CBSE Class 10 Pair of Linear Equations MCQ New Pattern By Clarify Knowledge

CBSE Class 10 Pair of Linear Equations MCQ New Pattern 2022

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CBSE Class 10 Pair of Linear Equations MCQ Table

1. The pairs of equations x+2y-5 = 0 and -4x-8y+20=0 have:

(a) Unique solution

(b) Exactly two solutions

(c) Infinitely many solutions

(d) No solution

Answer: (c) Infinitely many solutions

Explanation:

a₁/a₂ = 1/-4

b₁/b₂ = 2/-8 = 1/-4

c₁/c₂ = -5/20 = -¼

This shows:

a₁/a₂ = b₁/b₂ = c₁/c₂

Therefore, the pair of equations has infinitely many solutions.

2. If a pair of linear equations is consistent, then the lines are:

(a) Parallel

(b) Always coincident

(c) Always intersecting

(d) Intersecting or coincident

Answer: (d) Intersecting or coincident

Explanation: Because the two lines definitely have a solution.

3. The pairs of equations 9x + 3y + 12 = 0 and 18x + 6y + 26 = 0 have

(a) Unique solution

(b) Exactly two solutions

(c) Infinitely many solutions

(d) No solution

Answer: (d) No solution

Explanation: Given, 9x + 3y + 12 = 0 and 18x + 6y + 26 = 0

a₁/a₂ = 9/18 = 1/2

b₁/b₂ = 3/6 = 1/2

c₁/c₂ = 12/26 = 6/13

Since, a₁/a₂ = b₁/b₂ ≠ c₁/c₂

So, the pairs of equations are parallel and the lines never intersect each other at any point, therefore there is no possible solution.

4. If the lines 3x+2ky – 2 = 0 and 2x+5y+1 = 0 are parallel, then what is the value of k?

(a) 4/15

(b) 15/4

(c) ⅘

(d) 5/4

Answer: (b) 15/4

Explanation: The condition for parallel lines is:

a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Hence, 3/2 = 2k/5

k=15/4

5. If one equation of a pair of dependent linear equations is -3x+5y-2=0. The second equation will be:

(a) -6x+10y-4=0

(b) 6x-10y-4=0

(c) 6x+10y-4=0

(d) -6x+10y+4=0

Answer: (a) -6x+10y-4=0

Explanation: The condition for dependent linear equations is:

a₁/a₂ = b₁/b₂ = c₁/c₂

For option a,

a₁/a₂ = b₁/b₂ = c₁/c₂= ½

6.The solution of the equations x-y=2 and x+y=4 is:

(a) 3 and 1

(b) 4 and 3

(c) 5 and 1

(d) -1 and -3

Answer: (a) 3 and 1

Explanation: x-y =2

x=2+y

Substituting the value of x in the second equation we get;

2+y+y=4

2+2y=4

2y = 2

y=1

Now putting the value of y, we get;

x=2+1 = 3

Hence, the solutions are x=3 and y=1.

7. A fraction becomes 1/3 when 1 is subtracted from the numerator and it becomes 1/4 when 8 is added to its denominator. The fraction obtained is:

(a) 3/12

(b) 4/12

(c) 5/12

(d) 7/12

Answer: (c) 5/12

Explanation: Let the fraction be x/y

So, as per the question given,

(x -1)/y = 1/3 => 3x – y = 3…………………(1)

x/(y + 8) = 1/4 => 4x –y =8 ………………..(2)

Subtracting equation (1) from (2), we get

x = 5 ………………………………………….(3)

Using this value in equation (2), we get,

4×5 – y = 8

y= 12

Therefore, the fraction is 5/12.

8. The solution of 4/x+3y=14 and 3/x-4y=23 is:

(a) ⅕ and -2

(b) ⅓ and ½

(c) 3 and ½

(d) 2 and ⅓

Answer: (a) ⅕ and -2

Explanation: Let 1/x = m

4m + 3y = 14

3m – 4y = 23

By cross multiplication we get;

m/(-69-56) = y/(-42-(-92)) = 1/(-16-9)

m/-125=y/50=-1/25

m/-125 = -1/25 and y/50=-1/25

m=5 and y=-2

m=1/x or x=1/m = ⅕

9. Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Her speed of rowing in still water and the speed of the current is:

(a) 6km/hr and 3km/hr

(b) 7km/hr and 4km/hr

(c) 6km/hr and 4km/hr

(d) 10km/hr and 6km/hr

Answer: (c) 6km/hr and 4km/hr

Explanation: Let, Speed of Ritu in still water = x km/hr

Speed of Stream = y km/hr

Now, speed of Ritu, during,

Downstream = x + y km/h

Upstream = x – y km/h

As per the question given,

2(x+y) = 20

Or x + y = 10……………………….(1)

And, 2(x-y) = 4

Or x – y = 2………………………(2)

Adding both the equations, we get,

2x=12

x = 6

Putting the value of x in eq.1, we get,

y = 4

Therefore,

Speed of Ritu is still water = 6 km/hr

Speed of Stream = 4 km/hr

10. The angles of cyclic quadrilaterals ABCD are: A = (6x+10), B=(5x)°, C = (x+y)° and D=(3y-10)°. The value of x and y is:

(a) x=20° and y = 10°

(b) x=20° and y = 30°

(c) x=44° and y=15°

(d) x=15° and y=15°

Answer: (b) x=20° and y = 30°

Explanation: We know, in cyclic quadrilaterals, the sum of the opposite angles are 180°.

Hence,

A + C = 180°

6x+10+x+y=180 =>7x+y=170°

And B+D=180°

5x+3y-10=180 =>5x+3y=190°

By solving the above two equations we get;

x=20° and y = 30°.

11. The pair of equations x = a and y = b graphically represents lines which are

(a) parallel 

(b) intersecting at (b, a)

(c) coincident 

(d) intersecting at (a, b)

Answer: (d) intersecting at (a, b)

The pair of equations x = a and y = b graphically represents lines which are intersecting at (a, b).

12. The pair of equations 5x – 15y = 8 and 3x – 9y = 24/5 has

(a) one solution 

(b) two solutions 

(c) infinitely many solutions

(d) no solution

Answer: (c) infinitely many solutions

Explanation:

The given pair of equations are 5x – 15y = 8 and 3x – 9y = 24/5.

Comparing with the standard form,

a₁ = 5, b₁ = -15, c₁ = -8

a₂ = 3, b₂ = -9, c₂ = -24/5

a₁/a₂ = 5/3

b₁/b₂ = -15/-9 = 5/3

c₁/c₂ = -8/(-24/5) = 5/3

Thus, a₁/a₂ = b₁/b₂ = c₁/c₂

Hence, the given pair of equations has infinitely many solutions.

13. 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

Answer: (d) no solution

Explanation:

Given pair of equations are x + 2y + 5 = 0 and –3x – 6y + 1 = 0.

Comparing with the standard form,

a₁ = 1, b₁ = 2, c₁ = 5

a₂ = -3, b₂ = -6, c₂ = 1

a₁/a₂ = -1/3

b₁/b₂ = 2/-6 = -1/3

c₁/c₂ = 5/1

Thus, a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Hence, the given pair of equations has no solution.

14. 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

Answer: (d) no value

Explanation:

Given pair of equations are cx – y = 2 and 6x – 2y = 3.

Comparing with the standard form,

a₁ = c, b₁ = -1, c₁ = -2

a₂ = 6, b₂ = -2, c₂ = -3

a₁/a₂ = c/6

b₁/b₂ = -1/-2 = 1/2

c₁/c₂ = -2/-3 = ⅔

Condition for having infinitely many solutions is

a₁/a₂ = b₁/b₂ = c₁/c₂

c/6 = ½ = ⅔

Therefore, c = 3 and c = 4

Here, c has different values.

Hence, for no value of c the pair of equations will have infinitely many solutions.

15. If the lines representing the pair of linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 are coincident, then

(a) a₁/a₂ = b₁/b₂

(b) a₁/a₂ = b₁/b₂ = c₁/c₂

(c) a₁/a₂ ≠ b₁/b₂

(d) a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Answer: (b) a₁/a₂ = b₁/b₂ = c₁/c₂

If the lines representing the pair of linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 are coincident, then a₁/a₂ = b₁/b₂ = c₁/c₂.