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Sequences and Series · Question 5

JEE Main 2026 (Online) 6th April Evening ShiftMCQMedium📈 Topic in 42% papers+4−1
1

Consider the quadratic equation (n2-2n+2 )x2-3x+ (n2-2n+2 )2=0, n R. Let be the minimum value of the product of its roots and be the maximum value of the sum of its roots. Then the sum of the first six terms of the G.P., whose first term is and the common ratio is , is :

  1. 6137
  2. 12181
  3. 364243
  4. 1093729

Answer

(3) 364243

Solution

Let k=n2-2n+2=(n-1)2+1 k 1

Equation: kx2-3x+k2=0

Product of roots = k2k=k minimum value =1 (at n=1)

Sum of roots = 3k maximum value =3 (at n=1)

G.P.: first term a=1, common ratio r= = 13

S6= a (1-r6 )1-r= 1- 1729 23

S6= 728729 32= 364243

Answer: (3)

JEE Main 2026 (Online) 6th April Evening Shift

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2

The sum 1+ 12 (12+22 )+ 13 (12+22+32 )+ upto 10 terms is equal to :

  1. 130
  2. 155
  3. 3152
  4. 3252
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3

Let the sum of the first n terms of an A.P. be 3n2+5n. Then the sum of squares of the first 10 terms of the A.P. is:

  1. 10220
  2. 12860
  3. 15220
  4. 19780
JEE Main 2026 (Online) 5th April Morning Shift
4

n=110 ( 528n(n+1)(n+2) ) is equal to:

  1. 65
  2. 130
  3. 220
  4. 440
JEE Main 2026 (Online) 5th April Morning Shift
5

If the sum of the first 10 terms of the series 11+14 4+ 21+24 4 + 31+34 4+ 41+44 4+ is mn, (m, n)=1, then m+n is equal to :

  1. 256
  2. 264
  3. 276
  4. 284
JEE Main 2026 (Online) 5th April Evening Shift
6

Let A1, A2, A3, , A39 be 39 arithmetic means between the numbers 59 and 159. Then the mean of A25, A28, A31 and A36 is equal to :

  1. 129
  2. 136
  3. 131.50
  4. 134
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7

The first term of an A.P. of 30 non-negative terms is 103. If the sum of this A.P. is the cube of its last term, then its common difference is:

  1. 587
  2. 2583
  3. 1529
  4. 529
JEE Main 2026 (Online) 4th April Morning Shift
8

Let = 14+ 18+ 116+ and = 13+ 19+ 127+ . Then the value of (0.2) 5( )+(0.04) 5( ) is equal to :

  1. 4
  2. 5
  3. 8
  4. 25
JEE Main 2026 (Online) 4th April Evening Shift
9

Let A be the set of first 101 terms of an A.P., whose first term is 1 and the common difference is 5 and let B be the set of first 71 terms of an A.P., whose first term is 9 and the common difference is 7. Then the number of elements in A B, which are divisible by 3, is :

  1. 4
  2. 5
  3. 6
  4. 7
JEE Main 2026 (Online) 2nd April Morning Shift
10

If k=1n ak=6n3, then k=16 ( ak+1-ak36 )2 is equal to .

JEE Main 2026 (Online) 2nd April Morning Shift
11

Let a1, a2, a3, be an A.P. and g1=a1, g2, g3, be an increasing G.P. If a1=a2+g2=1 and a3+g3=4, then a10+g5 is equal to:

  1. 81
  2. 76
  3. 62
  4. 55
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