Q1. If $p_1$ and $p_2$ are two odd prime numbers such that $p_1 > p_2$, then $p_1^2 - p_2^2$ is:
Q2. If two positive integers $a$ and $b$ are written as $a = x^3y^2$ and $b = xy^3$, where $x, y$ are prime numbers, then $HCF(a, b)$ is:
Q3. The LCM of the smallest two-digit composite number and the smallest composite number is:
Q4. Assertion (A): The HCF of two numbers is 18 and their product is 3072. Then their LCM is 169. Reason (R): If $a, b$ are two positive integers, then $HCF \times LCM = a \times b$.
Q5. Explain why $3 \times 5 \times 7 + 7$ is a composite number.
Solve on white paper.
Q6. Find the LCM and HCF of 6 and 20 by the prime factorisation method.
Q7. Prove that $7\sqrt{5}$ is an irrational number.
Q8. Prove that $5 - \sqrt{3}$ is an irrational number, given that $\sqrt{3}$ is irrational.
Q9. The length, breadth and height of a room are 8 m 25 cm, 6 m 75 cm and 4 m 50 cm respectively. Find the length of the longest rod that can measure the three dimensions of the room exactly.
Q10. Prove that $\sqrt{2}$ is an irrational number. Hence, show that $3 - \sqrt{2}$ is an irrational number.
Q11. Case Study:
To enhance the reading skills of grade X students, the school nominates you and two of your friends to set up a class library. There are two sections- Section A and Section B of grade X. There are 32 students in section A and 36 students in section B.
(i) Express 36 as a product of its primes.
(ii) Find the HCF of 32 and 36.
(iii) What is the minimum number of books you will acquire for the class library, so that they can be distributed equally among students of Section A or Section B?