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JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE

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Q.1. Five numbers are in A.P., whose sum is 25 and product is 2520. If one of these five numbers is -1/2, then the greatest number amongst them is    (2020)
(1) 27
(2) 7
(3) 21/2
(4) 16
Ans.
(4)
Solution. Let the five terms of A.P. be a - 2d, a - d, a, a + d and a + 2d.
Therefore,
a - 2d + a - d + a + a + d + a + 2d = 25
⇒ 5a = 25 ⇒ a = 5
Now, (a - 2d)(a - d)a(a + d)(a + 2d) = 2520
⇒ (a2 - 4d2) (a2 - d2) = 504
⇒ (25 - 4d2)(25 - d2) = 504
⇒ 4d2 - 125d2 + 121 = 0
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
Now, none of the term of A.P. is -1/2 by using d = ±1. So, d = ± 11/2
So, the terms of A.P. are -6, -1/2, 5, 21/2 and 16. Hence, the greatest term among them is 16.

Q.2. Let a1, a2, a3,… be a G.P. such that a1 < 0, a1 + a2 = 4 and a3 + a4 = 16. If JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE then λ is equal to    (2020)
(1) -513
(2) -171
(3) 171
(4) 511/3
Ans.
(2)
Solution. Let the common ratio of G.P. be r, then
a1 + a2 = 4 ⇒ a1 + a1r = 4
⇒ a1(1 + r) = 4 ...(1)
And
a3 + a4 = 16 ⇒ a1r2 + a1r3 = 16
⇒ a1r2(r+1) = 16 ...(2)
From Eqs. (1) and (2), we get
r2 = 4 ⇒ r = -2 (since a1 < 0)
Now,
a1 + a2 = 4 ⇒  a1(1 + r) = 4 ⇒ a1 = -4
Therefore,
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE

Q.3. If a, b and c be three distinct real numbers in G.P. and a + b + c = xb, then x cannot be:    (2019)
(1) -2    
(2) -3
(3) 4    
(4) 2
Ans. 
(4)
Solution. 
∵ a, b, c, are in G.P.
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
Take square on both sides, we get
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
∵ a2 + c2 is positive and b2 = ac which is also positive.
Then, x2 - 2x - 1 would be positive but for x = 2, x2 - 2x - 1 is negative.
Hence, x cannot be taken as 2.

Q.4. Let a1, a2, ..... a30 be an JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE 
If a5 = 27 and S - 2T = 75, then a10 is equal to:    (2019)
(1) 52    
(2) 57
(3) 47    
(4) 42
Ans. 
(1)
Solution. 
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
Hence, a10 = a1 + 9d = 7 + 9 x 5 = 52

Q.5. The sum of the following series
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE

JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE up to 15 terms, is:    (2019)
(1) 7520    
(2) 7510
(3) 7830    
(4) 7820
Ans. 
(4)
Solution.
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEEJEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
Now, nth term of the series,
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
Hence, sum of the series up to 15 terms is,
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
= 7200 + 620
= 7820

Q.6. Let a, b and c be the 7th, 11th and 13th terms respectively of a non-constant A.P. If these are also the three consecutive terms of a G.P., then a/c is equal to:    (2019)
(1) 2    
(2) 1/2
(3) 7/13
(4) 4
Ans.
(4)
Solution. Let first term and common difference be A and D respectively.
∴ a = A + 6D, b = A+ 10D and c = A + 12D
Since, a, b, c are in G.P.
Hence, relation between a, b and c is,
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE

Q.7. Let a1, a2, a3, ..... a10 be in G.P. with ai > 0 for i = 1,2, .... 10 and S be the set of pairs (r,k), r,k ∈ N (the set of natural numbers) for which
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
Then the number of elements in S, is:    (2019)
(1) 4    
(2) infinitely many
(3) 2    
(4) 10
Ans. 
(2)
Solution. Let common ratio of G.P. be R
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
C1→ C1-C2, C2→ C2-C3
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
Hence, number of elements in S is infinitely many.

Q.8. The sum of an infinite geometric series with positive terms is 3 and the sum of the cubes of its terms is 27/19. Then the common ratio of this series is:    (2019)
(1) 1/3    
(2) 2/3
(3) 2/9    
(4) 4/9
Ans. 
(2)
Solution. Let the terms of infinite series are a, ar, ar2, or3, ...
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
 Since, sum of cubes of its terms is 27/19 that is sum of a3, a3r3....

JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE

Q.9. Let a1, a2, ..., be a G.P. If JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE    (2019)
(1) 54    
(2) 4(52)
(3) 53    
(4) 2(52)
Ans. 
(1)
Solution. Let a= a, a2 = ar, a3 = ar2 ... a10 = ar9
where r= common ratio of given G.P.
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
Now, JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE

Q.10. Let α and β be the roots of the quadratic equation x2 sinθ -x (sinθ cosθ + 1) + cosθ = 0 (0 < θ < 45°), and JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE is equal to:    (2019)
(1) JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
(2) JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
(3) JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
(4) JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
Ans.
(3)
Solution.
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE

Q.11. If 19th term of a non-zero A.P. is zero, then its (49th term): (29th term) is:    (2019)
(1) 4:1    
(2) 1:3
(3) 3:1    
(4) 2:1
Ans.
(3)
Solution. Let first term and common difference of AP be a and d respectively, then
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE   ...(1)
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
t49 : t29 =  3 : 1

Q.12. The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now form an A.P. Then the sum of the original three terms of the given G.P. is:    (2019)
(1) 36    
(2) 32
(3) 24    
(4) 28
Ans.
(4)
Solution. Lets three terms of a G.P. be JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
a3 = 512
a = 8
4 is added to each of the first and the second of three terms then three terms are,
 JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
Therefore, sum of three terms JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE

Q.13. JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEEThen A is equal to    (2019)
(1) 283    
(2) 301
(3) 303    
(4) 156
Ans. 
(3)
Solution.
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE

Q.14. If nC4, nC5 and nC6 are in A.P., then n can be:    (2019)
(1) 9    
(2) 14
(3) 11    
(4) 12
Ans.
(2)
Solution. Since nC4, nC5 and nCare in A.P.
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE

Q.15. If the sum of the first 15 terms of the series JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE is equal to 225 k then k is equal to:    (2019)
(1) 108    
(2) 27
(3) 54    
(4) 9
Ans.
(2)
Solution.
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
Let the general term of S be
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
⇒ k = 27

Q.16. If three distinct numbers a, b, c are in G.P. and the equations ax2 + 2bx + c = 0 and dx2 + 2ex + f = 0 have a common root, then which one of the following statements is correct?    (2019)
(1) JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
(2) d, e, f are in A.P.
(3) d, e, f are in G.P.
(4) JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
Ans.
(1)
Solution. Since a, b, c are in G.P.
∴ b2 = ac
Given equation is, ax2 + 2bx + c = 0
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
Also, given that ax2 + 2bx + c = 0 and dx2 + 2ex + f = 0 have a common root.
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE

Q.17. The sum JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE is equal to:    (2019)
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
Ans.
(3)
Solution. 
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
On subtracting equations (ii) by (i),
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE

Q.18. Let the sum of the first n terms of a non-constant A.P., a1, a2, a3, ..... be 50n + JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE A, where A is a constant. If d is the common difference of this A.P., then the ordered pair (d, a50) is equal to:    (2019)
(1) (50.50 + 46A)    
(2) (50, 50 + 45A)
(3) (A, 50 + 45A)    
(4) (A, 50 + 46A)
Ans.
(4)
Solution.
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE

Q.19. If the sum and product of the first three terms in an A,P. are 33 and 1155, respectively, then a value of its 11th term is:    (2019)
(1) -35    
(2) 25    
(3) -36    
(4) -25
Ans.
(4)
Solution. Let three terms of A.P. are a - d, a, a + d
Sum of terms is, a - d + a + a + d = 33 ⇒ a = 11
Product of terms is, (a - d)a(a + d)= 11(121 -d2)= 1155
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE

Q.20. The sum of the series 1+ 2 x 3+ 3 x 5 + 4 x 7 + ..... up to 11th term is:    (2019)
(1) 915    
(2) 946    
(3) 945    
(4) 916
Ans. 
(2)
Solution.
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
Hence required sum of the series = 1 + 945 = 946

Q.21. If a1, a2, a3, ... an are in A.P. and a1 + a4 + a7 + ....+ a16 = 114, then a1 + a6 + a11 + a16 is equal to:    (2019)
(1) 98    
(2) 76    
(3) 38    
(4) 64
Ans.
(2)
Solution.
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE

Q.22. The sum JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE up to 10th term, is:    (2019)
(1) 680    
(2) 600    
(3) 660    
(4) 620
Ans.
(3)
Solution.
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE

Q.23. The sumJEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE 

JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE is equal to:    (2019)
(1) 620    
(2) 1240    
(3) 1860    
(4) 660
Ans.
(1)
Solution.
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE

Q.24. Let a, b and c be in G.P. with common ratio r, where a≠ 0 and 0 < r≤ 1/2. If 3a, 7b and 15c are the first three terms of an A.P., then the 4th term of this A.P. is:    (2019)
(1) 2/3a
(2) 5a
(3) 7/3a
(4) a
Ans.
(4)
Solution. ∵ a, b, c are in G.P. ⇒ b = ar, c = ar2
∵ 3a, 7b, 15c are in A.P. ⇒ 3a, 7ar, 15ar2 are in A.P.
∴ 14ar = 3a + 15 ar2
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE

Q.25. For x ∈ R, let [x] denote the greatest integer ≤ x, then the sum of the series JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE    (2019)
(1) -153    
(2) -133    
(3) -131    
(4) -135
Ans.
(2)
Solution.
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE

Q.26. If α and β are the roots of the equation 375x2-25x-2=0, JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE    (2019)
(1) 21/346
(2) 29/358
(3) 1/12
(4) 7/116
Ans.
(3)
Solution. 

JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE

Q.27. Let Sn denote the sum of the first n terms of an A.P.
If S4 = 16 and S6 = -48, then S10 is equal to:    (2019)
(1) -260    
(2) -410    
(3) -320    
(4) -380
Ans. 
(3)
Solution.
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE

Q.28. If a1, a2, a3, .... are in A.P. such that a1 + a7 + a16 = 40, then the sum of the first 15 terms of this A.P. is:    (2019)
(1) 200    
(2) 280    
(3) 120    
(4) 150
Ans.
(1)
Solution. Let the common difference of the A.P. is ‘d’
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
Now, sum of first 15 terms of this A.P. is,
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE

Q.29. If α, β and γ are three consecutive terms of a non-constant G.P. such that the equations ax2+2βx+γ =0 and x2+x-1=0 have a common root, then α (β+γ) is equal to:    (2019)
(1) 0    
(2) αβ
(3) αγ
(4) βγ
Ans. 
(4)
Solution. 
∵ α, β, γ are three consecutive terms of a non-constant G.P.
∴ β2 = αγ
So roots of the equation ax2 +2βx + γ = 0 are
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
∵ ax2 + 2βx + γ = 0 and x+ x - 1 = 0 have a common root.
∴ this root satisfy the equation x2 + x - 1 = 0
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE

Q.30. Let a1 , a2 , a3 , ......, a49 be in A.P. such that JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE = 416 a9 + a43  = 66.
If a12 + a22 + .... + a172 = 140m then m is equal to:    (2018)
(1) 66
(2) 68
(3) 34
(4) 33
Ans.
(3)
Solution. a1 + a5 + a9 = 416 ⇒ a + 24d = 32 ......(i)
a9 + a43 = 66 ⇒ a + 25d = 33......(ii)
From (i) and (ii) d = 1 and a = 8
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE

Q.31. If x1 , x2 …. , xn and 1/h1, 1/h2 ...... 1/hn are two A. P.s such that x3 = h2 = 8 and x8 = h7 = 20 , then x5 . h10 equals:    (2018)
(1) 2650
(2) 2560
(3) 3200
(4) 1600
Ans. 
(2)
Solution. 
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE

Q.32. If b is the first term of an infinite G.P. whose sum is five, then b lies in the interval:    (2018)
(1) (0, 10)
(2) [10, ∞]
(3) (-10, 0)
(4) (-∞, -10]
Ans.
(1)
Solution. If b is the first term and r is the common ratio of an infinite G.P. then sum is 5
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
– 5 < 5 – b < 5
– 5 < 5 – b < 5
– 10 < – b < 0
0 < b < 10
b ∈ (0,10)

Q.33. Let 1/x1, 1/x2 ,..., 1/xn (xi ≠ 0 for i = 1, 2, & ., n) be in A.P. such that x1 = 4 and x21 = 20. If n is the least positive integer for which xn > 50, then JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE is equal to:    (2018)
(1) 3
(2) 1/8
(3) 13/4
(4) 13/8
Ans.
(3)
Solution.
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE

Q.34. The sum of the first 20 terms of the series JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE ..... is:    (2018)
(1) JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
(2) JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
(3) JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
(4) JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
Ans.
(3)
Solution.
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE

Q.35. For any three positive real numbers a, b and c, 9(25a2 +b2)+ 25(c2 - 3ac) = 15b(3a + c). Then    (2017)
(1) a, b and c are in G.P
(2) b, c and a are in G.P
(3) b, c and a are in A.P
(4) a, b and c are in A.P
Ans.
(3)
Solution. 9(25a2 + b2) + 25 (c2 - 3ac) = 15b (3a + c)
⇒ (15a)2 + (3b)2 + (5c)2 - 45ab - 15bc - 75ac = 0
⇒ (15a - 3b)2 + (3b - 5c)2 + (15a - 5c)2 = 0
It is possible when
15a - 3b = 0 and 3b - 5c = 0 and 15a - 5c= 0
15a = 3b= 5c
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
∴ b, c, a are in A.P.

Q.36. Let a, b, c ∈ R. If f(x) = ax2 + bx + c is such that a + b + c = 3 and JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEEthen JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE is equal to    (2017)
(1) 255
(2) 330
(3) 165
(4) 190
Ans.
(2)
Solution. As, f (x + y)= f (x) + f (y) + xy
Given, f (1) = 3
Putting, x = y = 1 ⇒ f (2) = 2f (1)+ 1 = 7
Similarly, x = 1,y = 2 ⇒ f (3) = f (1) + f (2) + 2 = 12
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
= 3 + 7 + 12 + 18 + ... = S (let)
Now, S = 3 + 7 + 12 + 18 + ... +tn
Again, S = 3 +7 + 12 + ... +tn -1 + tn
We get, tn = 3 + 4 + 5 + ... terms
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE

Q.37. If the arithmetic mean of two numbers and b,a, > b > 0. is five times their geometric mean, then JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE is equal to:    (2017)
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
Ans.
(4)
Solution.
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE

Q.38. If the sum of the first n terms of the series JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEEthen n equals:    (2017)
(1) 13
(2) 15
(3) 29
(4) 18
Ans. 
(2)
Solution.
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE

Q.39. If three positive numbers a, b and c are in A.P. such that abc = 8, then the minimum possible value of b is:    (2017)
(1) 42/3 
(2) 2
(3) 41/3
(4) 4
Ans. 
(2)
Solution. 
For a set of positive numbers, the arithmetic mean is never smaller than the geometric mean. The geometric mean of the three numbers is 3 √ 8 = 2 . Since the numbers are in AP, their arithmetic mean is b . Thus the smallest possible value for b is 2 (this is achievable when all three numbers are 2 - an AP with zero common difference)

Q.40. Let JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE then n is equal to:    (2017)
(1) 200
(2) 199
(3) 99
(4) 19
Ans.
(2)
Solution.
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
n + 1 = 200
n = 199

Q.41. If the sum of the first ten terms of the seriesJEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEEm, then m is equal to:    (2016)
(1) 102
(2) 101
(3) 100
(4) 99
Ans.
(2)
Solution.
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEEupto 10 terms
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEEupto 10 terms.
(8)2 + (12)2 + (16)2 + ……… up to 10 terms
Tn = [4 (n + 1)]2 where n varies from 1 to 10.
= 16(n2 + 2n + 1)
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEEupto 10 terms= JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
It is given that JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE= 16/5 m
∴ m = 505/5 = 101

Q.42. If the 2nd, 5th and 9th terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is:    (2016)
(1) 8/5
(2) 4/3
(3) 1
(4) 7/4
Ans.
(2)
Solution. t2 = a + d
t5 = a + 4d
t9 = a + 8d
Given  t2, t5, t9 are in G.P.
(a + 4d)2 = (a + d) (a + 8d)
a2 + 16d2 + 8ad = a2 + 8d2 + 9ad
8d2 - ad = 0
d(8d - a) = 0
As given non-constant AP. => d ≠ 0
∴ d = a/8 => a = 8d
so,  A.P. is  8d, 9d, 10d, …..
Common ratio of G.P. = JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE

Q.43. Let x, y, z be positive real numbers such that x + y + z = 12 and x3y4z5 = (0.1) (600)3. Then x3 + y3 + z3 is equal to    (2016)
(1) 270
(2) 258
(3) 216
(4) 342
Ans.
(3)
Solution. x + y + z = 12
x3yz5 = (0.1) (600)
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE

Q.44. The sum JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE is equal to    (2016)
(1) 10 × (11!)
(2) 101 × (10!)
(3) (11!)
(4) 11 × (11!)
Ans.
(1)
Solution.
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE

Q.45. Let a1, a2, a3,......., a,.......be in A.P. If a3 + a7+ a11+ a15 = 72 then the sum of its first 17 terms is equal to    (2016)
(1) 153
(2) 306
(3) 612
(4) 204
Ans.
(2)
Solution. a1, a2, a3,......., a,....... are in A.P.
a3 + a15 = a7 + a11 = a1 + a17 = 36
JEE Main Previous year questions (2016-20): Sequences and Series Notes | Study Mathematics For JEE - JEE 

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