Final answer:
A pair of linear equations 2kx+5y=7 and 6x-5y=11 has a unique solution if the determinant of their coefficients is not zero, which implies k must not equal -3 for a unique solution to exist. The correct answer is (2) k in not equal to -3.
Step-by-step explanation:
The pair of linear equations 2kx+5y=7 and 6x-5y=11 will have a unique solution if the determinant of the coefficient matrix is not equal to zero. In other words, linear equations have unique solutions if their slopes are different, meaning they are not parallel. We derive the coefficient matrix from the system as:
[2k 5]
[ 6 -5]
The determinant is (2k * -5) - (5 * 6), which simplifies to -10k - 30. Setting the determinant equal to zero gives us -10k - 30 = 0.
Dividing by -10 gives k + 3 = 0, or k = -3. Therefore, for the system to have a unique solution, k must not equal -3. This corresponds to option (2).