RS Aggarwal - Mathematics

Book: RS Aggarwal - Mathematics

Chapter: 10. Quadratic Equations

Subject: Maths - Class 10th

Q. No. 38 of Exercise 10E

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38

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 present age of son be x years

The present age of man = x2 years


One year ago age of son = (x – 1)years


age of man = (x2 – 1)years


According to given question


x2 – 1 = 8(x – 1)


x2 – 1 = 8x – 8


x2 – 8x + 7 = 0


Using the splitting middle term – the middle term of the general equation is divided in two such values that:


Product = a.c


For the given equation a = 1 b = – 8 c = 7


= 1.7 = 7


And either of their sum or difference = b


= – 8


Thus the two terms are – 7 and – 1


Sum = – 7 – 1 = – 8


Product = – 7. – 1 = 7


x2 – 7x – x + 7 = 0


x(x – 7) – 1(x – 7) = 0


(x – 7) (x – 1) = 0


x = 1 or x = 7


Man’s age cannot be 1 year


Thus x = 7


Thus the present age of son is 7years


The present age of man is 72 = 49years


38

Chapter Exercises

Exercise 10A
Exercise 10B
Exercise 10C
Exercise 10D
Exercise 10E
Exercise 10F

More Exercise Questions

1

The sum of a natural number and its square is 156. Find the number.
2

The sum of a natural number and its positive square root is 132. Find the number.
3

The sum of two natural numbers is 28 and their product is 192. Find the numbers.
4

The sum of the squares of two consecutive positive integers is 365. Find the integers.
5

The sum of the squares of two consecutive positive odd numbers is 514. Find the numbers.
6

The sum of the squares of two consecutive positive even numbers is 452. Find the numbers.
7

The product of two consecutive positive integers is 306. Find the integers.
8

Two natural numbers differ by 3 and their product is 504. Find the numbers.
9

Find two consecutive multiples of 3 whose product is 648.
10

Find two consecutive positive odd integers whose product is 483.
11

Find two consecutive positive even integers whose product is 288.
12

The sum of two natural numbers is 9 and the sum of their reciprocals is 1/2. Find the numbers.
13

The sum of two natural numbers is 15 and the sum of their reciprocals is 3/10. Find the numbers.
14

The difference of two natural numbers is 3 and the difference of their 3 reciprocals is 3/28.Find the numbers.
15

The difference of two natural numbers is 5 and the difference of their reciprocals is 5/14. Find the numbers.
16

The sum of the squares of two consecutive multiples of 7 is 1225. Find the multiples.
17

The sum of a natural number and its reciprocal is 65/8. Find the number.
18

Divide 57 into two parts whose product is 680.
19

Divide 27 into two parts such that the sum of their reciprocals is 3/20.
20

Divide 16 into two parts such that twice the square of the larger part exceeds the square of the smaller part by 164.
21

Find two natural numbers, the sum of whose squares is 25 times their sum and also equal to 50 times their difference.
22

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.
23

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.
24

A two – digit number is 4 times the sum of its digits and twice the product of its digits. Find the number.
25

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.
26

The denominator of a fraction is 3 more than its numerator. The sum of the fraction and its reciprocal is . Find the fraction.
27

The numerator of a fraction is 3 less than its denominator. If 1 is added to the denominator, the fraction is decreased by . Find the fraction.
28

The sum of a number and its reciprocal is . Find the number.
29

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 that he was short of 25 students. Find the number of students.
30

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.
31

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.
32

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?
33

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.
34

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.
35

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.
36

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?
37

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.
38

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.
39

The sum of the reciprocals of Meena's ages (in years) 3 years ago and 5 years hence is 1/3. Find her present age.
40

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.
41

The product of Tanvy's age (in years) 5 years ago and her age 8 years later is 30. Find her present age.
42

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.
43

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.
44

While boarding an aeroplane, a passenger got hurt. The pilot showing promptness and concern, made arrangements to hospitalize 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?
45

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.
46

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?
47

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.
48

A train covers a distance of 90 km at a uniform speed. Had the speed been 15 km/hr more, it would have taken 30 minutes less for the journey. Find the original speed of the train.
49

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.
50

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 km/hr.
51

A motor boat whose speed in still water is 18 km/hr, takes 1 hour more to go 24 km upstream than to return to the same spot. Find the speed of the stream.
52

The speed of a boat in still water is 8 km/hr. It can go 15 km upstream and 22 km downstream in 5 hours. Find the speed of the stream.
53

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.
54

A takes 10 days less than the 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.
55

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.
56

Two pipes running together can fill a tank in minutes. If one pipe takes 5 minutes more than the other to fill the tank separately, find the time in which each pipe would fill the tank separately.
57

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.
58

The length of a rectangle is twice its breadth and its area is 288 cm2. Find the dimensions of the rectangle.
59

The length of a rectangular field is three times its breadth. If the area of the field be 147 sq metres, find the length of the field.
60

The length of a hall is 3 meters more than its breadth. If the area of the hall is 238 sq metres, calculate its length and breadth.
61

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

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 m2. Find the width of the path.
63

The sum of the areas of two squares is 640 m2. If the difference in their perimeters be 64 m, find the sides of the two squares.
64

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65

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66

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67

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68

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69

The hypotenuse of a right – angled triangle is 20 meters. If the difference between the lengths of the other side’s be 4 meters, find the other sides.
70

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71

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