**Mathematical Reasoning**

### Page No.31.4 Ex.31.1

**Q1.**

**Answer :**

**(i)** Listen to me, Ravi!

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

**(ii)** Every set is a finite set.

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.

**(iii)** Two non-empty sets have always a non-empty intersection.

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

**(iv)** The pussy cat is black.

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

**(v)** Are all circles round?

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

**(vi)** All triangles have three sides.

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

**(vii)** Every rhombus is a square.

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.

**(viii)** x^{2} + 5 | x | + 6 = 0 has no real roots.

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

**(ix)** This sentence is a statement.

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

**(x)** Is the earth round?

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

**(xi)** Go!

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

**(xii)** The real number x is less than 2.

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

**(xiii)** There are 35 days in a month.

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

**(xiv)** Mathematics is difficult.

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

**(xv)** All real numbers are complex numbers.

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

**(xvi)** The product of (−1) and 8 is 8.

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

**Q2.**

**Answer :**

**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.

### Page No.31.7 Ex.31.2

**Q1.**

**Answer :**

**(i)** Negation of the given statement:

It is not true that Bangalore is the capital of Karnataka.

Or

Bangalore is not the capital of Karnataka.

**(ii)** 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.

**(iii)** Negation of the given statement:

It is not true that Ravish is honest.

Or

Ravish is not honest.

**(iv)** Negation of the given statement:

The earth is not round.

Or

It is not true that the earth is round.

**(v)** Negation of the given statement:

The sun is not cold.

Or

It is not true that the sun is cold.

### Page No.31.8 Ex.31.2

**Q2.**

**Answer :**

**(i)** Negation of the given statement:

Some birds do not sing.

Or

There exists a bird that does not sing.

**(ii)** Negation of the given statement:

Some integers are not prime.

Or

No even integer is prime.

**(iii)** Negation of the given statement:

All complex numbers are real numbers.

**(iv)** Negation of the given statement:

I will go to school.

**(v)** 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.

**(vi)** Negation of the given statement:

There exists a policeman who is not a thief.

Or

At least one policeman is not a thief.

**Q3.**

**Answer :**

**(i)** The number x is not a rational number.

The number x is not an irrational number.

The statements in this pair are the negation of each other.

**(ii)** The number x is not a rational number.

The number x is an irrational number.

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.

**Q4.**

**Answer :**

**(i)** p: For every positive real 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.

**(ii)** q: For every real number x, either x > 1 or x < 1.

~q: At least for one real number x, neither x > 1 nor x < 1.

**(iii)** r: There exists a number x such that 0 < x < 1.

~r: For every real number x, either x ≤ 0 or x < 1.

**Q5.**

**Answer :**

The given statements are not negation of each other because the negation of “a+b=b+a is true for every real number a and b” is “There exist real numbers a and b for which a+b≠b+a”.

### Page No.31.15 Ex.31.3

**Q1.**

**Answer :**

**(i)** The component statements of the given compound statement are:

1) The sky is blue.

2)The grass is green

**(ii)** The component statements of the given compound statement are:

1)The earth is round.

2)The sun is cold.

**(iii)** The component statements of the given compound statement are:

1) All rational numbers are real.

2) All real numbers are complex.

**(iv)** The component statements of the given compound statement are:

1) 25 is a multiple of 5.

2) 25 is a multiple of 8.

### Page No.31.16 Ex.31.3

**Q2.**

**Answer :**

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

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

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

**(iv)** Inclusive OR because a person could have both ration card as well as passport.

**Q3.**

**Answer :**

**(i)** 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.

**(ii)** 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.

**(iii)** 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.

**(iv)** The component statements of the given compound statement are:

1) x=2 is the root or the equation 3×2-x-10=0.

2) x=3 is the root or the equation 3×2-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=3 is the root or the equation 3×2-x-10=0” is false. Therefore, the compound statement is false.

**(v)** 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.

**Q4.**

**Answer :**

**(i)** True

Both component statements are true and the connective is “and”.

**(ii)** False

The first component statement “Delhi is in England” is false. Since the connective is “and” and one component statement is false, the compound statement is false.

**(iii)** False

The second component statement “2 plus 2 equals 5” is false. Since the connective is “and” and one component statement is false, the compound statement is false.

**(iv)** False

Both the component statements are false. Therefore, the compound statement is false.

### Page No.31.18 Ex.31.4

**Q1.**

**Answer :**

**(i)** Negation of the given statement:

There exists a number x such that x+3≥10.

**(ii)** Negation of the given statement:

For every x ϵ N, x+3≠10.

**Q2.**

**Answer :**

**(i)** Negation of the given statement:

Some students did not complete their homework.

**(ii)** Negation of the given statement:

There exists a number which is not equal to its square.

### Page No.31.23 Ex.31.5

**Q1.**

**Answer :**

**(i)** If you pay a subscription fee, then you can access the **website.**

**(ii)** If it rains, then there is a traffic **jam.**

**(iii)** If you want to log on to the server, then you need a **passport.**

**(iv)** If you want to be happy, then you will have to be rich.

**(v)** If it rains, only then the game is **cancelled.**

**(vi)** If it rains, then it is **cold.**

**(vii)** If it rains, then it is **cold.**

**(viii)** If it is cold, then it never rains.

**Q2.**

**Answer :**

**(i)** Converse of the given statement:

If you feel thirsty, then it is hot outside.

Contrapositive of the given statement:

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

**(ii)** Converse of the given statement:

If I go to a beach, then it is a sunny day.

Contrapositive of the given statement:

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

**(iii)** Converse of the given statement:

If a positive integer has no divisors other than 1 and itself, then it is prime.

Contrapositive of the given statement:

If a positive integer has some divisors other than 1 and itself, then it is not prime.

**(iv)** Converse of the given statement:

If you have winter clothes, then you live in Delhi.

Contrapositive of the given statement:

If you do not have winter clothes, then you do not live in Delhi.

**(v)** Converse of the given statement:

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

Contrapositive of the given statement:

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

**Q3.**

**Answer :**

**(i)** You watch television if and only if your mind is free.

**(ii)** A quadrilateral is a rectangle if and only if it is equiangular.

**(iii)** You get an A grade if and only if you do all the homework regularly.

**(iv)** The tumbler is half empty if and only if the tumbler is half full.

### Page No.31.24 Ex.31.5

**Q4.**

**Answer :**

**(i)** If Mohan is not poor, then he is not a poet.

**(ii)** If Max does not study, then he will not pass the test.

**(iii)** If she does not earn money, then she will not work.

**(iv)** If they do not drive the car, then there is no snow.

**(v)** If it rains, it is not cold.

**(vi)** If it did not snow, then Ravish does not ski.

**(vii)** If x is positive, then x is not less than zero.

**(viii)** If he does not win, then he does not have courage.

**(ix)** If you are not strong, then you cannot be a sailor.

**(x)** If he tires, then he will not win.

**(xi)** If x is even, then x^{2} is even.

### Page No.31.31 Ex.31.6

**Q1.**

**Answer :**

**(i)** p: 100 is a multiple of 4 and 5.

Since 100 is a multiple of 4 and 5, the statement is true.

Hence, it is a valid statement.

**(ii)** q: 125 is a multiple of 5 and 7.

Since 125 is a multiple of 5 but is not a multiple of 7, the statement is not true.

Hence, it is not a valid statement.

**(iii)** r: 60 is a multiple of 3 or 5.

Since 60 is a multiple of 3 and 5, the statement is true.

Hence, it is a valid statement.

**Q2.**

**Answer :**

**(i)** p: If x and y are odd integers, then x + y is an even integer.

Let q and r be two statements.

Here,

q: x and y are odd integers.

r: x + y is an even integer.

Let q be true.

Then, q is true.

Now,

x and y are odd integers.

x = 2m +1 and y = 2n + 1 for some integers m and n.

⇒x + y = (2m + 1) + (2n + 1)

⇒x + y = 2m + 2n + 2 = 2(m + n + 1)

So, x + y is an even integer.

Hence, the statement is true.

**ii)** p: If x and y are integers such that xy is even, then at least one of x and y is an even integer.

Let q and r be two statements.

Here,

q: xy is an even integer.

r: At least one of x and y is an even integer.

Let r be not true.

Then, r is not true.

It is false that at least one of x and y is an even integer.

Now,

x and y are odd integers.

x = 2m +1 and y = 2n + 1 for some integers m and n.

⇒xy =(2m + 1)(2n + 1)

⇒xy = 2(2mn + m + n) + 1

So, xy is not an even integer. Thus, xy is not true.

Hence, the statement is true.

**Q3.**

**Answer :**

p : “If x is a real number such thatx3 + x = 0then x is 0”.

Let q and r be the statements.

Here,

q: x is a real number such that x3 + x = 0.

r: x is 0.

**(i)** Direct method

Let q be true.

To obtain x3 + x = 0we have:

x(x2+1) = 0

or, x = 0

Thus, r is true.

Hence, “if q, then r” is a true statement.

**(ii)** Method of contrapositive

Let r not be true.

r is not 0.

If x(x2+1) ≠0, then q is not true.

Hence, “if ~q, then ~r” is a true statement.

**(iii)** Method of contradiction

Let q not be true.

Then,

~q is true

~(q ⇒r) is true.

q &~r is true

x is a real number such that x3+x=0

Then, x is not 0.

x = 0 and x≠0

This is a contradiction.

Hence, q is true.

**Q4.**

**Answer :**

Let q and r be statements.

Here,

q: If x is an integer and x^{2} is odd.

r: x is also odd.

Then, p is” if q, then r”.

Let r be false.

So, x is not an odd integer, i.e., x is an even integer.

Let:

x = 2n for some integer n

or, x2=4n2

In other words, x^{2} is an even integer.

Now, q is false.

Thus, r is false; this implies q is false.

Hence, p is a true statement.

**Q5.**

**Answer :**

The given statement can be rewritten as:

“The necessary and sufficient condition for integer n to be even is n^{2} must be even”.

Let p and q be the following statements.

p: The integer n is even.

q: n^{2} is even.

The given statement is “p if and only if q”.

To check its validity, we have to check the validity of the following statements:

(i) If p, then q.

(ii) If q, then p.

Checking the validity of “if p, then q”

“If the integer n is even, then n^{2} is even.”

Let us assume that n is even.

Then, n=2m, where m is an integer.

Thus, we have:

n2=4m2

Here, n^{2} is even.

Therefore, “if p, then q” is true.

The statement “if q, then p” is given by

“If n is an integer and n^{2} is even, then n is even”.

To check he validity of the statement, we will use the contrapositive method. So, let n be an integer. Then,

n is odd.

Here, n=2k+1 for some integer k.

⇒n2=4k2+2k+1

Then, n^{2} is an odd integer.

n^{2} is not an even integer.

Thus “if q, then p” and “p if and only if q” are true.

**Q6.**

**Answer :**

The given statement is of the form “if q, then r”.

q: All the angles of a triangle are equal.

r: The triangle is an obtuse-angled triangle.

Statement p has to be proved false.

For this purpose, we need to prove that if q, then ~r.

To show this, none of the angles of the triangle should be obtuse.

We know that the sum of all angles of a triangle is 180°. Therefore, if all three angles are equal, then each of them will measure 60°, which is not an obtuse angle.

In an equilateral triangle, the measure of all angles is equal. Thus, the triangle is not an obtuse-angled triangle.

Hence, it can be concluded that statement p is false.

**Q7.**

**Answer :**

**(i)** The given statement is false.

According to the definition of a chord, it should intersect the circumference of a circle at two distinct points.

**(ii)** The given statement is false.

If a chord is not the diameter of a circle, then the centre does not bisect that chord. In other words, the centre of a circle only bisects the diameter, which is the chord of the circle.

**(iii)** Equation of an ellipse:

x2a2+y2b2=1

If we put a = b = 1, then we obtain x2+y2=1, which is an equation of a circle. Therefore, a circle is a particular case of an ellipse.

Thus, the statement is true.

**(iv)** x > y

⇒ –x < –y (By the rule of inequality)

Thus, the given statement is true.

**(v)** 11 is a prime number and we know that the square root of any prime number is an irrational number. Therefore, 11 is an irrational number.

Thus, the given statement is false.

**Q8.**

**Answer :**

The argument used to check the validity of the given statement is false because it is given in the argument that x^{2} = π ^{2} is irrational and therefore x = π is irrational; however, according to the statement, x is rational.