Question No 46.For how many integral values of k will the equation x2 + kx + 144 = 0 have distinct integer roots?1)  72)  123)  144)  15

The equation given is a quadratic equation of the form ax^2 + bx + c = 0. The roots of the equation are given by the formula x = [-b ± sqrt(b^2 - 4ac)] / 2a. For the roots to be distinct, the discriminant (b^2 - 4ac) must be greater than 0. For the roots to be integers, the discriminant must also be a perfect square.

Step 1: Identify a, b, and c in the given equation. Here, a = 1, b = k, and c = 144.

Step 2: Set up the discriminant. In this case, it is k^2 - 41144 = k^2 - 576.

Step 3: For the roots to be distinct, k^2 - 576 > 0. This simplifies to k^2 > 576, so k must be greater than the square root of 576 or less than the negative square root of 576. The square root of 576 is 24, so k > 24 or k < -24.

Step 4: For the roots to be integers, k^2 - 576 must be a perfect square. The perfect squares less than 576 are 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529. These correspond to the values of k from -24 to 24, excluding 0.

Step 5: Count the number of valid k values. There are 24 values from -24 to -1, 24 values from 1 to 24, and 1 value of 0, for a total of 49 values.

So, none of the given options is correct. The correct answer should be 49.