CAT 2007 — QA Question 10
Answer the next 2 questions based on the information given below.
Cities A and B are in different time zones. A is located 3000 km east of B. The table below describes the schedule of an airline operating non-stop flights between A and B. All the times indicated are local and on the same day.
Assume that planes cruise at the same speed in both directions. However, the effective speed is influenced by a steady wind blowing from east to west at 50 km per hour.
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Consider all four digit numbers for which the first two digits are equal and the last two digits are also equal. How many such numbers are perfect squares?
Answer & solution
- A
3
- B
2
- C
4
- D
0
1
Let aabb (a ≠ 0, a and b being single digits) be a perfect square.
⇒ aabb = 1000a + 100a + 10b + b = 1100a + 11b = 11(100a + b)
Also, as aabb is a perfect square, it has to be a multiple of 121.
∴ aabb = 121K, where K is also perfect square.
For K = 4, aabb is a 3 digit number, while for K > 82, K is a 5 digit number.
For 81 ≥ K ≥ 9,
121 × 9 = 1089
121 × 16 = 1936
121 × 25 = 3025
121 × 36 = 4356
121 × 49 = 5929
121 × 64 = 7744
121 × 81 = 9801
∴ There is only one number 7744 of the form aabb, which is a perfect square.
Hence, option (e).