Find the values of the letters in each of the following and give reasons for the steps involved. \(\begin{matrix}\quad 1\ A \\ \times \ \quad A \\ \_ \_ \_ \_ \_ \_ \_ \\ \quad 9\ A \\ \_ \_ \_ \_ \_ \_ \_ \_ \end{matrix}\)

Find the values of the letters in each of the following and give reasons for the steps involved.\(\begin{matrix} \quad \quad A\ B \\ +\quad 3\ \ 7 \\ \_ \_ \_ \_ \_ \_ \_ \\ \quad 6\ A \\ \_ \_ \_ \_ \_ \_ \_ \_ \end{matrix}\)

If from a 2-digit number, we subtract the number formed by reversing its digits, then the result so obtained is a perfect cube. How many such numbers are possible? Write all of them.

More about Pythagorean triplets We have seen one way of writing Pythagorean triplets as 2m, m² - 1, m² + 1. A Pythagorean triplet a, b, c means a² + b² = c2. If we use two natural numbers m and nim. > n), and take a = m² - n², b = 2mn, c = m² + n², then we can see that c² = a² + b². Thus for different values of m and n with m > n we can generate natural numbers a, b, c such that they form Pythagorean triplets. For example: Take, m = 2, n = 1. Then, a = m² - n² = 3, b = 2mn = 4, c = m² + n² = 5, is a Pythagorean triplet. (Check it !) For, m = 3, n = 2, we get a = 5, b = 12, c = 13 which is again a Pythagorean triplet. Take some more values for m and nand generate more such triplets.

When water freezes its volume increases by 4%. What volume of water is required to make 221 cm³ of ice?

Ceremony Awards began in 1958. There were 28 categories to win an award. In 1993, there were 81 categories.
(i) The awards given in 1958 is what per cent of the awards given in 1993?
(ii) The awards given in 1993 is what per cent of the awards given in 1958?

Out of a swarm of bees, one fifth settled on a blossom of Kadamba, one third on a flower of Silindhiri, and three times the difference between these two numbers flew to the bloom of Kutaja. Only ten bees were then left from the swarm. What was the number of bees in the swarm? (Note, Kadamba, Silindhiri and Kutaja are flowering trees. The problem is from the ancient Indian text on algebra.)

In computing the area of a square. Shekhar used the formula for area of a square, while his friend Maroof used the formula for the perimeter of a square. Interestingly their answers were numerically same. Tell me the number of units of the side of the square they worked on.

The area of a square is numerically less than six times its side. List some squares in which this happens.

Is it possible to have a right circular cylinder to have volume numerically equal to its curved surface area? If yes state when.