## Chapter 10 – Quadratic Equations Exercise Ex. 10A

Which of the following are quadratic equations in x?

(i)_{}

(i)_{}is a quadratic polynomial

_{}= 0 is a quadratic equation

Which of the following are quadratic equations in x?

_{}

Clearly _{} is a quadratic polynomial

_{}is a quadratic equation.

Which of the following are quadratic equations in x?

_{} is a quadratic polynomial

_{ }= 0 is a quadratic equation

Clearly, is a quadratic equation

_{}is a quadratic equation

_{}is not a quadratic polynomial since it contains_{} in which power _{} of x is not an integer.

_{}is not a quadratic equation

And Being a polynomial of degree 2, it is a quadratic polynomial.

Hence, _{}is a quadratic equation.

And _{}being a polynomial of degree 3, it is not a quadratic polynomial

Hence, is not a quadratic equation

_{}is not a quadratic equation

Which

of the following are quadratic equations in x?

Which

of the following are quadratic equations in x?

Which

of the following are quadratic equations in x?

Which of the following are the roots of _{}

(i) -1

(ii) _{}

(iii) _{}

The given equation is _{}

(i) On substituting x = -1 in the equation, we get

_{}

(ii) On substituting _{} in the equation, we get

_{}

(iii) On substituting _{} in the equation _{}, we get

_{}

Find the value of k for which x = 1 is a root of the equation _{}

Since x = 1 is a solution of _{}it must satisfy the equation.

_{}

Hence the required value of k = -4

Find the value of a and b for which _{} and x = -2 are the roots of the equation _{}

Since _{} is a root of _{}, we have_{}

_{}

Again x = -2 being a root of _{}, we have

_{}

Multiplying (2) by 4 adding the result from (1), we get

11a = 44 _{} a = 4

Putting a = 4 in (1), we get

_{}

Solve each of the following quadratic equations:

3_{}– 243 = 0

_{}

Hence, 9 and -9 are the roots of the equation _{}

Solve the following quadratic equation:

_{}

_{}

Hence, _{}are the roots of _{}

Solve the following quadratic equation:

_{}

_{}

Hence, _{}are the roots of equation_{}

Solve the following quadratic equation:

_{}

_{}

Hence, _{}and 1 are the roots of the equation _{}.

Solve the following quadratic equation:

_{}

_{}

_{}are the roots of the equation _{}

Solve the following quadratic equation:

_{}

_{}

Hence, _{}are the roots of the given equation _{}

Solve the following quadratic equation:

_{}

_{}

Hence, _{}are the roots of given equation

Solve the following quadratic equation:

_{}

_{}

Hence, _{}are the roots of _{}

Solve the following quadratic equation:

_{}

_{}

Hence,_{ }are the roots of the given equation

Solve the following quadratic equation:

_{}

_{}

Hence, _{}are the roots of the given equation

Solve the following quadratic equation:

_{}

_{}

Hence, _{}are the roots of the given equation

Solve the following quadratic equation:

_{}

_{}

Hence, _{}are the roots of the given equation

Solve the following quadratic equation:

_{}

_{}

Hence, 1 and _{}are the roots of the given equation

Solve the following quadratic equation:

_{}

_{}

Hence, _{}are the roots of the given equation

Solve the following quadratic equation:

_{}

_{}

Hence, 2 and _{}are the roots of given equation

Solve the following quadratic equation:

_{}

_{}

Hence, _{}are the roots of the given equation

Solve the following quadratic equation:

_{}

_{}

Hence, _{}are the roots of given equation

Solve the following quadratic equation:

_{}

_{}

Hence, _{}are the roots of given equation

Solve the following quadratic equation:

_{}

_{}

Hence, _{}are the roots of given equation

Solve the following quadratic equation:

_{}

Putting _{}the given equation become

_{}

Case I:

_{}

Case II:

_{}

Hence, _{}are the roots of the given equation

Solve the following quadratic equation:

_{}

The given equation

_{}

Hence, _{}is the roots of the given equation

Solve the following quadratic equation:

_{}

_{}

Hence, -2,0 are the roots of given equation

Solve the following quadratic equation:

_{}

_{}

Hence, _{}are the roots of given equation

Solve the following quadratic equation:

_{}

_{}

Hence, 3 and 2 are roots of the given equation

## Chapter 10 – Quadratic Equations Exercise Ex. 10B

## Chapter 10 – Quadratic Equations Exercise Ex. 10C

## Chapter 10 – Quadratic Equations Exercise Ex. 10D

Show that the roots of the equation _{}are real for all real values of p and q.

The given equation is _{}

This is the form of _{}

_{}

Now _{}.

So, the roots of the given equation are real for all real value of p and q.

For what values of k are the roots of the quadratic equation _{}real and equal.

_{}

If the equation _{}has equal roots , prove that _{}

The given equation is _{}

_{}

For real and equal roots, we must have D = 0

_{}

If the roots of the equation _{}are real and equal, show that either a = 0 or _{}

The given equation is _{}

_{}

For real and equal roots , we must have D =0

_{}

## Chapter 10 – Quadratic Equations Exercise Ex. 10E

The sum of a natural number and its square is 156. Find the number.

The sum of a natural number and its positive square root is 132. Find the number.

The sum of two natural numbers is 28 and their product is 192. Find the numbers.

The sum of the squares of two consecutive positive integers is 365. Find the integers.

The sum of the squares of two consecutive positive odd numbers is 514. Find the numbers.

The sum of the squares of two consecutive positive even numbers is 452. Find the numbers.

The product of two consecutive positive integers is 306. Find the integers.

Two numbers differ by 3 and their product is 504. Find the numbers.

Let the required number be x and x – 3, then

_{}

Hence, the required numbers are (24,21) or (-21 and-24)

Find two consecutive multiples of 3 whose product is 648.

Find two consecutive positive odd integers whose product is 483.

Find two consecutive positive even integers whose product is 288.

The sum of the squares of two consecutive multiples of 7 is 1225. Find the multiples.

Divide 57 into two parts whose product is 680.

Divide 16 into two parts such that twice the square of the larger part exceeds the square of the smaller part by 164.

Let the smaller part and larger part be x, 16 – x

Then,

_{}

-42 is not a positive part

Hence, the larger and smaller parts are 10, 6 respectively

Find two natural numbers, the sum of whose squares is 25 times their sum and also equal to 50 times their difference.

Let the required number be x and y, hen

_{}

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

Let x, y be the two natural numbers and x > y

_{}——(1)

Also, square of smaller number = 4 larger number

_{}———(2)

Putting value of _{}from (1), we get

Thus, the two required numbers are 9 and 6

Three consecutive positive integers are such that the sum of the square of the first and the product of the other two is 46, find the integers.

Let the three consecutive numbers be x, x + 1, x + 2

Sum of square of first and product of the other two

_{}

_{}

_{}Required numbers are 4, 5 and 6

A two-digit number is 4 times the sum of its digits and twice the product of its digits. Find the numbers.

Let the tens digit be x and units digit be y

_{}

Hence, the tens digit is 3 and units digit is (2 3) is

Hence the required number is 36

A two-digit number is such that the product of its digits is 14. If 45 is added to the number, the digits interchange their places. Find the number.

Let the tens digit and units digits of the required number be x and y respectively

_{}

The ten digit is 2 and unit digit is 7

Hence, the required number is 27

The denominator of a fraction is 3 more than its numerator. The sum of the fraction and its reciprocal is_{}. Find the fraction.

Let the numerator and denominator be x, x + 3

Then,

_{}

Hence, numerator and denominator are 2 and 5 respectively and fraction is _{}

A teacher on attempting to arrange the students for mass drill in the form of a solid square found that 24 students were left. When he increased the size of the square by one student, he found he was short of 25 students. Find the number of students.

Let there be x rows and number of student in each row be x

Then, total number of students = _{}

_{}

Hence total number of student

_{}

Total number of students is 600

300 apples are distributed equally among a certain number of students. Had there been 10 more students, each would have received one apple less. Find the number of students.

Let the number of students be x, then

_{}

_{}

Hence the number of students is 50

In a class test, the sum of Kamal’s marks in Mathematics and English is 40. Had he got 3 marks more in Mathematics and 4 marks less in English, the product of the marks would have been 360. Find his marks in two subjects separately.

Let the marks obtained by Kamal in Mathematics and English be x and y

_{}

The marks obtained by Kamal in Mathematics and English respectively are (21,19) or (12,28)

Some students planned a picnic. The total budget for food was Rs.2000. But, 5 students failed to attend the picnic and thus the cost for food for each member increased by Rs.20. How many students attended the picnic and how much did each student pay for the food?

If the price of a book is reduced by Rs.5, a person can buy 4 more books for Rs.600. Find the original price of the book.

A person on tour has Rs.10800 for his expenses. If he extends his tour by 4 days, he has to cut down his daily expenses by Rs.90. Find the original duration of the tour.

In a class test, the sum of the marks obtained by P in mathematics and science is 28. Had he got 3 more marks in mathematics and 4 marks less in science, the product of marks obtained in the two subjects would have been 180. Find the marks obtained by him in the two subjects separately.

A man buys a number of pens for Rs.180. If he had bought 3 more pens for the same amount, each pen would have cost him Rs.3 less. How many pens did he buy?

A dealer sells an article for Rs.75 and gains as much per cent as the cost price of the article. Find the cost price of the article.

One year ago, a man was 8 times as old as his son. Now his age is equal to the square of his son’s age. Find their present ages.

Let the age of son be x and age of man = y

1 year ago

_{}

The sum of the ages of a boy and his brother is 25 years and the product of their ages in years is 126.Find their ages.

The product of Meena’s age 5 years ago and her age 8 years later is 30. Find her present age.

Let the present age of Meena be x

Then,

Hence the present age of Meena is 7 years

Two years ago, a man’s age was three times the square of his son’s age. In three years time, his age will be four times his son’s age. Find their present ages.

A truck covers a distance of 150 km at a certain average speed and then covers another 200 km at an average speed which is 20 km per hour more than the first speed. If the truck covers the total distance in 5 hours, find the first speed of the truck.

While boarding an aeroplane, a passenger got hurt. The pilot showing promptness and concern, made arrangements to hospitalise the injured and so the plane started late by 30 minutes. To reach the destination, 1500 km away, in time, the pilot increased the speed by 100 km/hour. Find the original speed of the plane.

Do you appreciate the values shown by the pilot, namely promptness in providing help to the injured and his efforts to reach in time?

A train covers a distance of 480 km at a uniform speed. If the speed had been 8 km/hr less then it would have taken 3 hours more to cover the same distance. Find the usual speed of the train.

A train travels at a certain average speed for a distance of 54 km and then travels a distance of 63 km at an average speed of 6 km/hr more than the first speed. If it takes 3 hours to complete the total journey, what is its first speed?

A train travels 180 km at a uniform speed. If the speed had been 9 km/hr more, it would have taken 1 hour less for the same journey. Find the speed of the train.

A train covers a distance of 90 km at a uniform speed. Had the speed been 15 kmph more, it would have taken 30 minutes less for the journey. Find the original speed of the train.

Let the original speed of the train be x km.hour

Then speed increases by 15 km/ph = (x + 15)km/hours

Then time taken at original speed = _{}

Then, time taken at in increased speed = _{}

_{}Difference between the two lines taken _{}

_{}

Then, original speed of the train = 45km / h

A passenger train takes 2 hours less for a journey of 300 km if its speed is increased by 5 km/hr from its usual speed. Find its usual speed.

The distance between Mumbai and Pune is 192 km. Travelling by the Deccan Queen, it takes 48 minutes less than another train. Calculate the speed of the Deccan Queen if the speeds of the two trains differ by 20 kmph.

Let the speed of the Deccan Queen = x kmph

The, speed of other train = (x – 20)kmph

Then, time taken by Deccan Queen = _{}

Time taken by other train = _{}

Difference of time taken by two trains is

_{}

Hence, speed of Deccan Queen = 80km/h

A motorboat whose speed is 18 km per hour in still water takes 1 hour more to go 24 km upstream than to return to the same point. Find the speed of the stream.

Let the speed of stream be x km/h

Speed of boat in still stream = 18 km/h

Speed of boat up the stream = 18 – x km/h

_{}Time taken by boat to go up the stream 24 km = _{}

_{}Time taken by boat to go down the stream = _{}

Time taken by the boat to go up the stream is 1 hour more that the time taken down the stream

_{}

_{}Speed of the stream = 6 km/h

The speed of a boat in still water is 8 kmph. It can go 15 km upstream and 22 km downstream in 5 hours. Find the speed of the stream.

Let the speed of the stream be x kmph

Then the speed of boat down stream = (8 + x) kmph

And the speed of boat upstream = (8 – x)kmph

Time taken to cover 15 km upstream = _{}

Time taken to cover 22 km downstream = _{}

Total time taken = 5 hours

_{}

Hence, the speed of stream is 3 kmph

A motorboat whose speed is 9 km/hr in still water, goes 15 km downstream and comes back in a total time of 3 hours 45 minutes. Find the speed of the stream.

Let the speed of the stream be = x km/h

Speed of boat in still waters = 9 km/h

Speed of boat down stream = 9 + x

_{}time taken by boat to go 15 km downstream = _{}

Speed of boat upstream = 9 – x

_{}time taken by boat to go 15 km of stream = _{}

_{}

A takes 10 days less than time taken by B to finish a piece of work. If both A and B together can finish the work in 12 days, find the time taken by B to finish the work.

Two pipes running together can fill a cistern in_{}minutes. If one pipe takes 3 minutes more than the other to fill it, find the time in which each pipe would fill the cistern.

Let the faster pipe takes x minutes to fill the cistern

Then, the other pipe takes (x + 3) minute

_{}

The faster pipe takes 5 minutes to fill the cistern

Then, the other pipe takes (5 + 3) minutes = 8 minutes

Two water taps together can fill a tank in 6 hours. The tap of larger diameter takes 9 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

The length of rectangle is twice its breadth and its area is 288 sq.cm. Find the dimensions of the rectangle.

Let the breadth of a rectangle = x cm

Then, length of the rectangle = 2x cm

_{}

Thus, breadth of rectangle = 12 cm

And length of rectangle = (2 12) = 24 cm

The length of a rectangular field is three times its breadth. If the area of the field be 147 sq.m, find the length of the field.

Let the breadth of a rectangle = x meter

Then, length of rectangle = 3x meter

_{}

Thus, breadth of rectangle = 7 m

And length of rectangle = (3 7)m = 21 m

The length of a hall is 3 metres more than its breadth. If the area of the hall is 238 sq.m, calculate its length and breadth.

Let the breadth of hall = x meters

Then, length of the hall = (x + 3) meters

_{}Area = length breadth = _{}

_{}

Thus, the breadth of hall is 14 cm

And length of the hall is (14 + 3) = 17 cm

The perimeter of a rectangular plot is 62 m and its area is 228 sq. metres. Find the dimensions of the plot.

A rectangular field is 16 m long and 10 m wide. There is a path of uniform width all around it, having an area of 120 m^{2}. Find the width of the path.

Let the width of the path be x meters,

Then,

Area of path = 16 10 – (16 – 2x) (10 – 2x) = 120

_{}

Hence the required width is 3 meter as x cannot be 10m

The sum of the areas of two squares is 640 m^{2}. If the difference in their perimeters be 64 m, find the sides of the two squares.

Let x and y be the lengths of the two square fields.

_{}

4x – 4y = 64

_{}x – y = 16——(2)

From (2),

x = y + 16,

Putting value of x in (1)

_{}

_{}Sides of two squares are 24m and 8m respectively

The length of a rectangle is thrice as long as the side of a square. The side of the square is 4 cm more than the width of the rectangle. Their areas being equal, find their dimensions.

Let the side of square be x cm

Then, length of the rectangle = 3x cm

Breadth of the rectangle = (x – 4) cm

_{}Area of rectangle = Area of square x

_{}

Thus, side of the square = 6 cm

And length of the rectangle = (3 6) = 18 cm

Then, breadth of the rectangle = (6 – 4) cm = 2 cm

A farmer prepares a rectangular vegetable garden of area 180 sq. m. With 39 metres of barbed wire, he can fence the three sides of the garden, leaving one of the longer sides unfenced. Find the dimensions of the garden.

Let the length = x meter

Area = length breadth = _{}

_{}

If ength of the rectangle = 15 m

_{}

Also, if length of rectangle = 24 m

_{}

The area of a right triangle is 600 cm^{2}. If the base of the triangle exceeds the altitude by 10 cm, find the dimensions of the triangle.

Let the altitude of triangle be x cm

Then, base of triangle is (x + 10) cm

_{}

Hence, altitude of triangle is 30 cm and base of triangle 40 cm

_{}

The area of a right-angled triangle is 96 sq m. If the base is three times the altitude, find the base.

Let the altitude of triangle be x meter

Hence, base = 3x meter

_{}

Hence, altitude of triangle is 8 cm

And base of triangle = 3x = (3 8) cm = 24 cm

The area of a right-angled triangle is 165 sq m. Determine its base and altitude if the latter exceeds the former by 7 metres.

Let the base of triangle be x meter

Then, altitude of triangle = (x + 7) meter

_{}

Thus, the base of the triangle = 15 m

And the altitude of triangle = (15 + 7) = 22 m

The hypotenuse of a right-angled triangle is 20 metres. If the difference between the lengths of the other sides be 4 m, find the other sides.

Let the other sides of triangle be x and (x -4) meters

By Pythagoras theorem, we have

_{}

Thus, height of triangle be = 16 cm

And the base of the triangle = (16 – 4) = 12 cm

The length of the hypotenuse of a right-angled triangle exceeds the length of the base by 2 cm and exceeds twice the length of the altitude by 1 cm. Find the length of each side of the triangle.

Let the base of the triangle be x

Then, hypotenuse = (x + 2) cm

_{}

Thus, base of triangle = 15 cm

Then, hypotenuse of triangle = (15 +2 )= 17 cm

And altitude of triangle = _{}

The hypotenuse of a right-angled triangle is 1 metre less than twice the shortest side. If the third side is 1 metre more than the shortest side, find the sides of the triangle.

Let the shorter side of triangle be x meter

Then, its hypotenuse = (2x – 1)meter

And let the altitude = (x + 1) meter

_{}

## Chapter 10 – Quadratic Equations Exercise Ex. 10F

Which

of the following is a quadratic equation?

Which

of the following is a quadratic equation?

Which

of the following is not a quadratic equation?

(a) Real

and equal

(b) Real

and unequal

(c) Imaginary

(d) None

of these

(a) Real,

unequal and rational

(b) Real,

unequal and irrational

(c) Real

and equal

(d) Imaginary

(a) Real,

unequal and rational

(b) Real,

unequal and irrational

(c) Real

and equal

(d) Imaginary

The

perimeter of a rectangle is 82 m and its area is 400 m^{2}. The

breadth of the rectangle is

(a) 25

m

(b) 20

m

(c) 16

m

(d) 9

m

The

length of a rectangular field exceeds its breadth by 8 m and the area of the

field is 240 m^{2}. The breadth of the field is

(a) 20

m

(b) 30

m

(c) 12

m

(d) 16

m

The

sum of two natural numbers is 8 and their product is 15. Find the numbers.