Ajaysaini231/JEE-Main-2025-Math huggingface dataset:
Question: The focus of the parabola $y^2 = 4x + 16$ is the centre of the circle $C$ of radius 5. If the values of $\lambda$, for which $C$ passes through the point of intersection of the lines $3x - y = 0$ and $x + \lambda y = 4$, are $\lambda_1$ and $\lambda_2$, $\lambda_1 < \lambda_2$, then $12\lambda_1 + 29\lambda_2$ is equal to:
Options:
Answer: 15
Question: Let $\alpha, \beta$ be the roots of the equation $x^2 - ax - b = 0$ with $\operatorname{Im}(\alpha) < \operatorname{Im}(\beta)$. Let $P_n = \alpha^n - \beta^n$. If $\mathbf{P}_3 = -5\sqrt{7}i$, $\mathbf{P}_4 = -3\sqrt{7}i$, $\mathbf{P}_5 = 11\sqrt{7}i$ and $\mathbf{P}_6 = 45\sqrt{7}i$, then $|\alpha^4 + \beta^4|$ is equal to:
Options:
Answer: 31