Pair of linear equation in two variables Class 10 (Questions)

📝 Pair of linear equation in two variables

Multiple Choice questions 

1. Graphically, the pair of equations

  `6x – 3y + 10 = 0`

     `2x – y + 9 = 0`

represents two lines which are

(A) intersecting at exactly one point. 

(B) intersecting at exactly two points.

(C) coincident. 

(D) parallel.

Ans: (D)

2. The pair of equations  `x + 2y + 5 = 0` and `–3x – 6y + 1 = 0` have

(A) a unique solution 

(B) exactly two solutions

(C) infinitely many solutions 

(D) no solution

Ans: (D)

3. If a pair of linear equations is consistent, then the lines will be

(A) parallel

(B) always coincident

(C) intersecting or coincident 

(D) always intersecting

Ans: (C)

4. The pair of equations `y = 0` and `y = –7` has

(A) one solution 

(B) two solutions

(C) infinitely many solutions 

(D) no solution

Ans: (D)

5. The pair of equations `x = a` and `y = b` graphically represents lines that are

(A) parallel 

(B) intersecting at `(b, a)`

(C) coincident 

(D) intersecting at `(a, b)`

Ans: (D)

6. For what value of k, do the equations  `3x – y + 8 = 0` and  `6x – ky = –16` represent coincident lines?

(A) `1/2`

(B) `1/2`

(C)  `2`

(D)  `–2`

Ans: (C)

7. If the lines given by `3x + 2ky = 2` and `2x + 5y + 1 = 0` are parallel, then the value of `k` is

(A) `–5/4`

(B) `2/5`

(C) `15/4`

(D) `3/2`

Ans: (C)

8. The value of c for which the pair of equations `cx – y = 2` and `6x – 2y = 3` will have infinitely many solutions is

(A)  3 

(B)  – 3 

(C)  –12 

(D)  no value

Ans: (D)

9. One equation of a pair of dependent linear equations is `–5x + 7y = 2`. The second equation can be

(A) `10x + 14y + 4 = 0` 

(B) `–10x – 14y + 4 = 0`

(C) `–10x + 14y + 4 = 0`

(D) `10x – 14y = –4`

Ans: (D)

10. A pair of linear equations that has a unique solution `x = 2, y = –3` is

(A)  `x + y = –1 ,  2x – 3y = –5`

(B) `2x + 5y = –11,  4 x + 10y = –22`

(C) `2x – y = 1` , `3x + 2y = 0`

(D)  `x – 4y –14 = 0`, `5x – y – 13 = 0`

Ans: (D)

11. If x = a, y = b is the solution of the equations `x – y = 2` and `x + y = 4`, then the values of a and b are, respectively

(A) 3 and 5 

(B) 5 and 3

(C) 3 and 1 

(D) –1 and –3

Ans: (C)

12. Aruna has only  ₹1 and  ₹2 coins with her. If the total number of coins that she has is 50 and the amount of money with her is  ₹75, then the number of ₹1 and  ₹2 coins are, respectively

(A) 35 and 15 

(B) 35 and 20

(C) 15 and 35 

(D) 25 and 25

Ans: (D)

13. The father’s age is six times his son’s age. Four years hence, the age of the father will be four times his son’s age. The present ages, in years, of the son and the father, are, respectively

(A) 4 and 24

(B) 5 and 30

(C) 6 and 36 

(D) 3 and 24

Ans: (C)

14. Which of these linear equations have a unique solution?

(a)

(b) 

(c) 

(d) 

Ans: (b)

15. Consider the graph shown

(a) These lines have a unique solution as they are intersecting at a point.

(b) These lines have infinitely many solutions as they lie in the same quadrant.

(c) These lines have a unique solution as the coefficient of x in both the equations is one.

(d) These lines have infinitely many solutions as they lie in the same quadrant.

Ans: (a)

16. Consider the equations as shown:

`9x + 6y = 5`

`3x + 2y = 7`

Ans: (Option 4)

17. The value of k for which the lines `5x+7y=3 and 15x + 21y = k` coincide is

(a) 9

(b) 5

(c) 7

(d) 18

Ans: (9)

18. The lines `x = a` and `y = b`, are

(a) intersecting

(b) parallel

(c) overlapping

(d) (None of these)

Ans: (a) Lines `x=a` is a line parallel to `y` axis and `y=b` is a line parallel to `x` axis. So they will intersect

19. Given below is the graph representing two linear equations by lines AB and CD respectively. What is the area of the triangle formed by these two lines and the line `x=0`?

(a) 3sq. units

(b) 4sq. units

(c) 6sq. units

(d) 8sq. units

Ans: (c)

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Case Studies 

Case Study 1

Amit is planning to buy a house and the layout is given below. The design and the measurement have been made such that areas of two bedrooms and kitchen together is 95 sq.m.

Based on the above information, answer the following questions:

1. Form the pair of linear equations in two variables from this situation.

2. Find the length of the outer boundary of the layout.

3. Find the area of each bedroom and kitchen in the layout.

4. Find the area of the living room in the layout.

5. Find the cost of laying tiles in the kitchen at the rate of  ₹50 per sq.m

Ans: (1)

Area of two bedrooms= `10x  sq.m`

Area of kitchen = `5y  sq.m`

`10x + 5y = 95`

`2x + y =19`

Also, `x + 2+ y = 15`

`x + y = 13`

Ans: (2)

 Length of outer boundary= 12 + 15 + 12 + 15= 54m 

Ans: (3)

On solving two equation part(i)

`x= 6m` and `y =7m`

area of bedroom = `5 × 6= 30m`

area of kitchen = `5 × 7= 35m`

Ans: (4)

 Area of living room = `(15× 7)-30 = 105-30 = 75  sq.m`

Ans: (5)

Total cost of laying tiles in the kitchen = `₹50 × 35 = ₹1750`

Case Study 2
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 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 8km, the charge paid is ₹91 and for a journey of 14km, the charge paid is  ₹145.

Refer situation 1

(i) If the fixed charges of auto-rickshaw be  `₹x` and the running charges be  `₹y` km/hr, the pair of linear equations representing the situation is

a) `x + 10y =110, x + 15y = 75`

b) `x + 10y =75, x + 15y = 110`

c) `10x + y =110, 15x + y = 75`

d) `10x + y = 75, 15 x + y =110`

Ans: (B)

(ii)  A person travels a distance of 50km. The amount he has to pay is

a)  ₹155

b)  ₹255

c)  ₹355

d)  ₹455

Ans: (c)

Refer situation 2

(iii) What will a person have to pay for travelling a distance of 30km?

a) ₹185

b) ₹289

c) ₹275

d) ₹305

Ans: (b)

(iv) The graph of lines representing the conditions are: (situation 2)

Ans: (c)

Case Study 3

A test consists of ‘True’ or ‘False’ questions. One mark is awarded for every correct answer while ¼ mark is deducted for every wrong answer. A student knew answers to some of the questions. Rest of the questions he attempted by guessing. He answered 120 questions and got 90 marks.

(i) If answer to all questions he attempted by guessing were wrong, then how many questions did he answer correctly?

(ii) How many questions did he guess?

(iii) If answer to all questions he attempted by guessing were wrong and answered 80 correctly, then how many marks he got?

(iv) If answer to all questions he attempted by guessing were wrong, then how many questions answered correctly to score 95 marks?

Ans: 

Let the no of questions whose answer is known to the student `x` and questions attempted by cheating be `y`

`x + y =120`

`x-1/4y =90`

solving these two

`x=96` and `y= 24`

(i) He answered 96 questions correctly.

(ii) He attempted 24 questions by guessing.

(iv) Marks = `80- ¼` of `40 =70`

(v) `x – ¼` of `(120-x) =95`

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1. A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km down-stream. Determine the speed of the stream and that of the boat in still water.

Ans: The speed of the boat in still water is 8 km/h and the speed of the stream is 3 km/h.