Enhanced Curriculum Support

SecA

1. What is the ratio of HCF to LCM of the smallest prime number and the smallest composite number?

2. If two positive integers a and b are written as a=x3y2 and b=xy3, where x, y are prime numbers, then the result obtained by dividing the product of the positive integers by the LCM (a, b) is (a) xy (b) xy2 (c) x3y3 (d) x2y2

3. If LCM(x, 18) = 36 and HCF(x, 18) = 2, then x is (a) 2 (b) 3 (c) 4 (d) 5

4. If sum of two numbers is 1215 and their HCF is 81, then the possible number of pairs of such numbers are (a) 2 (b) 3 (c) 4 (d) 5

5. The LCM of two prime numbers p and q (p > q) is 221. Find the value of 3p – q. (a) 4 (b) 28 (c) 38 (d) 48

6. The LCM of 23×32 and 22×33isa23b33c23×33d22×32

7. The HCF of two numbers is 18 and their product is 12960. Their LCM will be (a) 420 (b) 600 (c) 720 (d) 800

8. The prime factorisation of 3825 is (a) 3 × 52 × 21 (b) 32 × 52 × 35 (c) 32 × 52 × 17 (d) 32 × 25 × 17

9. The least number that is divisible by all the numbers from 1 to 10 (both inclusive) (a) 100 (b) 1000 (c) 2520 (d) 5040

10. If xy=180 and HCF(x,y)=3, then find the LCM(x,y).

11. The LCM of smallest two digit composite number and smallest composite number is a) 12 b) 4 c) 20 d) 44

SecB

1. Three bells ring at intervals of 4, 7 and 14 minutes. All three rang at 6 AM. When will they ring together again?(a) 6:07 AM (b) 6:14 AM (c) 6:28 AM (d) 6:25 AM

2. Prove that 2-3 is irrational, given that 3 is irrational.

3. The HCF and LCM of two numbers are 9 and 360 respectively. If one number is 45, find the other number.

4. Show that 7 − 5 is irrational, give that 5 is irrational.

5. If two positive integers p and q are written as p=a2 b3 and q=a3 b; a, b are prime numbers, then verify: LCM (p, q) × HCF (p, q) = pq

6. Given that 3 is irrational, show by contradiction that the sum of 3 and 2 is irrational. Show your steps.

7. 5 is an irrational number. Meera was asked to prove that (3 + 5) is an irrational number. Shown below are the steps of Meera's proof:

Step 1: Let (3 + 5) be a rational number. Then (3 + 5) can be written as p q , where p and q ( q ≠ 0) are co-primes.

Step 2 : Hence, 5 = ( pq - 3).

Step 3 : Since p and q are integers, pq−3is also an integer.

Step 4 : Since pq−3is an integer and every integer is a rational number, pq−3is a rational number. It implies that 5 is a rational number.

Step 5 : But this contradicts the fact that 5 is an irrational number.

Hence, (3 + 5) is an irrational number.

She made an error in one step due to which her subsequent steps were incorrect too. In which step did she make that error? Justify your answer.

8. Ajay has a box of length 3.2 m, breadth 2.4 m, and height 1.6 m. What is the length of the longest ruler that can exactly measure the three dimensions of the box? Show your steps and give valid reasons.

9. m is a positive integer. HCF of m and 450 is 25. HCF of m and 490 is 35. Find the HCF of m, 450 and 490. Show your steps.

10. Check whether the statement below is true or false.“The square root of every composite number is rational.”Justify your answer by proving rationality or irrationality as applicable.

SecC

1. Given that 3 is irrational, prove that 5 + 23 is irrational.

2. Given that 5 is irrational, prove that 25 − 3is an irrational number.

3. If HCF of 144 and 180 is expressed in the form 13m-16. Find the value of m.

4. Prove that 7 is irrational.

5. Prove that 12 is irrational.

6. Write two rational numbers each between the following pair: i) 3 and 10 ii) 7 and 64 iii) 15 and 6

7. The number 3837425721 is divided by a number between 5621 and 5912. State true or false for the below statements about the remainder and justify your answer. i) The remainder can be more than 5912. ii) The remainder cannot be less than 5621. iii) The remainder is always between 5621 and 5912.

8. Given 2 positive integers a and b expressed as powers of x and y, finding HCF(a,b) or LCM(a,b). x=a3 b2, y=a4 b5

SecD

1. On the two real numbers a = 2 + 5 and b = 3 - 7, perform the following operations:

i) Calculate the sum ( a + b ).

ii) Calculate the product ( ab ).

iii) Find the additive inverse of a.

iv) Rationalise 1b .

v) Verify whether the numbers a and b are rational or irrational. Provide a valid reason for your answer.

2. i) Find the LCM and HCF of 78, 91, and 195.

ii) Check whether LCM( a, b , c ) × HCF( a, b , c ) = a × b × c where a, b and c are natural numbers. Show your work

3. Find the HCF & LCM of 60 and 48 and verify the relationship- LCM x HCF = a x b

4. Using the prime factorization method, find the LCM and HCF of 72 and 120.

5. Prove that the number 2 + 3 is irrational.

6. Prove that there is no rational number whose square is 2.

7. Find the HCF and LCM of 84 and 132 using the prime factorization method and verify the relationship between them.

Problem1

Situation: A forester, Rajesh, wants to plant 66 apple trees, 88 banana trees and 110 mango trees in equal rows(in terms of number of trees). Also he wants to make distinct rows of trees (i.e. only the type of tree in the row).

Find the number of minimum rows required ? What value you learn from Rajesh?

Problem 2

Situation: An army contingent of 1000 members is to march behind an army band of 56 members in a parade. The two groups are to march in the same number of columns.

What is the maximum number of columns in which they can march?What value you learn from Amy soliders?

Q1

Prove that x2 – x is divisible by 2 for all positive integer x.

Q2

If m and n are odd positive integers, then m2+n2 is even, but not divisible by 4. Justify.

Q3

Find the smallest number which leaves remainder 8 and 12 when divided by 28 and 32 respectively.

Questions

1. Write whether every positive integer can be of the form 4q + 2, where q is an integer? Justify your answer.

2. The product of two consecutive positive integers is divisible by 2”. Is this statement true or false? Give reasons.

3. A positive integer is of the form 3q + 1, q being a natural number. Can you write its square in any form other than 3m + 1, i.e., 3m or 3m + 2 for some integer m? Justify your answer.

4. Find the least number that must be added to 1300 to make it a perfect square.

5. Find the HCF and LCM of 56 and 98 using the prime factorization method. Also, verify the relationship between HCF and LCM.

6. Use the Euclidean algorithm to find the HCF of the following:

(a) 84 and 132

(b) 144 and 240

7. Prove that the square root of any prime number is irrational.

8. Show that 2 + 3 is irrational.

9. Prove that there is no rational number whose square is 2.

10. Prove that 3 is irrational, and hence, 6is irrational.

Q1

In a school, buses are hired to take the students of Std-X on a field trip. There are 156, 208, and 260 students in groups A, B, and C respectively. The goal is to determine the minimum number of buses needed, with each bus accommodating the same number of students, and each group traveling separately.

Based on your understanding of the above case study,answer all the five questions below:

(1) Maximum number of students to be accommodated in each bus is:

(2) How many buses will be required for group A students?

(3) If group C will not be taken along and 26 students of group A refuse to join, then the minimum number of buses required to accommodate the remaining students is:

(4) If group A students will not be taken along, then the minimum number of buses required to accommodate the students of group B and group C is in the form of xy then the value of x and y are respectively and

(5) What is the minimum number of buses required for all the students?

Sol1

Solution

(1)Prime factorization:

• 156: 156 = 22 x 3 x 13

• 208: 208 = 24 x 13

• 206: 260 = 22 x 5 x 13

Common factors: 22 x 13 = 4 x 13 = 52

So, the maximum number of students per bus is 52.

(2)Number of buses required for group A students:

Number of buses for group A = 15252 = 3

(3)If group C is not taken and 26 students of group A refuse to join:

• Remaining students in group A: 156 - 26 = 130

• Number of buses for group A and B together:

• HCF of 130 and 208:

130: 130 = 2 x 5 13

208: 208 = 24 x 13

Common factor: 2 x 13 = 26

Number of buses for group A = 13026 = 5

Number of buses for group B = 20826= 8

Total number of buses = 5 + 8 =13

(4)If group A students will not be taken along:

Number of buses required for group B and group C together:

HCF of 208 and 260:

208: 24 x 13

260: 22 x 5 x 13

Common factor: 22 x 13 = 4 x13 = 52

Number of buses for group B = 20852 = 4

Number of buses for group C = 26052 = 5

Total number of buses = 4+5=9

The format of xy seems unclear, but the total buses are 9.

(5)Minimum number of buses required for all students:

Number of buses required for groups A, B, and C:

HCF of 156, 208, and 260 is 52.

Number of buses for group A = 15652 = 3

Number of buses for group B = 20852 = 4

Number of buses for group C = 26052 = 5

Total number of buses = 3 + 4 + 5 = 12

Question

ASSERTION AND REASONING:

1.Choose the correct option Statement

A (Assertion): If product of two numbers is 5780 and their HCF is 17, then their LCM is 340 Statement .

R( Reason): HCF is always a factor of LCM

(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A)

(b) Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of assertion (A)

(c) Assertion (A) is true but reason (R) is false.

(d) Assertion (A) is false but reason (R) is true.

2. Assertion (A): Product of HCF and LCM of THREE numbers is equal to the product of those numbers.

Reason (R): Product of HCF and LCM of TWO numbers is equal to the product of those numbers.

1 Both (A) and (R) are true and (R) is the correct explanation for (A).

2 Both (A) and (R) are true and (R) is not the correct explanation for (A).

3 (A) is false but (R) is true.

4 Both (A) and (R) are false.

Solution

1.Assertion (A): If the product of two numbers is 5780 and their HCF is 17, then their LCM is 340.

Explanation: Use the formula:

Product of the two numbers=HCF×LCM

Solving for LCM:

LCM=578017=340

Thus, Assertion (A) is true.

Reason (R): HCF is always a factor of LCM.

Explanation: While it is true that HCF is always a factor of LCM, this fact does not explain why the LCM is 340 in this specific case. The correct explanation is based on the calculation of LCM using HCF and the product of the numbers, not just the fact that HCF is a factor of LCM.

Answer: (b) Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of assertion (A).

2.Statement and Options

Assertion (A): Product of HCF and LCM of THREE numbers is equal to the product of those numbers.

Explanation: This statement is false. The property that the product of HCF and LCM equals the product of the numbers applies only to TWO numbers, not three.

Reason (R): Product of HCF and LCM of TWO numbers is equal to the product of those numbers.

Explanation: This statement is true. For any two numbers, the product of their HCF and LCM equals the product of the two numbers.

Answer: (3) (A) is false but (R) is true.