Chap 3 Pair of Linear Equation in Two Variables Page 21

CHAPTER 3

Pair of Linear Equation in Two Variables

OBJECTIVE QUESTION
6. For which value(s ) of p , will the lines represented by
5 the following pair of linear equations be parallel
1. The 2 digit number which becomes th of itself when
6
3x - y - 5 = 0 and 6x - 2y - p = 0
its digits are reversed. The difference in the digits of
the number being 1, then the two digits number is (a) all real values except 10 (b) 10
(a) 45 (b) 54 (c) 5/2 (d) 1/2


Sol : [Board Term-1 Delhi 2012]
(c) 36 (d) None of these


Sol : [Board 2022 Term-1 SQP STD]
7. x and y are 2 different digits. If the sum of the two
digit numbers formed by using both the digits is a
2. In a number of two digits, unit’s digit is twice the tens perfect square, then value of x + y is
digit. If 36 be added to the number, the digits are (a) 10 (b) 11
reversed. The number is
(c) 12 (d) 13
(a) 36 (b) 63

Sol : [Board 2022 Term-1 STD]

(c) 48 (d) 84


Sol : [Board 2022 Term-1 SQP Basic]
8. The value of k for which the system of linear equations
x + 2y = 3 , 5x + ky + 7 = 0 is inconsistent is
If 3x + 4y : x + 2y = 9 : 4 , then 3x + 5y : 3x − y is equal
(a) - 14 (b) 2
3.
to 3 5
(a) 4 : 1 (b) 1 : 4 (c) 5 (d) 10


Sol : [Board 2020 OD Standard]
(c) 7 : 1 (d) 1 : 7


Sol : [Board 2022 Term-1 STD]

9. The value of k for which the system of equations

x+y x−y x + y − 4 = 0 and 2x + ky = 3 , has no solution, is
4. The pair of equations 3 = 81, 81 = 3 has
(a) - 2 (b) ! 2
(a) no solution
(c) 3 (d) 2
(b) unique solution

Sol : [Board 2020 Delhi Standard]
(c) infinitely many solutions
(d) x = 2 1 , y = 1 7 10. The pair of linear equations 2kx + 5y = 7 , 6x − 5y = 11


Sol : 8 8 [Board 2022 Term-1 Basic]
has a unique solution, if
(a) k !- 3 (b) k ! 2
3
5. A fraction becomes 4 when 1 is added to both the (c) k ! 5 (d) k ! 2
9
numerator and denominator and it becomes 7

Sol : [Board Term-1 SQP 2012]
when 1 is subtracted from both the numerator and
denominator. The numerator of the given fraction is
11. The pair of equations x + 2y + 5 = 0 and
(a) 2 (b) 3
− 3x − 6y + 1 = 0 has
(c) 5 (d) 15 (a) a unique solution


Sol : [Board 2022 Term-1 Basic]
Page 22 Pair of Linear Equation in Two Variables Chap 3

(b) exactly two solutions 18. One equation of a pair of dependent linear equations
(c) infinitely many solutions − 5x + 7y = 2 The second equation can be
(a) 10x + 14y + 4 = 0
(d) no solution


Sol : [Board Term-1 Foreign 2012] (b) − 10x − 14y + 4 = 0
(c) − 10x + 14y + 4 = 0
12. If a pair of linear equations is consistent, then the (d) 10x − 14y =− 4
lines will be

Sol : [Board Term-1 OD 2013]

(a) parallel
(b) always coincident
19. If x = a and y = b is the solution of the equations
(c) intersecting or coincident x − y = 2 and x + y = 4 , then the values of a and b
(d) always intersecting are, respectively


Sol : [Board Term-1 OD 2013]
(a) 3 and 5 (b) 5 and 3
(c) 3 and 1 (d) - 1 and - 3


Sol : [Board Term-1 OD 2015]
13. The pair of equations y = 0 and y =− 7 has
(a) one solution
20. Aruna has only < 1 and < 2 coins with her. If the total
(b) two solutions
number of coins that she has is 50 and the amount of
(c) infinitely many solutions money with her is < 75, then the number of < 1 and
(d) no solution < 2 coins are, respectively


Sol : [Board Term-1 SQP 2017] (a) 35 and 15 (b) 35 and 20
(c) 15 and 35 (d) 25 and 25
14. For what value of k , do the equations 3x − y + 8 = 0

Sol : [Board Term-1 OD 2016]

and 6x − ky =− 16 represent coincident lines ?

(a) 1 (b) - 1 21. The father’s age is six times his son’s age. Four years
2 2
hence, the age of the father will be four times his son’s
(c) 2 (d) - 2 age. The present ages (in year) of the son and the


Sol : [Board Term-1 Foreign 2016]
father are, respectively.
(a) 4 and 24 (b) 5 and 30
15. The pair of equations x = a and y = b graphically (c) 6 and 36 (d) 3 and 24
represents lines which are

Sol : [Board Term-1 Delhi 2016]

(a) parallel
(b) intersecting at (b, a)

ONE MARK QUESTION
(c) coincident
(d) intersecting at (a, b)


Sol : [Board Term-1 Foreign 2011] 22. Find the value of k for which the system of linear
equations x + 2y = 3 , 5x + ky + 7 = 0 is inconsistent.


Sol : [Board 2020 OD Standard]
16. If the lines given by 3x + 2ky = 2 and 2x + 5y + 1 = 0
are parallel, then the value of k is
(a) - 5 (b) 2 23. Find the value of k for which the system of equations
4 5
x + y − 4 = 0 and 2x + ky = 3 , has no solution.
(c) 15 (d) 3

Sol : [Board 2020 Delhi Standard]


Sol : 4 2 [Board Term-1 OD 2011]

17. The value of c for which the pair of equations 24. For which value(s ) of p , will the lines represented by
cx − y = 2 and 6x − 2y = 3 will have is the following pair of linear equations be parallel ?
(a) 3 (b) - 3 3x - y - 5 = 0 , 6x - 2y - p = 0


Sol : [Board Term-1 OD 2017]
(c) - 12 (d) no value


Sol : [Board Term-1 SQP 2014]
Chap 3 Pair of Linear Equation in Two Variables Page 23

25. The 2 digit number which becomes 56 th of itself when 35. If ad ! bc, then find whether the pair of linear
its digits are reversed. If the difference in the digits of equations ax + by = p and cx + dy = q has no solution,
the number being 1, what is the two digits number? unique solution or infinitely many solutions.


Sol : [Board Term-1 Delhi 2011]

Sol : [Board Term-1 Delhi 2015]

36. Two lines are given to be parallel. The equation of one

26. In a number of two digits, unit’s digit is twice the tens
of the lines is 4x + 3y = 14 , then find the equation of
digit. If 36 be added to the number, the digits are
the second line.
reversed. What is the number ?

Sol : [Board 2007]


Sol : [Board Term-1 Delhi 2016]

27. If 3x + 4y : x + 2y = 9 : 4 , then find the value of 37. Find whether the lines represented by 2x + y = 3 and
3x + 5y : 3x − y . 4x + 2y = 6 are parallel, coincident or intersecting.


Sol : [Board Term-1 Foreign 2012]

Sol : [Board Term-1 Delhi 2016]

28. raction becomes 4 when 1 is added to both the 38. Given the linear equation 3x + 4y = 9 . Write another
numerator and denominator and it becomes 7 linear equation in these two variables such that the
when 1 is subtracted from both the numerator and geometrical representation of the pair so formed is:
denominator. What is the numerator of the given (1) intersecting lines
fraction ? (2) coincident lines.


Sol : [Board Term-1 Foreign 2016]


Sol : [Board Term-1 2016]

29. x and y are 2 different digits. If the sum of the two 39. Find the value(s) of k so that the pair of equations
digit numbers formed by using both the digits is a x + 2y = 5 and 3x + ky + 15 = 0 has a unique solution.
perfect square, then what is the value of x + y ?

Sol : [Board 2019 OD]


Sol : [Board Term-1 OD 2013]

40. If 2x + y = 23 and 4x − y = 19 , find the value of

If a pair of linear equations is consistent, then the
(5y - 2x) and ^ yx - 2h .
30.
lines will be intersecting or coincident. Justify.

Sol : [Board 2020 OD Standard]


Sol : [Board 2008]

41. Find whether the following pair of linear equation is

31. The pair of equations y = 0 and y =− 7 has no
consistent or inconsistent:
solution. Justify.


Sol : [Board Term-1 Foreign 2014] 3x + 2y = 8 , 6x − 4y = 9


Sol : [Board Term-1 2016]

32. If the equations kx − 2y = 3 and 3x + y = 5 represent

two intersecting lines at unique point, then the value
42. Is the system of linear equations 2x + 3y − 9 = 0 and
of k is ........... .


Sol : [Board Term-1 2011] 4x + 6y − 18 = 0 consistent? Justify your answer.


Sol : [Board Term-1 2012]

33. Find whether the pair of linear equations y = 0 and

y =− 5 has no solution, unique solution or infinitely 43. For what value of k , the pair of linear equations
many solutions. kx − 4y = 3, 6x − 12y = 9 has an infinite number of


Sol : [Board Term-1 OD 2011] solutions ?


Sol : [Board Term-1 2012]

34. If am = bl, then find whether the pair of linear

equations ax + by = c and lx + my = n has no 44. Solve the following pair of linear equations by
solution, unique solution or infinitely many solutions. substitution method:


Sol : [Board Term-1 OD 2015]
3x + 2y − 7 = 0 , 4x + y − 6 = 0


Sol : [Board Term-1 2015]
Page 24 Pair of Linear Equation in Two Variables Chap 3

45. Solve : 99x + 101y = 499 , 101x + 99y = 501 56. Solve for x and y :


Sol : [Board Term-1 2012]
ax + by = a + b and 3x + 5y = 4


Sol : 2 [Board Term-1 2011]
46. Solve graphically : 2x − 3y + 13 = 0 ; 3x − 2y + 12 = 0


Sol : [Board 2020 OD Basic]
57. Find whether the following pair of linear equations
has a unique solutions. If yes, find the solution :
47. Solve graphically : 2x + 3y = 2 , x − 2y = 8


Sol : [Board 2020 Delhi Basic] 7x - 4y = 49, 5x − 6y = 57 .


Sol : [Board Term-1 2017]

48. A fraction becomes 13 when 2 is subtracted from the

FIVE MARKS QUESTION
numerator and it becomes 12 when 1 is subtracted
from the denominator- Find the fraction.


Sol : [Board 2019 Delhi]

58. For what value of k , which the following pair of linear

49. Represent the following pair of linear equations equations have infinitely many solutions:
graphically and hence comment on the condition of 2x + 3y = 7 and ^k + 1h x + ^2k − 1h y = 4k + 1
consistency of this pair.

Sol : [Board 2019 Delhi]

x − 5y = 6 and 2x − 10y = 12 .


Sol : [Board Term-1 2011] 59. Find c if the system of equations
cx + 3y + ^3 − c h = 0; 12x + cy − c = 0 has infinitely
50. Determine the values of m and n so that the following many solutions?


Sol : [Board 2019 Delhi]
system of linear equation have infinite number of
solutions :
^2m − 1h x + 3y − 5 = 0 60. Determine graphically the coordinates of the vertices
3x + ^n − 1h y − 2 = 0 of triangle, the equations of whose sides are given by


Sol : [Board Term-1 2013] 2y − x = 8 , 5y − x = 14 and y − 2x = 1.


Sol : [Board 2020 Delhi Standard]

51. For what value of p will the following system of

equations have no solution ? 61. Determine graphically whether the following pair of
linear equations :
^2p − 1h x + ^p − 1h y = 2p + 1; y + 3x − 1 = 0


Sol : [Board Term-1 2011] 3x - y = 7 and 2x + 5y + 1 = 0 has :
(a) unique solution
52. Solve for x and y : (b) infinitely many solutions or
x + 2y =− 1 and x - y = 3 (c) no solution.
2 3 3

Sol : [Board Term-1 2015]


Sol : [Board Term-1 2015]

62. Aftab tells his daughter, ‘7 years ago, I was seven

53. Solve the following pair of linear equations :
times as old as you were then. Also, 3 years from now,
8x + 5y = 9 , 3x + 2y = 4 I shall be three times as old as you will be.’ Represent


Sol : [Board Term-1 2015]
this situation algebraically and graphically.


Sol : [Board Term-1 2015]
54. Solve for x and y :
x + 1 + y − 1 = 9 ; x − 1 + y + 1 = 8. 63. For Uttarakhand flood victims two sections A and B of
2 3 3 2


Sol : Board Term-1 Delhi 2011] class contributed Rs. 1,500. If the contribution of X-A
was Rs. 100 less than that of X-B, find graphically the
55. Given the linear equation 2x + 3y − 8 = 0 , write
amounts contributed by both the sections.
another linear equation in two variables such that the

Sol : [Board Term-1 2016]
geometrical representation of the pair so formed is :
(a) intersecting lines
64. Solve graphically the pair of linear equations :
(b) parallel lines
3x − 4y + 3 = 0 and 3x + 4y − 21 = 0
(c) coincident lines.


Sol : [Board Term-1 2014] Find the co-ordinates of the vertices of the triangular
Chap 3 Pair of Linear Equation in Two Variables Page 25

region formed by these lines and x -axis. Also, 73. For what values of a and b does the following pair
calculate the area of this triangle. of linear equations have infinite number of solution ?
2x + 3y = 7, a ^x + y h − b ^x − y h = 3a + b − 2


Sol : [Board Term-1 2015]

Sol : [Board Term-1 2015]

65. The cost of 2 kg of apples and 1kg of grapes on a day

was found to be Rs. 160. After a month, the cost of
4kg of apples and 2kg of grapes is Rs. 300. Represent 74. Find the value of p and q for which the system of
the situations algebraically and geometrically. equations represent coincident lines 2x + 3y = 7 ,


Sol : [Board Term-1 2013]
^p + q + 1h x + ^p + 2q + 2h y = 4 ^p + q h + 1


Sol : [Board Term-1 Delhi 2012]

66. Solve for x and y :

2x − y + 3 and 3x − 5y + 1 = 0


Sol : [Board Term-1 2015]

Word Problem

67. Draw the graphs of the equations x − y + 1 = 0 and

3x + 2y − 12 = 0. Determine the co-ordinates of the 75. In an election contested between A and B, A obtained
vertices of the triangle formed by these lines and the votes equal to twice the no. of persons on the electoral roll
X-axis and shade the triangular region. who did not cast their votes and this later number was


Sol : [Board Term-1 2013]
equal to twice his majority over B. If there were 1,8000
persons on the electoral roll. How many votes for B.
68. Solve the following pair of linear equations graphically:

Sol : [Board Term-1 2012]

x + 3y = 12, 2x − 3y = 12
Also shade the region bounded by the line 2x − 3y = 2 76. Sum of the ages of a father and the son is 40 years. If
and both the co-ordinate axes. father’s age is three times that of his son, then find


Sol : [Board Term-1 2013, 2012] their respective ages.


Sol : [Board Term-1 2015]

69. Solve the following pair of linear equations graphically:

x - y = 1, 2x + y = 8 77. Half the perimeter of a rectangular garden, whose
length is 4 m more then its width, is 36 m. Find the
Also find the co-ordinates of the points where the lines
dimensions of garden.
represented by the above equation intersect y - axis.

Sol : [Board Term-1 2013]


Sol : [Board Term-1 Delhi 2012]

70. Draw the graph of the following equations: 78. A part of monthly hostel charge is fixed and the
2x - y = 1, x + 2y = 13 remaining depends on the number of days one has
Find the solution of the equations from the graph and taken food in the mess. When Swati takes food for
shade the triangular region formed by the lines and 20 days, she has to pay Rs. 3,000 as hostel charges
the y -axis. whereas Mansi who takes food for 25 days Rs. 3,500


Sol : [Board Term-1 OD 2012] as hostel charges. Find the fixed charges and the cost
of food per day.


Sol : [Board Term-1 2016, 2015]
71. Solve the following pair of equations graphically:
2x + 3y = 12, x − y − 1 = 0 .
79. The present age of the father is twice the sum of the
Shade the region between the two lines represented by
ages of his 2 children. After 20 years, his age will be
the above equations and the X -axis.


Sol : [Board Term-1 2013] equal to the sum of the ages of his children. Find the
age of the father.


Sol : [Board Term-1 2012]
72. Solve x + y = 5 and 2x − 3y = 4 by elimination
method and the substitution method.


Sol : [Board Term-1 2015] 80. In the figure, ABCDE is a pentagon with BE z CD
and BC z DE . BC is perpendicular to CD. AB = 5 cm,
AE = 5 cm, BE = 7 cm, BC = x − y and CD = x + y.
Page 26 Pair of Linear Equation in Two Variables Chap 3

If the perimeter of ABCDE is 27 cm. Find the value 86. A man can row a boat downstream 20 km in 2 hours
of x and y , given x , y ! 0 . and upstream 4 km in 2 hours. Find his speed of
rowing in still water. Also find the speed of the stream.


Sol : [Board 2020 Delhi Standard]

87. A train covered a certain distance at a uniform speed.

If the train would have been 10 km/hr scheduled time.
And, if the train were slower by 10 km/hr, it would
have taken 3 hr more than the scheduled time. Find
the distance covered by the train.


Sol : [Board Term-1 Delhi 2012]

88. The ratio of incomes of two persons is 11:7 and the

Sol : [Board 2020 SQP Standard] ratio of their expenditures is 9:5. If each of them
manages to save Rs 400 per month, find their monthly
incomes.
81. In Figure, ABCD is a rectangle. Find the values of

Sol : [Board Term-1 2012]
x and y .

89. A and B are two points 150 km apart on a highway.

Two cars start A and B at the same time. If they
move in the same direction they meet in 15 hours. But
if they move in the opposite direction, they meet in 1
hours. Find their speeds.


Sol : [Board Term-1 2012]

Sol : [Board 2018]

90. If 2 is subtracted from the numerator and 1 is added
to the denominator, a fraction becomes 12 , but when
82. Seven times a two digit number is equal to four times 4 is added to the numerator and 3 is subtracted from
the number obtained by reversing the order of its the denominator, it becomes 32 . Find the fraction.
digits. If the difference of the digits is 3, determine

Sol : [Board Term-1 2012]

the number.


Sol : [Board Term-1 2017]
91. If a bag containing red and white balls, half the number
of white balls is equal to one-third the number of red
83. 4 chairs and 3 tables cost Rs 2100 and 5 chairs and balls. Thrice the total number of balls exceeds seven
2 tables cost Rs 1750. Find the cost of one chair and times the number of white balls by 6. How many balls
one table separately. of each colour does the bag contain ?


Sol : [Board Term-1 2015]

Sol : [Board Term-1 2012]

84. A chemist has one solution which is 50 ~ acid and a

second which is 25 ~ acid. How much of each should
be mixed to make 10 litre of 40 ~ acid solution.


Sol : [Board Term-1 2015]

85. It can take 12 hours to fill a swimming pool using two

pipes. If the pipe of larger diameter is used for four
hours and the pipe of smaller diameter for 9 hours,
only half of the pool can be filled. How long would it
take for each pipe to fill the pool separately?


Sol : [Board 2020 OD Standard]
Chap 3 Pair of Linear Equation in Two Variables Page 27

92. The area of a rectangle gets reduced by 9 square units, Find where the breakdown occurred and his original
if its length is reduced by 5 units and the breadth speed.
is increased by 3 units. The area is increased by 67

Sol : [Board Term-1 2013]

square units if length is increased by 3 units and

breadth is increased by 2 units. Find the perimeter of 100. The population of a village is 5000. If in a year, the
the rectangle. number of males were to increase by 5% and that of a


Sol : [Board Term-1 Delhi 2012]
female by 3% annually, the population would grow to
5202 at the end of the year. Find the number of males
93. The students of a class are made to stand in rows. If and females in the village.


Sol : [Board Term-1 2012]
3 students are extra in a row, there would be 1 row
less. If 3 students are less in a row, there would be 2
rows more. Find the number of students in the class. 101. A father’s age is three times the sum of the ages of


Sol : [Board Term-1 2012]
his two children. After 5 years his age will be two
times the sum of their ages. Find the present age of
94. The ages of two friends ani and Biju differ by 3 years. the father.
Ani’s father Dharam is twice as old as ani and Biju

Sol : [Board 2019 Delhi]

is twice as old as his sister Cathy. The ages of Cathy

and Dharam differ by 30 year. Find the ages of Ani 102. Sumit is 3 times as old as his son. Five years later he
and Biju. shall be two and a half times as old as his son. How


Sol : [Board Term-1 2012]
old is Sumit at present?


Sol : [Board 2019 OD]

95. One says, “Give me a hundred, friend! I shall then

become twice as rich as you.” The other replies, “If

COMPETENCEY BASED QUESTION
you give me ten, I shall be six times as rich as you.”
Tell me what is the amount of their (respective)
capital.


Sol : [Board Term-1 2012] 103. Lawn Service : Nitin and his sons run a lawn service,
which includes mowing, edging, trimming, and
aerating a lawn. His fixed cost includes insurance,
96. At a certain time in a deer, the number of heads and
his salary, and monthly payments on equipment, and
the number of legs of deer and human visitors were
amounts to Rs 4000 per month. The variable costs
counted and it was found that there were 39 heads
include gas, oil, hourly wages for his employees, and
and 132 legs.


Sol : [Board Term-1 2015] miscellaneous expenses, which run about Rs 75 per
lawn. The average charge for full service lawn care is
Rs 115 per visit.
97. The length of the sides of a triangle are
(i) How many lawns Nitin must service each month
2x + y2 , 53x + y + 12 and 23 x + 2y + 52 . If the triangle is
to break even ?
equilateral , find its perimeter.


Sol : [Board Term-1 2012] (ii) What is the revenue required to break even ?
(iii) What is the revenue if they get 90 services ?

98. When 6 boys were admitted and 6 girls left, the

percentage of boys increased from 60% to 75%. Find
the original no. of boys and girls in the class.


Sol : [Board Term-1 2015]

99. A cyclist, after riding a certain distance, stopped

for half an hour to repair his bicycle, after which he
completes the whole journey of 30 km at half speed in
5 hours. If the breakdown had occurred 10 km farther
off, he would have done the whole journey in 4 hours.
Page 28 Pair of Linear Equation in Two Variables Chap 3

104. Production of Frying Pan : Due to high market 106. Alumni Contributions : Alumni can help college sustain
demand, a manufacturer decides to introduce a new through their donations and voluntary help. Alumni
line of frying pan. By using existing factory space and can also be helpful in providing valuable financial,
retraining some employees, fixed costs are estimated intellectual and human resource. If a big chunk of
at Rs 84000/mo. The components to assemble and money that institutes require comes from alumni, it
test each frying pan are expected to run Rs 450 per will help those institutes remain competitive.
unit. If market research shows consumers are willing
to pay at least Rs 690 for this product, find
(i) How many units must be made and sold each
month to break even ?
(ii) What is the revenue required to break even ?

105. Theatre Productions : A play is a work of drama,

usually consisting mostly of dialogue between
characters and intended for theatrical performance
rather than just reading. Comedies are plays which
are designed to be humorous. Comedies are often Alumni association of NIT Kuruskhstra donated Rs
filled with witty remarks, unusual characters, and 100,000 to his alma mater. The college used the funds
strange circumstances. Certain comedies are geared to make a loan to a science student at 7% interest and
toward different age groups. a loan to a engineering student at 6% interest. That
year the college earned Rs 6350 in interest.
(i) How much was loaned to engineering student?
(ii) How much was loaned to science student?
(i) Rs 65000 was loaned to the engineering student.
(ii) Rs 35000 was loaned to the science student.

At a recent production of a comedy drama, the

Ravindra Rangmanch Theater brought in a total of
Rs 304950 in revenue. If adult tickets were Rs 90 and
children’s tickets were Rs 65, how many tickets of
each type were sold if 3800 tickets in all were sold?


Sol : 107. Airport Walkways : A moving walkway, also known as
an autowalk, is a slow-moving conveyor mechanism that
transports people across a horizontal or inclined plane
Chap 3 Pair of Linear Equation in Two Variables Page 29

over a short to medium distance. Moving walkways 109. CHEMISTRY : When you mix two or more substances
can be used by standing or walking on them. They are with different levels of concentration, the final solution
often installed in pairs, one for each direction. does not simply equate to the combined concentration
levels of the original ingredients. It depends on the
concentration of each solution.

As part of an algebra field trip, Jenish takes his

class to the airport to use their moving walkways
for a demonstration. The class measures the longest
walkway, which turns out to be 256 ft long. Using a
Rahman works as a chemist in Biolab Pvt Ltd at
stop watch, Jenish shows it takes him just 32 sec to
Jaipur. He has two solutions of hydrochloric acid in
complete the walk going in the same direction as the
stock: a 50% solution and an 80% solution. He want
walkway. Walking in a direction opposite the walkway,
to make 100 milliliters of a 68% solution? How much
it takes him 320 sec (10 times as long!). The next day
of each should be used to obtain 100 milliliters of a
in class, Jenish hands out a two question quiz:
68% solution?
(i) What is my (Jenish’s) normal walking speed?
(ii) What was the speed of the walkway in feet per
110. Nutrition : Pathmeda village near Sanchore has Gopal
second?
Govardhan Gaushala, the largest Gaushala in India,
spread over 200 acres. The gaushala takes care of
108. Nutrition : Shalvi wants to use milk and orange juice more than 18,000 cattle.
to increase the amount of calcium and vitamin A in
her daily diet. An ounce of milk contains 38 milligrams
of calcium and 56 micrograms of vitamin A. An ounce
of orange juice contains 5 milligrams of calcium and
60 micrograms of vitamin A. How many ounces of
milk and orange juice should she drink each day to
provide exactly 550 milligrams of calcium and 1,200
micrograms of vitamin A?

Cows of Pathmeda gaushala in an experiment are to

be kept on a strict diet. Each cow is to receive, among
other things, 20 grams of protein and 6 grams of fat.

The laboratory technician is able to purchase two

Page 30 Pair of Linear Equation in Two Variables Chap 3

food mixes of the following compositions: to fill an order for exactly 4,000 keyboards and exactly
Mix A has 10% protein and 6% fat, 4,000 screens?
Mix B has 20% protein and 2% fat.
Plant Keyboards Screens
How many grams of each mix should be used to obtain
the right diet for a single cow ? Bangalore 40 32
Bhiwadi 20 32
111. Gold Mixing : A jeweller has two bars of gold alloy in
stock, one of 12 carats and the other of 18 carats (24
carat gold is pure gold, 12 carat is 1224 pure, 18 carat
gold is 1824 pure, and so on). How many grams of each
alloy must be mixed to obtain 10 grams of 14 carat
gold?

112. BREAK-EVEN ANALYSIS : It costs a small recording

company Rs 176, 800 to prepare a compact disc. This
is a one-time fixed cost that covers recording, package
design, and so on. Variable costs, including such 114. NUTRITION : Orange trees thrive in warm,
things as manufacturing, marketing, and royalties, are Mediterranean climates where there is no threat
Rs 46 per CD. of frost. This full-sun plant produces the best fruit
(i) If the CD is sold to music shops for Rs 80 each, when provided with optimal moisture, light and
how many must be sold for the company to break nutrition, in the form of fertilization. Orange trees
even? require fertilization three times per year. You need
(ii) What is the break even revenue ? to increase fertilizer amounts as the tree ages and
becomes established. Complete nutrition is essential
for a healthy tree.

113. Production : Ridhima Electronics Pvt . Ltd is main A fruit grower can use two types of fertilizer in an
supplier for CASIO for electronics component. They orange grove, brand A and brand B. Each bag of
manufactures keyboards and screens for graphing brand A contains 8 pounds of nitrogen and 4 pounds
calculators at plants in Bangalore and Bhiwadi. The of phosphoric acid. Each bag of brand B contains 7
hourly production rates at each plant are given in the pounds of nitrogen and 7 pounds of phosphoric acid.
table. How many hours should each plant be operated Tests indicate that the grove needs 720 pounds of
Chap 3 Pair of Linear Equation in Two Variables Page 31

nitrogen and 500 pounds of phosphoric acid. difference of 16 seconds between the arrival of the two
(i) How many bags of brand A should be used to waves.
provide the required amounts of nitrogen and (i) How long did each wave travel ?
phosphoric acid? (ii) How far was the earthquake from the station ?
(ii) How many bags of brand B should be used to
provide the required amounts of nitrogen and
phosphoric acid?

115. Orbital Launches : In 2013 there was a total of 81

commercial and noncommercial orbital launches
worldwide. In addition, the number of noncommercial
orbital launches was twelve more than twice the
number of commercial orbital launches. Determine
the number of commercial and noncommercial orbital
launches in 2013.

Sol :

117. Architectural Wonders : When it was first constructed

in 1889, the Eiffel Tower in Paris, France, was the
tallest structure in the world. In 1975, the CN Tower
in Toronto, Canada, became the world’s tallest
structure. The CN Tower is 153 ft less than twice
the height of the Eiffel Tower, and the sum of their
heights is 2799 ft.


Sol :

(i) How tall is CN tower?

116. EARTH SCIENCE : An earthquake emits a primary (ii) How tall is Eiffel tower?


Sol :
wave and a secondary wave. Near the surface of the
Earth the primary wave travels at about 5 miles per
second and the secondary wave at about 3 miles per 118. TOWER OF PISA : To prove that objects of different
second. From the time lag between the two waves weights fall at the same rate, Galileo dropped two
arriving at a given receiving station, it is possible to objects with different weights from the Leaning Tower
estimate the distance to the quake. (The epicenter can of Pisa in Italy. The objects hit the ground at the
be located by obtaining distance bearings at three or same time.
more stations.) Suppose a station measured a time An object dropped off the top of Leaning Tower of
Pisa falls vertically with constant acceleration. If s
is the distance of the object above the ground (in
Page 32 Pair of Linear Equation in Two Variables Chap 3

feet) t seconds after its release, then s and t are (ii) Find her original speed.
related by an equation of the form s = a + bt2 where
a and b are constants. Suppose the object is 180 feet
120. At some point, it’s time to kick, or gently ease, kids off
above the ground 1 second after its release and 132
the parental gravy train. The circle graph shows the
feet above the ground 2 seconds after its release.
percentage of parents who think significant financial
support should end at various milestones.

(i) Find the constants a and b . The difference in the percentage who would end this
(ii) How high is the Leaning Tower of Pisa? support after completing college and after completing
(iii) How long does the object fall? high school is 6 %.
(i) Find the percentage of parents who would end
119. Jyoti Kumari is an Indian student from Sirhulli in financial support after a child completes college.
the rural Darbhanga district of Bihar. She came to (ii) Find the percentage of parents who would end
notice after she bicycled some 1,200 km with her financial support after a child completes high
injured father to reach their home village during school.
COVID-19 lockdowns in India. This act of bravery
was praised by the Senior Advisor to the President of 121. Cash Register Malfunction : You are the manager of
the United States, Ivanka Trump, and Prime Minister a shoe store. On Sunday morning you are going over
Narendra Modi. She was given a national award, and the receipts for the previous week’s sales. A total of
a Bollywood film was proposed to record her story. 320 pairs of cross-training shoes were sold. One style
sold for Rs 1135 and the other sold for Rs 1495. The
total receipts were Rs 420,480. The cash register that
was supposed to keep track of the number of each
type of shoe sold malfunctioned. Can you recover the
information? If so, how many of each type were sold?

Jyoti travelled 90 km every day to reach her home

town in Harbin. One day, when she started, after
riding a certain distance, she stopped for some time
to repair his bicycle. After which she completes the
whole journey of 90 km at half speed in 12 hours. If
the breakdown had occurred 10 km farther off, she
would have done the whole journey in 11 hours.

Sol :
(i) Find where the breakdown occurred.
Chap 3 Pair of Linear Equation in Two Variables Page 33

122. Hostel Life : Banasthali Vidyapith, is a fully residential they rowed back downstream to the drop in point in
women’s university offering courses from primary just 30 min. Use this information to find
to Ph.D. level. It offers a number of UG, PG, and (i) Find the speed of the current.
Doctoral level Programs under various Departments. (ii) Find the speed Mohinder and Aslam would be
Admission to the same is done on the basis of merit rowing in still water.
scored in qualifying examination, however, for some
courses, an aptitude test is also conducted at the
university level.

Swati is doing MSc. in biotechnology from Banastli

Vidyapith and lives in university hostel. A part of
monthly hostel charge is fixed and the remaining
depends on the number of days one has taken food
in the mess. When Swati takes food for 18 days, she
has to pay Rs. 5160 as hostel charges whereas Taniya
who takes food for 23 days Rs. 5760 as hostel charges.
(i) Find the fixed charges of hostel.

Sol :
(ii) Find the cost of food per day.


Sol : 125. Luxury Cruise : As India’s first domestic cruise liner,
Angriya has made many voyages on the Mumbai-Goa
123. Uniform motion with current : sea route, along the pristine Konkan Coast. It has
(R + C) t = d With the current given India and Indians a sense of pride and happiness,
while introducing the travelers to coral diversity and
(R − C) t = d Against the current
royal sea forts along the way.
The formula shown can be used to solve uniform
motion problems involving a current, where d
represents distance travelled, R is the rate of the
object with no current, C is the speed of the current,
and t is the time. Vibhur rows 9 km up river (against
the current) in 3 hr. It only took him 1 hr to row 5 km
downstream (with the current).
(i) How fast was the current?
(ii) How fast can he row in still water?


Sol :

124. Canoeing on a stream : On a recent camping trip, it Last year we enjoyed our summer vacation at Angariya
took Mohinder and Aslam 2 hr to row 4 mi upstream cruise. From Mumbai to the Goa, with the current
from the drop in point to the camp site. After a the trip took 70 hr. After a few days of fun in the sun,
leisurely weekend of camping, fishing, and relaxation, the ship leaves for Mumbai, against the current with
the return trip taking 82 hr.
Page 34 Pair of Linear Equation in Two Variables Chap 3

(i) Find the speed of the current. raise the money to fund your future. As you save that
(ii) Find the cruising speed of the ship. money, you have to invest it to enable it to grow.


Sol :

126. Point of No Return : The point during a flight at

which an aircraft is no longer capable of returning
to the airfield from which it took off due to fuel
considerations. Beyond this point the aircraft must
proceed to some other destination.

A recently retired couple needs Rs 120,000 per year

to supplement their Social Security. They have Rs
A plane carries enough fuel for 20 hours of flight at 1,500,000 to invest to obtain this income. They have
an airspeed of 150 miles per hour. How far can it fly decided on two investment options: AA bonds yielding
into a 30 mph headwind and still have enough fuel to 10% per annum and a fixed deposit yielding 5%.
return to its starting point? (i) How much should be invested in each to realize


Sol :
exactly Rs 120,000?
(ii) If, after 2 years, the couple requires Rs 140,000
127. Computing a Refund : The grocery store we use does per year in income, how should they reallocate
not mark prices on its goods. My wife went to this their investment to achieve the new amount?
store, purchased three 1-kg packages of almond and
two 500-gram packages cashew, and paid a total of Rs 129. Actual Number of Calories : University of Arkansas
1345. Not knowing that she went to the store, I also researchers discovered that we underestimate the
went to the same store, purchased two 1-kg packages number of calories in restaurant meals. The next time
of almond and three 500-gram packages cashew, and you eat out, take the number of calories you think you
paid a total of Rs 1145. Now we want to return two ate and double it.
1-kg packages of almond and two 500-gram packages
cashew. How much will be refunded?

The researchers concluded that this number should

be a more accurate estimate. The actual number of
calories in one portion of hamburger and fries and
two portions of pizza is 4240. The actual number of
calories in two portions of hamburger and fries and
one portion of pizza is 3980.


Sol : (i) Find the actual number of calories in one portions
of hamburger and fries.
(ii) Find the actual number of calories in one portions
128. Financial Planning : Planning for retirement starts
of pizza.
with thinking about your retirement goals and how
long you have to meet them. Then you need to look
at the types of retirement accounts that can help you 130. Presale Order : A wireless store owner takes presale
orders for a new smartphone and tablet. He gets 340
Chap 3 Pair of Linear Equation in Two Variables Page 35

preorders for the smartphone and 250 preorders for

the tablet. The combined value of the preorders is
Rs 27,050,000. The price of a smartphone and tablet
together is Rs 96500.
(i) How much does smartphone cost?
(ii) How much does tablet cost?
The cost of type A mask is Rs. 15 and of type B mask
is Rs. 20. In the month of April, 2020, the store sold
100 masks for total sales of Rs. 1650.
(i) How many masks of each type were sold in the
month of April?
(ii) If the store had sold 50 masks of each type, what
would be its sales in the month of April?
(iii) Due to great demand and short supply, the store
has increased the price of each type by Rs. 5 from
May 1, 2020. In the month of May, 2020, the store
sold 310 masks for total sales of Rs. 6875. How
many masks of each type were sold in the month
of May?
(iv) What percent of masks of each type sale was
increased in the month of May, compared with


Sol : the sale of month April?
(v) What extra profit did store earn by increasing
price in May month.
131. MASK : Masks are an additional step to help prevent
people from getting and spreading COVID-19. They
provide a barrier that keeps respiratory droplets from 132. Wilt Chamberlain : Wilton Norman “Wilt”
spreading. Wear a mask and take every day preventive Chamberlain was an American basketball player, and
actions in public settings. played in the NBA during the 1960s. At 7 feet 1 inch,
he was the tallest and heaviest player in the league
for most of his career, and he was one of the most
famous people in the game for many years. He is the
first and only basketball player to score 100 points in
an NBA game.

Due to ongoing Corona virus outbreak, Wellness

Medical store has started selling masks of decent
quality. The store is selling two types of masks
currently type A and type B .
In the 1961–1962 NBA basketball season, Wilt
Chamberlain of the Philadelphia Warriors made 30
baskets. Some of the baskets were free throws (worth
1 point each) and some were field goals (worth 2
points each). The number of field goals was 10 more
than the number of free throws.
Page 36 Pair of Linear Equation in Two Variables Chap 3

(i) How many field goals did he make ? (i) Which pair of linear equations does describe this
(ii) How many free throws did he make? situation ?
(iii) What was the total number of points scored? (ii) What is the length of the outer boundary of the
(iv) If Wilt Chamberlain played 5 games during this layout.
season, what was the average number of points (iii) What is the area of bedroom 1 ?
per game? (iv) What is the area of living room in the layout ?
(v) If Wilt Chamberlain played 10 games during this (v) What is the cost of laying tiles in Kitchen at the
season, what was the average number of points rate of Rs. 50 per sq. m ?
per game?

Sol :

133. Architect : An architect is a skilled professional who 134. Mr. RK Agrawal is owner of a famous amusement
plans and designs buildings and generally plays a key park in Delhi. The ticket charge for the park is Rs 150
role in their construction. Architects are highly trained for children and Rs 400 for adult.
in the art and science of building design. Since they
bear responsibility for the safety of their buildings’
occupants, architects must be professionally licensed.

Generally he does not go to park and it is managed by

team of staff. One day Mr Agrawal decided to random
check the park and went there. When he checked the
cash counter, he found that 480 tickets were sold and
Rs 134500 was collected.
(i) Let the number of children visited be x and the
number of adults visited be y . Which of the
following is the correct system of equations that
model the problem ?
(ii) How many children visited the park ?
(iii) How many adults visited the park?
Varsha is a licensed architect and design very
innovative house. She has made a house layout for her (iv) How much amount collected if 300 children and
client which is given below. In the layout, the design 350 adults visited the park?
and measurements has been made such that area of (v) One day total visited children and adults together
two bedrooms and kitchen together is 95 sq. m. is 750 and the total amount collected is Rs 212500.
What are the number of children and adults
visited the park ?


Sol :

135. Dipesh bought 3 notebooks and 2 pens for Rs. 80. His
friend Ramesh said that price of each notebook could
be Rs. 25. Then three notebooks would cost Rs.75,
the two pens would cost Rs. 5 and each pen could be
for Rs. 2.50. Another friend Amar felt that Rs. 2.50
Chap 3 Pair of Linear Equation in Two Variables Page 37

for one pen was too little. It should be at least Rs. 16. Last year we visited Jodhpur in a group of 25 friends.
Then the price of each notebook would also be Rs.16. When we went Mehrangarh fort we found following
fare for ride :

Ride Normal Hours Fare Peak Hours Fare

Horse Rs 50 3 Times
Elephant Rs 100 2 Times
Some people choose to ride on horse and rest choose
to ride on elephant.
(i) First day we rode in normal hours and we paid
Rs 1950 for ride. Let x be the number of horses
hired and y be the number elephants hired.
Which of the following is the correct system of
equation that model the problem ?
(ii) How many horses were hired ?
(iii) How many elephant were hired ?
Aditya also bought the same types of notebooks and (iv) Next day we rode in peak hours, then how much
pens as Dipesh. He paid 110 for 4 notebooks and 3 total fare was paid by our group?
pens. (v) What was the increase in total fare because of
(i) Whether the estimation of Ramesh and Amar is peak hours ride ?


Sol :
applicable for Aditya?
(ii) Let the cost of one notebook be x and that of
pen be y . Which of the following set describe the
given problem ?
(iii) What is the exact cost of the notebook?
(iv) What is the exact cost of the pen?
(v) What is the total cost if they purchase the same
type of 15 notebooks and 12 pens.

136. Jodhpur is the second-largest city in the Indian state

of Rajasthan and officially the second metropolitan
city of the state. Jodhpur was historically the capital
of the Kingdom of Marwar, which is now part of
Rajasthan. Jodhpur is a popular tourist destination,
featuring many palaces, forts, and temples, set in the
stark landscape of the Thar Desert. It is popularly
known as the “Blue City” among people of Rajasthan
and all over India. The old city circles the Mehrangarh
Fort and is bounded by a wall with several gates. The
city has expanded greatly outside the wall, though,
over the past several decades. Jodhpur is also known
for the rare breed of horses known as Marwari or
Malani, which are only found here.