Let Z1 Z2 and Z_3 be three complex numbers on the circle

Let Z1 Z2 and Z_3 be three complex numbers on the circle

Question 4:- Let $Z_1, Z_2$ and $Z_3$ be three complex numbers on the circle |Z| = 1 with $\arg(Z_1) = \dfrac{-\pi}{4},\arg(Z_2) = 0$ and $\arg(Z_3) = \dfrac{\pi}{4}$ . If $|Z_1.\bar{Z_2} + Z_2.\bar{Z_3} + Z_3.\bar{Z_1}|^2 = \alpha + \beta\sqrt{2}, \alpha,\beta \in Z$ , then the value of α² + β² is :

(1) 24 (2) 41

(3) 31 (4) 29 [JEE 22 JAN 2025]

Answer : (4) 29

Explanation:- Given, |z| = 1

⇒ $|Z_1| = 1, |Z_2| = 1, |Z_3| = 1$

$\arg(Z_1) = \dfrac{-\pi}{4},\arg(Z_2) = 0, \arg(Z_3) = \dfrac{\pi}{4}$

⇒ $Z_1 = |Z_1|e^{-i \pi/4}, Z_2 = |Z_2|e^{i(0)}, Z_3 = |Z_3|e^{i\pi/4}$

⇒ $Z_1 = e^{-i \pi/4},Z_2 = 1, Z_3 = e^{i\pi/4}$

$|Z_1.\bar{Z_2} + Z_2.\bar{Z_3} + Z_3.\bar{Z_1}|^2$

$= |e^{-i \pi/4}\times 1 + 1\times e^{-i \pi/4} + e^{i \pi/4} \times e^{i \pi/4}|^2$

$= |e^{-i \pi/4} + e^{-i \pi/4} + e^{i \pi/4}|^2$

$= |2e^{-i \pi/4} + i|^2$

$= |\sqrt{2}- \sqrt{2} i + i|^2$

= |√2 + i(1 – √2)|²

= (√2)² + (1 – √2)²

= 2 + 1 + 2 – 2√2 = 5 – 2√2 = α + β√2 (Compare)

α = 5, β = -2

⇒ α² + β² = (5)² + (-2)²

= 25 + 4 = 29

Question 2:- Let f : R → R be a twice differentiable function such that f(x + y) = f(x) f(y) for all x, y ∈ R. If f'(0) = 4a and f satisfies f”(x) – 3a f'(x) – f(x) = 0, a > 0, then the area of the region

R = {(x, y) | 0 ≤ y ≤ f(ax), 0 ≤ x ≤ 2} is :

(i) $e^2 - 1$ (ii) $e^4 + 1$

(iii) $e^4 - 1$ (iv) $e^2 + 1$ [JEE 22 JAN 2025]

Solution:- See full solution

Question 3:- Let the triangle PQR be the image of the triangle with vertices (1, 3), (3, 1) and (2, 4) in the line x + 2y = 2. If the centroid of ΔPQR is the point (α, β), then 15(α – β) is equal to :

(1) 24 (2) 19

(3) 21 (4) 22 [JEE 22 JAN 2025]

Answer:- See full answer

Question 5:- Using the principal values of the inverse trigonometric functions the sum of the maximum and the minimum values of $16((\sec^{-1} x)^2 + (\operatorname{cosec}^{-1} x)^2)$ is :

(1) 24 π² (2) 18 π²

(3) 31 π² (4) 22 π² [JEE 22 JAN 2025]

Answer :- See full answer

Question 6:- A coin is tossed three times. Let X denote the number of times a tail follows a head. If μ and σ² denote the mean and variance of X, then the value of 64(μ + σ²) is :

(i) 51 (ii) 48

(iii) 32 (iv) 64 [JEE 22 JAN 2025]

Answer:- See full solution

Question 7:- Let $a_1, a_2, a_3 . . .$ be a G.P. of increasing positive terms. If $a_1.a_5 = 28$ and $a_2 + a_4 = 29$ , the $a_6$ is equal to

(i) 628 (ii) 526

(iii) 784 (iv) 812 [JEE 22 JAN 2025]

Answer:- See full Answer

Question 8:- Let $L_1: \dfrac{x - 1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4}$ and $L_2 : \dfrac{x - 2}{3}=\dfrac{y-4}{4}=\dfrac{z-5}{5}$ be two lines. Then which of the following points lies on the line of the shortest distance between $L_1$ and $L_2$ ?

(i) (-5/3, -7, 1) (ii) (2, 3, 1/3)

(iii) (8/3, -1, 1/3) (iv) (14/3, -3, 22/3) [JEE 22 JAN 2025]

Answer:- See full solution

Question 9:- The product of all solutions of the equation $e^{5(\log_e x)^2 + 3} = x^8, x > 0$ is :

(i) $e^{8/5}$ (ii) $e^{6/5}$

(iii) $e^{2}$ (iv) $e$ [JEE 22 JAN 2025]

Answer :- See full solution

Question 10:- If $\displaystyle \Sigma_{r= 1}^n T_r = \dfrac{(2n - 1)(2n + 1)(2n + 3)(2n + 5)}{64}$ , then $\displaystyle \lim_{n \to \infty} {\Sigma _{r = 1}^n}\dfrac{1}{T_r}$ is equal to

(i) 1 (ii) 0

(iii) 2/3 (iv) 1/3 [JEE 22 JAN 2025]

Answer :- See full Answer