জব সল্যুশন [৯ম - ১৩তম গ্রেড]

Question: A man rows at 6 km/hr in still water and 4.5 km/hr against current. His rate along the current is:

Solution:
Speed in still water is 6 km/hr
Speed of the current is x km/hr

against current,
6 - x = 4.5
⇒ x = 6 - 4.5
∴ x = 1.5

along the current,
6 + x
= 6 + 1.5
= 7.5 km/hr

লাইভ পরীক্ষার প্রশ্নে 'z²' এর পরিবর্তে 'x²' এর মান জানতে চাওয়া হয়েছিল। 
প্রশ্ন অনুযায়ী সঠিক উত্তর - ঙ) None of these. 
---------------------------- 

Question: x, y and z are consecutive positive integers such that x < y < z . If the units digit of x² is 6 and the units digit of y² is 9, what is the units digit of x²?

Solution:
Given that x, y, and z are consecutive positive integers such that x < y < z, we are asked to find the units digit of z², based on the following information:
The units digit of x² is 6.
The units digit of y² is 9.

First, let's analyze the possible values of x and y by checking what numbers, when squared, have the given units digits:

For x², the units digit is 6. The numbers whose squares end in 6 are:
4² = 16(units digit is 6) and 6² = 36(units digit is 6).
Thus, x could be either 4 or 6.

For y², the units digit is 9. The numbers whose squares end in 9 are:
3² = 9 (units digit is 9) and 7² = 49(units digit is 9).
Thus, y could be either 3 or 7.

Since x < y, we will now check the possible pairs of x and y:
If x = 4, the next integer y would be 5, but the units digit of 5² is 5, not 9. So this case is not valid.
If x = 6, the next integer y would be 7, and indeed, the units digit of 7² is 9.
Thus, the valid pair is x = 6 and y = 7.

Since z is the next consecutive integer after y = 7, we have z = 8.
Now,
we compute the units digit of z²
8² = 64. The units digit of z² is 4.