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 x2−1=0 does not have a root lying between 0 and 2.
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(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 ∼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 x2−1=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 x2−1=0 x2=1 x=±1 One root of the equation x2−1=0 i.e., the root x=1 lies between 0 and 2. Thus the given statement is false.