CBSE Class 10 Mathematics Pair of Linear Equations in Two Variables VBQs

Question. If 217𝑥 + 131𝑦 = 913 and 131𝑥 + 227𝑦 = 827 then 𝑥 + 𝑦 =
a) 5
b) 6
c) 7
d) 18
Answer : A

Question. The pair of linear equations 3𝑥 + 5𝑦 = 3 𝑎𝑛𝑑 6𝑥 + 𝑘𝑦 = 8 do not have a solution if k
a) = 5
b) = 10
c) ≠10
d) ≠ 5
Answer : B

Question. The pair of equations 𝑦 = 0 𝑎𝑛𝑑 𝑦 = − 5 has
a) One solution
b) two solution
c) infinite no.of solution
d) no solution
Answer : B

Question. The pair linear equations 𝑥 − 𝑦 = 1 𝑎𝑛𝑑 𝑥 + 𝑘𝑦 = 5 has a unique solution when 𝑥 = 2,𝑦 = 1 𝑡ℎ𝑒𝑛 𝑘 =
a) -2
b) 3
c) -3
d) 4
Answer : B

Question. The pair of linear equations 3𝑥 + 7𝑦 = 𝑘; 12 𝑥 + 2𝑘𝑦 = 4𝑘 + 1 do not have any solution if
a) k = 7
b) k = 14
c) k = 21
d) k = 28
Answer : B

Question. The pair of equation 𝑥 = 𝑎,𝑦 = 𝑏 represent lines which are
a) Parallel
b) intersect at (b, a)
c) coincide
d) intersect at( a, b)
Answer : D

Question. The pair of linear equations 𝑘𝑥 + 4𝑦 = 5,3𝑥 + 2𝑦 = 5 is consistent only when
a) K = 9
b) k = -9
c) k ≠ -9
d) k ≠ 7
Answer : A

Case Based Questions

Read the following text and answer the following questions on the basis of the same:
It is common that governments revise travel fares from time to time based on various factors such as inflation (a general increase in prices and fall in the purchasing value of money) on different types of vehicles like auto, Rickshaws, taxis, Radio cab etc. The auto charges in a city comprise of a fixed charge together with the charge for the distance covered. Study the following situations

SITUATION 1: In a city A, for a journey of 10 km, the charge paid is ₹75 and for a journey of 15 km, the charge paid is ₹110.
SITUATION 2: In a city B, for a journey of 8 km, the charge paid is ₹91 and for a journey of 14 km, the charge paid is ₹145.

REFER SITUATION 1

Question. If the fixed charges of auto rickshaw be ₹ x and the running charges be ₹ y km/hr. then write the pair of linear equation representing the situation is …
Answer : X+10y=75, x+15y=75………

Question. A person travels a distance of 50 km. find the amount paid by Him…………
Answer : Rs. 355

REFER SITUATION 2

Question. What will a person have to pay for travelling a distance of 30 km?
Answer : Rs. 289

Question. The graph of lines representing the conditions are …..
Answer : Draw graph

Read the following text and answer the following questions on the basis of the same:
TOWER OF PISA : To prove that objects of different weights fall at the same rate, Galileo dropped two objects with different weights from the Leaning Tower of Pisa in Italy. The objects hit the ground at the same 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 feet) t seconds after its release, then s and t are related by an equation of the form s = a + bt2 where a and b are constants. Suppose the object is 180 feet above the ground 1 second after its release and 132 feet above the ground 2 seconds after its release.

Question. How long does the object fall?
Answer : 3.5 sec

Question. Find the constants a and b.
Answer : a =196 , b = -16

Question. How high is the Leaning Tower of Pisa?
Answer : 196 feet

Read the following text and answer the following questions on the basis of the same:
Place A and B are 100 km apart on a highway. One car starts from A and another form B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each other, they meet in 1 hour.

Question. What is the actual speed of the car?
Answer : 60km/h

Question. Assuming that the speed of first car and second car be u km/h and v km/h respectively. What is the relative speed of both cars while they are travelling in the same direction?
Answer : (u-v)km/h

Question. What is the actual speed of the other car?
Answer : 40km/h

Question. What is the relative speed of both cars while they are travelling towards each other?
Answer : (u+v)km/h

Read the following text and answer the following questions on the basis of the same:
Architect : An architect is a skilled professional who plans and designs buildings and generally plays a key role in their construction. Architects are highly trained in the art and science of building design. Since they bear responsibility for the safety of their buildings’ occupants, architects must be professionally licensed.
Varsha is a licensed architect and design very innovative house. She has made a house layout for her client which is given below. In the layout, the design and measurements has been made such that area of two bedrooms and kitchen together is 95 sq. m.

Question. What is the area of living room in the layout?
Answer : 75 sq.m

Question. What is the length of the outer boundary of the layout?
Answer : 54m

Question. What is the cost of laying tiles in Kitchen at the rate of Rs. 50 per sq.
Answer : Rs. 1750

Question. Which pair of linear equations does describe this situation?
Answer : 2x+y=19 and x+y=13

Question. What is the area of bedroom 1 ?
Answer : Area of bedroom = 30sq.m  Area of kitchen = 35sq.m

Read the following text and answer the following questions on the basis of the same:
John and Jivanti are playing with the marbles in the playground. They together have 45 marbles and John has 15 marbles more than Jivanti.

Question. If 45 is replaced by 55 in the above case discussed in the question, then the number of marbles Jivanti have……………..
Answer : 20

Question. The number of marbles Jivanti had……………
Answer : 15

Question. According to Question 3, the number of marbles John have:
Answer : 35

Question. The number of marbles John had………………
Answer : 30

Very Short Answer Type Questions

Question. A fraction becomes 9/11, if 2 is added to both the numerator and the denominator. If, 3 is added to both the numerator and the denominator it becomes 5/6. Find the fraction.
Answer : Fraction = 7/9

Question. Solve the following pair of linear equations by the elimination method and the substitution method :
(i) 3x + 4y = 10 and 2x – 2y = 2
(ii) 3x – 5y – 4 = 0 and 9x = 2y + 7
(iii) x/2 + 2y/3 = - 1 and x - (y - 3) = 3
Answer : (i) x = 2, y = 1 (ii) = 9/13, y = - (5/13) (iii) x = 2, y = –3

Question. On comparing the ratios a¹/a², b¹/b² and c¹/c², find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident:
(i) 5x – 4y + 8 = 0 , 7x + 6y – 9 = 0
(ii) 9x + 3y + 12 = 0, 18x + 6y + 24 = 0
(iii) 6x – 3y + 10 = 0, 2x – y + 9 = 0
Answer : (i) Intersect at a point (ii) Coincident (iii) Parallel

Question. Five years hence, the age of Jacob will be three times that of his son. Five years ago, Jacob’s age was seven times that of his son. What are their present ages?
Answer : Jacob’s age = 40 years, son’s age = 10 years.

Question. The coach of a cricket team buys 7 bats and 6 balls for Rs 3800. Later, she buys 3 bats and 5 balls for ₹ 1750. Find the cost of each bat and each ball.
Answer : cost of one bat = ₹500, cost of one ball = ₹50.

Question. Solve the following pair of linear equations by the substitution method :
(i) 3x – y = 3, 9x – 3y = 9
(ii) 0.2x + 0.3y = 1.3, 0.4x + 0.5y = 2.3
(iii) √2 x+ √3 y= 0, √3 x − √8 y = 0
Answer : (i) infinitely many solutions. (ii) x = 2, y = 3 (iii) x = 0, y = 0

Question. The difference between two numbers is 26 and one number is three times the other. Find them.
Answer : 39, 13.

Question. Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.
Answer : Length = 20 m and breadth = 16 m.

Question. The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.
Answer : 18

Question. The larger of two supplementary angles exceeds the smaller by 18 degrees. Find them.
Answer : 99, 81.

Question. A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid ₹ 27 for a book kept for seven days, while Susy paid ₹ 21 for the book she kept for five days. Find the fixed charge and the charge for each extra day.
Answer : Fixed charge per day = ₹ 15and the additional charge per day = ₹3.

Question. On comparing the ratios a¹/a², b¹/b² and c¹/c², find out whether the following pair of linear equations are consistent, or inconsistent.
(i) 5x – 3y = 11 ; – 10x + 6y = –22
(ii) 2x – 3y = 8 ; 4x – 6y = 9
(iii) (3/2)x + (5/3)y = 7 : 9x – 10y = 14
Answer : (i) Consistent (ii) inconsistent (iii) Consistent

Question. The taxi charges in a city consist of a fixed charge together with the charge for the distance covered. For a distance of 10 km, the charge paid is ₹ 105 and for a journey of 15 km, the charge paid is ₹ 155. What are the fixed charges and the charge per km? How much does a person have to pay for travelling a distance of 25 km?
Answer : Fixed charge = ₹5, charge per km = ₹10; travelling charge = ₹255.

Question. Meena went to a bank to withdraw ₹ 2000. She asked the cashier to give her ₹ 50 and ₹ 100 notes only. Meena got 25 notes in all. Find how many notes of ₹ 50 and ₹ 100 she received.
Answer : Number of Rs 50 = 10, Nnumber of Rs 100 notes = 15.

Question. Half the perimeter of a rectangular garden, whose length is 4 m more then its width, is 36 m. Find the dimensions of garden.
Answer : Length =20m and width =16m

Question. Given the linear equation 3x+4y = 9. Write another linear equation in these two variables such that the geometrical representation of the pair so formed is:
(1) intersecting lines (2) coincident lines.
Answer : One of the possible equation 3x-5y=10
One of the possible equation 6x+8y=18

Question. Find the value(s) of k for which the pair of linear equations kx + y = k2 and x + ky = 1 have infinitely many solutions.
Answer : K=1 or k= -1

Question. Two lines are given to be parallel. The equation of one of the lines is 4x + 3y = 14, then find the equation of the second line.
Answer : One of the possible solution 12x+9y=5

Question. For what value of p does the pair of linear equations given below has unique solution?
4 x+ py +8 = 0 and 2 x+2 y + 2 =0.
Answer : p≠4

Question. Determine graphically whether the following pair of linear equations :
3x - y = 7
2x + 5y + 1 = 0 has : unique solution infinitely many solutions or no solution.
Answer : Unique solution

Question. A fraction become 9/11 if 2 is added to both numerator and denominator. If 3 is added to both numerator and denominator it becomes 5/6 . find the fraction.
Answer : 79

Question. The ratio of incomes of two persons is 11:7 and the ratio of their expenditures is 9:5. If each of them manages to save Rs 400 per month, find their monthly incomes.
Answer : 2200 and1400

Question. 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.
Answer : x=6 ,y=4

Question. 2 man and 7 boys can do a piece of work in 4 days. It is done by 4 men and 4 boys in 3 days. How long would it take for one man or one boy to do it ?
Answer :15 days

Question. Draw the graph of the following equations:
2x - y = 1, x + 2y = 13 ,Find the solution of the equations from the graph and shade the triangular region formed by the lines and the y -axis.
Answer : Draw the graph

Question. Solve graphically: 2x − 3y + 13 = 0; 3x − 2y + 12 = 0
Answer : X= -2 and y=3

Question. Find the value of k for which the following pair of equations has no solution :
x + 2y = 3, (k-1)x+(k+1)y=(k+2)
Answer : K=3

Question. Is the system of linear equations 2 x+3 y − 9 =0 and 4 x+6 y − 18 =0 consistent? Justify your answer.
Answer : Consistent

Question. Solve x + y = 5 and 2x − 3y = 4 by elimination method and the substitution method.
Answer : X=19/5 and y=6/5

Question. Solve the following pair of linear equations graphically: x - y = 1, 2x + y = 8. Also find the co-ordinates of the points where the lines represented by the above equation intersect y - axis.
Answer : Draw the graph

Question. Solve : 99x + 101y = 499, 101x + 99y = 501
Answer : X=2 , y=3