A continuous random variable, X, has the following probability density function - HSC - SSCE Mathematics Advanced - Question 23 - 2020 - Paper 1

To find the value of k, we need to ensure that the total area under the probability density function (PDF) equals 1.

The area can be computed using the definite integral:

$\int_{0}^{k} \sin x \, dx = 1$

Calculating the integral:

$\int \sin x \, dx = -\cos x + C$

Thus,

$\int_{0}^{k} \sin x \, dx = [-\cos x]_{0}^{k} = -\cos k + \cos 0 \implies -\cos k + 1 = 1$

This simplifies to:

$-\cos k + 1 = 1 \implies -\cos k = 0 \implies \cos k = 0$

The value of k that satisfies this equation is:

$k = \frac{\pi}{2}$