3 Marks Question

Question 13 Marks

Write the negation of the following statements and check whether the resulting statements are true:
(i) The sum of 2 and 5 is 9 .
(ii) Every natural number is greater than zero.

Answer

(i) The negation of the given statement is :
"It is false that the sum of 2 and 5 is 9 ".
We know that, $2+5=7 \neq 9$. So, negation of given statement is true.
(ii) The negation of the given statement is :
"It is false that every natural number is greater than 0 ".
We know that, all natural numbers are greater than 0 So, negation of given statement is true.

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Question 23 Marks

Identify the quantifiers and write negation of the following statements.
(i) For all even integers, $x, x^2$ is also even.
(ii) There exists a number which is multiple of 6 and 9 .

Answer

(i) The quantifier is 'for All'.
The negation of the statement is :
"There exists an even integer $x$ such that $x^2$ is not even".

(ii) The quantifier is "There exists".
The negation of the statement is :
"There does not exist a number which is a multiple of both 6 and 9.

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Question 33 Marks

Write the component statements of the following compound statements and check whether the compound statement is true or false.
(i) $2+4=6$ or $2+4=7$
(ii) A rectangle is a quadrilateral or five-sided polygon

Answer

(i) The components of compound statement are :
$
\begin{array}{l}
p: 2+4=6 \\
q: 2+4=7
\end{array}
$
We observe, $p$ is true and $q$ is false and both the statements are connected with 'or'. Hence, the compound statement is true.

(ii) The components of compound statement are :
$p$ : A rectangle is a quadrilateral.
$q$ : A rectangle is five-sided polygon.
We observe, $p$ is true and $q$ is false and both the statements are connected with 'or'. Hence, the compound statement is true.

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Question 43 Marks

Check whether the following sentences are statements? Give reason your answer.
(i) $3+x=5$
(ii) Every set is a finite set.
(iii) The sun is a star.
(iv) $x^2-3 x+2=0$.

Answer

(i) This is not a statement. But if numerical value is assigned to $x$, then it becomes a statement.
For $x=2$, this is a true statement and for $x=3$ it is false statement.
(ii) This sentence is always false, because there are sets which are not finite. Hence, it is a statement.
(iii) Since, sun is a star (it is a scientific fact). So, the given sentence is always true. Hence it is a statement.
(iv) This is not a statement. But for $x=1$ or $x=2$, this is a true statement.

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Applied Maths 3 Marks Question

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