Question

Which of the following is/are counter example(s) of the statement \[{{x}^{2}}-7x+10 > 0\] for all real x?
(a) 2
(b) 3
(c) 4
(d) 5
A. only (a) and (b)
B. only (b) and (c)
C. All (a), (b), (c) and (d)
D. None of these.

Answer 1

Hint: In this problem, we have to find which of the given following numbers are the counter examples of the given statement \[{{x}^{2}}-7x+10 > 0\]. We should first know that counterexamples are values of x for which the given statement does not hold True. We can first find the factor for the given equation. We can then substitute the given following values to check for the conditions to be satisfied and find the answer.

Complete step by step solution:
Here we have to find which of the given following numbers are the counter examples of the given statement \[{{x}^{2}}-7x+10 > 0\].
We know that counterexamples are values of x for which the given statement does not hold True.
We can now write the given equation in factored form, we get
\[\begin{align}
  & \Rightarrow \left( x-5 \right)\left( x-2 \right) > 0 \\
 & \Rightarrow x > 5,x > 2....(1) \\
\end{align}\]
We can now substitute the given options (a) in (1), we get
\[\Rightarrow 2 > 5,2 > 2\]
Here, we can see that the above inequation is not true as 2 is not greater than 5 and 2.
We can now substitute the given options (b) in (1), we get
\[\Rightarrow 3 < 5,3 < 2\]
Here, the above inequation is not true as 3 is not greater than 5.
We can now substitute the given options (c) in (1), we get
\[\Rightarrow 4 < 5,4 < 2\]
Here, 4 is not greater than 5, so it is not true.
We can now substitute the given options (d) in (1), we get
\[\Rightarrow 5 < 5,5 < 2\]
Here, it is not true as 5 is not greater than 5 (as it is equal).
 The numbers given in the options, 2, 3, 4, 5 are counter examples of the given statement as they are all not greater than 5(does not satisfy the condition).
Therefore, the answer is option C. All (a), (b), (c) and (d).

Note: We should always remember that counterexamples are values of x for which the given statement does not hold True. We should substitute the given values in the given statement if the values does not satisfy the statement then it must be a counter example