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Formation of a Differential Equation from a General Solution

Assertion : ...

Question

Assertion : $3 x - 4 y = 7$ and $6 x - 8 y = k$ have infinite number of solution if $k = 14$ Reason: $a_{1} x + b_{1} y + c_{1} = 0$ and $a_{2} x + b_{2} y + c_{2} = 0$ have a unique solution if $\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$

A

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

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B

Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

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C

Assertion is correct but Reason is incorrect

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D

Assertion is incorrect but Reason is correct

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Solution

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
Let $3 x - 4 y = 7 . . . . . . (1)$
$6 x - 8 y = k . . . . . . (2)$
If $k = 14$ in equation (2) $\Rightarrow 6 x - 8 y = 14$
Now, $(2) \div 2 \Rightarrow 3 x - 4 y = 7 . . . . . . (3)$
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.
The correct answer is B.

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Similar questions

Q. Assertion (A)
The system of equations

3 x - 2 y + 1 = 0 and 6 x - 4 y + 3 = 0

is inconsistent.
Reason (R)
The system of equations

a_{1} x + b_{1} y + c_{1} = 0 and a_{2} x + b_{2} y + c_{2} = 0

has a unique solution when

\frac{a_{1}}{b_{2}} \neq \frac{b_{1}}{b_{2}} .

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.

Q. Assertion (A)
The system of equations

x + y - 8 = 0 and x - y - 2 = 0

has a unique solutions.
Reason (R)
The system of equations

a_{1} x + b_{1} y + c_{1} = 0 and a_{2} x + b_{2} y + c_{2} = 0

has a unique solution when

\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} .

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.

Q. Assertion (A)
The system of equations

2 x + 3 y + 5 = 0 and 4 x + ky + 7 = 0

is inconsistent when k = 6.
Reason (R)
The system of equations

a_{1} x + b_{1} y + c_{1} = 0 and a_{2} x + b_{2} y + c_{2} = 0

is inconsistent when

\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}} .

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.

3 x + 4 y + 5 = 0

6 x + k y + 9 = 0

represent parallel lines if

k = 8

a_{1} x + b_{1} y + c_{1} = 0

a_{2} x + b_{2} y + c_{2} = 0

represent parallel lines if

\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}

Which is the correct answer

Q. Assertion: The compressibility factor for

H_{2}

H e

(1 + \frac{P b}{R T})

Reason: The compressibility factor for

H_{2}

H e

can be derived from Van der Waal's equation.

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