Class X Session 2023-24 Question 22
MATH SAMPLE QUESTION PAPER Class X Session 2023-24 MATHEMATICS STANDARD (Code No.041) SECTION B - Question 22 Prove that √2 is an irrational number. Proof: Let us assume √2 is a rational number. Then, there exist positive integers a and b such that \[\sqrt{2} =\frac{p}{q} where,\; p\; and\; q,\; are \;co-prime\; i.e. \;their\; HCF \;is \;1\] \[ OR \; (\sqrt{2})^2 =\frac{p^2}{q^2} \] \[OR\; 2 =\frac{p^2}{q^2} \] \[OR\; 2 \times q^2 = p^2 \] This implies that 𝑝^2 is even because it is equal to 2𝑞^2. From this, we can conclude that 𝑝 must also be even because the square of an odd number is odd, and the square of an even number is even. Let 𝑝 = 2k, where k is arbitrary integer , Substituting 𝑝 = 2k into the equation 2q^2 = p^2 we get Substituting 𝑝 = 2k into the equation 2q^2 = p^2 \[\Rightarrow 2q^2 = (2k)^2 \] \[\Rightarrow 2q^2 = 4(k)^2 \] \[\Rightarrow q^2 = 2(k)^2 \] This implies that 𝑞^2 is even. Therefore, 𝑞 must also be even. Now, we have shown that both 𝑝 and q ar