JEE Advanced 2022 — Set 2

Let 𝛼 and 𝛽 be real numbers such that − 𝜋 4 < 𝛽< 0 < 𝛼< 𝜋 4 . If sin(𝛼+ 𝛽) = 1 3 and cos(𝛼−𝛽) = 2 3 , then the greatest integer less than or equal to ( sin 𝛼 cos 𝛽+ cos 𝛽 sin 𝛼+ cos𝛼 sin 𝛽+ sin 𝛽 cos 𝛼 ) 2 is _____________.

If 𝑦(𝑥) is the solution of the differential equation 𝑥𝑑𝑦−(𝑦2 −4𝑦)𝑑𝑥= 0 for 𝑥> 0, 𝑦(1) = 2, and the slope of the curve 𝑦= 𝑦(𝑥) is never zero, then the value of 10 𝑦(√2 ) is _____________.

The greatest integer less than or equal to ∫log2(𝑥3 + 1) 2 1 𝑑𝑥 + ∫ (2𝑥−1) 1 3 log2 9 1 𝑑𝑥 is _____________.

The product of all positive real values of 𝑥 satisfying the equation 𝑥(16(log5 𝑥)3−68 log5 𝑥) = 5−16 is __________ . SECTION 1 (Maximum marks: 24) • This section contains EIGHT (08) questions. • The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 TO 9, BOTH INCLUSIVE. • For each question, enter the correct integer corresponding to the answer using the mouse and the on­ screen virtual numeric keypad in the place designated to enter the answer. • Answer to each question will be evaluated according to the following marking scheme: Full Marks            :   +3 If ONLY the correct integer is entered; Zero Marks          :     0 If the question is unanswered; Negative Marks  :   −1   In all other cases.

If 𝛽= lim 𝑥→0 𝑒𝑥3 −(1 −𝑥3) 1 3 + ((1 −𝑥2) 1 2 −1) sin 𝑥 𝑥sin2 𝑥 , then the value of 6𝛽 is ___________.

Let 𝛽 be a real number. Consider the matrix 𝐴= ( 𝛽 0 1 2 1 −2 3 1 −2 ) . If 𝐴7 −(𝛽−1)𝐴6 −𝛽𝐴5 is a singular matrix, then the value of 9𝛽 is ___________.

Consider the hyperbola 𝑥2 100 −𝑦2 64 = 1 with foci at 𝑆 and 𝑆1, where 𝑆 lies on the positive x-axis. Let 𝑃 be a point on the hyperbola, in the first quadrant. Let ∠𝑆𝑃𝑆1 = 𝛼, with 𝛼< 𝜋 2 . The straight line passing through the point 𝑆 and having the same slope as that of the tangent at 𝑃 to the hyperbola, intersects the straight line 𝑆1𝑃 at 𝑃1. Let 𝛿 be the distance of 𝑃 from the straight line 𝑆𝑃1, and 𝛽= 𝑆1𝑃. Then the greatest integer less than or equal to βδ 9 sin 𝛼 2 is _____________.

Consider the functions 𝑓, 𝑔∶ℝ→ℝ defined by 𝑓(𝑥) = 𝑥2 + 5 12 and 𝑔(𝑥) = { 2 (1 −4|𝑥| 3 ) , |𝑥| ≤3 4 , 0 , |𝑥| > 3 4 . If 𝛼 is the area of the region {(𝑥, 𝑦) ∈ℝ× ℝ∶ |𝑥| ≤3 4 , 0 ≤𝑦≤min{𝑓(𝑥), 𝑔(𝑥)} } , then the value of 9𝛼 is _____________.

Let 𝛼= ∑sin2𝑘(𝜋 6) ∞ 𝑘=1 . Let 𝑔∶[0, 1] →ℝ be the function defined by 𝑔(𝑥) = 2𝛼𝑥+ 2𝛼(1−𝑥) . Then, which of the following statements is/are TRUE ?

Let 𝑧̅ denote the complex conjugate of a complex number 𝑧. If 𝑧 is a non-zero complex number for which both real and imaginary parts of ( 𝑧̅ )2 + 1 𝑧2 are integers, then which of the following is/are possible value(s) of |𝑧| ?