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Matrix-Match Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

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PASSAGE - 1
 Let Vr 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= Vr + 1 – Vr – 2 and Q= Tr + 1 – Tr for r = 1, 2, ... 

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

(a)Matrix-Match Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

(b) Matrix-Match Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

(c) Matrix-Match Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

(d) Matrix-Match Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. Sol. 

(b) V+  V2 + .... Matrix-Match Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Matrix-Match Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Matrix-Match Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Matrix-Match Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Matrix-Match Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

 

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= Vr +1 -Vr- 2

Matrix-Match Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

= 3r 2 + 2r+1

Tr = (r + 1) (3r – 1)

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

∴ Tr 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) ∴ Qr +1 - Qr = Tr +2 - Tr +1 - (Tr+1-Tr)            = Tr + 2 - 2Tr+1+Tr
= (r+ 3)(3r + 5) - 2(r + 2)(3r + 2)+ (r + 1)(3r -1)

∵Qr + 1 – Q=  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 An – 1 and Hn – 1 have arithmetic, geometric and harmonic means as An, Gn, Hrespectively. 

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

Ans. Sol. 

(c)   Given  Matrix-Match Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

also  Matrix-Match Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Matrix-Match Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Matrix-Match Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Similarly we can prove

AnHn = An–1Hn–1 = An–2 Hn–2 = .... =  A1H1

⇒ AnHn = ab

∴     G1= G2 = G3 ....= ab

⇒ G1 = G2 = G3 ... Matrix-Match Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

 

 

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

Ans. Sol. 

(a) We have

Matrix-Match Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Matrix-Match Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Matrix-Match Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

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

 

 

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

Ans. Sol. 

(b) We have An Hn = ab Matrix-Match Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Matrix-Match Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE   ∴ H< H2 < H3 < .......

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