JEE (Advanced) 2022 Paper-1

Considering only the principal values of the inverse trigonometric functions, the value of

$$\frac{3}{2} \cos^{-1} \sqrt{\frac{2}{2+\pi^2}} + \frac{1}{4} \sin^{-1} \frac{2\sqrt{2}\pi}{2+\pi^2} + \tan^{-1} \frac{\sqrt{2}}{\pi}$$

is ______.

Let $$\alpha$$ be a positive real number. Let $$f: \mathbb{R} \to \mathbb{R}$$ and $$g: (\alpha, \infty) \to \mathbb{R}$$ be the functions defined by

$$f(x) = \sin\left(\frac{\pi x}{12}\right)$$ and $$g(x) = \frac{2\log_e(\sqrt{x} - \sqrt{\alpha})}{\log_e(e^{\sqrt{x}} - e^{\sqrt{\alpha}})}$$.

Then the value of $$\lim_{x \to \alpha^+} f(g(x))$$ is ______.

In a study about a pandemic, data of 900 persons was collected. It was found that

190 persons had symptom of fever,

220 persons had symptom of cough,

220 persons had symptom of breathing problem,

330 persons had symptom of fever or cough or both,

350 persons had symptom of cough or breathing problem or both,

340 persons had symptom of fever or breathing problem or both,

30 persons had all three symptoms (fever, cough and breathing problem).

If a person is chosen randomly from these 900 persons, then the probability that the person has at most one symptom is ______.

Let $$z$$ be a complex number with non-zero imaginary part. If

$$\frac{2 + 3z + 4z^2}{2 - 3z + 4z^2}$$

is a real number, then the value of $$|z|^2$$ is ______.

Let $$\bar{z}$$ denote the complex conjugate of a complex number $$z$$ and let $$i = \sqrt{-1}$$. In the set of complex numbers, the number of distinct roots of the equation

$$\bar{z} - z^2 = i(\bar{z} + z^2)$$

is ______.

Let $$l_1, l_2, \ldots, l_{100}$$ be consecutive terms of an arithmetic progression with common difference $$d_1$$, and let $$w_1, w_2, \ldots, w_{100}$$ be consecutive terms of another arithmetic progression with common difference $$d_2$$, where $$d_1 d_2 = 10$$. For each $$i = 1, 2, \ldots, 100$$, let $$R_i$$ be a rectangle with length $$l_i$$, width $$w_i$$ and area $$A_i$$. If $$A_{51} - A_{50} = 1000$$, then the value of $$A_{100} - A_{90}$$ is ______.

The number of 4-digit integers in the closed interval [2022, 4482] formed by using the digits 0, 2, 3, 4, 6, 7 is ______.

Let $$ABC$$ be the triangle with $$AB = 1$$, $$AC = 3$$ and $$\angle BAC = \frac{\pi}{2}$$. If a circle of radius $$r > 0$$ touches the sides $$AB$$, $$AC$$ and also touches internally the circumcircle of the triangle $$ABC$$, then the value of $$r$$ is ______.

Consider the equation

$$\int_1^e \frac{(\log_e x)^{1/2}}{x\left(a - (\log_e x)^{3/2}\right)^2} dx = 1, \quad a \in (-\infty, 0) \cup (1, \infty).$$

Which of the following statements is/are TRUE?

Let $$a_1, a_2, a_3, \ldots$$ be an arithmetic progression with $$a_1 = 7$$ and common difference 8. Let $$T_1, T_2, T_3, \ldots$$ be such that $$T_1 = 3$$ and $$T_{n+1} - T_n = a_n$$ for $$n \geq 1$$. Then, which of the following is/are TRUE?