প্রশ্ন: a2 - b2 + 2b - 1 এর একটি উৎপাদক-

সমাধান:
a2 - b2 + 2b - 1
= a2 - (b2 - 2b + 1)
= a2 - (b - 1)2
= {a + (b - 1)}{a - (b - 1)}
= (a + b - 1)(a - b + 1)

Question: In a mixture 60 litres, the ratio of milk and water 2 : 1. If this ratio is to be 1 : 2, then the quantity of water to be further added is-

Solution:
Quantity of milk = 60 × (2/3) litres = 40 litres.
Quantity of water in it = (60 -  40) litres = 20 litres.
New ratio = 1 : 2
Let quantity of water to be added further be x litres.
Then,
milk : water = 40/(20 + x)
Now,
40/(20 + x) = 1/2
⇒ 20 + x = 80
∴ x = 60.
Quantity of water to be added = 60 litres.

প্রশ্ন: x + 1/x = 2 হলে x4 + 1/x4 এর মান কত?

সমাধান:
দেওয়া আছে,
x + (1/x) = 2
⇒ {x + (1/x)}2 = 22 [বর্গ করে] [(a + b)2 = a2 + 2ab + b2]
⇒ x2 + 2 × x × (1/x) + (1/x)2 = 4
⇒ x2 + (1/x2) = 4 - 2
⇒ {x2 + (1/x2)}2 = 22 [আবার বর্গ করে]
⇒ (x2)2 + 2 × x2 × (1/x2) + (1/x2)}2 = 4
⇒ x4 + 1/x4 = 4 - 2
∴ x4 + 1/x4 = 2