JEE Advanced (Matrix Match): Sequences & Series | Chapter-wise Tests for JEE Main & Advanced PDF Download

PASSAGE - 1
 Let V_r denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r – 1).
 Let  T_r = V_{r + 1} – V_{r – 2} and Q_r = T_{r + 1} – T_r for r = 1, 2, ... 

1. The sum V₁ + V₂ + ... + V_n is (2007 -4 marks)

(a)

(b) 

(c) 

(d) 

Ans. Sol. 

(b) V₁ +  V₂ + ....

2. Tr is always (2007 -4 marks)
 (a) an odd number
 (b) an even number
 (c) a prime number
 (d) a composite number 

Ans. Sol. 

(d) T_r = V_{r +1} -V_{r- 2}

= 3r ² + 2r+1

T_r = (r + 1) (3r – 1)

For each r, T_r has two different factors other than 1 and itself.

∴ T_r is always a composite number.

3. Which one of the following is a correct statement ? (2007 -4 marks)
 (a) Q₁, Q₂, Q₃, ... are in A.P. with common difference 5
 (b) Q₁, Q₂, Q₃, ... are in A.P. with common difference 6
 (c) Q₁, Q₂, Q₃, ... are in A.P. with common difference 11
 (d) Q₁= Q₂ = Q₃ = ....

Ans. Sol. 

(b) ∴ Q_{r +1} - Q_r = T_{r +2} - T_{r +1} - (T_r+1-T_r)            = T_{r + 2} - 2T_r+1+Tr
= (r+ 3)(3r + 5) - 2(r + 2)(3r + 2)+ (r + 1)(3r -1)

∵Q_{r + 1} – Q_r =  6 (r + 1) + 5 – 6r – 5 = 6 (constant)

∴ ∵1, Q₂, Q₃, .... are in AP with common difference 6.

PASSAGE -2
 Let A₁, G₁, H₁ denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For n ≥ 2, Let A_{n – 1} and H_{n – 1} have arithmetic, geometric and harmonic means as A_n, G_n, H_n respectively. 

4. Which one of the following statements is correct ? (a) G₁ > G₂ > G₃ > ... (2007 -4 marks) (b) G₁ < G₂ < G₃ < ... (c) G₁ = G₂ = G₃ = ... (d) G₁ < G₃ < G₅ < ... and G₂ > G₄ > G₆ > .... 

Ans. Sol. 

(c)   Given 

also  

Similarly we can prove

AnHn = A_n–1H_n–1 = A_n–2 H_n–2 = .... =  A₁H₁

⇒ A_nHn ₌ ab

∴     G₁² = G₂²  = G₃²  ....= ab

⇒ G₁ = G₂ = G₃ ... 

5. Which one of the following statements is correct ? (a) A₁ > A₂ > A₃ > ... (2007 -4 marks) (b) A₁ < A₂ < A₃ < ... (c) A₁ > A₃ > A₅ > ... and A₂ < A₄ < A₆ < ... (d) A₁ < A₃ < A₅ < ... and A₂ > A₄ > A₆ > ... 

Ans. Sol. 

(a) We have

⇒ A_n <  A_n–1 or   A_n–1 >  A_n ∴ We can conclude that A₁ > A₂ > A₃ > ....

6. Which one of the following statements is correct ? (a) H₁ > H₂ > H₃ > ... (2007 -4 marks) (b) H₁ < H₂ < H₃ < ... (c) H₁ > H₃ > H₅ > ... and H₂ < H₄ < H₆ < ... (d) H₁ < H₃ < H₆ < ... and H₂ > H₄ > H₆ > ...

Ans. Sol. 

(b) We have A_n H_n = ab

   ∴ H₁ < H₂ < H₃ < .......