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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If the perimeter of a rectangular plot is $68 m$ and length of its diagonal is $26 m$ , If its area is $k m^{2}$ , then find the value of $k$ .

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Important Questions on Quadratic Equations in One Variable

HARD

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 5 - Quadratic Equations in One Variable>Exercise 5.5>Q 19

If the sum of two smaller sides of a right-angled triangle is

17 cm

and the perimeter is

30 cm

, if the area of the triangle is

k m^{2}

, then find the value of

k

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10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 5 - Quadratic Equations in One Variable>Exercise 5.5>Q 20

The hypotenuse of a grassy land in the shape of a right triangle is

1

metre more than twice the shortest side. If the third side is

7

metres more than the shortest side, find the sides of the grassy land.

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10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 5 - Quadratic Equations in One Variable>Exercise 5.5>Q 21

Mohini wishes to fit three rods together in the shape of a right triangle. If the hypotenuse is

2 cm

longer than the base and

4 cm

longer than the shortest side, find the lengths of the rods.

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10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 5 - Quadratic Equations in One Variable>Exercise 5.5>Q 22

In a P.T. display,

480

students are arranged in rows and columns. If there are

4

more students in each row than the number of rows, find the number of students in each row.

HARD

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 5 - Quadratic Equations in One Variable>Exercise 5.5>Q 23

In an auditorium, the number of rows was equal to the number of seats in each row. If the number of rows is doubled and the number of seats in each row is reduced by

5

, then the total number of seats is increased by

375

. How many rows were there?

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Understanding ICSE Mathematics Class 10>Chapter 5 - Quadratic Equations in One Variable>Exercise 5.5>Q 24

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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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.

If the perimeter of a rectangular plot is 68 m and length of its diagonal is 26 m, If its area is k m2, then find the value of k.

Important Questions on Quadratic Equations in One Variable

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If the perimeter of a rectangular plot is $68 m$ and length of its diagonal is $26 m$ , If its area is $k m^{2}$ , then find the value of $k$ .