JEE Advanced 2022 — Set 1

Considering only the principal values of the inverse trigonometric functions, the value of 3 2 cos−1 √ 2 2 + π2 + 1 4 sin−1 2√2 π 2 + π2 + tan−1 √2 π is __________ .

Let 𝛼 be a positive real number. Let 𝑓: ℝ→ℝ and 𝑔: (𝛼, ∞) →ℝ be the functions defined by 𝑓(𝑥) = sin(𝜋𝑥 12) and 𝑔(𝑥) = 2 loge ( √𝑥−√𝛼 ) loge ( 𝑒√𝑥−𝑒√𝛼 ) . Then the value of lim 𝑥→𝛼+ 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 2 + 3𝑧+ 4𝑧2 2 −3𝑧+ 4𝑧2 is a real number, then the value of |𝑧|2 is _____________.

Let 𝑧̅ denote the complex conjugate of a complex number 𝑧 and let 𝑖= √−1 . In the set of complex numbers, the number of distinct roots of the equation 𝑧̅ −𝑧2 = 𝑖(𝑧̅ + 𝑧2) is _____________.

Let 𝑙1, 𝑙2, … , 𝑙100 be consecutive terms of an arithmetic progression with common difference 𝑑1, and let 𝑤1, 𝑤2, … , 𝑤100 be consecutive terms of another arithmetic progression with common difference 𝑑2 , where 𝑑1𝑑2 = 10. For each 𝑖= 1, 2, … , 100, let 𝑅𝑖 be a rectangle with length 𝑙𝑖, width 𝑤𝑖 and area 𝐴𝑖. If 𝐴51 −𝐴50 = 1000, then the value of 𝐴100 −𝐴90 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 ∠𝐵𝐴𝐶= π 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 _____________.

A medium having dielectric constant K >1 fills the space between the plates of a parallel plate capacitor. The plates have large area, and the distance between them is d. The capacitor is connected to a battery of voltage V, as shown in Figure (a). Now, both the plates are moved by a distance of from their original positions, as shown in Figure (b). Figure (a) Figure (b) In the process of going from the configuration depicted in Figure (a) to that in Figure (b), which of the following statement(s) is(are) correct?

Let 𝑎1, 𝑎2, 𝑎3, … be an arithmetic progression with 𝑎1 = 7 and common difference 8. Let 𝑇1, 𝑇2, 𝑇3, … be such that 𝑇1 = 3 and 𝑇𝑛+1 −𝑇𝑛= 𝑎𝑛 for 𝑛≥1. Then, which of the following is/are TRUE ?