Find two consecutive odd positive integers, sum of whose squares is 394 .

Let one of the number be x which is an odd positive integer then the other number be x + 2 . Then according to question, we have x 2 + x + 2 2 = 394 ⇒ x 2 + x 2 + 4 x + 4 = 394 ⇒ 2 x 2 + 4 x - 390 = 0 ⇒ x 2 + 2 x - 195 = 0 ⇒ x 2 + 15 x - 13 x - 195 = 0 ⇒ x x + 15 - 13 x + 15 = 0 ⇒ x - 13 x + 15 = 0 ⇒ x - 13 = 0 or x + 15 = 0 ⇒ x = 13 [ x ≠ - 15 , since we have let x to be is an odd positive integer] The other number is x + 2 = 13 + 2 = 15 Hence, the required numbers are 13 , and 15

The difference of the squares of two positive numbers is 45 . The square of the smaller number is four times the larger number. Find the numbers.

Consider, the greater number be x and the smaller number be y . According to the question: x 2 - y 2 = 45 … … … ( 1 ) y 2 = 4 x … … … ( 2 ) From ( 1 ) and ( 2 ) , we get: x 2 - 4 x = 45 ⇒ x 2 - 4 x - 45 = 0 ⇒ x 2 - 9 x + 5 x - 45 = 0 ⇒ x ( x - 9 ) + 5 ( x - 9 ) = 0 ⇒ ( x - 9 ) ( x + 5 ) = 0 ⇒ x - 9 = 0 or x + 5 = 0 ⇒ x = 9 ( ∵ x is a natural number) Putting the value of x in equation ( 2 ) , we get: y 2 = 36 ⇒ y = 6 Hence, the required numbers are x = 9 and y = 6 .

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10th ICSE

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At an annual function of a school, each student gives gift to every other student. If the number of gifts is $1980$ , find the number of students.

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Important Questions on Quadratic Equations in One Variable

MEDIUM

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 5 - Quadratic Equations in One Variable>Exercise 5.5>Q 25

A bus covers a distance of

240 km

at a uniform speed. Due to heavy rain its speed gets reduced by

10 km / h

and as such it takes two hrs longer to cover the total distance. Assuming the uniform speed to be

' x' km / h,

form an equation and solve it to evaluate

' x' .

HARD

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 5 - Quadratic Equations in One Variable>Exercise 5.5>Q 26

The speed of an express train is

x km / h

and the speed of an ordinary train is

12 km / h

less than that of the express train. If the ordinary train takes one hour longer than the express train to cover a distance of

240 km

, find the speed of the express train.

HARD

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 5 - Quadratic Equations in One Variable>Exercise 5.5>Q 27

A car covers a distance of

400 km

at a certain speed. Had the speed been

12 km / h

more, the time taken for the journey would have been

1

hour

40

minutes less. Find the original speed of the car.

MEDIUM

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 5 - Quadratic Equations in One Variable>Exercise 5.5>Q 28

An aeroplane covered a distance of at an average speed of

x km / h

. On the return journey, the speed was increased by

40 km / h

. Write down an expression for the time taken for: the onward journey. If the return journey took

30

minutes less than the onward journey, write down an equation in

x

and find its value.

MEDIUM

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 5 - Quadratic Equations in One Variable>Exercise 5.5>Q 28

An aeroplane covered a distance of at an average speed of

x km / h

. On the return journey, the speed was increased by

40 km / h

. Write down an expression for the time taken for: the return journey. If the return journey took

30

minutes less than the onward journey, write down an equation in

x

and find its value.

HARD

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 5 - Quadratic Equations in One Variable>Exercise 5.5>Q 29

The distance by road between two towns

A

and

B

, is

216 km

, and by rail it is

208 km

. A car travels at a speed of

x km / h

, and the train travels at a speed which is

16 km / h

faster than the car. Calculate: The time taken by the car, to reach town

B

from

A

, in terms of

x

MEDIUM

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 5 - Quadratic Equations in One Variable>Exercise 5.5>Q 29

The distance by road between two towns

A

and

B

, is

216 km

, and by rail it is

208 km

. A car travels at a speed of

x km / h

, and the train travels at a speed which is

16 km / h

faster than the car. Calculate: The time taken by the train, to reach town

B

from

A

, in terms of

x

MEDIUM

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 5 - Quadratic Equations in One Variable>Exercise 5.5>Q 29

The distance by road between two towns

A

and

B

, is

216 km

, and by rail it is

208 km

. A car travels at a speed of

x km / h

, and the train travels at a speed which is

16 km / h

faster than the car. Calculate: If the train takes

2

hours less than the car, to reach town

B

, obtain an equation in

x

, and solve it.

At an annual function of a school, each student gives gift to every other student. If the number of gifts is 1980, find the number of students.

Important Questions on Quadratic Equations in One Variable

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At an annual function of a school, each student gives gift to every other student. If the number of gifts is $1980$ , find the number of students.