2 Marks Questions

Question 12 Marks

Prove by direct method that for any real numbers x, y if x = y, then x² = y².

Answer

Let p: $\text{x} = \text{y},\text{x},\text{y}\in\text{R}$
On squaring both sides we have
x² = y²: q (say)
⇒ p = q
Hence, proved.

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Question 22 Marks

Using contrapositive method prove that if n 2 is an even integer, then n is also an even integers.

Answer

Let p: n² is an even integer.
q: n is also an even integer.
Let ~p is true i.e., n is not an even integer.
⇒ n² is not an even integer. [Since square of an odd integer is odd]
⇒ ~p is true.
Therefore, ~q is true which provides that ~p is true.
Hence proved.

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Question 32 Marks

State whether the following sentences are statements are not:

The angles opposite to equal sides of a triangle are equal.
The moon is a satellite of earth.
May God bless you!
Asia is a continent.
How are you?

Answer

It is a statement.
It is a statement,
It is not a statement, since it is an exclamations.
It is a statement.
It is not a statement, since it is a question.

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Question 42 Marks

Prove by direct method that for any integer ‘n’, n³ - n is always even.
[Hint: Two cases (i) n is even, (ii) n is odd.]

Answer

Case I: When n s even.
Let $\text{n} = 2\text{k}, \text{k}\in\text{N}$
$\Rightarrow\text{n}^3-\text{n}=(2\text{k})^3-(2\text{k})=2\text{k}(4\text{k}^2-1)=2\lambda,$ Where $\lambda = \text{k}(4\text{k}^2-1)$
Thus, (n³ - n) is even when n is even.
Case II: When n is odd.
Let $\text{n} = 2\text{k}+1, \text{k}\in\text{N}$
$\Rightarrow\text{n}^3-\text{n}=(2\text{k}+1)^3 -(2\text{k}+1)$
$=(2\text{k}+1)[(2\text{k}+1)^2-1]$
$=(2\text{k}+1)[4\text{k}^2+1+4\text{k}-1]$
$=(2\text{k}+1)[4\text{k}^2+4\text{k}]$
$=4\text{k}(2\text{k}+1)(\text{k}+1)=2\mu,$ Where $\mu=2\text{k}(\text{k}+1)(2\text{k}+1)$
Thus, n³ - n is even when n is odd.
So, n³ - n is always even.

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