MATHS 1 Marks Question

If the Banana tree stays warm for a month, then it will bloom.

The day that is being referred to is not evident from the sentence. Hence, it is not a statement.

This sentence is subjective in the sense that for some people, mathematics can be easy and for some others, it can be difficult. Hence, it is not a statement.

If you get a job, then your credentials are good.

This is not the same as statement.
Thus, the given statements are not the negation of each other.

It is an order. Therefore, it is not a statement.

Chennai is not the capital of Tamil Nadu.

The negation of the first statement is “the number x is not a rational number”. This means that the number x is an irrational number, which is the same as the second statement.
Therefore, the given statements are negations of each other.

Statement r can be written as follows.
If you can access the website, then you pay a subscription fee.

This sentence is sometimes correct and sometimes incorrect. For example, the square of 2 is an even number. However, the square of 3 is an odd number. Hence, it is not a statement.

Every natural number is not an integer.

The given statement p is false.
According to the definition of chord, it should intersect the circle at two distinct points.

Statement p can be written as follows.
If you log on to the server, then you have a password.

The product of (–1) and 8 is (–8). Therefore, the given sentence is incorrect. Hence, it is a statement.

The given statement q is false.
If the chord is not the diameter of the circle, then the centre will not bisect that chord.
In other words, the centre of a circle only bisects the diameter, which is the chord of the circle.

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

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

The negation of statement q is as follows.
There exists a cat that does not scratch.

Statement q can be written as follows.
If it rains, then there is a traffic jam.

The negation of statement s is as follows.
There does not exist a number x, such that 0 < x < 1.

The quantifier is “There exists”.
The negation of this statement is as follows.
There exists a state in India which does not have a capital.

The quantifier is “For every”.
The negation of this statement is as follows.
There exist a real number x such that x is not less than x + 1.

The number 2 is not greater than 7.

The negation of statement r is as follows.
There exists a real number x, such that neither x > 1 nor x < 1.

If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

Here, “or” is inclusive since a person can have both a ration card and a passport to apply for a driving license.

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

x > y
⇒ –x < –y (By a rule of inequality)
Thus, the given statement s is true.

Here, “or” is exclusive because it is not possible for the Sun to rise and the moon to set together.

The negation of the first statement is “the number x is a rational number”.
This is same as the second statement. This is because if a number is not an irrational number, then it is a rational number.
Therefore, the given statements are negations of each other.

Here, “or” is exclusive because all integers cannot be both positive and negative.

All real numbers can be written as a × 1 + 0 × i. Therefore, the given sentence is always correct. Hence, it is a statement.

The sum of 5 and 7 is 12, which is greater than 10. Therefore, this sentence is always correct. Hence, it is a statement.

This sentence is sometimes correct and sometimes incorrect. For example, squares and rhombus have sides of equal lengths. However, trapezium and rectangles have sides of unequal lengths. Hence, it is not a statement.

All triangles are equilateral triangles.

The negation of statement
is as follows.
There exists real number x and y for which x + y ≠ y + x.

This sentence is incorrect because the maximum number of days in a month is 31. Hence, it is a statement.

The quantifier is “There exists”.
The negation of this statement is as follows.
There does not exist a number which is equal to its square.

This sentence is correct and hence, it is a statement.

The negation of statement p is as follows.
There exists a positive real number x, such that x – 1 is not positive.

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.

p: Number 7 is prime.
q: Number 7 is odd.

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

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

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

It is a false assertive sentence, so it is a false statement.

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

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

p only if q ⇒ If you do not study, then you will fail.

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.

If x is nether positive nor negative then x = 0.

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

There exist similar triangles which are not congruent.

If xy is not positive integer, then x or y is not negative integer.

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.

If tomorrow is Tuesday, then today is Monday.

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.

q is necessary for p ⇒ If any number is prime, then its square is not prime.

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 2x = 3y then x : y = 3 : 2.

p: 0 is less than every positive integer.
q: 0 is less than every negative integer.

We know that is either true or false but not both simultaneously.
y + 9 = 7, here that the value of y is not given. So it is true for y = -2 and false for any other value of y. Hence, it is not a statement.

2 is a prime number.

p: Two line is a plane intersect at one point.
q: Two lines in a plane are parallel.

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

If he tires, then he will not win.

p ⟷ q: A triangle is an equilateral triangle if and only if all three sides of triangle are equal.

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 if p ⇒ If take the dinner, then you will get sweet dish.

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.

2 + 7 ≠ 6.

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

p: Rahul passed in Hindi.
q: Rahul passed in English.
p ∧ q: Rahul passed in Hindi and English.

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

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

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.

p: 100 is divisible by 3.
q: 100 is divisible by 11.
r: 100 is divisible by 5.

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

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

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.

If the triangle is equilateral, then all three angles of the triangle are equal.

Violet are not blue.

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

p is sufficient for q ⇒ If an integer is divisible by 5, then its unit digits are 0 or 5.

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.

If the opposite angles of a quadrilateral are supplementary, then S is cyclic.

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 you must visit Taj Mahal, then you go to Agra.

If p, then q ⇒ If the number is odd, then its square is odd number.

Let p: 6 is divisible by 2.
q: 6 is divisible by 3.
Then, the negation of the given compound statement is:
~(p ∧ q): 6 is not divisible by 2 or it is not divisible by 3

Let p: All prime integers are even.
q: All prime integers are odd.
Then, the negation of the given compound statement is given by ~ (p v q):
All prime integers are not even and all prime integers are not odd.

If it rains, then there is a traffic jam.

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

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

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.

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

Every real number is not an irrational number.

Negation of everyone is not everyone. So, negation of the given statement should be “Not everyone in India speaks Hindi”, “Someone in India does not speaks Hindi”, “At least one person in India who does not speaks Hindi”. “No person in India speaks Hindi” cannot be the negation of the given statement as this means no one speaks Hindi which does not means not everyone.

p: A rectangle is a quadrilateral.
q: A rectangle is a 5-sides polygon.

p: Plants use sunlight for photosynthesis.
q: Plants use water for photosynthesis.
r: Plants use carbon-dioxide for photosynthesis.

The number 17 is not prime.

p: x is an odd integer.
q: (x + 1) is an odd integer.
p v q: Either x or (x + 1) is an odd integer.

If it rains, then it is cold.

p: Students can take Hindi as an optional paper.
q: Students can take English as an optional paper.
p v q: Students can take Hindi or English as an optional paper.

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.

Let p:|x| is equal to x.
q:|x| is equal to -x.
Then, the negation of the given compound statement is:
~(p v q): |x| is not equal to JC and it is not equal to -x.

If the triangle is right triangle, then the sum of the squares of two sides of a triangle is equal to the square of third side.

Let p: All rational numbers are real.
q: All rational numbers are complex.
~p: All rational numbers are not real.
~q; All rational numbers are not complex.
Then, the negation of the given compound statement is:
~(p ∧ q): All rational numbers are not real or not complex.
[~(p ∧ q) = ~p v ~ q]

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

If n is not an integer, then it is not a natural number.

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.

If x 2 > 1 then x is not a real number such that 0 < x < 1.

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.

If the ratio of corresponding sides of two triangles are equal, then triangles are similar.

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

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

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.

p: x is even integers.q: y is even integers.
p ∧ q: x andy are even integers.

If x ≠ 3, then x ≠ y or y ≠ 3.

Let p: A triangle has 3-sides.
q: A triangle has 4-sides.
Then, the negation of the given compound statement is:
~(p v q): A triangle has neither 3-sides nor 4-sides.

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.

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

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

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.

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

p ⟷ q: The unit digit of on integer is zero, if and only if it is divisible by 5.

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.