D) Both (B) and (C).

Question 4. Referring to the Case Study above, suppose the architect needed to find the square root of 1024. Which method would likely be most efficient?

(A) Repeated subtraction

(B) Listing factors

(C) Long division method

(D) Estimation

Question 5. Referring to the Case Study above, the architect also considers a smaller square area of $0.64$ square meters. What is the side length of this smaller square?

(A) $0.8$ meters

(B) $0.08$ meters

(C) 8 meters

(D) $\sqrt{0.64}$ meters

Question 1. Case Study: Stacking Boxes and Packaging

A packaging company uses cubic boxes. One type of box has a side length of 5 cm. They also have larger cubic containers used for shipping. One such container has a volume of 1728 cubic cm. They need to find the side length of this large container and determine if certain quantities of smaller items can be packed into perfect cubic arrangements.

What is the volume of the small box with a side length of 5 cm?

(A) $5^3$ cubic cm

(B) $5 \times 5 \times 5$ cubic cm

(C) 125 cubic cm

(D) All of the above

Question 2. Referring to the Case Study above, what is the side length of the large cubic container with a volume of 1728 cubic cm?

(A) $\sqrt{1728}$ cm

(B) 12 cm

(C) 13 cm

(D) $\sqrt[3]{1728}$ cm

Question 3. Referring to the Case Study above, if the company receives 216 small cubic items, can they arrange them to form a single larger perfect cube?

(A) Yes, because 216 is a perfect cube ($6^3$).

(B) No, because it's an even number.

(C) Yes, because the sum of digits is 9.

(D) Cannot be determined without more information.

Question 4. Referring to the Case Study above, another container has a volume of 8000 cubic cm. What is its side length?

(A) 20 cm

(B) 200 cm

(C) 80 cm

(D) 40 cm

Question 5. Referring to the Case Study above, if the company needs to pack items that measure $0.5$ cm on each side into a perfect cubic arrangement, and they have a total volume of $0.125$ cubic cm occupied by these items, what would be the side length of the perfect cube formed by these items?

(A) $\sqrt{0.125}$ cm

(B) $\sqrt[3]{0.125}$ cm

(C) $0.5$ cm

(D) Both (B) and (C)

Question 1. Case Study: Budgeting and Quick Calculations

Mr. Patel is planning a small party. He expects around 48 guests. He estimates that food per person will cost about $\textsf{₹} 210$, decorations will cost around $\textsf{₹} 1500$, and music will cost a fixed amount of $\textsf{₹} 5000$. He wants to quickly estimate the total cost to see if it fits his budget of $\textsf{₹} 15000$. He uses rounding to simplify his calculations.

Estimate the number of guests by rounding 48 to the nearest ten.

(A) 40

(B) 50

(C) 45

(D) 55

Question 2. Referring to the Case Study above, estimate the cost of food per person by rounding $\textsf{₹} 210$ to the nearest hundred.

(A) $\textsf{₹} 200$

(B) $\textsf{₹} 300$

(C) $\textsf{₹} 210$

(D) $\textsf{₹} 250$

Question 3. Referring to the Case Study above, estimate the total cost of food by multiplying the estimated number of guests by the estimated cost of food per person (using the rounding from the previous two questions).

(A) $50 \times \textsf{₹} 200 = \textsf{₹} 10000$

(B) $48 \times \textsf{₹} 210$

(C) $50 \times \textsf{₹} 210$

(D) $40 \times \textsf{₹} 200$

Question 4. Referring to the Case Study above, estimate the total cost of the party including decorations ($\textsf{₹} 1500$) and music ($\textsf{₹} 5000$) using the estimated food cost ($\textsf{₹} 10000$).

(A) $\textsf{₹} 10000 + \textsf{₹} 1500 + \textsf{₹} 5000 = \textsf{₹} 16500$

(B) $\textsf{₹} 10000 + \textsf{₹} 1500 + \textsf{₹} 5000 = \textsf{₹} 17500$

(C) $\textsf{₹} 10000 + \textsf{₹} 2000 + \textsf{₹} 5000 = \textsf{₹} 17000$

(D) $\textsf{₹} 10000 + \textsf{₹} 1000 + \textsf{₹} 5000 = \textsf{₹} 16000$

Question 5. Referring to the Case Study above, does the estimated total cost of the party ($\textsf{₹} 16500$) fit within Mr. Patel's budget of $\textsf{₹} 15000$? What is the conclusion based on the estimate?

(A) Yes, the estimate is less than the budget.

(B) No, the estimate is more than the budget. The party might be over budget.

(C) The estimate exactly matches the budget.

(D) Estimation is not useful for budgeting decisions.

Question 1. Case Study: Calculations Using Logarithms

Before electronic calculators were common, logarithms were widely used to simplify complex calculations involving multiplication, division, and powers. Mr. Verma is teaching his students how to use logarithm tables (base 10) for such calculations. They know that $\log_{10} 2 \approx 0.3010$ and $\log_{10} 3 \approx 0.4771$. They practice calculating values like $\log_{10} 6$ and $\log_{10} 5$.

Using the given values, calculate $\log_{10} 6$.

(A) $\log_{10} (2+3) = \log_{10} 2 + \log_{10} 3 = 0.3010 + 0.4771 = 0.7781$

(B) $\log_{10} (2 \times 3) = \log_{10} 2 + \log_{10} 3 = 0.3010 + 0.4771 = 0.7781$

(C) $\log_{10} 6 = \log_{10} 2 \times \log_{10} 3 = 0.3010 \times 0.4771 \approx 0.1435$

(D) $\log_{10} 6 = 6 \times \log_{10} 1 = 6 \times 0 = 0$

Question 2. Referring to the Case Study above, calculate $\log_{10} 5$ using the given values.

(A) $\log_{10} (10-5) = \log_{10} 10 - \log_{10} 5$ (Not useful)

(B) $\log_{10} (5 \times 1) = \log_{10} 5 + \log_{10} 1$ (Not useful)

(C) $\log_{10} (10/2) = \log_{10} 10 - \log_{10} 2 = 1 - 0.3010 = 0.6990$

(D) $\log_{10} (2+3)$ (Incorrect rule)

Question 3. Referring to the Case Study above, Mr. Verma asks the students to find the number whose logarithm (base 10) is approximately $0.4771$. This requires finding the:

(A) Logarithm of $0.4771$

(B) Characteristic of $0.4771$

(C) Mantissa of $0.4771$

(D) Antilogarithm of $0.4771$

Question 4. Referring to the Case Study above, the number whose logarithm (base 10) is approximately $0.4771$ is close to which integer?

(A) 2

(B) 3

(C) 4

(D) 5

Question 5. Referring to the Case Study above, if a student needs to calculate $\log_{10} 20$, they can use the laws of logarithms. How can this be calculated using $\log_{10} 2$ and $\log_{10} 10$?

(A) $\log_{10} (20) = \log_{10} (2 \times 10) = \log_{10} 2 + \log_{10} 10 = 0.3010 + 1 = 1.3010$

(B) $\log_{10} (20) = \log_{10} (2 \times 10) = \log_{10} 2 \times \log_{10} 10 = 0.3010 \times 1 = 0.3010$

(C) $\log_{10} (20) = 2 \times \log_{10} 10 = 2 \times 1 = 2$

(D) $\log_{10} (20) = 10 \times \log_{10} 2 = 10 \times 0.3010 = 3.010$

Question 1. Case Study: Clocks and Calendars

In everyday life, we often deal with cycles. A clock cycles every 12 or 24 hours. Days of the week cycle every 7 days. These cycles can be modeled using modulo arithmetic. For example, to find the time 15 hours after 9 AM, we can use modulo 12 or modulo 24. To find the day of the week after 100 days, we can use modulo 7.

If the time is 9 AM, what time will it be after 15 hours? Use modulo 12 arithmetic.

(A) $9 + 15 = 24$. $24 \pmod{12} = 0$. This corresponds to 12 AM (midnight).

(B) $9 + 15 = 24$. $24 \pmod{12} = 12$. This corresponds to 12 PM (noon).

(C) $9 + 15 = 24$ hours. Since it's AM, it will be 12 AM of the next day.

(D) $9 + 15 = 24$. $24 \div 12 = 2$ with remainder 0. The remainder 0 corresponds to 12.

Question 2. Referring to the Case Study above, if today is Thursday, what day of the week will it be after 50 days? Use modulo 7 arithmetic (assuming Sunday=0, Monday=1, ..., Saturday=6 or Monday=1, ..., Sunday=7).

Let's use Monday=1, Tuesday=2, ..., Sunday=7. Thursday is day 4.

(A) $50 \pmod 7$. $50 = 7 \times 7 + 1$. The remainder is 1.

(B) The day will be 1 day after Thursday.

(C) The day will be Friday.

(D) The day will be Monday (if starting count from 0).

Question 3. Referring to the Case Study above, the times 9 AM and 9 PM are 12 hours apart. In modulo 12 arithmetic, how are 9 and 21 (representing 9 PM in 24-hour format) related?

(A) $21 \equiv 9 \pmod{12}$

(B) $21 - 9 = 12$, which is divisible by 12.

(C) They have the same remainder when divided by 12 (both have remainder 9).

(D) All of the above.

Question 4. Referring to the Case Study above, the property $a \equiv b \pmod m$ and $c \equiv d \pmod m \implies a+c \equiv b+d \pmod m$ is used when calculating the day of the week after a number of days. If today is Monday (day 1), and you want to find the day after 10 days, you calculate $1 + 10 = 11$. Then $11 \pmod 7$. If you instead added 5 days, waited, then added another 5 days, you would use this property. $1+5 \equiv 6 \pmod 7$, $6+5 = 11 \equiv 4 \pmod 7$. What is the day corresponding to remainder 4?

(A) Monday

(B) Wednesday

(C) Thursday

(D) Friday

Question 5. Referring to the Case Study above, what is the remainder when the number of hours in a day (24) is divided by the number of hours in a 12-hour cycle?

(A) 0

(B) 12

(C) 2

(D) 1

Question 1. Case Study: Planning a Trip

Mr. and Mrs. Singh are planning a road trip. The total distance to their destination is 1250 km. They plan to cover $\frac{2}{5}$ of the distance on the first day and the remaining distance on the second day. The car consumes petrol at a rate of 1 liter for every 15 km, and the price of petrol is $\textsf{₹} 102.50$ per liter. They also need to buy snacks for the trip; they decide to spend $\textsf{₹} 750.50$ on snacks.

How much distance do they plan to cover on the first day?

(A) $\frac{2}{5} \times 1250$ km

(B) 500 km

(C) 250 km

(D) 750 km

Question 2. Referring to the Case Study above, how much distance is left to be covered on the second day?

(A) $1250 - (\text{distance covered on day 1})$ km

(B) $1250 - 500 = 750$ km

(C) $1 - \frac{2}{5} = \frac{3}{5}$ of the total distance

(D) Both (B) and (C).

Question 3. Referring to the Case Study above, how many liters of petrol will the car consume for the entire trip of 1250 km?

(A) $1250 \times 15$ liters

(B) $1250 \div 15$ liters

(C) Approximately 83.33 liters

(D) 80 liters

Question 4. Referring to the Case Study above, what is the total cost of petrol for the trip?

(A) $(\text{total liters consumed}) \times (\text{price per liter})$

(B) $83.33 \times \textsf{₹} 102.50$

(C) Approximately $\textsf{₹} 8541.33$

(D) $\textsf{₹} 83.33 + \textsf{₹} 102.50$

Question 5. Referring to the Case Study above, what is the total estimated expenditure for the trip, including petrol and snacks?

(A) Total petrol cost + Total snack cost

(B) $\textsf{₹} 8541.33 + \textsf{₹} 750.50$

(C) Approximately $\textsf{₹} 9291.83$

(D) $\textsf{₹} 8541.33 \times \textsf{₹} 750.50$