MATHS 1 Marks Question

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".

Let p: 35 is a prime number.
q: 35 is a composite number.
Then, the negation of the given compound statement is:
~(p v q): 35 is not a prime number and it is not a composite number.

Quantifier are the phrases like ‘There exists’ and ‘For every1, ‘For all’ etc.
There exists.

p: Chandigarh is Capital of Haryana.
q: Chandigarh is Capital of U.P.

We know that is either true or false but not both simultaneously.
Since it is a question. Hence, it is not a statement.

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

p: A number is divisible by 2.
q: A number is divisible by 3.
p v q: A number is either divisible by 2 or 3.

If the rectangle R is rhombus, then it is square.

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.

Here the given statement is the form p q which has the truth value T whenever either p or q or both have the truth value T. Hence, it is a true statement and its component statements are
p: 57 is divisible by 2. (False).
q: 57 is divisible by 3. (True).

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

p ⟷ q: A natural number is odd if and only if it is not divisible by 2.

If natural number ‘n’ is not divisible by 2 or 3, then n is not divisible by 6.

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.

The weather will not be cold, if it does not snow.

We know that is either true or false but not both simultaneously.
It is false. Hence, it is a statement.

Cow does not have four legs.

We know that is either true or false but not both simultaneously.
It is true. Hence, it is a statement.

If it rains, only then the game is cancelled.

p: Chennai is in India.
q: Chennai is the Capital of Tamil Nadu.

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.

Let p: All real numbers are rationals.
q: All real numbers are irrationals.
Then, the negation of the given compound statement is:
~(p v q): All real numbers are not rational and all real numbers are not irrational.
[~(p v q) = ~p ∧ ~ q]

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.

Quantifier are the phrases like ‘There exists’ and ‘For every1, ‘For all’ etc.For all.

Area of a circle is not same as the perimeter of the circle.

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.

A leap year does not have 366 days.

Quantifier are the phrases like ‘There exists’ and ‘For every1, ‘For all’ etc.
For all.

q implies p ⇒ If a, b, c are in A.P then 2b = a + c.

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

If the triangle is not equilateral, then all three sides of the triangle are not equal.

p: 2 is factor of 12.
q: 3 is factor of 12.
r: 6 is factor of 12.
p ∧ q ∧ r: 2, 3 and 6 are factors of 12