Class X Chapter 4 – Quadratic Equations Maths
Exercise 4.1Question 2:Represent the following situations in the form of quadratic equations.
(i) The area of a rectangular plot is 528 m2. The length of the plot (in
metres) is one more than twice its breadth. We need to find the length
and breadth of the plot.
(ii) The product of two consecutive positive integers is 306. We need
to find the integers.
(iii) Rohan’s mother is 26 years older than him. The product of their
ages (in years) 3 years from now will be 360. We would like to find
Rohan’s present age.
(iv) A train travels a distance of 480 km at a uniform speed. If the
speed had been 8 km/h less, then it would have taken 3 hours more to
cover the same distance. We need to find the speed of the train.
Answer:
(i) Let the breadth of the plot bex m.
Hence, the length of the plot is (2x + 1) m.
Area of a rectangle = Length × Breadth∴ 528 = x (2x + 1)
(ii) Let the consecutive integers bex and x + 1.
It is given that their product is 306.∴(iii) Let Rohan’s age be x.
Hence, his mother’s age =x + 26
3 years hence,
Rohan’s age =x + 3
Mother’s age =x + 26 + 3 = x + 29
It is given that the product of their ages after 3 years is 360.
(iv) Let the speed of train bex km/h.
Time taken to travel 480 km =
In second condition, let the speed of train = km/h
It is also given that the train will take 3 hours to cover the same
distance.
Therefore, time taken to travel 480 km = hrs
Speed × Time = Distance
Exercise 4.2
Question 1:Find the roots of the following quadratic equations by factorisation:
Answer:
Roots of this equation are the values for which = 0∴ = 0 or = 0
i.e.,x = 5 or x = -2
Roots of this equation are the values for which = 0∴ = 0 or = 0
i.e.,x = -2 or x =
Roots of this equation are the values for which = 0∴ = 0 or = 0
i.e.,x = or x =
Roots of this equation are the values for which = 0
Therefore,
i.e.,
Roots of this equation are the values for which = 0
Therefore,
i.e.,Question 2:(i) John and Jivanti together have 45 marbles. Both of them lost 5
marbles each, and the product of the number of marbles they now
have is 124. Find out how many marbles they had to start with.
(ii) A cottage industry produces a certain number of toys in a day. The
cost of production of each toy (in rupees) was found to be 55 minus
the number of toys produced in a day. On a particular day, the total
cost of production was Rs 750. Find out the number of toys produced
on that day.
Answer:
(i) Let the number of John’s marbles bex.
Therefore, number of Jivanti’s marble = 45 -xAfter losing 5 marbles,
Number of John’s marbles =x - 5
Number of Jivanti’s marbles = 45 -x - 5 = 40 - xIt is given that the product of their marbles is 124.
Either = 0 or x - 9 = 0
i.e.,x = 36 or x = 9
If the number of John’s marbles = 36,
Then, number of Jivanti’s marbles = 45 - 36 = 9
If number of John’s marbles = 9,
Then, number of Jivanti’s marbles = 45 - 9 = 36
(ii) Let the number of toys produced bex.∴ Cost of production of each toy = Rs (55 - x)
It is given that, total production of the toys = Rs 750
Either = 0 orx - 30 = 0
i.e.,x = 25 or x = 30
Hence, the number of toys will be either 25 or 30.
Question 4:Which of the following are APs? If they form an A.P. find the common
differenced and write three more terms.
(i) 2, 4, 8, 16 …
(ii)
(iii) - 1.2, - 3.2, - 5.2, - 7.2 …
(iv) - 10, - 6, - 2, 2 …
(v)
(vi) 0.2, 0.22, 0.222, 0.2222 ….
(vii) 0, - 4, - 8, - 12 …
(viii)
(ix) 1, 3, 9, 27 …
(x)a, 2a, 3a, 4a …
(xi)a, a2, a3, a4 …
(xii)
(xiii)
(xiv) 12, 32, 52, 72 …
(xv) 12, 52, 72, 73 …
Answer:
(i) 2, 4, 8, 16 …
It can be observed thata2 - a1 = 4 - 2 = 2
a3 - a2 = 8 - 4 = 4a4 - a3 = 16 - 8 = 8
i.e.,ak+1- ak is not the same every time. Therefore, the given
numbers are not forming an A.P.
(ii)
It can be observed that
i.e.,ak+1- ak is same every time.
| Therefore, | and the given numbers are in A.P. |
| Three more terms are |
It can be observed thata2 - a1 = (-3.2) - (-1.2) = -2a3 - a2 = (-5.2) - (-3.2) = -2a4 - a3 = (-7.2) - (-5.2) = -2
i.e., ak+1- ak is same every time. Therefore, d = -2
The given numbers are in A.P.
Three more terms area5 = - 7.2 - 2 = -9.2a6 = - 9.2 - 2 = -11.2a7 = - 11.2 - 2 = -13.2
(iv) -10, -6, -2, 2 …
It can be observed thata2 - a1 = (-6) - (-10) = 4a3 - a2 = (-2) - (-6) = 4a4 - a3 = (2) - (-2) = 4
i.e.,ak+1 - ak is same every time. Therefore, d = 4
The given numbers are in A.P.
Three more terms area5 = 2 + 4 = 6a6 = 6 + 4 = 10a7 = 10 + 4 = 14
(v)
It can be observed that
i.e.,ak+1 - ak is same every time. Therefore,
The given numbers are in A.P.
Three more terms are
(vi) 0.2, 0.22, 0.222, 0.2222 ….
It can be observed thata2 - a1 = 0.22 - 0.2 = 0.02a3 - a2 = 0.222 - 0.22 = 0.002a4 - a3 = 0.2222 - 0.222 = 0.0002
i.e.,ak+1 - ak is not the same every time.
Therefore, the given numbers are not in A.P.
(vii) 0, -4, -8, -12 …
It can be observed thata2 - a1 = (-4) - 0 = -4a3 - a2 = (-8) - (-4) = -4a4 - a3 = (-12) - (-8) = -4
i.e.,ak+1 - ak is same every time. Therefore, d = -4
The given numbers are in A.P.
Three more terms area5 = - 12 - 4 = -16a6 = - 16 - 4 = -20a7 = - 20 - 4 = -24
(viii)
It can be observed that
i.e., ak+1 - ak is same every time. Therefore, d = 0
The given numbers are in A.P.
Three more terms are
(ix) 1, 3, 9, 27 …
It can be observed thata2 - a1 = 3 - 1 = 2a3 - a2 = 9 - 3 = 6a4 - a3 = 27 - 9 = 18
i.e.,ak+1 - ak is not the same every time.
Therefore, the given numbers are not in A.P.
(x)a, 2a, 3a, 4a …
It can be observed thata2 - a1 = 2a - a = a
a3 - a2 = 3a - 2a = a
a4 - a3 = 4a - 3a = a
i.e., ak+1 - ak is same every time. Therefore, d = aThe given numbers are in A.P.
Three more terms area5 = 4a + a = 5a
a6 = 5a + a = 6a
a7 = 6a + a = 7a(xi) a, a2, a3, a4 …
It can be observed thata2 - a1 = a2 - a = a (a - 1)a3 - a2 = a3 - a2 = a2 (a - 1)a4 - a3 = a4 - a3 = a3 (a - 1)
i.e.,ak+1 - ak is not the same every time.
Therefore, the given numbers are not in A.P.
(xii)
It can be observed that
i.e.,ak+1 - ak is same every time.
Therefore, the given numbers are in A.P.
And,
Three more terms are
(xiii)
It can be observed that
i.e.,ak+1 - ak is not the same every time.
Therefore, the given numbers are not in A.P.
(xiv) 12, 32, 52, 72 …
Or, 1, 9, 25, 49 …..
It can be observed thata2 - a1 = 9 - 1 = 8a3 - a2 = 25 - 9 = 16a4 - a3 = 49 - 25 = 24
i.e.,ak+1 - ak is not the same every time.
Therefore, the given numbers are not in A.P.
(xv) 12, 52, 72, 73 …
Or 1, 25, 49, 73 …
It can be observed thata2 - a1 = 25 - 1 = 24a3 - a2 = 49 - 25 = 24
a4 - a3 = 73 - 49 = 24
i.e.,ak+1 - ak is same every time.
Therefore, the given numbers are in A.P.
And,d = 24
Three more terms area5 = 73+ 24 = 97a6 = 97 + 24 = 121a7 = 121 + 24 = 145Question 3:Find two numbers whose sum is 27 and product is 182.
Answer:
Let the first number bex and the second number is 27 - x.
Therefore, their product =x (27 - x)
It is given that the product of these numbers is 182.
Either = 0 orx - 14 = 0
i.e.,x = 13 or x = 14
If first number = 13, then
Other number = 27 - 13 = 14
If first number = 14, then
Other number = 27 - 14 = 13
Therefore, the numbers are 13 and 14.Question 4:Find two consecutive positive integers, sum of whose squares is 365.
Answer:
Let the consecutive positive integers bex and x + 1.
Eitherx + 14 = 0 or x - 13 = 0, i.e., x = -14 or x = 13
Since the integers are positive,x can only be 13.∴ x + 1 = 13 + 1 = 14
Therefore, two consecutive positive integers will be 13 and 14.Question 5:The altitude of a right triangle is 7 cm less than its base. If the
hypotenuse is 13 cm, find the other two sides.
Answer:
Let the base of the right triangle bex cm.
Its altitude = (x - 7) cm
Either x - 12 = 0 or x + 5 = 0, i.e., x = 12 or x = -5
Since sides are positive,x can only be 12.
Therefore, the base of the given triangle is 12 cm and the altitude of
this triangle will be (12 - 7) cm = 5 cm.Question 6:A cottage industry produces a certain number of pottery articles in a
day. It was observed on a particular day that the cost of production of
each article (in rupees) was 3 more than twice the number of articles
produced on that day. If the total cost of production on that day was
Rs 90, find the number of articles produced and the cost of each
article.
Answer:
Let the number of articles produced bex.
Therefore, cost of production of each article = Rs (2x + 3)
It is given that the total production is Rs 90.
Either 2x + 15 = 0 or x - 6 = 0, i.e., x = or x = 6
As the number of articles produced can only be a positive integer,
therefore,x can only be 6.
Hence, number of articles produced = 6
Cost of each article = 2 × 6 + 3 = Rs 15
Exercise 4.3
Question 1:Find the roots of the following quadratic equations, if they exist, by the
method of completing the square:
Answer:
Question 2:Find the roots of the quadratic equations given in Q.1 above by
applying the quadratic formula.
Answer:
Question 3:Find the roots of the following equations:
Answer:
Question 4:The sum of the reciprocals of Rehman’s ages, (in years) 3 years ago
and 5 years from now is . Find his present age.
Answer:
Let the present age of Rehman bex years.
Three years ago, his age was (x - 3) years.
Five years hence, his age will be (x + 5) years.
It is given that the sum of the reciprocals of Rehman’s ages 3 years
ago and 5 years from now is .
However, age cannot be negative.
Therefore, Rehman’s present age is 7 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.
Answer:
Let the marks in Maths bex.
Then, the marks in English will be 30 -x.
According to the given question,
If the marks in Maths are 12, then marks in English will be 30 - 12 =
18
If the marks in Maths are 13, then marks in English will be 30 - 13 =
17Question 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.
Answer:
Let the shorter side of the rectangle bex m.
Then, larger side of the rectangle = (x + 30) m
However, side cannot be negative. Therefore, the length of the shorter
side will be
90 m.
Hence, length of the larger side will be (90 + 30) m = 120 mQuestion 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.
Answer:
Let the larger and smaller number bex and y respectively.
According to the given question,
However, the larger number cannot be negative as 8 times of the
larger number will be negative and hence, the square of the smaller
number will be negative which is not possible.
Therefore, the larger number will be 18 only.
Therefore, the numbers are 18 and 12 or 18 and -12.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.
Answer:
Let the speed of the train bex km/hr.
Time taken to cover 360 km hr
According to the given question,
However, speed cannot be negative.
Therefore, the speed of train is 40 km/hQuestion 9:Two water taps together can fill a tank in hours. The tap of larger
diameter takes 10 hours less than the smaller one to fill the tank
separately. Find the time in which each tap can separately fill the tank.
Answer:
Let the time taken by the smaller pipe to fill the tank bex hr.
Time taken by the larger pipe = (x - 10) hr
Part of tank filled by smaller pipe in 1 hour =
Part of tank filled by larger pipe in 1 hour =
It is given that the tank can be filled in hours by both the pipes
together. Therefore,
Time taken by the smaller pipe cannot be = 3.75 hours. As in this
case, the time taken by the larger pipe will be negative, which is
logically not possible.
Therefore, time taken individually by the smaller pipe and the larger
pipe will be 25 and 25 - 10 =15 hours respectively.Question 10:An express train takes 1 hour less than a passenger train to travel 132
km between Mysore and Bangalore (without taking into consideration
the time they stop at intermediate stations). If the average speeds of
the express train is 11 km/h more than that of the passenger train,
find the average speed of the two trains.
Answer:
Let the average speed of passenger train bex km/h.
Average speed of express train = (x + 11) km/h
It is given that the time taken by the express train to cover 132 km is
1 hour less than the passenger train to cover the same distance.
Speed cannot be negative.
Therefore, the speed of the passenger train will be 33 km/h and thus,
the speed of the express train will be 33 + 11 = 44 km/h.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.
Answer:
Let the sides of the two squares bex m and y m. Therefore, their
perimeter will be 4x and 4y respectively and their areas will be x2 andy2 respectively.
It is given that
4x - 4y = 24x - y = 6x = y + 6
However, side of a square cannot be negative.
Hence, the sides of the squares are 12 m and (12 + 6) m = 18 m
Exercise 4.4
Question 1:Find the nature of the roots of the following quadratic equations.
If the real roots exist, find them;
(I) 2x2 -3x + 5 = 0
(II)
(III) 2x2 - 6x + 3 = 0
Answer:
We know that for a quadratic equationax2 + bx + c = 0, discriminant
isb2 - 4ac.(A) If b2 - 4ac > 0 → two distinct real roots
(B) Ifb2 - 4ac = 0 → two equal real roots
(C) Ifb2 - 4ac < 0 → no real roots
(I) 2x2 -3x + 5 = 0
Comparing this equation withax2 + bx + c = 0, we obtaina = 2, b = -3, c = 5
Discriminant =b2 - 4ac = (- 3)2 - 4 (2) (5) = 9 - 40
= -31
Asb2 - 4ac < 0,
Therefore, no real root is possible for the given equation.
(II)
Comparing this equation withax2 + bx + c = 0, we obtain
Discriminant
= 48 - 48 = 0
Asb2 - 4ac = 0,
Therefore, real roots exist for the given equation and they are equal to
each other.
And the roots will be and .
Therefore, the roots are and .
(III) 2x2 - 6x + 3 = 0
Comparing this equation withax2 + bx + c = 0, we obtaina = 2, b = -6, c = 3
Discriminant =b2 - 4ac = (- 6)2 - 4 (2) (3)
= 36 - 24 = 12
Asb2 - 4ac > 0,
Therefore, distinct real roots exist for this equation as follows.
Therefore, the roots are or .Question 2:Find the values of k for each of the following quadratic equations, so
that they have two equal roots.
(I) 2x2 + kx + 3 = 0
(II)kx (x - 2) + 6 = 0
Answer:
We know that if an equationax2 + bx + c = 0 has two equal roots, its
discriminant
(b2 - 4ac) will be 0.
(I) 2x2 + kx + 3 = 0
Comparing equation withax2 + bx + c = 0, we obtaina = 2, b = k, c = 3
Discriminant =b2 - 4ac = (k)2- 4(2) (3)
=k2 - 24
For equal roots,
Discriminant = 0k2 - 24 = 0k2 = 24
(II)kx (x - 2) + 6 = 0
orkx2 - 2kx + 6 = 0
Comparing this equation withax2 + bx + c = 0, we obtaina = k, b = -2k, c = 6
Discriminant =b2 - 4ac = (- 2k)2 - 4 (k) (6)
= 4k2 - 24kFor equal roots,b2 - 4ac = 0
4k2 - 24k = 0
4k (k - 6) = 0
Either 4k = 0 or k = 6 = 0k = 0 or k = 6
However, ifk = 0, then the equation will not have the terms ‘x2’ and
‘x’.
Therefore, if this equation has two equal roots,k should be 6 only.Question 3:Is it possible to design a rectangular mango grove whose length is
twice its breadth, and the area is 800 m2?
If so, find its length and breadth.
Answer:
Let the breadth of mango grove bel.
Length of mango grove will be 2l.
Area of mango grove = (2l) (l)
= 2l2Comparing this equation with al2 + bl + c = 0, we obtaina = 1 b = 0, c = 400
Discriminant =b2 - 4ac = (0)2 - 4 × (1) × (- 400) = 1600
Here,b2 - 4ac > 0
Therefore, the equation will have real roots. And hence, the desired
rectangular mango grove can be designed.
However, length cannot be negative.
Therefore, breadth of mango grove = 20 m
Length of mango grove = 2 × 20 = 40 mQuestion 4:Is the following situation possible? If so, determine their present ages.
The sum of the ages of two friends is 20 years. Four years ago, the
product of their ages in years was 48.
Answer:
Let the age of one friend bex years.
Age of the other friend will be (20 -x) years.
4 years ago, age of 1st friend = (x - 4) years
And, age of 2nd friend = (20 - x - 4)
= (16 -x) years
Given that,
(x - 4) (16 - x) = 48
16x - 64 - x2 + 4x = 48
-x2 + 20x - 112 = 0x2 - 20x + 112 = 0
Comparing this equation withax2 + bx + c = 0, we obtaina = 1, b = -20, c = 112
Discriminant =b2 - 4ac = (- 20)2 - 4 (1) (112)
= 400 - 448 = -48
Asb2 - 4ac < 0,
Therefore, no real root is possible for this equation and hence, this
situation is not possible.Question 5:Is it possible to design a rectangular park of perimeter 80 and area
400 m2? If so find its length and breadth.
Answer:
Let the length and breadth of the park bel and b.
Perimeter = 2 (l + b) = 80l + b = 40
Or,b = 40 - lArea = l × b = l (40 - l) = 40l - l240l - l2 = 400l2 - 40l + 400 = 0
Comparing this equation withal2 + bl + c = 0, we obtaina = 1, b = -40, c = 400
Discriminate =b2 - 4ac = (- 40)2 -4 (1) (400)
= 1600 - 1600 = 0
Asb2 - 4ac = 0,
Therefore, this equation has equal real roots. And hence, this situation
is possible.
Root of this equation,
Therefore, length of park,l = 20 m
And breadth of park,b = 40 - l = 40 - 20 = 20 m
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