CBSE 10th Maths Question Paper 2025 (Set 4)

Given $7 \cos^2 \theta + 3 \sin^2 \theta = 4$. Using $\sin^2 \theta = 1 - \cos^2 \theta$, we get $7 \cos^2 \theta + 3(1 - \cos^2 \theta) = 4$, which simplifies to $4 \cos^2 \theta + 3 = 4$, so $4 \cos^2 \theta = 1$, $\cos^2 \theta = 1/4$, $\cos \theta = 1/2$. Therefore, $\theta = 60^\circ$. Choice C is correct.

The prime numbers between 1 and 30 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. The only even prime number is 2. The total number of outcomes is 30, and the favorable outcome is 1 (the number 2). Thus, the probability is $1/30$. Choice A is correct.

For roots $\alpha = 7$ and $\beta = 1/7$, the quadratic equation is $x^2 - (\alpha + \beta)x + (\alpha \beta) = 0$. Here $\alpha + \beta = 7 + 1/7 = 50/7$ and $\alpha \beta = 7 \times 1/7 = 1$. The equation is $x^2 - (50/7)x + 1 = 0$, which multiplies to $7x^2 - 50x + 7 = 0$. Choice A is correct.

First, find the LCM of 16, 20, and 50. $16 = 2^4$, $20 = 2^2 \times 5$, $50 = 2 \times 5^2$. LCM $= 2^4 \times 5^2 = 16 \times 25 = 400$. To make 400 a perfect square that contains the factors of the LCM, we look for the smallest square multiple. $400$ is already $20^2$. However, check divisibility: $400/16=25$, $400/20=20$, $400/50=8$. Since 400 is a perfect square and divisible by all, 400 is the answer, but 3600 is also a perfect square divisible by these. Checking options, 3600 is the only square listed that is a multiple of 400. Choice C is correct.

The length of the diameter is the distance between $(5, 2)$ and $(5, -2)$. $d = \sqrt{(5-5)^2 + (2 - (-2))^2} = \sqrt{0^2 + 4^2} = 4$. The radius is $d/2 = 4/2 = 2$. Choice D is correct.