Consider the two sets :
A = {m \[ \in \] R : both the roots of
x2
-- (m + 1)x + m + 4 = 0 are real}
and B = [--3, 5).
Which of the following is not true?
Solutions · 1
Official solution
As roots are real so, \[D \ge 0\]
\[{(m + 1)^2} - 4(m + 4) \ge 0\]
\[ \Rightarrow {m^2} - 2m - 15 \ge 0\]
\[ \Rightarrow \] \[(m - 5)(m + 3) \ge 0\]
\[m\, \in \,(−∝, - \[3] \] \cup \[ [5, \] \propto )A=(−\[ \propto \], −3] \[ \cup \] [5, \[ \propto \])
Given B = [−3, 5)
Now, let's examine the options.
Option A : A \cap B = {--3}
The intersection of sets A and B would be the set of elements common to both sets. In this case, the only common element is -3. So, option A is true.
Option B : B -- A = (--3, 5)
The subtraction (or difference) of sets A from B is the set of elements that are in B but not in A. B is [--3, 5), and A is (-\infty , -3] U [5, \infty ). Subtracting A from B would leave an open interval (-3, 5), not including -3 and 5. So, option B is also true.
Option C : A \cup B = R
The union of sets A and B is the set of elements that are in A, or B, or both. Here, A U B would cover all real numbers. So, option C is true.
Option D : A - B = (-\infty , -3) \cup (5, \infty )
The subtraction (or difference) of set B from A is the set of elements that are in A but not in B. B is [--3, 5), and A is (-\infty , -3] U [5, \infty ). Subtracting B from A would leave (-\infty , -3) U [5, \infty ), not including -3 and 5. But according to the convention for writing intervals, it should be (-\infty , -3) U (5, \infty ). So, option D is not true.