Chapter 4 Quadratic Equations Ex 4.3

Chapter 4 Quadratic Equations Ex 4.3

Question 1.
Find the roots of the following quadratic equations, if they exist, by the method of completing the square:
(i) 2x² – 7x + 3 = 0
(ii) 2x² + x – 4 = 0
(iii) 4x² + 4√3x + 3 = 0
(iv) 2x² + x + 4 = 0
Solution:

Question 2.
Find the roots of the quadratic equations by applying the quadratic formula.
(i) 2x² – 7x + 3 = 0
(ii) 2x² – x + 4 = 0
(iii) 4x² – 4√3x + 3 = 0
(iv) 2x² – x + 4 = 0
Solution:
(i) 2x² – 7x + 3 = 0
This is of the form ax² + bx + c = 0,
where a = 2, b = -7 and c = 3

(ii) 2x² – x + 4 = 0
This is of the form ax² + bx + c = 0,
where a = 2, b = 1 and c = – 4

(iii) 4x² – 4√3x + 3 = 0

(iv) 2x² – x + 4 = 0

Question 3.
Find the roots of the following equations:


Solution:

Question 4.

Solution:
Let the present age of Rehman be x years
3 years ago Rehman’s age was = (x – 3) years
5 years from now Rehman’s age will be = (x + 5) years

Question 5.
In a class test, the sum of Shefali’s marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product of their marks would have been 210. Find her marks in the two subjects.
Solution:
Let the marks secured by Shefali in Mathematics = x Then,
According to question,

Question 6.
The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field.
Solution:
Let the shorter side of rectangle = x m
Then, longer side = (x + 30) m and diagonal = (x + 60) m
In ΔABC, (x + 60)² = (x + 30)² + (x)² [Pythagoras Theorem]

Question 7.
The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.
Solution:
Let the smaller number = x
and the larger number = y
According to question,

Question 8.
A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.
Solution:
Total distance travelled = 360 km
Let uniform speed be x km/h
Then, increased speed = (x + 5) km/h
According to question,

Question 9.

Solution:
Let larger pipe fills the tank in x hours and the smaller pipe fills the tank in y hours.
The tank filled by the larger pipe in 1 hour = 1x

Question 10.
An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bengaluru (without taking into consideration the time they stop at intermediate stations). If the average speed of the express train is 11 km/h more than that of the passenger train, find the average speed of the two trains.
Solution:
Let the speed of passenger train = x km/h
Then, the speed of express train be (x + 11) km/h
According to question,

Question 11.
Sum of the areas of two squares is 468 m². If the difference of their perimeters is 24 m, find the sides of the two squares.
Solution:
Let the sides of two squares be x m and y m
According to question,