maths problems
Basic forms of quadratic equations
1. Standard form of quadratic equation
ax² + bx + c = 0
2. Quadratic Formula (roots nikalne ka general formula)
x = [ -b ± √(b^2 - 4ac) ] / (2a)
3. Discriminant (D) ka formula
D = b² - 4ac
4. Nature of roots
• Agar D > 0 → Do alag-alag real roots
• Agar D = 0 → Do equal real roots
• Agar D < 0 → No real roots
5. Sum of roots (alpha + beta)
alpha + beta = -b / a
6. Product of roots (alpha × beta)
alpha × beta = c / a
7. Equation from given roots
x²- (alpha + beta)x + alpha × beta = 0
Q1. The product of two consecutive positive integers is 132.Find the integers.
Q2. The sum of the squares of two consecutive natural numbers is 365.Find the numbers.
Q3. The area of a rectangular garden is 48 square meters.Its length is 4 meters more than its width.
Find the length and width of the garden.
Q4. A train covers a distance of 120 km at a certain speed.If the speed were 10 km per hour more, it would take 1
hour less.
Find the original speed of the train.
Q5. The product of two positive numbers is 60 and their sum is 17.Find the numbers.
Q6. A motorboat travels 30 km upstream and 30 km downstream.The speed of the current is 2 km per hour and the
total time taken is 8 hours.
Find the speed of the boat in still water.
Q7. A square has area 169 square units.
Find the length of each side using a quadratic equation.
Q8. The difference between the squares of two consecutive even numbers is 52.
Find the numbers.
Q9. The age of a father is 4 times the age of his son.After 5 years, the product of their ages will be 375.Find their
present ages.
Q10. A rectangular field has perimeter 64 meters and area 240 square meters.
Find its length and width.
, Q11. The sum of the reciprocals of two consecutive positive integers is 11 over 30.
Find the integers.
Q12. The product of the ages of two sisters is 336.
After 4 years, the product of their ages will be 480.
Find their present ages.
Q13. A rectangular plot has an area of 528 square meters.If the length is 8 meters more than twice the width, find the
dimensions.
Q14. A cyclist covers 90 km at a certain speed.
If he increases his speed by 5 km per hour, he takes 1 hour less.Find his original speed.
Q15. The sum of two positive numbers is 20.
The sum of their squares is 208.Find the numbers.
Q16. A boat travels 40 km downstream and 40 km upstream in 9 hours.The speed of the stream is 3 km per hour.Find
the speed of the boat in still water.
Q17. The area of a square playground is 784 square meters.If its side is increased by 4 meters, find the increase in
area.
Q18. The difference between two positive numbers is 4 and their product is 165.
Find the numbers.
Q19. A train travels 360 km at a uniform speed.
If the speed was 15 km per hour more, it would take 2 hours less.
Find the original speed.
Q20. The sum of the areas of two squares is 468 square units.If the side of the larger square is 6 units more than the
side of the smaller square,
find the sides of both squares.
Basic forms of quadratic equations
1. Standard form of quadratic equation
ax² + bx + c = 0
2. Quadratic Formula (roots nikalne ka general formula)
x = [ -b ± √(b^2 - 4ac) ] / (2a)
3. Discriminant (D) ka formula
D = b² - 4ac
4. Nature of roots
• Agar D > 0 → Do alag-alag real roots
• Agar D = 0 → Do equal real roots
• Agar D < 0 → No real roots
5. Sum of roots (alpha + beta)
alpha + beta = -b / a
6. Product of roots (alpha × beta)
alpha × beta = c / a
7. Equation from given roots
x²- (alpha + beta)x + alpha × beta = 0
Q1. The product of two consecutive positive integers is 132.Find the integers.
Q2. The sum of the squares of two consecutive natural numbers is 365.Find the numbers.
Q3. The area of a rectangular garden is 48 square meters.Its length is 4 meters more than its width.
Find the length and width of the garden.
Q4. A train covers a distance of 120 km at a certain speed.If the speed were 10 km per hour more, it would take 1
hour less.
Find the original speed of the train.
Q5. The product of two positive numbers is 60 and their sum is 17.Find the numbers.
Q6. A motorboat travels 30 km upstream and 30 km downstream.The speed of the current is 2 km per hour and the
total time taken is 8 hours.
Find the speed of the boat in still water.
Q7. A square has area 169 square units.
Find the length of each side using a quadratic equation.
Q8. The difference between the squares of two consecutive even numbers is 52.
Find the numbers.
Q9. The age of a father is 4 times the age of his son.After 5 years, the product of their ages will be 375.Find their
present ages.
Q10. A rectangular field has perimeter 64 meters and area 240 square meters.
Find its length and width.
, Q11. The sum of the reciprocals of two consecutive positive integers is 11 over 30.
Find the integers.
Q12. The product of the ages of two sisters is 336.
After 4 years, the product of their ages will be 480.
Find their present ages.
Q13. A rectangular plot has an area of 528 square meters.If the length is 8 meters more than twice the width, find the
dimensions.
Q14. A cyclist covers 90 km at a certain speed.
If he increases his speed by 5 km per hour, he takes 1 hour less.Find his original speed.
Q15. The sum of two positive numbers is 20.
The sum of their squares is 208.Find the numbers.
Q16. A boat travels 40 km downstream and 40 km upstream in 9 hours.The speed of the stream is 3 km per hour.Find
the speed of the boat in still water.
Q17. The area of a square playground is 784 square meters.If its side is increased by 4 meters, find the increase in
area.
Q18. The difference between two positive numbers is 4 and their product is 165.
Find the numbers.
Q19. A train travels 360 km at a uniform speed.
If the speed was 15 km per hour more, it would take 2 hours less.
Find the original speed.
Q20. The sum of the areas of two squares is 468 square units.If the side of the larger square is 6 units more than the
side of the smaller square,
find the sides of both squares.
