MATHS 1 Marks Question

It is a true declarative sentence because a figure that has three sides is a triangle. Thus, it is a true statement.

Negation of the given statement:
All complex numbers are real numbers.

If it rains, then it is cold.

P: For every positive number x, the number (x - 1) is also positive.
P: At least for one positive real number x, the number (x - 1) is not positive.

Let q and r be the statements given by
q: x and y are odd integers.
r: x + y is an even integer.
Then, the given statement is
if q, then r,
Direct Metflod: Let q be true. Then,
q is true.
⇒ x and y are odd integers
⇒ x = 2m + 1, y = 2n + 1 for some integers m, n
⇒ x + y = (2m + 1) + (2n + 1)
⇒ x + y = (2m + 2n + 2)
⇒ x + y = 2 (m + n + 1)
⇒ x + y is an even integer
⇒ r is true.
Thus, q is true ⇒ r is true.
Hence, ''if q, then r'' is a true statement.

Consider a triangle ABC with all angles equal. Then each angle of the triangle is equal to 60".
Hence, ABC is not an obtuse angle triangle.
Therefore the following statement is false.
p: "if all the angles of a triangle are equal, then the triangle is an obtuse angled triangle".

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

If x is positive, then x is not less than zero.

Negation of the given statement:
There exists a policeman who is not a thief.
Or
At least one policeman is not a thief.

The statements in this pair are not the negation of each other because both statements are the same. Both the statements convey that x is an irrational number.

If Mohan is not poor, then he is not a poet.

1. I won the trophy!

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

2. Please fetch me a glass of water.

It is an imperative sentence. In other words, it can be expressed either as a request or as a command. Therefore, it not a statement.

3. Can you do this work for me?

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

Converse:
If you feel thirsty, then it is hot outside.
Contrapositive:
If you do not feel thirsty, then it is not hot outside.

If it rains, only then the game is cancelled.

If it rains, it is not cold.

Negation of the given statement:
The earth is not round.
Or
It is not true that the earth is round

The component statements of the given compound statement are:
25 is a multiple of 5.
25 is a multiple of 8.

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

The statem ant is:
"100 ism ultiple of 4 and 5"
We know that 100 is a multiple of 4 as well as 5. So, p is true statement.
Hence, the statement is true i.e. the statement "p" is a valid statement.

The component statements of the given compound statement are:
All rational numbers are real.
All real numbers are complex.

If you are not strong, then you cannot be a sailor.

Negation of the given statement:
I will go to school.

Let r and s be two statements given by
r: xy is an even integer.
s: At least one of x and y is an even integer
Lets be not true. Then,
s is not true
⇒ Both x and y are odd integers
Let x = 2n + 1 and y = 2m + 1 for some integers n and m. Then,
⇒ xy = (2n + 1)(2m + 1) for some integers n and m.
⇒ xy = 4nm + 2(n + m) + 1 for some integers n and m,
⇒ xy is an odd integer
⇒ xy is not an even integer
⇒ -r is true
Thus, -s is trua es -r is true
Hence, the given statement is true.

If you want to be happy, then you will have to be rich.

r: There exists a number x such that 0 < x < 1.
r: For every real number x, either x ≤ 0 or x < 1.

It is a true declarative sentence, so it is a statement.

Mathematics could be easy for some people, so this sentence may or may not be true. So, it is not a statement.

Exclusive OR because a lady can give a birth to a baby who is either a boy or a girl.

The tumbler is half empty if and only if the tumbler is half full.

You get an A grade if and only if you do all the homework regularly.

False. Because square roots of prime num bars are irrational num bars.

The component statements of the given compound statement are:

1. All rational numbers are real.
2. All real numbers are not complex.

The compound statement is false because all real numbers are complex. The connective used is "and". So, even if one component statement is false, the compound statement is false.

Negation of the given statement:
It is not true that Ravish is honest.
Or
Ravish is not honest

Negation of the given statement:
It is not true that Bangalore is the capital of Karnataka.
Or
Bangalore is not the capital of Karnataka.

Negation of the given statement:
Both the diagonals of a rectangle do not have the same length.
Or
Both the diagonals of a rectangle have different lengths.

The component statements of the given compound statement are:

1. The sand heats up quickly in the sun.
2. Sand does not cool down fast at night.

The compound statement uses "and" as the connective. For the compound statement to be true, both the component statements must be true. The second component statement "Sand does not cool down fast at night" is false. Sand cools down fast at night. Therefore, the compound statement is false.

The argument used to check the validity of the given statement is not correct because it does not produce a contradiction.

Converse:
If you have winter clothes, then you live in Delhi.
Contrapositive:
​If you do not have winter clothes, then you do not live in Delhi.

False. Because, no radius of a circle is its chord.

The component statements of the given compound statement are:
The sky is blue.
The grass is green.

Negation of the given statement:
Some students did not complete their homework.

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

If it is cold, then it never rains.

If it did not snow, then Ravish does not ski.

If he tires, then he will not win.

Inclusive OR is used because a person can have both passport as well as voter registration card.

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

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

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.

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

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

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

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.

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.

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

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

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

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

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

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.

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

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

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

If it rains, then there is a traffic jam.

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

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

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

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

If it rains, then it is cold.

The component statements of the given compound statement are:

1. To enter into a public library, children need an identity card from the school.
2. 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.

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

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

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.

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

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

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

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.

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.

The component statements of the given compound statement are:

1. Square of an integer is positive.
2. 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.

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.

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

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

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

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

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.