Advertisements
Advertisements
प्रश्न
| 200 logs are stacked in the following manner: 20 logs in the bottom row, 19 in the next row, 18 in the next row and so on. |
Based on the above information, answer the following questions:
1. In how many rows these 200 logs are placed?
(a) 25
(b) 20
(c) 16
(d) 14
2. The number of logs in the top row is:
(a) 1
(b) 5
(c) 3
(d) 8
3. The number of logs in the 8th row from the bottom is:
(a) 14
(b) 11
(c) 12
(d) 13
4. Total number of logs in the first six rows from the bottom is:
(a) 105
(b) 95
(c) 85
(d) 75
5. The number of logs in the 5th row from the top is:
(a) 8
(b) 9
(c) 7
(d) 10
Advertisements
उत्तर
1. The correct option is (c)
Reasoning:
Here, \[a = 20\], \[d = -1\] and \[S_n = 200\].
Using \[S_n = \frac{n}{2}[2a + (n - 1)d]\]:
\[200 = \frac{n}{2}[40 - (n - 1)\]
\[\implies 400 = n(41 - n)\]
\[\implies n^2 - 41n + 400 = 0\]
Factoring: \[(n - 16)(n - 25) = 0\].
If \[n = 25\], the number of logs in the 25th row would be \[a_{25} = 20 + 24(-1) = -4\], which is impossible.
Hence, \[n = 16\].
Options (a), (b) and (d) are rejected as \[n = 25\] yields negative logs, and 20 or 14 do not satisfy the quadratic equation.
2. The correct option is (b)
Reasoning:
The top row is the 16th row from the bottom.
Number of logs in 16th row: \[a_{16} = a + 15d = 20 + 15(-1) = 20 - 15 = 5\].
Options (a), (c) and (d) are incorrect counts for the 16th row.
3. The correct option is (d)
Reasoning:
Number of logs in the 8th row from the bottom is \[a_8 = a + 7d = 20 + 7(-1) = 13\].
Options (a), (b) and (c) result from miscalculating the term \[a + 7d\].
4. The correct option is (a)
Reasoning:
Total logs in first 6 rows: \[S_6 = \frac{6}{2}[2(20) + (6 - 1)(-1)] = 3[40 - 5] = 3 \times 35 = 105\].
Direct check: \[20 + 19 + 18 + 17 + 16 + 15 = 105\].
Options (b), (c) and (d) are incorrect sums.
5. The correct option is (b)
Reasoning:
There are 16 rows in total.
The 5th row from the top corresponds to row number \[16 - 5 + 1 = 12\text{th}\] row from the bottom.
Number of logs in the 12th row from the bottom: \[a_{12} = a + 11d = 20 + 11(-1) = 20 - 11 = 9\].
Alternatively, starting from top row with 5 logs and increasing by 1 per row: 1st from top = 5, 2nd = 6, 3rd = 7, 4th = 8, 5th = 9.
Options (a), (c) and (d) are off by one or more positions.
