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Q1. Which of the following is not a statement?

• Mathematics is an easy subject
• The product of 8 and 12 is 100
• All integers are real numbers
• India is a part of SAARC

Solution

Whether Mathematics is an easy subject or not is an individual's perception. So it can't be taken as true universally. Thus, it's neither true nor false. Hence it's not a statement.

Q2. A compound statement with an ‘Or’ is false when both the component statements are ……

• True
• False
• Same
• Different

Solution

A compound statement with an ‘Or’ is false when both the component statements are false.

Q3. Two pairs of statement are: p: If a quadrilateral is a rectangle, then its opposite sides are equal. q: If opposite sides of a quadrilateral are equal, then the quadrilateral is a rectangle. The combined statement of these pairs using “If and only if” is:

• A quadrilateral is a square if and only if its opposite sides are equal.
• A quadrilateral is not a rectangle if and only if its opposite sides are equal.
• A quadrilateral is a rectangle if and only if its all sides are equal.
• A quadrilateral is a rectangle if and only if its opposite sides are equal.

Solution

A quadrilateral is a rectangle if and only if its opposite sides are equal.

Q4. If p and q are mathematical statements, then in order to show that the statement “p and q” is true, we need to show that:

• The statement p is true and the statement q is true
• The statement p is true and the statement q is false
• The statement p is false and the statement q is true.
• The statement p is true and the statement q is not true

Solution

If p and q are mathematical statements, then in order to show that the statement “p and q” is true, we need to show that “The statement p is true and the statement q is true”.

Q5. Component statements of the compound statement “The apple is a fruit and the horse is an animal” are:

• p: The apple is a vegetable, q: The horse is a living being.
• p: The apple is a fruit, q: The horse is not an animal.
• p: The apple is not a fruit, q: The horse is an animal.
• p: The apple is a fruit, q: The horse is an animal.

Solution

Component statements of the compound statement “The apple is a fruit and the horse is an animal” are: p: The apple is a fruit, q: The horse is an animal.

Q6. The negation of the statement “The sum of 5 and 3 is 7” is:

• The sum of 5 and 3 is 5.
• The sum of 5 and 3 is not equal to 8
• The sum of 5 and 3 is not equal to 7
• The sum of 5 and 3 is 9

Solution

The negation of the statement “The sum of 5 and 3 is 7” is “The sum of 5 and 3 is not equal to 7”.

Q7. Which of the following sentences is not a statement?

• 6 is less than 4
• Every set is an infinite set
• The sun is the only star
• Science is an interesting subject

Solution

This sentence is subjective in the sense that for those who like science, it may be interesting but for others it may not be. This means that this sentence is not always true. Hence it is not a statement.

Q8. Write the statement “A quadrilateral is a rhombus if its diagonals bisect each other at 90°” in the form “if then”

• If the diagonals of a quadrilateral do not bisect each other at 90°, then it is rhombus.
• If the diagonals of a quadrilateral bisect each other at 90°, then it is rhombus.
• If the diagonals of a quadrilateral bisect each other at 90°, then it is not a rhombus.
• If the diagonals of a quadrilateral do not bisect each other at 90°, then it is not a rhombus.

Solution

If the diagonals of a quadrilateral bisect each other at 90°, then it is rhombus.

Q9. The quantifier used in the statement “For every real number x, x is less than x + 5” is:

• And
• Or
• There exist
• For every

Solution

The quantifier used in the statement “For every real number x, x is less than x + 1” is for every.

Q10. Write the  truth value of the converse of the given statement."If the opposite angles of a quadrilateral are equal then it is a parallelogram."

• True
• False
• Neither
• Can not be deciphered

Solution

True The converse of the given statement is  If a quadrilateral is a parallelogram then its opposite angles  are equal." which is True

Q11. The component statements 'p' and 'q' of the given compound statement 'All integers are either even or odd.; are

• P : All integers are even,q: All integers are  odd
• P : All integers are not  even,q: All integers are   notodd
• P : All  non integers are even,q: All  non integers are  odd
• P : All integers are even,q: All integers are  not odd

Solution

The two statements that go to make up the given compound statement All integers are either even or odd are p : All integers are even,q: All integers are r odd

Q12. The component statements are: p: You are wet when it rains. q: You are wet when you are in river. The compound statement of these component statements using appropriate connective is:

• You are wet when it rains or you are in a river.
• You are wet when it rains and you are in a river
• You are wet when you are in river and it rains.
• You are not wet when you are in river or it rains.

Solution

The compound statement is “You are wet when it rains or you are in a river”.

Q13. A statement which is either true or false but not both is called a ______.

• Mathematically acceptable statement
• Ambiguous statement
• Implicative  statement
• Subjective statement

Solution

A sentence is called a mathematically acceptable statement if it is either true or false but not both.

Q14. A sentence is called a mathematically accepted statement if

• it's true
• it's false
• it's neither true or false
• it's  either true or false but not both.

Solution

A Mathematical  statement which is either true or false that is  there is no ambiguity about it. Hence the answer.

Q15. Write the compound statement of the following component statements: p: A rectangle is a quadrilateral. q: The opposite sides of a rectangle are equal.

• A rectangle is not a quadrilateral and its opposite sides are equal.
• A rectangle is a quadrilateral and its opposite sides are equal.
• A rectangle is a quadrilateral and its opposite sides are not equal.
• A rectangle is a figure and its opposite sides are of different length.

Solution

The compound statement is “A rectangle is a quadrilateral and its opposite sides are equal”.

Q16. The connecting word used in the compound statement “Cube of an integer is positive or negative” is:

• And
• Or
• There exist
• For all

Solution

The connecting word used in the compound statement “Square of an integer is positive or negative” is or.

Q17. In order to prove the statement “If p then q” we need to show that:

• By assuming that p is true, prove that q must be false.
• By assuming that p is false, prove that q must be true
• By assuming that p is true, prove that q must be true.
• By assuming that p is not true, prove that q must be true.

Solution

In order to prove the statement “If p then q” we need to show that “By assuming that p is false, prove that q must be true”.

Q18. The converse of the statement “If a number n is odd, then n² is odd” is:

• If a number n² is even, then n is even.
• If a number n² is odd, then n is even.
• If a number n² is odd, then n is odd.
• If a number n² is even, then n is odd.

Solution

The converse of the statement “If a number n is odd, then n² is odd” is “If a number n² is odd, then n is odd”.

Q19. In order to prove the statement “p if and only if q”, we need to show that:

• If p is false, then q is true and if q is false, then p is true
• If p is true, then q is false and if q is true, then p is false
• If p is true, then q is true and if q is true, then p is true
• If p is true, then q is not true and if q is not true, then p is true

Solution

In order to prove the statement “p if and only if q”, we need to show that “If p is true, then q is true and if q is true, then p is true”.

Q20. The negation of the statement “p: For every real number x, x³ > x².” is:

• There exists a real number x such that x³ < x².
• There exists real number c such that x³ > x.
• There exists real number c such that x² > x.
• There exists real number c such that x³ = x².

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