Question

Assertion(A):the linear equation x-2y-3=0 and3x+4y-20=0 have exactly one solution.

Reason(R): the linear equation 2x+3y-9=0 and 4x+6y-18=0 have a unique solution.

a) Both assertion (A )and reason (R) are true and reason (R) is the correct explanation of

assertion (A).

b) Both assertion (A )and reason (R) are true But reason (R) is the not the correct explanation

of assertion (A).

c) Assertion (A ) is true But reason (R) is False.
d) Assertion (A ) is False But reason (R) is True.​

Answer 1

Answer:

c

Step-by-step explanation:

step-by-step answer:

Let us first consider the assertion. It says that the linear equations x−2y−3=0 and 3x+4y−20=0 have exactly one solution.

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=1110

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

x−2×1110−3=0

⟹x=2210+3=22+3010=5210=265

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−18=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.

Hence, option C is correct.

Answer 2

Option (c) Assertion (A) is true but reason (R) is false.

Solution: For a pair of linear equations ax+by+c=0 and px+qy+r=0 to have an unique solution, the necessary condition to be satisfied is:

$\frac{a}{p} ≠ \frac{b}{q}$

In assertion, a= 1, b= -2, p= 3 and q=4.

a/p = 1/3

b/q= -2/4= -1/2

Since both are not equal to each other, they have exactly one solution.

In reasoning, a= 2, b= 3, c= -9, p= 4, q= 6 and r= -18

a/p= 2/4= 1/2

b/q= 3/6= 1/2

c/r = -9/-18= 1/2

Since both are equal, they do not have a unique solution. Here, a/p = b/q = c/r which is a condition for infinite solutions.

Therefore, this pair of linear equations have infinite solutions.