Out of all the patients in a hospital 89% are found to be suffering from heart ailment and 98% are suffering from lungs infection. If K% of them are suffering from both ailments, then K can not belong to the set :

Question

If the set $R = {(a, b) : a + 5b = 42, a, b \in \mathbb{N}}$ has $m$ elements and $\sum_{n=1}^m (1 + i^n) = x + iy$, where $i = \sqrt{-1}$, then the value of $m + x + y$ is:

Answer 1

To solve this problem, we need to determine the range of possible values for the percentage $ K $ of patients suffering from both heart ailments and lung infections. We are given:

89% of the patients are suffering from heart ailments.
98% of the patients are suffering from lung infections.
We need to find the value of $ K $ such that the set of possible values does not include $ K $.

Let's denote:

$ H = 89\%$ representing the percentage of patients with heart ailments.
$ L = 98\%$ representing the percentage of patients with lung infections.
$ K $ representing the percentage of patients with both ailments.

The principle of inclusion-exclusion for set theory gives us:

\[ |H \cup L| = |H| + |L| - |H \cap L| \]

In terms of percentages:

\[ \text{Total Percentage with at least one ailment} = H + L - K \]

Since the total percentage cannot exceed 100%:

\[ 89 + 98 - K \leq 100 \]

Simplifying gives:

\[ K \geq 187 - 100 = 87 \]

This implies that $ K $ must be at least 87%. Now let's rule out the sets:

\{84, 86, 88, 90\} — $87$ and higher is not ruled out, as 88 and 90 are possible.
\{80, 83, 86, 89\} — $87$ and higher is not ruled out, as 89 is possible.
\{79, 81, 83, 85\} — All values are below 87, so this set cannot include $ K $.
\{84, 87, 90, 93\} — $87$ and higher is not ruled out, as 87, 90, and 93 are possible.

Therefore, the correct answer is that the percentage of patients suffering from both ailments, $ K $, cannot belong to the set \{79, 81, 83, 85\}.

Answer 2