Rd Sharma Xi 2020 2021 _volume 2 Solutions for Class 12 Commerce Maths Chapter 31 Mathematical Reasoning are provided here with simple step-by-step explanations. These solutions for Mathematical Reasoning are extremely popular among Class 12 Commerce students for Maths Mathematical Reasoning Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rd Sharma Xi 2020 2021 _volume 2 Book of Class 12 Commerce Maths Chapter 31 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rd Sharma Xi 2020 2021 _volume 2 Solutions. All Rd Sharma Xi 2020 2021 _volume 2 Solutions for class Class 12 Commerce Maths are prepared by experts and are 100% accurate.

#### Page No 31.14:

#### Question 1:

Find the component statements of the following compound statements:

(i) The sky is blue and the grass is green.

(ii) The earth is round or the sun is cold.

(iii) All rational numbers are real and all real numbers are complex.

(iv) 25 is a multiple of 5 and 8.

#### 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 31.14:

#### Question 2:

For each of the following statements, determine whether an inclusive "OR" or exclusive "OR" is used. Give reasons for your answer.

(i) Students can take Hindi or Sanskrit as their third language.

(ii) To entry a country, you need a passport or a voter registration card.

(iii) A lady gives birth to a baby boy or a baby girl.

(iv) To apply for a driving licence, you should have a ration card or a passport.

#### 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.

#### Page No 31.14:

#### Question 3:

Write the component statements of the following compound statements and check whether the compound statement is true or false:

(i) To enter into a public library children need an identity card from the school or a letter from the school authorities.

(ii) All rational numbers are real and all real numbers are not complex.

(iii) Square of an integer is positive or negative.

(iv) *x* = 2 and *x* = 3 are the roots or the equation 3*x*^{2} − *x* − 10 = 0.

(v) The sand heats up quickly in the sun and does not cool down fast at night.

#### 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 $3{x}^{2}-x-10=0$.

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

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 $3{x}^{2}-x-10=0$" 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.

#### Page No 31.14:

#### Question 4:

Determine whether the following compound statements are true or false:

(i) Delhi is in India and 2 + 2 = 4.

(ii) Delhi is in England and 2 + 2 = 4.

(iii) Delhi is in India and 2 + 2 = 5.

(iv) Delhi is in England and 2 + 2 =5.

#### 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 31.16:

#### Question 1:

Write the negation of each of the following statements:

(i) For every *x* ϵ *N*, *x* + 3 < 10

(ii) There exists *x* ϵ *N*, *x* + 3 = 10

#### Answer:

(i) Negation of the given statement:

There exists a number *x* such that $x+3\ge 10$.

(ii) Negation of the given statement:

For every *x *ϵ N, $x+3\ne 10$.

#### Page No 31.16:

#### Question 2:

Negate each of the following statements:

(i) All the students completed their homework.

(ii) There exists a number which is equal to its square.

#### 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 31.21:

#### Question 1:

Write each of the following statements in the form "if *p*, then *q*".

(i) You can access the website only if you pay a subscription fee.

(ii) There is traffic jam whenever it rains.

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

(iv) It is necessary to be rich in order to be happy.

(v) The game is cancelled only if it is raining.

(vi) It rains only if it is cold.

(vii) Whenever it rains it is cold.

(viii) It never rains when it is cold.

#### 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.

#### Page No 31.21:

#### Question 2:

State the converse and contrapositive of each of the following statements:

(i) If it is hot outside, then you feel thirsty.

(ii) I go to a beach whenever it is a sunny day.

(iii) A positive integer is prime only if it has no divisors other than 1 and itself.

(iv) If you live in Delhi, then you have winter clothes.

(v) If a quadrilateral is a parallelogram, then its diagonals bisect each other.

#### 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.

#### Page No 31.21:

#### Question 3:

Rewrite each of the following statements in the form "*p* if and only if *q*".

(i) *p* : If you watch television, then your mind is free and if your mind is free, then you watch television.

(ii) *q* : If a quadrilateral is equiangular, then it is a rectangle and if a quadrilateral is a rectangle, then it is equiangular.

(iii) *r* : For you to get an A grade, it is necessary and sufficient that you do all the homework regularly.

(iv) *s* : If a tumbler is half empty, then it is half full and if a tumbler is half full, then it is half empty.

#### 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 31.21:

#### Question 4:

Determine the contrapositive of each of the following statements:

(i) If Mohan is a poet, then he is poor.

(ii) Only if Max studies will he pass the test.

(iii) If she works, she will earn money.

(iv) If it snows, then they do not drive the car.

(v) It never rains when it is cold.

(vi) If Ravish skis, then it snowed.

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

(viii) If he has courage he will win.

(ix) It is necessary to be strong in order to be a sailor.

(x) Only if he does not tire will he win.

(xi) If *x* is an integer and *x*^{2} is odd, then *x* is odd.

#### 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 31.28:

#### Question 1:

Check the validity of the following statements:

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

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

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

#### 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.

#### Page No 31.28:

#### Question 2:

Check whether the following statement are true or not:

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

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

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

$\Rightarrow $*x *+ *y* = (2*m* + 1) + (2*n** *+ 1)

$\Rightarrow $*x* + *y* = 2*m* + 2*n* + 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*: *x**y* 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* = 2*m* +1 and *y *= 2*n* + 1 for some integers *m *and* n.*

$\Rightarrow $*xy* =(2*m* + 1)(2*n* + 1)

$\Rightarrow $xy = 2(2*mn* + *m *+ *n*) + 1

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

Hence, the statement is true.

#### Page No 31.29:

#### Question 3:

Show that the statement

*p* : *"If x is a real number such that* *x*^{3} + *x* = 0, *then x is 0"*

is true by

(i) direct method

(ii) method of contrapositive

(iii) method of contradition.

#### Answer:

p : "If *x* is a real number such that${x}^{3}+x=0$then x is 0".

Let q and r be the statements.

Here,

q: *x* is a real number such that ${x}^{3}+x=0$.

r: *x* is 0.

(i) Direct method

Let q be true.

To obtain ${x}^{3}+x=0$we have:

^{$x({x}^{2}+1)=0$}

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 $x({x}^{2}+1)\ne 0$, 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 $\Rightarrow $r)* is true.

*q *&$~$*r *is true

*x* is a real number such that *${x}^{3}+x=0$*

Then, x is not 0.

*x* = 0 and *x$\ne $*0

This is a contradiction.

Hence, q is true.

#### Page No 31.29:

#### Question 4:

Show that the following statement is true by the method of contrapositive

*p : "If x is an integer and **x*^{2} i*s odd, then x is also odd"*

#### Answer:

Let *q *and *r *be statements.

Here,

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

*.*

r:

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,

*${x}^{2}=4{n}^{2}$*

In other words,

*x*is an even integer.

^{2}^{ }Now,

*q*is false.

Thus,

*r*is false; this implies

*q*is false.

Hence,

*p*is a true statement.

#### Page No 31.29:

#### Question 5:

Show that the following statement is true

*"The integer n is even if an only if **n*^{2} *is even"*

#### 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*is even.

^{2}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*is even.

^{2}*"*

Let us assume that

*n*is even.

Then, $n=2m$, where

*m*is an integer.

Thus, we have:

*${n}^{2}=4{m}^{2}$*

Here,

*n*is even.

^{2 }Therefore,

*"*if

*p,*then

*q"*is true.

The statement "if

*q,*then

*p"*is given by

"If

*n*is an integer and

*n*is even, then

^{2}^{ }*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.*

*$\Rightarrow $${n}^{2}=4{k}^{2}+2k+1$*

Then

*, n*is an odd integer

^{2}*.*

nis not an even integer.

n

^{}^{2 }Thus "if

*q,*then

*p"*and "

*p*if and only if

*q"*are true.

#### Page No 31.29:

#### Question 6:

By giving a counter example, show that the following statement is not true.

*p : "If all the angles of a triangle are equal, then the triangle is an obtuse angled triangle".*

#### 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.

#### Page No 31.29:

#### Question 7:

Which of the following statements are true and which are false? In each case give a valid reason for saying so

(i) *p* : Each radius of a circle is a chord of the circle.

(ii) *q* : The centre of a circle bisects each chord of the circle.

(iii) *r* : Circle is a particular case of an ellipse.

(iv)* s* : If *x* and *y* are integers such that *x* > *y*, then − *x* < − *y*.

(v) *t* : $\sqrt{11}$ is a rational number.

#### 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:

$\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$

If we put *a* = *b* = 1, then we obtain ${x}^{2}+{y}^{2}=1$, 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, $\sqrt{11}$ is an irrational number.

Thus, the given statement is false.

#### Page No 31.29:

#### Question 8:

Determine whether the argument used to check the validity of the following statement is correct:

*p* : "If *x*^{2} is irrational, then *x* is rational"

The statement is true because the number *x*^{2} = π^{2} is irrational, therefore *x* = π is irrational.

#### 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.

#### Page No 31.29:

#### Question 1:

*Mark the correct answer in each of the following:*

Which of the following is a statement?

(a) Rose are black

(b) Mind your business

(c) Be punctual

(d) Do not tell lies

#### Answer:

Not all roses are black

i.e. Roses are black is a false statement

Hence, Roses are black is a statement

Hence, the correct answer is option A.

#### Page No 31.29:

#### Question 2:

*Mark the correct answer in each of the following:*

The negation of the statement "It is raining and weather is cold", is

(a) It is not raining and weather is cold.

(b) it is raining or weather is not cold.

(c) It is not raining or weather is not cold.

(d) It is not raining and weather is not cold.

#### Answer:

For statement = It is raining and weather is cold.

*~ *negative statement is

It is not raining or weather is not cold. (since negation of and is or)

Hence, the correct answer is option C.

#### Page No 31.29:

#### Question 3:

*Mark the correct answer in each of the following:*

Which of the following is the converse of the statement?

"If Billu secure good marks, then he will get a bicycle."

(a) If Billu will not get bicycle, then he will not secure good marks.

(b) If Billu will get a bicycle, then he will secure good marks.

(c) If Billu will get a bicycle, then he will not secure good marks.

(d) If Billu will not get a bicycle, then he will secure good marks.

#### Answer:

For the statement,

If Billu secure good marks, then he will get a bicycle.

Let *p* and* q* be two statements given by

*p*: Billu secure good marks.

*q*: Billu will get a bicycle.

Given* p *⇒ *q*

for* q *⇒ *p,*

i.e. convert statement will be

If Billu will a get a bicycle, then he will secure good makes

Hence, the correct answer is option B.

#### Page No 31.30:

#### Question 4:

*Mark the correct answer in each of the following:*

Which of the following is a statement?

(a) *x* is a real number

(b) Switch off the fan

(c) 6 is a natural number

(d) Let me go

#### Answer:

a) *x *is a real number;

The sentence is not always true, it depends an value of *x*.

Hence, it is not a statement

b) Switch off the fan;

This sentence is not specific, i.e. no information regarding which fan.

Hence, it is not a statement

c) 6 is a natural number; is a true statement

d) Let me go,

This sentence is not specific, have not a statement.

Hence, the correct answer is option C.

#### Page No 31.30:

#### Question 5:

*Mark the correct answer in each of the following:*

Which of the following is not a statement?

(a) Smoking is injurious to health.

(b) 2 + 2 = 4

(c) 2 is the only prime number

(d) Come here

#### Answer:

a) Smoking is injurious to health;

This sentence is always True;

i.e. It is a true statement.

b) 2 + 2 = 4;

i.e. It is a true sentence.

c) 2 is the only prime number.

This sentence is false because infinitely many other prime numbers exist.

i.e. It is a false statement

d) Come here,

This sentence is not specific.

Hence, not a statement.

Hence, the correct answer is option D.

#### Page No 31.30:

#### Question 6:

*Mark the correct answer in each of the following:*

Which of the following sentences are not statements?

(a) The angles opposite to equal sides of a rectangle are equal.

(b) The moon is a setellite of the Earth

(c) May god bless you!

(d) Asia is a continent.

#### Answer:

a) The angle opposite to equal sides of a rectangle are equal;

This sentence is always True;

i.e. It is a true statement.

b) The moon is a satellite of the earth;

which is a universal truth

i.e. It is a true statement.

c) May god bless you;

This sentence is not specific about the person,

Hence, it is not a statement.

d) Asia is a continent;

which is a universal truth

i.e. it is a true statement.

Hence, the correct answer is option C.

#### Page No 31.30:

#### Question 7:

*Mark the correct answer in each of the following:*

The connective in the statement "2 + 7 > 9 or 2 + 7 < 9" is

(a) and

(b) or

(c) >

(d) <

#### Answer:

For given statement,

2 + 7 > 9 or 2 + 7 < 9

or is connective i.e or is connecting the two statements

2 + 7 > 9 and 2 + 7 < 9

Hence, the correct answer is option B.

#### Page No 31.30:

#### Question 8:

*Mark the correct answer in each of the following:*

The contrapositive of the statement "If *p*, then *q*", is

(a) If *q*, then *p*

(b) If *p*, then ~ *q*

(c) If ~ *q*, then ~ *p*

(d) If ~ *p*, then ~ *q*

#### Answer:

If *p* then *q*

Means *p *⇒ *q *

center positive from means not *q* then not *p*.

i.e. ~*q* ⇒ ~*p*

i.e.* *~*q* then ~*p*

Hence, the correct answer is option C.

#### Page No 31.30:

#### Question 9:

*Mark the correct answer in each of the following:*

The negation of the statement "7 is greater than 8", is

(a) 7 is equal to 8

(b) 7 is not greater than 8

(c) 8 is less than 7

(d) none of these

#### Answer:

For given statement;

''7 is greater than 8" i.e. 7 > 8

negation is 7 is not greater than 8

i.e. 7 ⋡ 8

Hence, the correct answer is option B.

#### Page No 31.30:

#### Question 10:

*Mark the correct answer in each of the following:*

The negation of the statement "A circle is an ellipse", is

(a) An ellipse is a circle

(b) An ellipse is not a circle

(c) A circle is not an ellipse

(d) A circle is an ellipse

#### Answer:

For given statement;

" A circle is an ellipse"

i.e. circle ⇒ ellipse

Negation is circle ⇒ ellipse

i.e. A circle is not an ellipse.

Hence, the correct answer is option C.

#### Page No 31.30:

#### Question 11:

*Mark the correct answer in each of the following:*

The negation of the statement "101 is not a multiple of 3" is

(a) 101 is a multiple of 3

(b) 101 is a multiple of 2

(c) 101 is an odd number

(d) 101 is an even number

#### Answer:

For given statement;

"101 is not a multiple of 3

i.e. 101 ≠ 3*k; *for any* k*.

negation is 101 is a multiple of 3

i.e. 101 = 3*k *; for some *k*.

Hence, the correct answer is option A.

#### Page No 31.30:

#### Question 12:

*Mark the correct answer in each of the following:*

The negation of the statement "72 is divisible by 2 and 3", is

(a) 72 is not divisible by 2 or 72 is not divisible by 3.

(b) 72 is not divisible by 2 and 72 is not divisible by 3.

(c) 72 is divisible by 2 and 72 is not divisible by 3.

(d) 72 is not divisible by 2 and 72 is divisible by 3.

#### Answer:

For given statement,

*p*: "72 is divisible by 2 and 3"

~*p*: 72 is not divisible by 2 or 72 is not divisible by 3.

Hence, the correct answer is option is A.

#### Page No 31.30:

#### Question 13:

*Mark the correct answer in each of the following:*

The negation of the statement "Rajesh or Rajni lived in Bangalore", is

(a) Rajesh did not live in Bangalore or Rajni lives in Bangalore

(b) Rajesh lives in Bangalore and Rajni did not live in Bangalore

(c) Rajesh did not live in Bangalore and Rajni did not live in Bangalore

(d) Rajesh did not live in Bangalore or Rajni did not live in Bangalore.

#### Answer:

For given statement,

*p*: " Rajni or Rajesh lived in Bangalore"

~*p*: Rajesh did not live in Bangalore and Rajni did not live in Bangalore.

Hence, the correct answer is option C.

#### Page No 31.30:

#### Question 14:

*Mark the correct answer in each of the following:*

The negation of the statement "Plants take in CO_{2} and give out O_{2}", is

(a) Plants do not take CO_{2} and do not give out O_{2}

(b) Plants do not take CO_{2} or do not give out O_{2}

(c) Plants take in CO_{2} and do not give out O_{2}

(d) Plants take in CO_{2} or do not give out O_{2}

#### Answer:

For given statement;

*p*: "Plants take in CO_{2} and give out O_{2} ''.

~*p*: "Plants do not take CO_{2} or do not give out O_{2}".

Hence, the correct answer is option B.

#### Page No 31.3:

#### Question 1:

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

(i) Listen to me, Ravi !

(ii) Every set is a finite set.

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

(iv) The cat pussy is black.

(v) Are all circles round?

(vi) All triangles have three sides.

(vii) Every rhombus is a square.

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

(ix) This sentence is a statement.

(x) Is the earth round?

(xi) Go !

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

(xiii) There are 35 days in a month.

(xiv) Mathematics is difficult.

(xv) All real numbers are complex numbers.

(xvi) The product of (−1) and 8 is 8.

#### 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 $-1\times 8=-8$; therefore, the statement is false.

#### Page No 31.3:

#### Question 2:

Give three examples of sentences which are not statements. Give reasons for the answers.

#### 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 31.31:

#### Question 15:

*Mark the correct answer in each of the following:*

The negation of the statement "The product of 3 and 4 is 9", is

(a) It is false that the product of 3 and 4 is 9.

(b) The product of 3 and 4 is 12

(c) The product of 3 and 4 is not 12

(d) It is false that the product of 3 and 4 is not 9.

#### Answer:

For given statement;

*p*: "The product of 3 and 4 is 9"

~*p*: It is false that the product of 3 and 4 is 9.

Hence, the correct answer is option A

#### Page No 31.31:

#### Question 16:

*Mark the correct answer in each of the following:*

Which of the following is not a negation of "A natural number is greater than zero".

(a) A natural number is not greater than zero

(b) It is false that a natural number is greater than zero

(c) It is false that a natural number is not greater than zero

(d) None of these

#### Answer:

For given statement;

*p*: " A natural number is greater than zero".

~*p*: A natural number is not greater than zero.

∴ Not negation of *p* is the statement, It is false that a natural number is not greater than zero.

Hence, the correct answer is option C.

#### Page No 31.31:

#### Question 17:

*Mark the correct answer in each of the following:*

The connective in the statement "Earth revolves round the Sun and Moon is a satellite of earth" is

(a) or

(b) Earth

(c) Sun

(d) and

#### Answer:

For given statement;

*p*: ''Earth revolves around the sun and moon is a satellite of earth''.

Since Earth revolves around the sun has no influence on the other statement.

∴ The two statements are connected by and.

Hence, the correct answer is option D.

#### Page No 31.31:

#### Question 18:

*Mark the correct answer in each of the following:*

The converse of the statement "If *x* > *y*, then *x* + *a* > *y* + *a*", is

(a) If *x* < *y*, then *x* + *a* < *y* + *a*

(b) If *x* + *a* > *y* + *a*, then *x* > *y*

(c) If *x* < *y*, then *x* + *a* > *y* + *a*

(d) If *x* > *y*, then *x* + *a* < *y* + *a*

#### Answer:

For the statement;

*p*: ''If *x* >* **y*, then *x *+ *a* > *y *+ *a'' *

i.e.* x* > *y *⇒ *x *+ *a* > *y *+ *a*

Converse of *p* is

*x *+ *a* > *y *+ *a* ⇒ *x* > *y *

i.e. If *x *+ *a* > *y *+ *a* then *x* > *y*

Hence, the correct answer is option B.

#### Page No 31.31:

#### Question 19:

*Mark the correct answer in each of the following:*

The converse of the statement "If sun is not shining, then sky is filled with clouds", is

(a) If sky is filled with clouds, then sun is not shining.

(b) If sun is shining, then sky is filled with clouds.

(c) If sky is clear, then sun is shining.

(d) If sun is not shining, then sky is not filled with clouds.

#### Answer:

For given statement;

*p*: ''if sun is not shining, then sky is filled with clouds''.

Converse of *p*; if sky is filled with clouds, then sun is not shining.

Hence, the correct answer is option A.

#### Page No 31.31:

#### Question 20:

*Mark the correct answer in each of the following:*

The statement "If *x*^{2} is not even, then *x* is not even", is converse of the statement

(a) If *x*^{2} is odd, then *x* is even

(b) If *x* is not even, then *x*^{2} is not even

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

(d) If *x* is odd, then *x*^{2} is even

#### Answer:

For given statement;

*p*: if *x*^{2} is not even, then *x* is not even.

i.e. *p*:* x*^{2} is not even ⇒ *x* is not even.

Converse of *p*; *x *is not even ⇒ *x*^{2} is not even.

i.e. if *x* is not even, then *x*^{2} is not even.

Hence, the correct answer is option B.

#### Page No 31.31:

#### Question 21:

*Mark the correct answer in each of the following:*

Which of the following statement is a conjunction?

(a) Ram and Shyam are friends

(b) Both Ram and Shyam are tall

(c) Both Ram and Shyam are enemies

(d) none of the above

#### Answer:

None of the statement is a conjunction, because no two compound statement formed are joined.

Hence, the correct answer is option D.

#### Page No 31.31:

#### Question 22:

*Mark the correct answer in each of the following:*

Which of the following is conditional *p* → *q*?

(a) *q* is sufficient for *p*

(b) *p* is necessary for *q*

(c) *p* only if *q*

(d) if *q*, then *p*

#### Answer:

*p* only if *q* is same* p* → *q*.

Hence, the correct answer is option C.

#### Page No 31.31:

#### Question 23:

*Mark the correct answer in each of the following:*

The contrapositive of the statement "If Chandigarh is capital of Punjab, then Chandigarh is in India", is

(a) If Chandigarh is not in India, then Chandigarh is not the capital of Punjab.

(b) If Chandigarh is in India, then Chandigarh is the capital of Punjab.

(c) If Chandigarh is not capital of Punjab, then Chandigarh is not the capital of India.

(d) If Chandigarh is capital of Punjab, then Chandigarh is not in India.

#### Answer:

For given statement;

*p*: If Chandigarh is capital of Punjab, then

*q*: Chandigarh is in India

i.e. *p* ⇒ *q*

Contrapositive form of *p* is; ~*q* ⇒ ~*p*.

i.e. If Chandigarh is not in India, then Chandigarh is not the capital of Punjab.

Hence, the correct answer is option A.

#### Page No 31.31:

#### Question 24:

*Mark the correct answer in each of the following:*

The contrapositive of the statement "If 7 is greater than 5, then 8 is greater than 6", is

(a) It 8 is greater than 6, then 7 is greater than 5

(b) If 8 is not greater than 6, then 7 is greater than 5

(c) If 8 is not greater than 6, then 7 is not greater than 5

(d) If 8 is greater than 6, then 7 is not greater than 5

#### Answer:

For given statement;

*p*: If 7 is greater than 5, then

*q* 8 is greater than 6

i.e. *p* ⇒ *q* is given.

contrapositive form is ~*q* ⇒ ~*p*

i.e. If 8 is not greater than 6, then 7 is not greater than 5.

Hence, the correct answer is option C.

#### Page No 31.31:

#### Question 25:

*Mark the correct answer in each of the following:*

Which of the following statements is not the negation of the statement "All triangles are equilateral triangles".

(a) It is false that all triangles are equilateral triangles

(b) There exists a triangle which is not an equilateral triangle

(c) Not all triangles are equilateral triangles

(d) None of these

#### Answer:

For given statement;

*p*: All triangle are equilateral triangles.

~*p*: Not all triangle are equilateral triangles. or,

~*p*: It is false that all the triangle are equilateral triangles. or,

~*p* There exist a triangle which is not an equilateral triangle.

Hence, the correct answer is option D.

#### Page No 31.32:

#### Question 26:

Which of the following is a tautology?

(a) *p* ⇒ (*q *∨ ~ *q*)

(b) *p* ⇔ (~*q* ∧ *p*)

(c) *p* ⇔ (*p *∧ ~ *p*)

(d) (*p* ∧ ~ *p*) ∧ *q*

#### Answer:

*p* ⇒ (*q *∨ ∼ *q*) is a tautological statement, if assertion is true in all possible interpretation.

Hence, the correct answer is option A.

#### Page No 31.32:

#### Question 27:

*Mark the correct answer in each of the following:*

Which of the following is a contradiction?

(a) (*p* ∨ *q*) ⇔ (*p* ∧ *q*)

(b) (*p* ∨ *q*) ⇒ (*p* ∧ *q*)

(c) (*p* ⇒ *q*) ∨ (*q* ⇒ *p*)

(d) (~*q*) ∧ (*p* ∧ *q*)

#### Answer:

Since (∼*q*) ∧ (*p *∧ *q*)

States that both *q *and ~*q* are occurring

Simultaneously; which is a contradiction.

Hence, the correct answer is option D.

#### Page No 31.32:

#### Question 28:

*Mark the correct answer in each of the following:*

The contrapositive of the statement ~ *p* ⇒ (*p* ∧ ~*q*) is

(a) *p* ⇒ (~ *p* ∨ *q*)

(b) *p* ⇒ (*p* ∧ *q*)

(c) *p* ⇒ (~ *p *∧ *q*)

(d) (~*p* ∨ *q*) ⇒ *p*

#### Answer:

For given statement;

∼*p* ⇒ (*p*∧ ∼*q*)

has contrapositive from given by

∼(*p*∧ ∼*q*) ⇒ ∼ (∼*p*)

i.e. (∼*p*∨ ∼(∼*q*)) ⇒ *p* (By De-Morgan's law)

i.e. (∼*p *∨ *q*) ⇒ *p*

Hence, the correct answer is option D.

#### Page No 31.32:

#### Question 29:

*Mark the correct answer in each of the following:*

If *p* is negation of *q*, then (*p* ⇒ *q*) v (*q* ⇒ *p*) is a

(a) tautology

(b) contradiction

(c) contingency

(d) none of these

#### Answer:

Given ~*p *= *q*

Then, (*p* ⇒ *q*) ∨ (*q* ⇒ *p*)

is (*p* ⇒ ~*p*) ∨ (~*p* ⇒ *p*)

i.e. both the possibility of occurrence is observed.

Hence, statement is a tautology statement.

Hence, the correct answer is option A.

#### Page No 31.32:

#### Question 30:

*Mark the correct answer in each of the following:*

If the truth value of *p* ∨ *q* is true, then the truth value of ~ *p* ∧ *q *is

(a) false if *p* is true

(b) true if *p* is true

(c) false if *q* is true

(d) true if *q* is true

#### Answer:

If truth value of *p *∨ *q* is true

i.e. either *p* is true or *q* is true or both *p* and* q* are true.

Then If *p* is true, ~*p *∧ *q* is false.

If *q* is true; ~*p *∧ *q*; cannot be judged.

Hence, the correct answer is option A.

#### Page No 31.32:

#### Question 1:

The property $p\wedge (q\vee r)\cong (p\wedge q)\vee (p\wedge q)$ is called_________________.

#### Answer:

*p *∧ (*q *∨ *r*) ≅ (*p *∧ *q*) ∨ (*p *∧ *r*) is called

$\frac{\mathrm{distributive}\mathrm{law}}{{o}{n}{c}{e}{}q\mathrm{and}r}\mathrm{since}\mathrm{it}\mathrm{distributes}P$

i.e. *p* ∧ (*q* ∨ *r*) ≅ (*p *∧ *q*) ∨ (*p *∧ *r*) is called __distributive law.__

#### Page No 31.32:

#### Question 2:

The property $~(p\wedge q)\cong ~p\vee ~q$ is called _____________________.

#### Answer:

For

~(*p *∧ *q*) ≅ ~*p* ∨ ~*q*

is called De Morgan's Law

#### Page No 31.32:

#### Question 3:

$~(p\vee ~q)\cong \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_.$

#### Answer:

~(*p *∨ ~*q*) = (~*p* ∧ ~(~*q*))

i.e. ~(*p *∧ ~*q*) = (~*p* ∧ *q*) (By De- Morgan's Law)

i.e. ~*p *∧ *q *is the answer

#### Page No 31.32:

#### Question 4:

$(p\vee ~q)\wedge q$ is equivalent to __________________.

#### Answer:

Given, (*p *∨ ~*q*) ∧ *q*

= (*p *∧ *q*) ∨ (~*q *∧ *q*) [By distributive law]

= (*p *∧ *q*) ∨ *ϕ*

= *p *∧ *q*

Hence, (*p *∨ ~*q*) ∧ *q* = *p *∧ *q.*

#### Page No 31.32:

#### Question 5:

$~[~p\vee (~p\iff q\left)\right]$ is equivalent to _______________.

#### Answer:

ans

#### Page No 31.32:

#### Question 6:

The negation of $~(p\wedge q)\vee r$ is ______________.

#### Answer:

Given : ~(*p *∧ *q*) ∨ *r*

Negation of above statement

is ~(~(*p *∧ *q*) ∨ *r*)

i.e. ~(~(*p *∧ *q*)) ∧ ~*r* (By Demorgan's Law)

i.e. (*p *∧ *q*) ∧ ~*r*

Hence, ~(*p *∧ *q*) ∨ *r* is equivalent to (*p *∧ *q*) ∧ ~*r*

#### Page No 31.32:

#### Question 7:

p ⇒ q ≅ _______________.

#### Answer:

*P* ⇒ *q* is logically equivalent to ~*q* ⇒ ~*p *

i.e. either *q* is true.

or *q* is false i.e. ~*q* ⇒ ~*p*

∴ ~*p* is true

∴* p* ⇒ *q* is logically equivalent to *q* ∨ ~*p*

i.e. ∼*p *∨ *q*.

#### Page No 31.32:

#### Question 8:

*p ⇔ q *≅ __________________.

#### Answer:

*p *⇔* q *given

i.e.* p* is true if *q* is true

or *p* is false if *q* is false

i.e.* p* ⇔ *q* is equivalent to (*p *∧ *q*) ∨ (~*p *∧ ~*q*)

i.e. both *p* and *q* are true or false simultaneously

∴* p* ⇔ *q* ≅ (*p *∧ *q*) ∨ (~*p *∧ ~*q*)

#### Page No 31.32:

#### Question 9:

If *p* ∨ ∼ *q* is false (F), then *q* is ___________________.

#### Answer:

Given, *p *∨ ~*q* is false

i.e. both *p* and ~*q* are false

i.e. *p* is false but *q* is true

i.e. *q* is true (T)

#### Page No 31.32:

#### Question 10:

The contrapositive of the statement p ⇒ ∼q is ________________________.

#### Answer:

For given statement;

*p* ⇒ ~*q *

Contrapositive form is ~(~*q*) ⇒ ~*p*

i.e. *q *⇒ ~*p *

∴ Contrapositive form *p* ⇒ ~*q* is *q *⇒ ~*p*

#### Page No 31.32:

#### Question 11:

$~[~p\wedge (p\iff q\left)\right]$ is equivalent to _______________________.

#### Answer:

~[~*p *∧ (*p* ⇔ *q*))

Since *p* ⇔ *q* is equivalent to if *p* than *q *and if *q* than *p*.

or *p* ⇔ *q* is equivalent to ~*q* ⇒ ~*p* and ~*p* ⇒ ~*q *

∴ ~[~*p *∧ {(~*q *⇒ ~*p*) ∧ (~*p *⇒ ~*q*)}]

= ~[~*p *∧ (~*p *⇒ ~*q*)]

= ~[~*p *∧ (~*q*)] (∴ ~*p* is True)

= *p *∨ *q* (By De-Morgan's Law)

#### Page No 31.32:

#### Question 12:

The statement $p\Rightarrow (p\vee q)$ is a ________________.

#### Answer:

For given statement,

*p *⇒ *p *∨ *q*

both the possibility are present

*∴ p *⇒ *p *∨ *q* is a tautology

#### Page No 31.32:

#### Question 13:

The statement $(p\iff q)\iff (~p\iff ~q)$ is a __________________________.

#### Answer:

For given statement,

(*p *⇔ *q*) ⇔ (~*p *⇔ ~*q*)

both the possibility of statements is possible

∴ (*p *⇔ *q*) ⇔ (~*p *⇔ ~*q*) is tautology

#### Page No 31.32:

#### Question 14:

$p\vee ~(p\Rightarrow ~q)$ is equivalent to ____________________.

#### Answer:

For given statement;

*p *∨ ~(*p *⇒ ~*q*)

i.e. *p *∨ ~(*p *⇒ ~*q*)

i.e. *p *∨ (*p *⇒ ~*q*)

i.e.* p *∨ ~(*p *⇒ ~*q*) is equivalent to* p* and *q*.

#### Page No 31.6:

#### Question 1:

Write the negation of the following statements:

(i) Banglore is the capital of Karnataka.

(ii) It rained on July 4, 2005.

(iii) Ravish is honest.

(iv) The earth is round.

(v) The sun is cold.

#### 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 31.7:

#### Question 2:

(i) All birds sing.

(ii) Some even integers are prime.

(iii) There is a complex number which is not a real number.

(iv) I will not go to school.

(v) Both the diagonals of a rectangle have the same length.

(vi) All policemen are thieves.

#### 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.

#### Page No 31.7:

#### Question 3:

Are the following pairs of statements are negation of each other:

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

The number *x* is not an irrational number.

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

The number *x *is an irrational number.

#### 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.

#### Page No 31.7:

#### Question 4:

Write the negation of the following statements:

(i) *p* : For every positive real number *x*, the number (*x* − 1) is also positive.

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

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

#### 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 $x1$ nor $x1$.

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

~*r*: For every real number *x*, either $x\le 0$ or $x1$.

#### Page No 31.7:

#### Question 5:

Check whether the following pair of statements are negation of each other. Give reasons for your answer.

(i) *a* + *b* = *b* + *a* is true for every real number *a* and *b*.

(ii) There exist real numbers *a* and *b* for which *a* + *b* = *b* + *a*.

#### 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 $a+b\ne b+a$".* *

View NCERT Solutions for all chapters of Class 15