1If 11 is added to the square of a natural number, the sum is 36. Find the number.
2If 11 is subtracted from the square of a number, the remainder is 25. Find the number.
3If 7 is subtracted from double the square of a positive number, the remainder is 91. Find the number.
4If 2 is subtracted from the square of a natural number, the remainder is 7. Find the number.
5If 11 is subtracted from the square of a number and the remainder is 89, find the number.
6If 17 is subtracted from the square of a number, the remainder is 55. Find the number.
7If 3 is subtracted from double the square of a positive number, the remainder is 285. Find the number.
8If the sum of a number and its square is 72, find the number.
9If the product of two consecutive even numbers is 80, find the numbers.
10If the product of two consecutive odd numbers is 225, find the numbers.
11If the sum of a number and its reciprocal is \( \frac{10}{3} \), find the number.
12If the sum of two natural numbers is 21 and the sum of their squares is 261, find the numbers.
13If the age difference between two brothers is 4 years and the product of their ages is 221, find their ages.
14The sum of the present ages of two brothers is 22 and the product of their ages is 120. Find their present ages.
15The age difference between two sisters is 3 years and the product of their ages is 180. Find their present ages.
16The present ages of a father and his son are 40 years and 13 years. How many years ago was the product of their ages 198?
17The present ages of a mother and her daughter are 34 years and 4 years. After how many years will the product of their ages be 400?
18The present ages of a father and his son are 35 years and 1 year. After how many years will the product of their ages be 240?
19The present ages of a husband and wife are 35 years and 27 years. How many years ago was the product of their ages 425?
20The hypotenuse of a right-angled triangle is 25 m. If the difference between its other two sides is 17 m, find the lengths of those sides.
21The hypotenuse of a right-angled triangle is 6 m more than double its shortest side. If the remaining side is 2 m less than the hypotenuse, find the lengths of all the sides.
22Calculate the length and breadth of a rectangular land whose area is \( 150 \text{ m}^2 \) and perimeter is 50 m.
23Calculate the length and breadth of a rectangular land whose area is \( 54 \text{ m}^2 \) and perimeter is 30 m.
24Calculate the area of a rectangle whose length is 24 m and whose diagonal is 16 m more than its breadth.
25The area and perimeter of a rectangular land are \( 2000 \text{ m}^2 \) and 180 m. If the land is to be made a square, by what percentage should its length or breadth be reduced?
26A two-digit number is equal to four times the sum of its digits and also to three times the product of its digits. Find the number.
27An institute planned to share 180 pencils equally among the students enrolled in grade one. On the day, 5 students were absent, so each student got 3 more pencils. (a) How many students were enrolled in grade one? (b) How many pencils did each student receive in total?
28Three groups built volleyball courts. The first court has area \( 128 \text{ m}^2 \) and perimeter 48 m, the second \( 162 \text{ m}^2 \) and 54 m, the third \( 200 \text{ m}^2 \) and 60 m. Find the length and breadth of each and decide which court is suitable for playing volleyball.
29Why must the roots be checked against the situation after solving a quadratic word problem? Give an example.
30The length of a rectangle is 3 cm more than its breadth, and its area is \( 70 \text{ cm}^2 \). Find its length and breadth.