1 Marks Question

Question 511 Mark

Determine the contrapositive of the following statements:
Only if Max studies will he pass the test.

Answer

If Max does not study, then he will not pass the test.

Question 521 Mark

Rewrite the following statements in the form "p if and only if q".
p: If you watch television, then your mind is free and if your mind is free, then you watch television.

Answer

You watch television if and only if your mind is free.

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Question 531 Mark

Show that the following statement is true by the method of contrapositive
p: "If $x$ is an integer and $x^2$ is odd, then $x$ is also odd"

Answer

Let $q$ and r be the statements given by
$q:$ lf $x$ is an integer and $x^2 $ is odd
$r: x$ is an odd integer.
Then, $p: "$ lf $q,$ thenr."
If possible, let $r$ be false. Then,
$r$ is false
$\Rightarrow x$ is not an odd integer
$\Rightarrow x$ is an even integer
$\Rightarrow x = (2n)$ for some integer $n$
$\Rightarrow x^2 = 4n^2$
$\Rightarrow x^2$^ is an even integer
$\Rightarrow q$ is false.
Thus, r is false $\Rightarrow q$ is false.
Hence, $p:$ "if $q,$ then $r" $ is a true statement.

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Question 541 Mark

Show that the statement:
$p:$ "If $x$ is a real number such that $x^3 + x = 0,$ then x is $0"$ is true by.
Method of contradition.

Answer

Let $q$ and $r$ be the statements given
$q: x$ is a real number such that $x^3 + x = 0.$
$r: x$ is $0.$
Then, $p:$ if $q,$ then $r.$
Metnod of contradiction: If possible, let $p$ be not true. Then,
$p$ is not true
$\Rightarrow $ -pis true
$\Rightarrow -(p \Rightarrow r)$ is true
$\Rightarrow q$ and $-r$ is true
$\Rightarrow x$ is a real number such that $x^3 + x = 0$ and $x = 0$
$\Rightarrow\text{x}=0$ and $\text{x}\not=0$
This a contradiction.
Hence, $p$ is true.

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Question 551 Mark

Determine the contrapositive of the following statements:
If $x$ is an integer and $x ^2$ is odd, then $x$ is odd.

Answer

If $x$ is even, then $x^2$ is even.

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Question 561 Mark

Write the negation of the following statements:
q: For every real number x, either x > 1 or x < 1.

Answer

q: For every real number x, either x > 1 or x < 1.
q: At least for one real number x, neither x > 1x > 1 nor x < 1x < 1.

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Question 571 Mark

Write the negation of the following statements:
For every $\text{x}\in\text{N},\text{x}+3<10$

Answer

The Negation of the statement:
For every $\text{x}\in\text{N},\text{x}+3<10$
There exists $\text{x}\in\text{N}$ such that $\text{x}+3\geq10$

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Question 581 Mark

Some even integers are prime.

Answer

Negation of the given statement:
Some integers are not prime.
Or
No even integer is prime.

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Question 591 Mark

Find the component statements of the following compound statements:
The earth is round or the sun is cold.

Answer

The component statements of the given compound statement are:
The earth is round.
The sun is cold.

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Question 601 Mark

Which of the following statements are true and which are false? In each case give a valid reason for saying so:
s: If x and y are integers such that x > y, then -x < -y.

Answer

True. Because, for any two integers, if x - y is positive then -(x - y) is negative.

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Question 611 Mark

Write the negation of the following statements:
It rained on July 4, 2005.

Answer

Negation of the given statement:
It is not true that it rained on July 4, 2005.
Or
It did not rain on July 4, 2005.

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Question 621 Mark

Check the validity of the following statements:
r: 60 is a multiple of 3 or 5.

Answer

The statement is:
r: 60 ism ultiple of 3 or 5
is a com pound statement of the following statements:
p: 60 is multiple of 3
q: 60 is multiple of 5
Suppose q is false. That is, 60 is not a multiple of 5. Clearly p is true.
Thus, if we assume that q is false, then p is true.
Hence, the statement is true i.e. the statement "r" is a valid statement.

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Question 631 Mark

Find out the following sentences are statements and which are not. Justify your answer.
Two non-empty sets have always a non-empty intersection.

Answer

It is a false assertive sentence. Two non-empty sets with no common elements can have an empty intersection. Therefore, it is a statement.

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Question 641 Mark

All birds sing.

Answer

Negation of the given statement:
Some birds do not sing.
Or
There exists a bird that does not sing.

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Question 651 Mark

For the following statements, determine whether an inclusive "OR" or exclusive "OR" is used. Give reasons for your answer.
Students can take Hindi or Sanskrit as their third language.

Answer

Exclusive OR is used because students can opt for either Hindi or Sanskrit as their third language.

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Question 661 Mark

Find out the following sentences are statements and which are not. Justify your answer.
The cat pussy is black.

Answer

It is a declarative sentence, which may be true or false but cannot be both at the same time, so it is a statement.

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Question 671 Mark

Find out the following sentences are statements and which are not. Justify your answer.
This sentence is a statement.

Answer

Without knowing the sentence, we cannot decide whether it is true or false. So, it is not a statement.

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Question 681 Mark

Find out the following sentences are statements and which are not. Justify your answer.
Every rhombus is a square.

Answer

It is not true that every rhombus is a square because some rhombi may have all angles other than 90. So, it is a false statement.

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Question 691 Mark

Which of the following statements are true and which are false? In each case give a valid reason for saying so:
r: Circle is a particular case of an ellipse.

Answer

True. Because a circle is an ellipse that has equal axes.

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Question 701 Mark

Find out the following sentences are statements and which are not. Justify your answer.
The real number x is less than 2.

Answer

We cannot decide whether this sentence is true or false without knowing the value of x. So, it is not a statement.

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Question 711 Mark

Determine the contrapositive of the following statements:
If she works, she will earn money.

Answer

If she does not earn money, then she will not work.

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Question 721 Mark

Write the following statements in the form "if p, then q".
There is traffic jam whenever it rains.

Answer

If it rains, then there is a traffic jam.

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Question 731 Mark

Write the following statements in the form "if p, then q".
It is necessary to have a passport to log on to the server.

Answer

It is necessary to have a passport to log on to the server.

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Question 741 Mark

Write the component statements of the following compound statements and check whether the compound statement is true or false:
$x = 2$ and$ x = 3$ are the roots or the equation $3x^2 − x − 10 = 0.$

Answer

The component statements of the given compound statement are:

$x = 2$ is the root or the equation $3x^2 - x - 10 = 0.$
$x = 3$ is the root or the equation $3x^2 - x - 10 = 0.$

The connective used is "and". So, both component statements must be true for the compound statement to be true. The statement $"x = 3x = 3$ is the root or the equation $3x^2 - x - 10 = 0"$ is false. Therefore, the compound statement is false.

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Question 751 Mark

Write the following statements in the form "if p, then q".
You can access the website only if you pay a subscription fee.

Answer

If you pay a subscription fee, then you can access the website.

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Question 761 Mark

Negate the following statements:
There exists a number which is equal to its square.

Answer

Negation of the given statement:
There exists a number which is not equal to its square.

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Question 771 Mark

For the following statements, determine whether an inclusive "OR" or exclusive "OR" is used. Give reasons for your answer.
To apply for a driving licence, you should have a ration card or a passport.

Answer

Inclusive OR because a person could have both ration card as well as passport.

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Question 781 Mark

Write the following statements in the form "if p, then q".
It rains only if it is cold.

Answer

If it rains, then it is cold.

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Question 791 Mark

Show that the statement:
p: "If $x$ is a real number such that $x^3 + x = 0$, then $x$ is $0"$ is true by.
Method of contrapositive

Answer

Let $q$ and $r$ be the statements given $q : x$ is a real number such that $x ^3+ x =0$.
$r : x$ is $0$ .
Then, $p$ : if $q$, then $r$.
Method of contrapositive: Let $r$ be not true. Then, $r$ is not true.
$\Rightarrow x \neq 0, x \in R$
$\Rightarrow x\left(x^2+1\right) \neq 0, x \in R$
$\Rightarrow q$ is not true
Thus, $-r=-q$.
Hence, $p: q \Rightarrow r$ is true.

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Question 801 Mark

Write the component statements of the following compound statements and check whether the compound statement is true or false:
To enter into a public library children need an identity card from the school or a letter from the school authorities.

Answer

The component statements of the given compound statement are:

To enter into a public library, children need an identity card from the school.
To enter into a public library, children need a letter from the school authorities.

The compound statement is true because both component statements are true.

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Question 811 Mark

Rewrite the following statements in the form "p if and only if q".
q: If a quadrilateral is equiangular, then it is a rectangle and if a quadrilateral is a rectangle, then it is equiangular.

Answer

A quadrilateral is a rectangle if and only if it is equiangular.

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Question 821 Mark

Determine the contrapositive of the following statements:
If it snows, then they do not drive the car.

Answer

If they do not drive the car, then there is no snow.

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Question 831 Mark

Check the validity of the following statements:
q: 125 is a multiple of 5 and 7.

Answer

The statement is:
"125 is multiple of 5 and 7"
Since 125 is a multiple of 5 but it is not a multiple of 7. So, q is not a true statement i.e. the statement "q" is not a valid statement.

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Question 841 Mark

Write the negation of the following statements:
The sun is cold.

Answer

Negation of the given statement:
The sun is not cold.
Or
It is not true that the sun is cold.

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Question 851 Mark

Which of the following statements are true and which are false? In each case give a valid reason for saying so:
q: The centre of a circle bisects each chord of the circle.

Answer

False. Because, a chord does not have to pass through the centre.

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Question 861 Mark

Are the following pairs of statements are negation of each other:
The number x is not a rational number.
The number x is not an irrational number.

Answer

The given statement in this pair are the negation of each other.

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Question 871 Mark

State the converse and contrapositive of the following statements:
I go to a beach whenever it is a sunny day.

Answer

Converse:
If I go to a beach, then it is a sunny day.
Contrapositive:
If I do not go to a beach, then it is not a sunny day.

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Question 881 Mark

State the converse and contrapositive of the following statements:
If a quadrilateral is a parallelogram, then its diagonals bisect each other.

Answer

Converse:
If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
Contrapositive:
If the diagonals of a quadrilateral do not bisect each other, then it is not a parallelogram.

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Question 891 Mark

Write the negation of the following statements:
There exists $\text{x}\in\text{N},\text{x}+3=10$

Answer

The Negation of the statement:
There exists $\text{x}\in\text{N},\text{x}+3=10$
is
For every $\text{x}\in\text{N},\text{x}+3\not=10$

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Question 901 Mark

Write the component statements of the following compound statements and check whether the compound statement is true or false:
Square of an integer is positive or negative.

Answer

The component statements of the given compound statement are:

Square of an integer is positive.
Square of an integer is negative.

The compound statement is true because the first statement is true. Since the connective used is "or" and one of the component statements is true, the compound statement is true.

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Question 911 Mark

State the converse and contrapositive of the following statements:
A positive integer is prime only if it has no divisors other than 1 and itself.

Answer

Converse:
If a positive integer has no divisors other than 1 and itself, then it is prime.
Contrapositive:
If a positive integer has some divisors other than 1 and itself, then it is not prime.

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Question 921 Mark

Find out the following sentences are statements and which are not. Justify your answer.
The product of (-1) and 8 is 8.

Answer

It is an assertive sentence; therefore, it is a statement. But -1 × 8 = -8 therefore, the statement is false.

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Question 931 Mark

Determine the contrapositive of the following statements:
If he has courage he will win.

Answer

If he does not win, then he does not have courage.

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Question 941 Mark

Find out the following sentences are statements and which are not. Justify your answer.
All real numbers are complex numbers.

Answer

It is true because we can write a real number as x + 0 i. So, it is a true statement.

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Question 951 Mark

Find out the following sentences are statements and which are not. Justify your answer.
Go!

Answer

It is an exclamatory sentence, so it is not a statement.

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Question 961 Mark

Find out the following sentences are statements and which are not. Justify your answer.
Every set is a finite set.

Answer

It is a false assertive sentence because there are some sets that are infinite like the set of all real numbers. Therefore, it is a statement.

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MATHS 1 Marks Question