JEE (Advanced) 2020 Paper-2

Two fair dice, each with faces numbered 1,2,3,4,5 and 6, are rolled together and the sum of the numbers on the faces is observed. This process is repeated till the sum is either a prime number or a perfect square. Suppose the sum turns out to be a perfect square before it turns out to be a prime number. If 𝑝 is the probability that this perfect square is an odd number, then the value of 14𝑝 is _____

Let the function $$f:[0, 1] \rightarrow R$$ be defined by $$f(x) = \frac{4^x}{4^x + 2}$$. Then the value of $$f\left(\frac{1}{40}\right)+f\left(\frac{2}{40}\right)+f\left(\frac{3}{40}\right)+.....+f\left(\frac{39}{40}\right) - f\left(\frac{1}{2}\right)$$ is__________.

Let $$f: R \rightarrow R$$ be a differentiable function such that its derivative $$𝑓′$$ is continuous and $$f(\pi) = -6$$. If $$F:[0, \pi] \rightarrow R$$ is defined by $$F(x) = \int_{0}^{x}f(t)dt $$, and if
$$\int_{0}^{\pi}\left(f'(x) + F(x)\right) \cos x dx = 2 $$, then the value of f(0) is_______

Let the function $$f:(0, \pi) \rightarrow R$$ be defined by
$$f(\theta) = (\sin \theta + \cos \theta)^2 + (\sin \theta - \cos \theta)^4$$.
Suppose the function 𝑓 has a local minimum at $$\theta$$ precisely when $$\theta \in \left\{\lambda_1 \pi, ...., \lambda_r \pi\right\}$$ where $$0 < \lambda_1 < ... < \lambda_r < 1$$. Then the value of $$\lambda_1 + ... + \lambda_r$$ is _________