Question

(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 $2$.

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Solution

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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 $∼r$

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 $180_{∘}$. Therefore if all the three angles are equal then each of them is of measure $60_{∘}$ 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 $x_{2}−1=0$

$x_{2}=1$

$x=±1$

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.

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