Assertion & Reason Test: Pair of Linear Equations in Two Variables - Class 10 MCQ

Given that, ∠C = 3∠B = 2(∠A + ∠B)

3∠B = 2(∠A + ∠B)

3∠B = 2∠A + 2∠B

∠B = 2∠A

2 ∠A − ∠B = 0 ...(i)

We know that the sum of the measures of all angles of a triangle is 180°.

Therefore, ∠A + ∠B + ∠C = 180°

∠A + ∠B + 3∠B = 180°

∠A + 4∠B = 180° ...(ii)

Multiplying equation (i) by 4, we obtain

8∠A − 4∠B = 0 ...(iii)

Adding equations (ii) and (iii), we obtain

9∠A = 180°

∠A = 20°

∴ Assertion is correct.

In case of reason: Given that, x, y and 40° are the angels of a triangle.

x + y + 40° = 180°

[Since the sum of all the angels of a triangle is 180°.]

⇒ x + y = 140° ...(i)

Also, x – y = 30° ...(ii)

On adding Eqs. (i) and (ii), we get

2x = 170°

⇒ x = 170/2

∴ x = 85°

On putting x = 85° in Eq. (i), we get

85° + y = 140°

y = 140° – 85° = 55°

∴ y = 55°

Hence, the required values of x and y are 85° and 55°, respectively.

∴ Reason is correct.

Hence, both assertion and reason are correct but reason is not the correct explanation for assertion.

Let x − 2y − 3 = 0--- (1) And, 3x + 4y − 20 = 0---(2)

In order to solve these equations, let us multiply the first equation by 3.

3(x − 2y − 3) = 3 × 0

⟹ 3x − 6y − 9 = 0 ----(3)

Subtracting equation 3 from equation 2, we get,

3x + 4y − 20 − (3x − 6y − 9) = 0

Removing the brackets, we get,

3x + 4y − 20 − 3x + 6y + 9 = 0

⟹ 10y − 11 = 0

⟹ y = 11 / 10

Now, in order to find the value of x, substituting the value of y in equation 1, we get,

Thus, the pair of linear equations given possess exactly one solution (unique solution).

Hence, the assertion is correct.

Now, let us consider the reason. It says that the linear equations 2x + 3y − 9 = 0 and 4x + 6y − 8 = 0 have a unique solution.

Let 2x + 3y − 9 = 0---(1)

And, 4x + 6y − 18 = 0---(2)

In order to solve these equations, let us multiply the first equation by 2.

2(2x + 3y − 9) = 2 × 0

⟹4x + 6y − 18=0---(3)

As, equation 2 and 3 are same thus, thus the two linear equations given to us are coincident possessing infinitely many solutions.

Thus, the reason is not correct.

Thus, Assertion is correct but the Reason is incorrect.

So, Reason is correct For Assertion, we have, a₁ = k, b₁ = -1, c₁ = -2, a₂ = 6, b₂ = -2 and c₂ = -3

So, Assertion is not correct.

By elimination method,

x + y = 5 ...(i)

2x – 3y = 4 ...(ii)

Multiplying equation (i) by (ii), we obtain

2x + 2y = 10 ...(iii)

Subtracting equation (ii) from equation (iii), we obtain

5y = 6

y = 6/5 …(iv)

Substituting the value in equation (i), we obtain

∴ x = 19/5 , y = 6/5

∴ Assertion is correct.

In case of reason:

By elimination method,

3x + 4y = 10 ...(i)

2x – 2y = 2 ...(ii)

Multiplying equation (ii) by 2, we obtain

4x – 4y = 4 ...(iii)

Adding equations (i) and (iii), we obtain

7x = 14

x = 2 ...(iv)

Substituting in equation (i), we obtain

6 + 4y = 10

4y = 4

y = 1

Hence, x = 2, y = 1

∴ Reason is correct.

Hence, both assertion and reason are correct but reason is not the correct explanation for assertion.

6x − 8y = k......(2)

If k = 14 in equation (2) ⇒ 6x − 8y = 14

Now, (2) ÷ 2 ⇒ 3x − 4y = 7......(3)

Therefore, the lines represented by (1) and (3) shows that the lines are coincident and there are infinitely many solutions.

Hence, the Assertion is correct.

Clearly, the Reason is correct.

Since both the Assertion and Reason are correct but Reason is not the correct explanation for Assertion.

∴ 2 x 3 + 1 – q² – 3 = 0

⇒ 4 – q² = 0

⇒ q² = 4 ⇒ q = ± 2

So, both A and R are correct and R explains A.

Let cost of 1 chair be ₹ x and cost of 1 table be ₹ y

According to the question,

4x +3y = 2100 ...(i)

and 5x +2y = 1750 ...(ii)

Multiplying eqn. (i) by 2 and eqn. (ii) by 3, we get

8x +6y = 4200 ...(iii)

15x + 6y = 5250 ...(iv)

eqn. (iv) – eqn. (iii)

⇒ 7x = 1050

∴ x = 150

Substituting the value of x in (i) we get y = 500

Thus, the cost of one chair and one table are ₹ 150 and ₹ 500 respectively.

∴ Assertion is correct.

In case of reason:

Let age of father and son be x and y respectively.

Then, x + y = 40 ...(i)

and x = 3y ...(ii)

By solving eqns. (i) and (ii), we get

x = 30 and y = 10

Thus, the ages of father and son are 30 years and 10 years.

∴ Reason is incorrect.

Hence, Assertion is correct but reason is incorrect.

So, both A and R are correct but R does not explain A.

So, Assertion is false. We know that if the lines are intersecting, then it has unique solution.

So, Reason is true.

6x – ky = –16 ...(ii)

For coincident lines,

So, k = 2.

∴ Assertion is correct.

In case of reason:

For parallel lines (or no solution)

∴ Reason is incorrect.

Hence, assertion is correct but reason is incorrect.

6x − 2y − 3 = 0......(2)

For the equation to be inconsistent

⇒ a₁/a₂ = b₁/b₂ ≠ c₁/c₂ ​

⇒ k/6 = -1/-2 ≠ -2/-3

∴ k = 3

Therefore the Assertion is correct.

Clearly, the Reason is correct.

Since both the Assertion and Reason are correct and Reason is a correct explanation of Assertion.

So, Reason is correct

For Assertion, we have, a₁ = 5, b₁ = 4, c₁ = 6, a₂ = 10, b₂ = 8 and c₂ = -12

So, the pair of linear equations has no solution and hence Assertion is correct.

But reason (R) is not the correct explanation of assertion (A).

x – 2y = 8 ...(i)

5x – 10 = c ...(ii)

As a₁/a₂ = b₁/b₂ , so system of linear equations can never have unique solution.

∴ Assertion is incorrect.

In case of reason:

The equation of one line is 4x + 3y = 14.

We know that if two lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 are parallel, then

or

Hence, one of the possible, second parallel line is 12x + 9y = 5.

∴ Reason is correct Hence, assertion is incorrect but reason is correct.

From the given equations, we have

9/18 = 3/6 = 12/24

1/2 = 1/2 = 1/2 i.e, a₁/a₂ = b₁/b₂ = c₁/c₂

So, both A and R are correct and R explains A.

We know that the system of linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 has infinitely many solutions if a₁/a₂ = b₁/b₂ = c₁/c₂.

So, Reason is not correct For Assertion, we have, a₁ = 3, b₁ = -4, c₁ = -7, a₂ = 6, b₂ = -8 and c₂ = -k