Let a1 a2 a3 . . . be a G.P. of increasing positive terms

Let a1 a2 a3 . . . be a G.P. of increasing positive terms

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 (iii) 784

Explanation:- $a_1, a_2, a_3 . . .$ be a G.P. of increasing positive terms.

Let the common ratio of G.P. = r

$a_1.a_5 = 28$

⇒ $a.ar^4 = 28$

⇒ $a^2r^4 = 28$ . . . (i)

$a_2 + a_4 = 29$

⇒ $ar + ar^3 = 29$

⇒ $ar(1 + r^2) = 29$

⇒ $a^2r^2(1 + r^2)^2 = (29)^2$ . . . (ii)

Divided equation (i) by (ii)

$\dfrac{r^2}{(1 + r)^2} = \dfrac{28}{29\times 29}$

Taking square root both side

$\dfrac{r}{1 + r^2} = \dfrac{\sqrt{28}}{29}$

⇒ r = √28

From equation (i)

$a^2r^4 = 28$ . . . (i)

⇒ $a^2(\sqrt{28})^4 = 28$

⇒ $a^2(28)^2 = 28$

⇒ $a = \dfrac{1}{\sqrt{28}}$

∴ $a_6 = ar^5 = \dfrac{1}{\sqrt{28}}\times (28)^2\sqrt{28}$

= 784

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 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 : 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