Question

For which value of \( k \), there will be an infinite number of solutions for the pair of linear equations \( x + ky = 1 \) and \( kx + y = k^2 \)?

• A. +1
• B. \(\pm 1\)
• C. -1
• D. 5

Hint:

For infinite solutions in a system of linear equations, the ratio of the coefficients of \( x \) and \( y \) in both equations must be the same. If the equations are proportional, the lines represented by them will coincide, resulting in infinite solutions.

Answer 1

The Correct Option is B

We are given the pair of linear equations: \[ x + ky = 1 \quad \text{(1)} \] \[ kx + y = k^2 \quad \text{(2)} \]
Step 1: Condition for infinite solutions.
For a system of two linear equations to have an infinite number of solutions, the two equations must be dependent. This means the ratio of the coefficients of the variables \( x \) and \( y \) in both equations must be the same. That is, the equations must be proportional. In other words, for equations (1) and (2) to have infinite solutions, the ratio of the coefficients of \( x \) and \( y \) should satisfy the following condition: \[ \frac{1}{k} = \frac{k}{1} \] This is the condition for the two lines to be identical.
Step 2: Solving the equation.
Now, let’s solve the equation: \[ \frac{1}{k} = \frac{k}{1} \] By cross-multiplying: \[ 1 \times 1 = k \times k \] \[ k^2 = 1 \]
Step 3: Finding the value of \( k \).
Taking the square root of both sides: \[ k = \pm 1 \]
Step 4: Conclusion.
Thus, for \( k = \pm 1 \), the system of equations will have infinite solutions. The correct answer is (B).