For how many integral values of k the equation x2 4x k 0 where k is an integer has real roots a...........

Mathematics (NDA) Quadratic Equations

For how many integral values of k, the equation x²- 4x + k = 0, where k is an integer has real roots and both of them lie in the interval (0, 5) ?

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2023 NDA -1 Math

To determine the integral values of k for which the quadratic equation has roots in the range (0, 5), consider:

- The quadratic \(x^2 - 4x + k = 0\) has real roots if its discriminant \(D = 16 - 4k \geq 0\).

- To ensure both roots lie between 0 and 5:

- The sum of roots, given by 4, satisfies \(0 < 4 < 5\).

- Each root must be positive: \(k < 4\).

- Each root must be less than 5, giving another constraint for k.

By solving these constraints, we find k can be 1, 2, 3, or 4.

- Option 1: Only has 3 values, not enough.

- Option 2: Correct; matches the 4 values (1, 2, 3, 4).

- Option 3 and 4: Offer more values than possible.

?? Option 2 is correct.

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