1.Given the equations (i) a² + 11a + 30 = 0 and (ii) b² + 6b + 5 = 0, determine the relationship between the values of a and b.
2.Given the equations (i) a² - 9a + 20 = 0 and (ii) 2b² - 5b - 12 = 0, determine the correct relationship between the values of a and b.
3.Given the quadratic equations (i) a² - 7a + 12 = 0 and (ii) b² - 3b + 2 = 0, what can be concluded about the relationship between the values of a and b?
4.A man can purchase a certain number of notebooks for Rs. 300. If the price of each notebook increases by Rs. 5, he would be able to buy 10 fewer notebooks for the same total amount. What is the original price of each notebook?
5.Determine the quadratic equation whose solutions are the reciprocals of the roots of the equation 2x² + 5x + 3 = 0.
6.If a and b are the roots of the quadratic equation x² + 8x + 4 = 0, what is the value of a/b plus b/a?
7.Given that a and b are the solutions of the quadratic equation x² - 9x + 20 = 0, what is the value of a² + b² + ab?
8.The sum of the squares of three consecutive even natural numbers equals 1460. What are these numbers?
9.A quadratic equation x² - 12x + a = 0 has roots where one root is three times the other. What is the value of a?
10.Two consecutive positive integers have the property that the sum of their squares is 91 more than their product. What are these integers?
11.Given the quadratic equation 2x² - 5x + b = 0, if its roots are in the ratio 2:3, what is the value of b?
12.What are the sum and product of the roots for the quadratic equation x² + 20x + 3 = 0?
13.Given that the roots of a quadratic equation are 20 and -7, what is the corresponding quadratic equation?
14.What type of roots does the quadratic equation 3x² - 12x + 10 = 0 have?
15.Determine the solutions for the quadratic equation: 2x² + 3x - 9 = 0.
16.What are the solutions to the quadratic equation x² + 2x - 15 = 0?
17.A carpet was ordered with its length and width in the ratio 3:2. Later, the dimensions were changed so that the length and width were in the ratio 7:3, while the perimeter remained unchanged. What is the ratio of the areas of the carpet before and after the modification?
18.A circle has a radius of 21 cm and a central angle measuring 135°. What is the perimeter of the corresponding sector?
19.A rectangle's length is two-fifths of the radius of a circle. The radius of this circle is equal to the side length of a square whose area measures 1225 square units. If the rectangle's breadth is 10 units, what is its area in square units?
20.Calculate the area of a rhombus with each side measuring 25 cm and one diagonal equal to 30 cm.