LINEAR EQUATIONS IN TWO VARIABLES - MCQ - EM worksheet | Live ... - Free Printable

Let’s go through each question one by one, solve them step by step, and find the correct answers.

---

1) If linear equation in two variables x + 2y = 3 and 2x + 4y = k coincides then the value of ‘k’ is,

→ For lines to coincide, their ratios must be equal:
a₁/a₂ = b₁/b₂ = c₁/c₂

Here:
Equation 1: x + 2y - 3 = 0 → a₁=1, b₁=2, c₁=-3
Equation 2: 2x + 4y - k = 0 → a₂=2, b₂=4, c₂=-k

Check ratio:
1/2 = 2/4 = (-3)/(-k)
→ 1/2 = 3/k → cross multiply: k = 6

✔ Answer: B) 6

---

2) 2x + 3y - 9 = 0 and 4x + 6y - 18 = 0 This pair of linear equations have ----- solutions.

→ Check ratios:
a₁/a₂ = 2/4 = 1/2
b₁/b₂ = 3/6 = 1/2
c₁/c₂ = -9/-18 = 1/2

All ratios equal → lines coincide → infinite solutions

✔ Answer: D) Infinite

---

3) In the equation x + y = 7, if x = 3 then the value of ‘y’ is,

→ Plug x = 3 into equation:
3 + y = 7 → y = 7 - 3 = 4

✔ Answer: B) 4

---

4) If a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 are the pair of coinciding lines then the ratios of their coefficients are

→ Coinciding lines mean all three ratios are equal:
a₁/a₂ = b₁/b₂ = c₁/c₂

✔ Answer: C) a₁/a₂ = b₁/b₂ = c₁/c₂

---

5) If the pair of equations 3x + 2ky = 2 and 2x + 5y + 1 = 0 are parallel then the value of ‘k’ is

→ Parallel lines: a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Write both in standard form:

Eq1: 3x + 2ky - 2 = 0 → a₁=3, b₁=2k, c₁=-2
Eq2: 2x + 5y + 1 = 0 → a₂=2, b₂=5, c₂=1

Set a₁/a₂ = b₁/b₂:
3/2 = (2k)/5 → Cross multiply: 3×5 = 2×2k → 15 = 4k → k = 15/4

Now check if c₁/c₂ is different:
c₁/c₂ = -2/1 = -2
a₁/a₂ = 3/2 → not equal to -2 → so yes, parallel

✔ Answer: A) 15/4

---

6) The pair of equations 2x – 5y + 4 = 0 and 2x + y – 8 = 0 has

→ Compare slopes or use determinant method.

a₁=2, b₁=-5; a₂=2, b₂=1

Check if a₁/a₂ = b₁/b₂?
2/2 = 1, -5/1 = -5 → Not equal → lines intersect → unique solution

✔ Answer: C) A unique solution

---

7) The values of ‘x’ and ‘y’ when a point lies on the linear equation 2x – 3y = 12

→ Plug in options:

A) x=0, y=-3 → 2(0) -3(-3)=9 ≠12 ✘
B) x=2, y=3 → 4 - 9 = -5 ≠12 ✘
C) x=3, y=-2 → 6 - (-6)=12 ✔
D) x=-2, y=3 → -4 -9 = -13 ≠12 ✘

✔ Answer: C) x = 3, y = -2

---

8) Identify the wrong statement with respect to a pair of linear equations

A) If lines are parallel there is no solution → TRUE
B) If the lines are perpendicular to each other, there is no solution → FALSE! Perpendicular lines still intersect at one point → unique solution
C) Many solutions if the lines coincide → TRUE
D) Unique solution if they intersect → TRUE

So B is wrong.

✔ Answer: B) If the lines are perpendicular to each other, there is no solution

---

9) In the equation 2x – y = 5 if y = 1 then the value of ‘x’ is,

→ Plug y=1:
2x - 1 = 5 → 2x = 6 → x = 3

✔ Answer: A) 3

---

10) The cost of 5 pens and 7 pencils is Rs 50. Which of the following equation describes the above statement,

→ Let pen = x, pencil = y
Then: 5x + 7y = 50

✔ Answer: B) 5x + 7y = 50

---

11) The solution of the equation x – y = 2 and x + y = 4 is

Add both equations:
(x - y) + (x + y) = 2 + 4 → 2x = 6 → x = 3
Plug into x + y = 4 → 3 + y = 4 → y = 1

Solution: (3,1)

✔ Answer: C) 3, 1

---

12) The line represented by 6x – 8y + 8 = 0 and 6x – 6y + 16 = 0

Compare ratios:

a₁=6, b₁=-8, c₁=8
a₂=6, b₂=-6, c₂=16

a₁/a₂ = 6/6 = 1
b₁/b₂ = -8/-6 = 4/3 → not equal → so not parallel or coincident → must intersect

✔ Answer: A) Intersects

---

13) If the equations 4x + Py +8 = 0 and 2x + 2y + 2 = 0 have unique solution then the value of ‘P’ is,

For unique solution: a₁/a₂ ≠ b₁/b₂

a₁=4, a₂=2 → 4/2 = 2
b₁=P, b₂=2 → P/2

So: 2 ≠ P/2 → Multiply both sides by 2: 4 ≠ P → So P can be anything except 4

✔ Answer: B) Except 4

---

14) x + y = 20 and x – y = 4 solve for x and y, and substitute the values of x and y in y = mx + 3 then the value of ‘m’ is,

First, solve system:

Add equations:
(x+y) + (x-y) = 20+4 → 2x = 24 → x=12
Then from x+y=20 → 12+y=20 → y=8

Now plug into y = mx + 3:
8 = m*12 + 3 → 8 - 3 = 12m → 5 = 12m → m = 5/12

✔ Answer: A) 5 / 12

---

Final Answer:
1) B
2) D
3) B
4) C
5) A
6) C
7) C
8) B
9) A
10) B
11) C
12) A
13) B
14) A