CBSE 10th Maths Question Paper 2024 (Set 3)

The smallest prime number is 2. The smallest odd composite number is 9 (since 1 is neither prime nor composite, 3 is prime, 5 is prime, 7 is prime, 9 is 3x3). The LCM of 2 and 9 is 18. However, checking the options provided: The prime factorization of 2 = 2, and 9 = 3^2. LCM = 2 * 9 = 18. Since 18 is not an option, but 9 is, checking the smallest odd composite number again: 1 is not composite, 3 is prime, 5 is prime, 7 is prime, 9 is composite. LCM(2, 9) = 18. Given the options provided, there is a discrepancy in the question or options. Assuming a potential error in the source material, 18 would be correct. If the question intended the smallest composite (4) and prime (2), LCM is 4. Given 18 isn't an option, and 9 is, 9 is an incorrect value for LCM. Re-evaluating: Smallest prime = 2. Smallest odd composite = 9. LCM = 18. None of the options A, B, C, or D match 18. If the question implies odd prime factors, the logic fails. Selecting 'D' as 18 is the mathematical answer.

The sum of the first n natural numbers is n(n+1)/2. The mean is [n(n+1)/2] / n = (n+1)/2. Setting (n+1)/2 = 9/5 * n, we get 5n + 5 = 18n. This yields 13n = 5, n = 5/13, which is not an integer. Re-reading: If mean is 9/5, then (n+1)/2 = 9/5 => n+1 = 18/5 = 3.6 => n=2.6. If the question meant (n+1)/2 = (9/5) * 5/9, it's not standard. Solving (n+1)/2 = 5 yields n = 9. Wait, if the mean is (n+1)/2 = 5, n=9. If the formula given is 9/5, and we test n=9, (9+1)/2 = 5. The equation 9/5 * n does not align. Assuming the question intended the mean of first n numbers is 5, then n=9.

Given 5 tan θ - 12 = 0, we have tan θ = 12/5. In a right triangle, tan θ = opposite/adjacent = 12/5. Using Pythagoras, hypotenuse = sqrt(12^2 + 5^2) = sqrt(144 + 25) = sqrt(169) = 13. Thus, sin θ = opposite/hypotenuse = 12/13. There seems to be a mismatch in the provided options; however, if the ratio is 12:5, sin is 12/13. Given option C is 13/5, this is 1/sin. Re-checking: 12/13 is the correct value.

The sequence is 18, 50, 98. These are 2*9, 2*25, 2*49. These are 2*(3^2), 2*(5^2), 2*(7^2). The next odd number squared is 9^2 = 81. Therefore, the 4th term is 2 * 81 = 162. Option C is 162.

By Thales Theorem (BPT), AD/DB = AE/EC. Given AD=2.4, DB=4, AE=2. So, 2.4/4 = 2/EC. EC = (2 * 4) / 2.4 = 8 / 2.4 = 80/24 = 10/3 cm. AC = AE + EC = 2 + 10/3 = (6+10)/3 = 16/3. Since option B is 10/3, it identifies the segment EC, not AC. The length of AC is 16/3.