Question

By giving a counter example show that the following statements are not true
(i) p : If all the angles of a triangle are equal then the triangle is an obtuse angled triangle.
(ii) q : The equation $x^2-1$ $=0$ does not have a root lying between $0$ and $2$.

Answer 1

(i) The given statement is of the form "if $q$ then $r$"
q : All the angles of a triangle are equal
r : The triangle is an obtuse-angled triangle
The given statement p has to be proved false. For this purpose it has to be proved that if $q$ then $\sim r$
To show this angles of a triangle are required such that none of them is an obtuse angle.
It is known that the sum of all angles of a triangle is ${180}^{\circ }$. Therefore if all the three angles are equal then each of them is of measure ${60}^{\circ }$ which is not an obtuse angle.
In an equilateral triangle the measure of all angles is equal However the triangle is not an obtuse-angled triangle.
Thus it can be concluded that the given statement $p$ is false.
(ii) The given statement is as follows,
q : The equation $x^2-1=0$ does not have a root lying between $0$ and $2$.
This statement has to be proved false. To show this a counter example is required.
Consider $x^2-1=0$
$x^2=1$
$x=\pm 1$
One root of the equation $x^2-1=0$ i.e., the root $x=1$ lies between $0$ and $2$.
Thus the given statement is false.