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.
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.
(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.
(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.
