CBSE Class 12 - Model Question Paper

SECTION B — Very Short Answer Type Questions [2 Marks Each]

2 Marks Q1. Find the HCF and LCM of 120 and 144 using the prime factorization method.
2 Marks Q2. Given that $HCF(306, 657) = 9$, find $LCM(306, 657)$.
2 Marks Q3. Explain why $7 \times 11 \times 13 + 13$ is a composite number.
2 Marks Q4. Can two numbers have 15 as their HCF and 175 as their LCM? Give reasons.
2 Marks Q5. Find the smallest number which when divided by 28 and 32 leaves remainders 8 and 12 respectively.
2 Marks Q6. Check whether $12^n$ can end with the digit 0 for any natural number $n$.
2 Marks Q7. Find the HCF of 96 and 404 by the prime factorization method. Hence, find their LCM.
2 Marks Q8. Two bells toll at intervals of 12 and 18 minutes respectively. If they toll together at 12:00 PM, when will they toll together again?
2 Marks Q9. Find the HCF of 81 and 237 and express it in the form $81x + 237y$.
2 Marks Q10. Is it possible for the HCF and LCM of two numbers to be 18 and 380 respectively? Justify.
2 Marks Q11. Prove that $\sqrt{2}$ is irrational.
2 Marks Q12. Prove that $\sqrt{3}$ is irrational.
2 Marks Q13. Show that $5 - \sqrt{3}$ is an irrational number.
2 Marks Q14. Prove that $3 + 2\sqrt{5}$ is irrational.
2 Marks Q15. Show that $\frac{1}{\sqrt{2}}$ is irrational.
2 Marks Q16. Prove that $7\sqrt{5}$ is irrational.
2 Marks Q17. Show that $2 - 3\sqrt{5}$ is irrational.
2 Marks Q18. Prove that $\sqrt{p} + \sqrt{q}$ is irrational, where $p$ and $q$ are primes.
2 Marks Q19. Prove that $5 - 2\sqrt{3}$ is an irrational number.
2 Marks Q20. Show that $\frac{3}{\sqrt{7}}$ is not a rational number.
2 Marks Q21. Without actually performing the long division, state whether $\frac{13}{3125}$ has a terminating or non-terminating repeating decimal expansion.
2 Marks Q22. Write the decimal expansion of $\frac{7}{80}$ without actual division.
2 Marks Q23. State the condition on the denominator $q$ such that a rational number $\frac{p}{q}$ has a terminating decimal expansion.
2 Marks Q24. Without dividing, show that $\frac{129}{2^2 \times 5^7 \times 7^5}$ is a non-terminating repeating decimal.
2 Marks Q25. Find the decimal representation of $\frac{14587}{1250}$.
2 Marks Q26. If $\frac{p}{q}$ is a rational number, what can you say about $q$ if its decimal expansion is non-terminating repeating?
2 Marks Q27. Determine if $\frac{17}{8}$ has a terminating decimal expansion. If yes, find it.
2 Marks Q28. After how many decimal places will the decimal expansion of $\frac{23}{2^4 \times 5^3}$ terminate?
2 Marks Q29. Write down the prime factorization of the denominator of the rational number $0.125$
2 Marks Q30. Is $\frac{6}{15}$ a terminating decimal? Justify your answer.
2 Marks Q31. Find the HCF of 180, 252, and 324 using prime factorization.
2 Marks Q32. The product of two numbers is 1600 and their HCF is 5. Find their LCM.
2 Marks Q33. Can the product of two irrational numbers be rational? Give an example.
2 Marks Q34. Find the largest number that divides 2053 and 967 and leaves a remainder of 5 and 7 respectively.
2 Marks Q35. If $n$ is an odd integer, show that $n^2 - 1$ is divisible by 8.
2 Marks Q36. Find the HCF of the smallest prime number and the smallest composite number.
2 Marks Q37. Prove that the product of a non-zero rational and an irrational number is irrational.
2 Marks Q38. Express 0.666... in the form $p/q$.
2 Marks Q39. Write the condition for a rational number to have a non-terminating repeating decimal expansion.
2 Marks Q40. Check if $6^n$ can end with the digit 0 for any natural number $n$.
2 Marks Q41. Find the LCM of $2^3 \times 3^2$ and $2^2 \times 3^3$.
2 Marks Q42. State the Fundamental Theorem of Arithmetic.
2 Marks Q43. If $a = x^3 y^2$ and $b = x y^3$, find the HCF of $a$ and $b$.
2 Marks Q44. Find the smallest number which when divided by 15, 20, and 30 leaves a remainder of 4 in each case.
2 Marks Q45. Prove that $n^2 - n$ is divisible by 2 for every positive integer $n$.
2 Marks Q46. Can $\frac{x}{y}$ be an irrational number if $x$ and $y$ are both irrational?
2 Marks Q47. Find the HCF of $2^3 \times 3^2 \times 5$ and $2^2 \times 3^3 \times 5^2$
2 Marks Q48. Explain the steps to represent $0.\bar{7}$ as a rational number.
2 Marks Q49. Find the prime factorization of 3825.
2 Marks Q50. If the HCF of 65 and 117 is expressible in the form $65n - 117$, find the value of $n$.

SECTION C — Short Answer Type Questions [3 Marks Each]

3 Marks Q1. Show that any number of the form $6^n$, where $n$ is a natural number, can never end with the digit 0.
3 Marks Q2. Prove that the product of any three consecutive positive integers is divisible by 6.
3 Marks Q3. Find the HCF and LCM of 306 and 657 and verify that $HCF \times LCM = \text{Product of the two numbers}$.
3 Marks Q4. Given that $HCF(210, 55) = 5$, find the LCM of 210 and 55.
3 Marks Q5. Explain why $17 \times 11 \times 13 + 13$ and $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5$ are composite numbers.
3 Marks Q6. Find the largest number that divides 2053 and 967 and leaves a remainder of 5 and 7 respectively.
3 Marks Q7. Find the smallest number which when divided by 20, 25, 35, and 40 leaves a remainder of 14, 19, 29, and 34 respectively.
3 Marks Q8. Two tankers contain 850 liters and 680 liters of petrol respectively. Find the maximum capacity of a container which can measure the petrol of either tanker in exact number of times.
3 Marks Q9. In a seminar, the number of participants in Hindi, English, and Mathematics are 60, 84, and 108 respectively. Find the minimum number of rooms required if in each room the same number of participants are to be seated and all of them being in the same subject.
3 Marks Q10. If $n$ is a positive integer, show that $n^2 - n$ is divisible by 2.
3 Marks Q11. Find the HCF of 441, 567, and 693 using the prime factorization method.
3 Marks Q12. Check if $4^n$ can end with the digit 0 for any natural number $n$.
3 Marks Q13. Prove that for any positive integer $n$, $n^3 - n$ is divisible by 6.
3 Marks Q14. Find the HCF of 1260 and 7344 using prime factorization.
3 Marks Q15. If the HCF of 210 and 55 is expressible in the form $210(5) + 55y$, find $y$.
3 Marks Q16. Prove that $\sqrt{2} + \sqrt{3}$ is an irrational number.
3 Marks Q17. Prove that $5 - 2\sqrt{3}$ is irrational, given that $\sqrt{3}$ is irrational.
3 Marks Q18. Show that $\frac{1}{\sqrt{2}}$ is irrational.
3 Marks Q19. Prove that $\sqrt{5}$ is an irrational number.
3 Marks Q20. Prove that $7\sqrt{5}$ is irrational.
3 Marks Q21. Prove that $3 + 2\sqrt{5}$ is irrational.
3 Marks Q22. Show that $\frac{2\sqrt{3}}{5}$ is an irrational number.
3 Marks Q23. Prove that $\sqrt{3} + \sqrt{5}$ is irrational.
3 Marks Q24. Prove that $6 + \sqrt{2}$ is irrational.
3 Marks Q25. Prove that $\sqrt{p} + \sqrt{q}$ is irrational, where $p$ and $q$ are distinct prime numbers.
3 Marks Q26. Show that $3 - \sqrt{5}$ is an irrational number.
3 Marks Q27. Prove that $\frac{5}{\sqrt{3}}$ is an irrational number.
3 Marks Q28. Prove that $2\sqrt{3} - 1$ is an irrational number.
3 Marks Q29. Prove that $4 - 5\sqrt{2}$ is an irrational number.
3 Marks Q30. If $a$ and $b$ are odd positive integers, prove that $a^2 + b^2$ is even but not divisible by 4.
3 Marks Q31. Without actually performing the long division, show that $\frac{987}{10500}$ is a non-terminating repeating decimal.Write the denominator of the rational number $\frac{257}{5000}$ in the form $2^m \times 5^n$, and hence write its decimal expansion without actual division.
3 Marks Q32. Prove that for any natural number $n$, $12^n$ cannot end with the digit 0 or 5.
3 Marks Q33. Find the smallest number which when divided by 28 and 32 leaves remainders 8 and 12 respectively.
3 Marks Q34. The length, breadth, and height of a room are 8 m 25 cm, 6 m 75 cm, and 4 m 50 cm respectively. Determine the longest rod which can measure the three dimensions of the room exactly.
3 Marks Q35. Prove that the square of any positive integer is of the form $4q$ or $4q + 1$ for some integer $q$.
3 Marks Q36. Show that one and only one of $n, n+2, n+4$ is divisible by 3.
3 Marks Q37. Find the HCF of 1656 and 4025 by prime factorization.
3 Marks Q38. If two positive integers $a$ and $b$ are written as $a = x^3 y^2$ and $b = x y^3$, where $x, y$ are prime numbers, then find $HCF(a, b)$ and $LCM(a, b)$.
3 Marks Q39. Express 0.2353535... as a rational number in the form $p/q$.
3 Marks Q40. Express 0.4777... in the form $p/q$.
3 Marks Q41. State the Fundamental Theorem of Arithmetic and use it to find the HCF of 26 and 91.
3 Marks Q42. Prove that the product of a non-zero rational and an irrational number is always irrational.
3 Marks Q43. Find the prime factorization of 32760.
3 Marks Q44. Can the number $6^n$, $n$ being a natural number, end with the digit 5? Give reasons
3 Marks Q45. If $d$ is the HCF of 56 and 72, find $x, y$ satisfying $d = 56x + 72y$.
3 Marks Q46. Prove that $n^2 - 1$ is divisible by 8, if $n$ is an odd positive integer.
3 Marks Q47. Explain why $(7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1) + 5$ is a composite number.
3 Marks Q48. Show that any positive odd integer is of the form $4q + 1$ or $4q + 3$.
3 Marks Q49. Find the LCM of the smallest prime number and the smallest composite number.
3 Marks Q50. Write the denominator of the rational number $\frac{257}{5000}$ in the form $2^m \times 5^n$, and hence write its decimal expansion without actual division

SECTION D — Case-Based/Source-Based Integrated Questions [4 Marks Each]

4 Marks Q1. Case Study: The Seminar Hall
In a seminar, the number of participants in Hindi, English, and Mathematics are 60, 84, and 108 respectively.

a) Find the maximum number of participants that can be accommodated in each room if each room has the same number of participants and each room contains participants of only one subject. (2 Marks)

b) What is the total number of rooms required? (2 Marks)
4 Marks Q2. Case Study: The Morning Bell
Three bells toll at intervals of 9, 12, and 15 minutes respectively. If they start tolling together at 8:00 AM:

a) After how much time will they next toll together? (2 Marks)

b) How many times will they toll together in the next 3 hours? (2 Marks)
4 Marks Q3. Case Study: The Circular Track
Ravi and Shikha are running on a circular track. Ravi takes 12 minutes to complete one round, while Shikha takes 18 minutes.

a) If they start at the same point and time, after how many minutes will they meet at the starting point? (2 Marks)

b) How many rounds will Ravi have completed when they meet? (2 Marks)
4 Marks Q4. Case Study: Packaging Sweets
A sweet seller has 420 kaju barfis and 130 badam barfis. He wants to stack them in such a way that each stack has the same number and they take up the least area of the tray.

a) What is the number of barfis that can be placed in each stack for this purpose? (2 Marks)

b) How many stacks will be formed in total? (2 Marks)
4 Marks Q5. Case Study: The Math Quiz
A quiz competition is organized with two groups, A and B. Group A has 48 members and Group B has 64 members. They need to be divided into groups of equal size for a game.

a) What is the maximum number of members each group can have? (2 Marks)

b) If the groups are formed with this size, how many total groups will participate? (2 Marks)
4 Marks Q6. Case Study: Irrigation System
A farmer has two rectangular fields of dimensions 120m x 80m and 150m x 100m. He wants to divide these into square plots of the same size.

a) Find the largest possible side length of the square plot. (2 Marks)

b) How many such plots can be made from the first field? (2 Marks)
4 Marks Q7. Case Study: Prime Factorization in Cryptography
A security system uses two large prime numbers $p = 17$ and $q = 19$ to generate a key.

a) Find the value of $p \times q$. (2 Marks)

b) If the system requires a number $N$ such that $N$ is the smallest number divisible by both $p$ and $q$, what is $N$? (2 Marks)
4 Marks Q8. Case Study: Rational vs Irrational Patterns
Analyze the following sequence of numbers: $0.12, 0.12122, 0.121221222...$

a) (i) Identify which of these are rational. (2 Marks)

b) (ii) Prove that $0.121221222...$ is irrational. (2 Marks)
4 Marks Q9. Case Study: The Library Collection
A library has 336 English books and 240 Science books. They need to be arranged in shelves where each shelf has the same number of books of a single subject.

a) Find the maximum number of books per shelf. (2 Marks)

b) How many shelves are needed for the Science books? (2 Marks)
4 Marks Q10. Case Study: Clockwork Mechanism
Two gears in a machine have 24 and 36 teeth respectively.

a) After how many rotations of the smaller gear will they return to their original relative position? (2 Marks)

b) What is the LCM of the number of teeth? (2 Marks)
4 Marks Q11. Case Study: The School Assembly
To prepare for the annual day, a school decides to arrange students in rows. There are 72 students from Grade 9 and 96 students from Grade 10. They need to be arranged in rows such that each row has the same number of students and all students in a row belong to the same grade.

a) What is the maximum number of students that can be in each row? (1 Marks)

b) How many rows will be formed for Grade 9 students? (1 Marks)

c) How many rows will be formed for Grade 10 students? (1 Marks)

d) If the total number of students was 168, and they were arranged in rows of 12, how many rows would there be in total? (1 Marks)
4 Marks Q12. Case Study: The Running Track
Two runners, Ravi and Shikha, start running on a circular track from the same point at the same time. Ravi completes one lap in 18 minutes, while Shikha completes one lap in 24 minutes.

a) After how many minutes will they meet again at the starting point? (1 Marks)

b) How many laps would Ravi have completed when they first meet at the start? (1 Marks)

c) How many laps would Shikha have completed when they first meet at the start? (1 Marks)

d) If a third runner, Amit, joins and completes a lap in 12 minutes, when will all three meet at the starting point? (1 Marks)
4 Marks Q13. Case Study: The Prime Factorization
A student is given a number $N = 2^3 \times 3^2 \times 5^1$ and another number $M = 2^2 \times 3^3 \times 7^1$.

a) Find the HCF of $N$ and $M$. (1 Marks)

b) Find the LCM of $N$ and $M$ (1 Marks)

c) Verify if $\text{HCF} \times \text{LCM} = N \times M$. (1 Marks)

d) Is $N$ a composite number? Explain why. (1 Marks)
4 Marks Q14. Case Study: The Rationals and Irrationals
A teacher asks students to classify numbers based on their decimal expansion and properties.

a) Show that $5 - \sqrt{3}$ is an irrational number, given that $\sqrt{3}$ is irrational. (1 Marks)

b) Without performing long division, determine the decimal expansion of $\frac{13}{125}$. (1 Marks)

c) Is $0.121221222...$ a rational or irrational number? Give a reason. (1 Marks)

d) Find the smallest number which is divisible by both 306 and 657. (1 Marks)
4 Marks Q15. Case Study: Gift Distribution
A shopkeeper has 120 liters of oil and 180 liters of ghee. He wants to fill them into tins of equal capacity so that each tin is completely filled.

a) What is the maximum capacity of each tin? (1 Marks)

b) How many tins of oil will be required? (1 Marks)

c) How many tins of ghee will be required? (1 Marks)

d) If the shopkeeper decides to use 10-liter tins, how many total tins will he need for both oil and ghee? (1 Marks)
4 Marks Q16. Case Study: The Marathon Route
A marathon organizer wants to place markers at equal intervals along a 400m track and a 600m path.

a) What is the greatest distance (in meters) between markers so that they are placed at equal intervals on both paths? (1 Marks)

b) How many markers are needed for the 400m track? (1 Marks)

c) How many markers are needed for the 600m path? (1 Marks)

d) If the organizer decides to place markers every 50 meters, will all markers align at the same positions? (1 Marks)
4 Marks Q17. Case Study: Prime Number Patterns
Given a number $X = p^2q^3$ and $Y = p^3q$, where $p$ and $q$ are prime numbers.

a) What is the HCF of $X$ and $Y$ in terms of $p$ and $q$? (1 Marks)

b) What is the LCM of $X$ and $Y$ in terms of $p$ and $q$? (1 Marks)

c) If $p=2$ and $q=3$, what is the numerical value of $X$? (1 Marks)

d) For these values, does the product $HCF \times LCM$ equal the product $X \times Y$? (1 Marks)
4 Marks Q18. Case Study: Decimal Expansions
Students are analyzing the fraction $\frac{17}{80}$.

a) Is the decimal expansion terminating or non-terminating? (1 Marks)

b) Write the prime factorization of the denominator. (1 Marks)

c) After how many decimal places will the expansion terminate? (1 Marks)

d) Perform the division to find the exact decimal value. (1 Marks)
4 Marks Q19. Case Study: Proofs and Logic
Consider the statement: "The product of a non-zero rational number and an irrational number is always irrational."

a) Is the statement true or false? (1 Marks)

b) Give an example to support your answer. (1 Marks)

c) Is $7 \times \sqrt{5}$ rational or irrational? (1 Marks)

d) Is $\sqrt{2} + \sqrt{3}$ rational or irrational? (1 Marks)
4 Marks Q20. Case Study: Clockwork
Three electronic bells toll at intervals of 9, 12, and 15 minutes respectively. They all toll together at 10:00 AM.

a) Find the LCM of 9, 12, and 15. (1 Marks)

b) After how many minutes will the bells toll together again? (1 Marks)

c) At what time will they next toll together? (1 Marks)

d) How many times will they toll together between 10:00 AM and 1:00 PM? (1 Marks)

SECTION E — Long Answer Type Questions [5 Marks Each]

5 Marks Q1. Prove that $\sqrt{5}$ is an irrational number.
5 Marks Q2. Show that $3 + 2\sqrt{5}$ is irrational, given that $\sqrt{5}$ is irrational.
5 Marks Q3. Prove that $\sqrt{3} + \sqrt{5}$ is an irrational number.
5 Marks Q4. If $p$ is a prime number, prove that $\sqrt{p}$ is irrational.
5 Marks Q5. Prove that $\frac{1}{\sqrt{5}}$ is irrational.
5 Marks Q6. Prove that $5 - \sqrt{3}$ is irrational.
5 Marks Q7. Show that $2\sqrt{3} - 1$ is an irrational number.
5 Marks Q8. Prove that $\frac{3}{2\sqrt{5}}$ is irrational.
5 Marks Q9. Prove that $\sqrt{p} + \sqrt{q}$ is irrational, where $p$ and $q$ are distinct prime numbers.
5 Marks Q10. Show that for any positive integer $n$, $\sqrt{n}$ is either a rational or an irrational number.
5 Marks Q11. Two tankers contain 850 liters and 680 liters of petrol respectively. Find the maximum capacity of a container which can measure the petrol of either tanker in exact number of times.
5 Marks Q12. Find the HCF and LCM of 306 and 657 and verify that $\text{HCF} \times \text{LCM} = \text{Product of the two numbers}$.
5 Marks Q13. Show that any number of the form $4^n$, where $n$ is a natural number, can never end with the digit zero.
5 Marks Q14. Explain why $7 \times 11 \times 13 + 13$ and $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5$ are composite numbers.
5 Marks Q15. Find the largest number that divides 2053 and 967 and leaves a remainder of 5 and 7 respectively.
5 Marks Q16. Three sets of English, Mathematics, and Science books containing 336, 240, and 96 books respectively have to be stacked in such a way that all the books are stored subject-wise and the height of each stack is the same. Find the number of stacks.
5 Marks Q17. Prove that the product of three consecutive positive integers is divisible by 6.
5 Marks Q18. Find the smallest number which when increased by 17 is exactly divisible by 520 and 468.
5 Marks Q19. Using the Fundamental Theorem of Arithmetic, find the HCF and LCM of 96 and 404.
5 Marks Q20. A merchant has 120 liters of oil of one kind, 180 liters of another kind, and 240 liters of a third kind. He wants to sell the oil by filling the three kinds of oil in tins of equal capacity. What should be the greatest capacity of such a tin?

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CBSE Class 10 Maths Chapter 1 Model Questions

CBSE CLASS X – MODEL QUESTION PAPER

SECTION B — Very Short Answer Type Questions [2 Marks Each]

SECTION C — Short Answer Type Questions [3 Marks Each]

SECTION D — Case-Based/Source-Based Integrated Questions [4 Marks Each]

SECTION E — Long Answer Type Questions [5 Marks Each]

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