RS Aggarwal Solutions for Class 11 Chapter 29 Mathematical Reasoning

Exercise 29A

Question 1. Which of the following sentences are statements? In case of a statement mention whether it is true or false.
(i) The sun is a star.
(ii) √7 is an irrational number.
(iii) The sum of 5 and 6 is less than 10.
(iv) Go to your class.
(v) Ice is always cold.
(vi) Have you ever seen the Red Fort?
(vii) Every relation is a function.
(viii) The sum of any two sides of a triangle is always greater than the third side.
(ix) May God bless you!
Answer: (i) "The sun is a star" qualifies as a statement. It represents a scientifically established fact that the sun is indeed a star, making this sentence always true. Hence it is a statement, and it is true.

Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both.

(ii) "√7 is an irrational number." An irrational number refers to any number that cannot be expressed as a ratio of two integers. √7 cannot be expressed as a ratio of two integers, therefore √7 is an irrational number; so the sentence is always true. Hence it is a statement, and it is true.

Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both.

(iii) The sentence is false because the sum of 5 and 6 does not equal less than 10. Adding 5 and 6 gives 11, which is not less than 10. Hence it is a statement. The statement is true.

Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both.

(iv) The sentence "Go to your class" represents an instruction. Hence it is not a statement.

Note: A sentence which is in the form of an order, exclamation and question is not a statement.

(v) "Ice is always cold" qualifies as a statement. It is scientifically verified that ice is always cold, and so the sentence is always true.

Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both.

(vi) The sentence "Have you ever seen the Red Fort?" is a question, hence it is not a statement.

Note: A sentence which is in the form of an order, exclamation and question is not a statement.

(vii) The sentence "Every relation is a function" qualifies as a statement. Some relations exist that are not functions. Therefore the sentence is false. Hence it is a statement, and it is false.

Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both.

(viii) The sentence "The sum of any two sides of a triangle is always greater than the third side" qualifies as a statement. This is because the sum of any two sides of a triangle is always greater than the third side. Hence the statement is true.

(ix) The sentence "May God bless you!" is an exclamation. Hence it is not a statement.

Note: A sentence which is in the form of an order, exclamation and question is not a statement.

Exam Tip: Always classify a sentence first - check if it is a question, command, exclamation, or declarative statement. Only declarative statements that have a definite true or false value qualify as mathematical statements.

Question 2. Which of the following sentences are statements? In case of a statement, mention whether it is true or false.
(i) Paris is in France.
(ii) Each prime number has exactly two factors.
(iii) The equation x² + 5|x| + 6 = 0 has no real roots.
(iv) (2 + √3) is a complex number.
(v) Is 6 a positive integer?
(vi) The product of -3 and -2 is -6.
(vii) The angles opposite the equal sides of an isosceles triangle are equal.
(viii) Oh! It is too hot.
(ix) Monika is a beautiful girl.
(x) Every quadratic equation has at least one real root.
Answer: (i) The sentence "Paris is in France" qualifies as a statement. Paris is found in France, making the given sentence true, so it is a statement. The statement is true.

Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both.

(ii) The sentence "Each prime number has exactly two factors" qualifies as a statement. It is a mathematically proven fact that each prime number has exactly two factors, so the given sentence is true. Hence it is a statement. The statement is true.

Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both.

(iii) The sentence "The equation x² + 5|x| + 6 = 0 has no real roots" qualifies as a statement. The equation x² + 5|x| + 6 = 0 does not have real roots.

Case 1: (x ≥ 0)
\( |x| = x: (x \geq 0) \)
\( x^2 + 5|x| + 6 = 0 \)
\( x^2 + 5x + 6 = 0 \)
\( (x + 2)(x + 3) = 0 \)
\( x = -2 \text{ and } x = -3 \)
But we assumed x ≥ 0. So it is a contradiction.

Case 2: (x < 0)
\( |x| = -x: (x < 0) \)
\( x^2 + 5|x| + 6 = 0 \)
\( x^2 - 5x + 6 = 0 \)
\( (x - 2)(x - 3) = 0 \)
\( x = 2 \text{ and } x = 3 \)
But we assumed x < 0. So it is a contradiction.

So, there are no real roots for the equation x² + 5|x| + 6 = 0
So, the given sentence is true, and it is a statement.

Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both.

(iv) The sentence "(2 + √3) is a complex number" qualifies as a statement. A number that can be expressed in the form "a + ib" qualifies as a complex number. Since (2 + √3) cannot be expressed in "a + ib" form, 2 + √3 is not a complex number. So the given sentence is a statement, and it is false.

Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both.

(v) The sentence "Is 6 a positive integer?" is a question, so it is not a statement.

Note: A sentence which is in the form of an order, exclamation and question is not a statement.

(vi) The sentence "The product of -3 and -2 is -6" qualifies as a statement. The product of -3 and -2 is 6 not -6, the given sentence is false. Hence the given sentence is a statement. This statement is false.

Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both.

(vii) The sentence given qualifies as a statement. It is mathematically proven that the angles opposite to the equal sides of an isosceles triangle are equal. So the given sentence is true, and it is a statement.

Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both.

(viii) The sentence "Oh! It is too hot" is not a statement. It represents an exclamation, and hot is subjective - it is not a fact, and it is an opinion. So, the given sentence is not a statement.

Note: A sentence which is in the form of an order, exclamation and question is not a statement.

(ix) The sentence "Monica is a beautiful girl" is not a statement. The given sentence expresses an opinion; this can be true in some cases, false in others. So, the given sentence is not a statement.

Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both.

(x) The given sentence qualifies as a statement. Not every quadratic equation will have a real root. So the given sentence is false. It is a statement. This statement is false.

Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both.

Exam Tip: Test each component logically and systematically. Check whether the sentence type (declarative, question, command, exclamation) can produce a definite truth value, and verify that truth value using mathematical logic or definitions.

Question 3. Which of the following statements are true and which are false? In each case give a valid reason for your answer.
(i) p: √11 is an irrational number
(ii) q: Circle is a particular case of an ellipse.
(iii) r: Each radius of a circle is a chord of the circle
(iv) S: The center of a circle bisects each chord of the circle
(v) t: If a and b are integers such that a < b, then –a > -b.
(vi) y: The quadratic equation x² + x + 1 = 0 has no real roots
Answer: (i) p: "√11 is an irrational number" is a TRUE statement. An irrational number refers to any number that cannot be expressed as a ratio of two integers. √11 cannot be expressed as a ratio of two integers, so √11 is an irrational number.

(ii) q: "Circle is a particular case of an ellipse" is a TRUE statement. A circle represents a special case of an ellipse where the radius stays the same at all points. The equation of an ellipse is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). When a = b, we get the equation of the circle, \( x^2 + y^2 = 1 \).

(iii) r: "Each radius of a circle is a chord of the circle" is a FALSE statement. A chord connects the circle at two points, but a radius touches the circle at only one point. So the radius is not a chord of the circle.

(iv) S: "The center of a circle bisects each chord of the circle" is a FALSE statement. Only the diameter of a circle gets bisected by the center of the circle. Apart from the diameter, no other chords are bisected by the center of the circle. The center lies only on the diameter of the circle.

(v) t: "If a and b are integers such that a < b, then –a > -b" is a TRUE statement. When a < b, then –a > -b holds true according to the rule of inequality.

(vi) y: "The quadratic equation x² + x + 1 = 0 have no real roots" is a TRUE statement. The standard form of a quadratic equation is ax² + bx + c = 0. If b² - 4ac < 0, there is no real solution. In the given equation; x² + x + 1 = 0: a = 1; b = 1; c = 1. So b² - 4ac = 1 - 4 × 1 × 1 = -3 < 0. Thus, there is no real root.

Exam Tip: For true/false statements, always provide mathematical justification - either verify the claim directly or give a counterexample. Understanding definitions (like chord, radius, ellipse) is essential for geometry statements.

Question 4. Write the negation of each of the following statements:
(i) Every natural number is greater than 0.
(ii) Both the diagonals of a rectangle are equal.
(iii) The sum of 4 and 5 is 8.
(iv) The number 6 is greater than 4.
(v) Every natural number is an integer.
(vi) The number -5 is a rational number
(vii) All cats scratch.
(viii) There exists a rational number x such that x² = 3.
(ix) All students study mathematics at the elementary level.
(x) Every student has paid the fees.
(xi) There is some integer k for which 2k = 6.
(xii) None of the students in this class has passed.
Answer: (i) The negation of the given statement is: It is false that every natural number is greater than 0.

(Or)

Every natural number is not greater than 0.

(Or)

There exists a natural number which is not greater than 0.

(ii) The negation of the given statement is: It is false that both the diagonals of a rectangle are equal.

(Or)

There exists at least one rectangle whose both the diagonals are not equal.

(iii) The negation of the given statement is: It is false that the sum of 4 and 5 is 8.

(Or)

The sum of 4 and 5 is not 8.

(iv) The negation of the given statement is: It is false that the number 6 is greater than 4.

(Or)

The number 6 is not greater than 4.

(v) The negation of the given statement is: It is false that every natural number is an integer.

(Or)

Every natural number is not an integer.

(Or)

There exists at least one natural number which is not an integer.

(vi) The negation of the given statement is: It is false that the number -5 is a rational number.

(Or)

The number -5 is not a rational number.

(vii) The negation of the given statement is: It is false that all cats scratch.

(Or)

There exists a cat which does not scratch.

(viii) The negation of the given statement is: It is false that there exists a rational number x such that x² = 3.

(Or)

There does not exist a rational number x such that x² = 3

(ix) The negation of the given statement is: It is false that all students study mathematics at the elementary level.

(Or)

It is not the case that all students study mathematics at the elementary level.

(x) The negation of the given statement is: It is false that every student has paid the fees.

(Or)

It is not the case that every student has paid the fees.

(Or)

There exists at least a student who does not pay the fees.

(xi) The negation of the given statement is: It is false that there is some integer k for which 2k = 6.

(Or)

It is not the case there is some integer k for which 2k = 6

(xii) The negation of the given statement is: It is false that none of the students in this class has passed.

(Or)

It is not the case that none of the students of this class has passed.

Exam Tip: When negating universal statements (every, all), convert to existential form (there exists). When negating existential statements (there exists, some), convert to universal form (all, every). Pay careful attention to logical connectives and quantifiers.

Exercise 29B

Question 1. Split each of the following into simple sentences and determine whether it is true or false.
(i) A line is straight and extends indefinitely in both the directions.
(ii) A point occupies a position, and its location can be determined.
(iii) The sand heats up quickly in the sun and does not cool down fast at night.
(iv) 32 is divisible by 8 and 12.
(v) x = 1 and x = 2 are the roots of the equation x² - x - 2 = 0.
(vi) 3 is rational, and √3 is irrational.
(vii) All integers are rational numbers, and all rational numbers are not real numbers.
(viii) Lucknow is in Uttar Pradesh, and Kanpur is in Uttarakhand.
Answer: (i) p: A line is straight.

q: A line extends indefinitely in both the directions.

Both the simple sentences are TRUE; therefore, the given sentence is TRUE.

(ii) p: A point occupies a position.

q: Its location can be determined.

Both the simple sentences are TRUE; therefore, the given sentence is TRUE.

(iii) p: The sand heats up quickly in the sun.

q: The sand does not cool down fast at night.

The first sentence is TRUE and the second sentence is FALSE.

Therefore the given sentence is FALSE.

(iv) p: 32 is divisible by 8.

q: 32 is divisible by 12.

First sentence is TRUE. Second sentence is FALSE.

Therefore the given sentence is FALSE.

(v) p: x = 1 is a root of the equation x² - x - 2 = 0

q: x = 2 is a root of the equation x² - x - 2 = 0

So, the first sentence is FALSE. Second sentence is TRUE.

Therefore, the given sentence is FALSE.

(vi) p: 3 is rational.

q: √3 is irrational.

First sentence is TRUE. Second sentence is TRUE.

Both the sentences are TRUE; therefore the given sentence is TRUE.

(vii) p: All integers are rational numbers.

q: All rational numbers are not real numbers.

First sentence is TRUE. Second sentence is FALSE.

Therefore, the given sentence is FALSE.

(viii) p: Lucknow is in Uttar Pradesh.

q: Kanpur is in Uttarakhand.

First sentence is TRUE. Second sentence is FALSE.

Therefore, the given sentence is FALSE

Exam Tip: When a compound sentence uses "and," both components must be true for the entire statement to be true. If even one component is false, the whole statement is false. Always evaluate each simple sentence independently first.

Question 2. Split each of the following into simple sentences and determine whether it is true or false. Also, determine whether an 'inclusive or' or 'exclusive or' is used.
(i) The sum of 3 and 7 is 10 or 11.
(ii) (1 + i) is a real or a complex number.
(iii) Every quadratic equation has one or two real roots.
(iv) You are wet when it rains, or you are in a river.
(v) 24 is a multiple of 5 or 8.
(vi) Every integer is rational or irrational.
(vii) For getting a driving license, you should have a ration card or a passport.
(viii) 100 is a multiple of 6 or 8.
(ix) Square of an integer is positive or negative.
(x) Sun rises or Moon sets.
Answer: (i) p: The sum of 3 and 7 is 10 or 11.

q: The sum of 3 and 7 is 10.

r: The sum of 3 and 7 is 11.

First sentence is TRUE. Second sentence is FALSE. Therefore, the given sentence is FALSE. The "or" used is 'Exclusive or'.

(ii) p: (1 + i) is a real or a complex number.

q: (1 + i) is a real number.

r: (1 + i) is a complex number.

First sentence is TRUE. Second sentence is FALSE. The "or" used is 'Exclusive or'.

(iii) p: Every quadratic equation has one or two real roots.

q: Every quadratic equation has one real root.

r: Every quadratic equation has two real roots.

Both the sentences are FALSE. So, the compound sentence itself is FALSE.

(iv) p: You are wet when it rains, or you are in a river.

q: You are wet when it rains.

r: You are wet when you are in a river.

Both the component sentences are TRUE. The "or" used is 'Inclusive or' because you can get wet either when it rains or when you are in the river.

(v) p: 24 is a multiple of 5 or 8.

q: 24 is a multiple of 5.

r: 24 is a multiple of 8.

First sentence is FALSE. Second sentence is TRUE. The "or" used is 'Exclusive or'.

(vi) p: Every integer is rational or irrational.

q: Every integer is rational.

r: Every integer is irrational.

The first sentence is TRUE. The second sentence is TRUE. But both cannot be TRUE at the same time. The "or" used is 'Exclusive or'.

(vii) p: For getting a driving license you should have a ration card or a passport.

q: For getting a driving license you should have a ration card.

r: For getting a driving license you should have a passport.

Both sentences are TRUE. The "or" used is 'Inclusive or', because one can get a driving license with ration card or with passport or when they have both.

(viii) p: 100 is a multiple of 6 or 8.

q: 100 is a multiple of 6.

r: 100 is a multiple of 8.

Both the sentences are FALSE. So the compound sentence itself is FALSE.

(ix) p: Square of an integer is positive or negative.

q: Square of an integer is positive.

r: Square of an integer is negative.

First sentence is TRUE. Second sentence is FALSE. The "or" used is 'Exclusive or'.

(x) p: Sun rises or Moon sets.

q: Sun rises.

r: Moon sets.

Here both the sentences are TRUE, but only one occurs at a time. So, the "or" used is 'Exclusive or'.

Exam Tip: In "exclusive or," exactly one of the two statements is true. In "inclusive or," at least one (possibly both) statements can be true. Think about whether the two conditions can happen together to determine which type of "or" is used.

Question 3. Find the truth set in case of each of the following open sentences defined on N:
(i) x + 2 < 10
(ii) x + 5 < 4
(iii) x + 3 > 2
Answer: (i) The open sentence x + 2 < 10 is defined on N; the set of natural numbers.

N: {1, 2, 3, 4…}

x = 1 → x + 2 = 3 < 10

x = 2 → x + 2 = 4 < 10

x = 3 → x + 2 = 5 < 10

x = 4 → x + 2 = 6 < 10

x = 5 → x + 2 = 7 < 10

x = 6 → x + 2 = 8 < 10

x = 7 → x + 2 = 9 < 10

x = 8 → x + 2 = 10

So, ∃ x ∈ N, such that x + 2 < 10

x = {1, 2, 3, 4, 5, 6, 7} satisfies x + 2 < 10.

So, the truth set of open sentence x + 2 < 10 defined on N is,

{1, 2, 3, 4, 5, 6, 7}

(ii) The open sentence x + 5 < 4 is defined on N; the set of natural numbers.

N: {1, 2, 3, 4…}

x = 1 → 1 + 5 = 6 > 4

So, the truth set of open sentence x + 5 < 4 defined on N is an empty set, {}.

(iii) The open sentence x + 3 > 2 is defined on N; the set of natural numbers.

N: {1, 2, 3, 4…}

x = 1 → x + 3 = 4 > 2

x = 2 → x + 3 = 5 > 2

x = 3 → x + 3 = 6 > 2

x = 4 → x + 3 = 7 > 2

x = 5 → x + 3 = 8 > 2

x = 6 → x + 3 = 9 > 2

And so on...

So, ∃ x ∈ N, such that x + 3 > 2

x = {1, 2, 3, 4, 5, 6, 7....} satisfies x + 3 > 2.

So, the truth set of open sentence x + 3 > 2 defined on N is an infinite set as there is infinite natural numbers satisfying the equation x + 3 > 2.

{1, 2, 3, 4, 5, 6, 7….}

Exam Tip: An open sentence becomes true or false only when you substitute values. Test values systematically to find all solutions. The truth set can be empty (no solutions), finite (limited solutions), or infinite (unlimited solutions).

Question 4. Let A = [2, 3, 5, 7]. Examine whether the statements given below are true or false.
(i) ∃ x ∈ A such that x + 3 > 9.
(ii) ∃ x ∈ A such that x is even.
(iii) ∃ x ∈ A such that x + 2 = 6.
(iv) ∀ x ∈ A, x is prime.
(v) ∀ x ∈ A, x + 2 < 10.
(vi) ∀ x ∈ A, x + 4 ≥ 11
Answer: A = [2, 3, 5, 7] (given in the question).

The given statement is: ∃ x ∈ A such that x + 3 > 9.

So, we need to see whether there exists 'x' which belongs to 'A', such that x + 3 > 9.

When x = 7 ∈ A,

x + 3 = 7 + 3 = 10 > 9

So, ∃ x ∈ A and x + 3 > 9.

So, the given statement is TRUE.

(ii) A = [2, 3, 5, 7] (given in the question).

The given statement is ∃ x ∈ A such that x is even.

So, we need to see whether there exists 'x' which belongs to 'A', such that x is even.

In the set A = [2, 3, 5, 7]

x = 2, is an even number and 2 ∈ A.

So, ∃ x ∈ A and x is even.

So, the given statement is TRUE.

(iii) A = [2, 3, 5, 7] (given in the question).

The given statement is ∃ x ∈ A such that x + 2 = 6.

So, we need to see whether there exists 'x' which belongs to 'A', such that x + 2 = 6.

When x = 4 ∉ A,

x + 2 = 4 + 2 = 6

So, there is no element in A for which x + 2 = 6. Thus the given statement is FALSE.

(iv) A = [2, 3, 5, 7] (given in the question).

The given statement is ∀ x ∈ A, x is prime.

So, we need to see whether all elements of 'A' are prime numbers.

In the set A = [2, 3, 5, 7], all the numbers 2, 3, 5, and 7 are prime numbers.

So, ∀ x ∈ A, x is prime.

So, the given statement is TRUE.

(v) A = [2, 3, 5, 7] (given in the question).

The given statement is ∀ x ∈ A, x + 2 < 10.

So, we need to see whether for all elements of 'A', x + 2 < 10.

When x = 2 ∈ A, x + 2 = 4 < 10 ✓

When x = 3 ∈ A, x + 2 = 5 < 10 ✓

When x = 5 ∈ A, x + 2 = 7 < 10 ✓

When x = 7 ∈ A, x + 2 = 9 < 10 ✓

So, ∀ x ∈ A, x + 2 < 10. The given statement is TRUE.

(vi) A = [2, 3, 5, 7] (given in the question).

The given statement is ∀ x ∈ A, x + 4 ≥ 11

So, we need to see whether for all elements of 'A', x + 4 ≥ 11.

When x = 2 ∈ A, x + 4 = 6 < 11 ✗

So, not all elements of A satisfy x + 4 ≥ 11. The given statement is FALSE.

Exam Tip: For existential statements (∃), you only need to find one element that satisfies the condition. For universal statements (∀), every element must satisfy the condition - finding even one exception makes the statement false.

Question 1. (ii) ∃ x ∈ A such that x is even.
Answer: Let us check each element of A = {2, 3, 5, 7}. The number 2 is an even integer. Since there is at least one element in A that satisfies the condition of being even, the statement is true.
In simple words: We need to find just one number in the set that is even. Since 2 is even, the statement holds.

Exam Tip: For existential statements (∃), you only need to find one example that works. One counterexample is enough to prove the statement true.

Question 2. (iii) A = [2, 3, 5, 7]. The given statement is: ∃ x ∈ A such that x + 2 = 6.
Answer: We need to check if any element of A, when we add 2 to it, gives 6. Testing each value: when x = 2, we get 4; when x = 3, we get 5; when x = 5, we get 7; when x = 7, we get 9. None of these equal 6. Therefore, no such element exists in A, making the statement false.
In simple words: We check if adding 2 to any number in the set gives 6. None of the results match 6, so the statement is false.

Exam Tip: Always test all elements systematically when checking existential claims. If even one element fails to satisfy the condition, the statement is false.

Question 3. (iv) A = [2, 3, 5, 7]. The given statement is: ∀ x ∈ A, x is prime.
Answer: We verify whether all elements in A are prime numbers. The set A contains 2, 3, 5, and 7, each of which is only divisible by 1 and itself. All four values are indeed prime numbers. Therefore, the statement is true.
In simple words: A prime number has no divisors except 1 and itself. All four numbers in the set fit this definition, so the statement is true.

Exam Tip: For universal statements (∀), every single element must satisfy the condition. If even one element fails, the statement is false.

Question 4. (v) A = [2, 3, 5, 7]. The given statement is: ∀ x ∈ A, x + 2 < 10.
Answer: We check whether adding 2 to each element yields a value less than 10. For x = 2, we get 4 < 10; for x = 3, we get 5 < 10; for x = 5, we get 7 < 10; for x = 7, we get 9 < 10. In every case, the sum is less than 10. Thus, the statement is true.
In simple words: We add 2 to each number and check if the result is smaller than 10. All results satisfy this, so the statement is true.

Exam Tip: For universal claims involving inequality, verify the boundary case carefully - in this problem, the largest value (7 + 2 = 9) still satisfies x + 2 < 10.

Question 5. (vi) A = [2, 3, 5, 7]. The given statement is: ∀ x ∈ A, x + 4 ≥ 11.
Answer: We test whether adding 4 to each element gives a value at least 11. For x = 2, we get 6, which is not ≥ 11; for x = 3, we get 7, which is not ≥ 11; for x = 5, we get 9, which is not ≥ 11; for x = 7, we get 11, which is ≥ 11. Since the condition fails for three out of four elements, the statement is false.
In simple words: Most numbers in the set, when increased by 4, do not reach 11. Because not all elements satisfy the condition, the statement is false.

Exam Tip: A universal statement fails completely if even one element does not satisfy the condition. Check the smallest elements first, as they are most likely to violate the inequality.

Exercise 29C

Question 6. Rewrite the following statement in five different ways conveying the same meaning. If a given number is a multiple of 6, then it is a multiple of 3.
Answer: Five equivalent rewordings of the statement are:

I. A given number being a multiple of 6 means it must be a multiple of 3.

II. A given number is a multiple of 6 only when it is a multiple of 3.

III. For a number to be a multiple of 6, it is essential that the number is a multiple of 3.

IV. If a number is a multiple of 3, then it is sufficient to conclude that it is a multiple of 6.

V. Whenever a number is not a multiple of 3, it cannot be a multiple of 6.
In simple words: Different ways to phrase the same logical relationship help clarify the meaning. Each version highlights a different aspect - implication, necessity, sufficiency, or contrapositive - while keeping the mathematical content identical.

Exam Tip: Master the conversions: "if p then q" becomes "p only if q," "q if p," "p is sufficient for q," "q is necessary for p," and "not q implies not p" (contrapositive). These forms are interchangeable.

Question 7. Write each of the following statements in the form 'if .... then':
(i) A rhombus is a square only if each of its angles measures 90°.
(ii) When a number is a multiple of 9, it is necessarily a multiple of 3.
(iii) You get a job implies that your credentials are good.
(iv) Atmospheric humidity increase only if it rains
(v) If a number is not a multiple of 3, then it is not a multiple of 6.
Answer:
(i) If a rhombus has each angle measuring 90°, then it is a square.

(ii) If a number is a multiple of 9, then it is a multiple of 3.

(iii) If you obtain a job, then your credentials are strong.

(iv) If it rains, then atmospheric humidity rises.

(v) If a number is a multiple of 6, then it is a multiple of 3.
In simple words: The "if - then" form clearly shows the cause (hypothesis) and effect (conclusion). This structure makes logical relationships easy to identify and compare.

Exam Tip: When converting "only if" or "necessarily" statements, remember that "p only if q" becomes "if p then q," and "necessarily" indicates sufficiency of the first condition for the second.

Question 8. Write the converse and contrapositive of each of the following:
(i) If x is a prime number, then x is odd.
(ii) If a positive integer n is divisible by 9, then the sum of its digits is divisible by 9.
(iii) If the two lines are parallel, then they do not intersect in the same plane.
(iv) If the diagonal of a quadrilateral bisect each other, then it is a parallelogram.
(v) If A and B are subsets of X such that A ⊆ B, then (X - B) ⊆ (X - A)
(vi) If f(2) = 0, then f(x) is divisible by (x - 2).
(vii) If you were born in India, then you are a citizen of India.
(viii) If it rains, then I stay at home.
Answer:
(i) Converse: If x is an odd number, then x is prime. Contrapositive: If x is not an odd number, then x is not prime.

(ii) Converse: If the sum of digits of a positive integer n is divisible by 9, then n is divisible by 9. Contrapositive: If the sum of digits of a positive integer n is not divisible by 9, then n is not divisible by 9.

(iii) Converse: If two lines do not intersect in the same plane, then they are parallel. Contrapositive: If two lines intersect in the same plane, then they are not parallel.

(iv) Converse: If a quadrilateral is a parallelogram, then its diagonals bisect each other. Contrapositive: If a quadrilateral is not a parallelogram, then its diagonals do not bisect each other.

(v) Converse: If (X - B) ⊆ (X - A), then A and B are subsets of X such that A ⊆ B. Contrapositive: If (X - B) is not a subset of (X - A), then either A is not a subset of B, or at least one of A or B is not a subset of X.

(vi) Converse: If f(x) is divisible by (x - 2), then f(2) = 0. Contrapositive: If f(x) is not divisible by (x - 2), then f(2) ≠ 0.

(vii) Converse: If you are a citizen of India, then you were born in India. Contrapositive: If you are not a citizen of India, then you were not born in India.

(viii) Converse: If I stay at home, then it rains. Contrapositive: If I do not stay at home, then it does not rain.
In simple words: The converse flips the hypothesis and conclusion but may not be true. The contrapositive also flips and negates both parts - and is logically equivalent to the original statement, so it has the same truth value.

Exam Tip: Always remember that the contrapositive of a true statement is also true, but the converse may be false. This distinction is critical in proofs and logical reasoning problems.

Question 9. Given below are some pairs of statements. Combine each pair using if and only if:
(i) p: If a quadrilateral is equiangular, then it is a rectangle.
q: If a quadrilateral is a rectangle, then it is equiangular.

(ii) p: If the sum of the digits of a number is divisible by 3, then the number is divisible by 3.
q: If a number is divisible by 3, then the sum of its digits is divisible by 3.

(iii) p: A quadrilateral is a parallelogram if its diagonals bisect each other.
q: If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

(iv) p: If f(a) = 0, then (x - a) is a factor of polynomial f(x).
q: If (x - a) is a factor of polynomial f(x), then f(a) = 0.

(v) p: If a square matrix A is invertible, then |A| is nonzero.
q: If A is a square matrix such that |A| is nonzero, then A is invertible.
Answer:
(i) A quadrilateral forms a rectangle if and only if it is equiangular.

(ii) A number is divisible by 3 if and only if the sum of its digits is divisible by 3.

(iii) A quadrilateral qualifies as a parallelogram if and only if its diagonals bisect each other.

(iv) The expression (x - a) serves as a factor of polynomial f(x) if and only if f(a) = 0.

(v) A square matrix A is invertible if and only if |A| is nonzero.
In simple words: "If and only if" combines two statements that imply each other - when one is true, the other must be true, and vice versa. This biconditional relationship shows complete equivalence.

Exam Tip: "If and only if" (iff) requires that both the statement and its converse are true. In mathematics, this creates a necessary and sufficient condition - very useful for definitions and characterizations.

Question 10. Write each of the following using 'if and only if':
(i) In order to get A grade, it is necessary and sufficient that you do all the homework regularly.
(ii) If you watch television, then your mind is free, and if your mind is free, then you watch television.
Answer:
(i) You obtain an A grade if and only if you complete all your homework on a regular basis.

(ii) You watch television if and only if your mind becomes free.
In simple words: The phrase "necessary and sufficient" translates directly to "if and only if." Similarly, when two conditional statements point to each other (if p then q, and if q then p), they form a biconditional relationship.

Exam Tip: Watch for language cues: "necessary and sufficient," paired conditionals, and "both directions true" all signal the need for an "if and only if" statement.

Exercise 29D

Question 11. Let p: If x is an integer and x² is even, then x is even. Using the method of contrapositive, prove that p is true.
Answer: To prove using the contrapositive method, we establish that if x is not even, then x² is not even.

Assume x is an odd (or not even) integer. We can write x = 2n + 1 for some integer n.

Squaring both sides: x² = (2n + 1)² = 4n² + 4n + 1

Notice that 4n² and 4n are both even. When we add 1 to an even number, the result becomes odd (not even).

Therefore, x² = 4n² + 4n + 1 is odd, which means x² is not even.

This confirms the contrapositive: if x is not even, then x² is not even. Since the contrapositive is true, the original statement p is also true.
In simple words: We assume x is odd and show that x² must also be odd. An odd number squared stays odd because odd times odd is always odd. This proves our statement works backwards, which guarantees it works forwards too.

Exam Tip: The contrapositive method is powerful because proving "not q implies not p" is logically equivalent to proving "p implies q." Choose contrapositive when the negated statement is easier to work with.

Question 12. Consider the statement: q: For any real numbers a and b, a² = b² ⇒ a = b. By giving a counter-example, prove that q is false.
Answer: We demonstrate this with specific numbers. Take a = +5 and b = -5. Computing the squares: a² = (+5)² = 25 and b² = (-5)² = 25. Thus a² = b², and the condition is satisfied. However, +5 ≠ -5, so a ≠ b. This single example shows that the statement is false - having equal squares does not guarantee equal values.
In simple words: Two different numbers (5 and -5) can have the same square (25). So squaring does not produce unique results, and the statement fails.

Exam Tip: One counterexample is enough to disprove a universal claim. Choose counterexamples that are simple and clearly expose the flaw in the statement.

Question 13. By giving a counter-example, show that the statement is false: p: If n is an odd positive integer, then n is prime.
Answer: Take the odd positive integer n = 9. Although 9 is odd, it is divisible by 3. A prime number has only 1 and itself as factors. Since 9 has 3 as a factor, it is not prime despite being an odd positive integer. This example shows the statement is false.
In simple words: Not all odd numbers are prime. The number 9 is odd but can be divided by 3, so it is not prime.

Exam Tip: For statements about integers, test small composite numbers (like 9, 15, 21) that are odd but clearly not prime - they often serve as quick counterexamples.

Question 14. Use contradiction method to prove that: p: \( \sqrt{3} \) is irrational is a true statement.
Answer: We assume the opposite - that \( \sqrt{3} \) is rational. If it were rational, we could express it as \( \sqrt{3} = \frac{p}{q} \) where p and q are co-prime integers (sharing no common factors).

Squaring both sides: \( 3 = \frac{p^2}{q^2} \), which gives \( p^2 = 3q^2 \).

This means p² is divisible by 3. If p² is divisible by 3, then p itself must be divisible by 3 (since 3 is prime). So we can write p = 3c for some integer c, giving p² = 9c².

Substituting back: \( 9c^2 = 3q^2 \), which simplifies to \( 3c^2 = q^2 \).

This means q² is divisible by 3, so q is also divisible by 3.

But now both p and q are divisible by 3, contradicting our assumption that they are co-prime. This contradiction shows our initial assumption was wrong. Therefore, \( \sqrt{3} \) cannot be rational, making it irrational.
In simple words: If we assume \( \sqrt{3} \) is rational, we end up proving that both numerator and denominator share a common factor (3). This breaks the definition of a rational number written in lowest terms, so our assumption must be false.

Exam Tip: Proof by contradiction works by assuming the negation and deriving an impossibility. This method is especially effective for irrational number proofs - aim to show that p and q cannot both be co-prime.

Question 15. By giving a counter-example, show that the following statement is false: p: If all the sides of a triangle are equal, then the triangle is obtuse angled.
Answer: Consider an equilateral triangle where all three sides are equal. In such a triangle, all angles are also equal. Since the sum of angles in any triangle is 180°, each angle measures 180° ÷ 3 = 60°. An obtuse angle is one that measures between 90° and 180°. Since all angles of an equilateral triangle are 60°, they are acute angles (between 0° and 90°), not obtuse. This counterexample shows that a triangle with all equal sides need not be obtuse angled.
In simple words: Equal sides in a triangle produce equal angles of 60° each. These are sharp angles, not wide angles, so the triangle is not obtuse.

Exam Tip: When disproving geometric statements, use well-known special shapes (equilateral, isosceles right triangle) as counterexamples - they are easy to analyze and visualize.