NCERT Solutions Class 11^{th} Maths Chapter 14: Mathematical ReasoningExercise 14.11. Which of the following sentences are statements? Give reasons for your answer.
SOLUTION (i) The maximum number of days in a month is 31. Therefore, the sentence is incorrect. Thus, this sentence is a statement. (ii) The sentence is subjective. It is true if a person does not like mathematics but it is false otherwise. Therefore, this sentence is not a statement. (iii) The sum of 5 and 7 is 12 which is greater than 10. Therefore, the sentence is universally correct. Thus, this sentence is a statement. (iv) This sentence is only true for the squares of even numbers. For odd numbers this sentence is incorrect. Therefore, this sentence is not a statement. (v) The sides of quadrilaterals like square rhombus are equal but it is not the case for rectangles, parallelograms, trapeziums etc. Therefore, this sentence is not a statement. (vi) The sentence is an order or command. Therefore, it is not a statement. (vii) The product of (1) and 8 is (8). Therefore, the sentence is always false. Thus, this sentence is a statement. (viii) It is known that the sum of interior angles of a triangle is always 180°. Therefore, this sentence is a statement. (ix) The sentence can be true on a windy day, but when the weather changes it will become false. Therefore, this sentence is not a statement. (x) All real numbers can be represented as complex numbers (a × 1 + 0 × i). Therefore, this sentence is a statement. 2. Give three examples of sentences which are not statements. Give reasons for the answers. SOLUTION I. All sides of a triangle are equal. This sentence is true if the triangle is and equilateral triangle but false otherwise. Hence, it is not a statement. II. Vegetables taste bad. This sentence is subjective. Some people might find vegetables tasty and others might not. Hence, it is not a statement. III. It is raining outside. This sentence can be true or false depending upon the weather. Hence, it is not a statement. Exercise 14.21. Write the negation of the following statements:
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2. Are the following pairs of statements negations of each other:
SOLUTION (i) The negation of first statement is "The number x is a rational number" which is the same as "The number x is not an irrational number" because a number can only be rational if it is not irrational. Therefore, the statements are negations of each other. (ii) The negation of first statement is "The number x is not a rational number" which is the same as "The number x is an irrational number" because a number can only be irrational if it is not rational. Therefore, the statements are negations of each other. 3. Find the component statements of the following compound statements and check whether they are true or false.
SOLUTION (i) The components of given statement are:
Both of the component statements are true. (ii) The components of given statement are:
Both of the component statements are false. (iii) The components of given statement are:
Component statements A and B are false whereas C is true. Exercise 14.31. For each of the following compound statements first identify the connecting words and then break it into component statements.
SOLUTION (i) In this compound statement, 'and' is the connecting word. The component statements are:
(ii) In this compound statement, 'or' is the connecting word. The component statements are:
(iii) In this compound statement, 'and' is the connecting word. The component statements are:
(iv) In this compound statement, 'and' is the connecting word. The component statements are:
2. Identify the quantifier in the following statements and write the negation of the statements.
SOLUTION (i) "There exists" acts as the quantifier in the given statement. Negation of the statement will be: There does not exist a number which is equal to its square (ii) "For every" acts as the quantifier in the given statement. Negation of the statement will be: For at least one real number x, x is not less than x + 1. (iii) "There exists" acts as the quantifier in the given statement. Negation of the statement will be: There exists a state in India which does not have a capital. 3. Check whether the following pair of statements are negation of each other. Give reasons for your answer.
SOLUTION The negation of statement (i) will be: There exist real numbers x and y for which x + y ≠ y + x This is different from the statement (ii). Therefore, the pair of statements is not negations of each other. 4. State whether the "Or" used in the following statements is "exclusive "or" inclusive. Give reasons for your answer.
SOLUTION (i) We know that it is not possible for the sun to rise and the moon to set simultaneously. Therefore, the use of "or" in this statement is exclusive. (ii) We know that both a ration card and a passport are acceptable when applying for a driving licence. Therefore, the use of "or" in this statement is inclusive. (iii) We know that it is not possible for all integers to be positive and negative simultaneously. Therefore, the use of "or" in this statement is exclusive. Exercise 14.41. Rewrite the following statement with "ifthen" in five different ways conveying the same meaning. If a natural number is odd, then its square is also odd. SOLUTION
2. Write the contrapositive and converse of the following statements.
SOLUTION (i) Contrapositive of the given statement will be: If a number x is not odd, then x is not a prime number Converse of the given statement will be: If a number x is odd, then it is a prime number (ii) Contrapositive of the given statement will be: If two lines intersect in the same plane, then the two lines are not parallel Converse of the given statement will be: If two lines do not intersect in the same plane, then they are parallel (iii) Contrapositive of the given statement will be: If something does not have a low temperature, then it is not cold Converse of the given statement will be: If something is at a low temperature, then it is cold (iv) Contrapositive of the given statement will be: If you know how to reason deductively, then you can comprehend geometry Converse of the given statement will be: If you do not know how to reason deductively, then you cannot comprehend geometry (v) This statement can be rewritten as "if x is an even number, then x is divisible by 4". Contrapositive of this statement will be: If x is not divisible by 4, then x is not an even number Converse of this statement will be: If x is divisible by 4, then x is an even number 3. Write each of the following statements in the form "ifthen"
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4. Given statements in (a) and (b). Identify the statements given below as contrapositive or converse of each other. a) If you live in Delhi, then you have winter clothes.
b) If a quadrilateral is a parallelogram, then its diagonals bisect each other.
SOLUTION (a) (i) This statement is contrapositive of the statement (a) (ii) This statement is converse of the statement (a) (b) (i) This statement is contrapositive of the statement (b) (ii) This statement is converse of the statement (b) Exercise 14.51. Show that the statement p: "If x is a real number such that x^{3} + 4x = 0, then x is 0" is true by SOLUTION Let us assume that q: x is a real number such that x^{3}+ 4x = 0 r: x is 0 (i) Let statement q be true Therefore, x^{3} + 4x = 0 x(x^{2} + 4) = 0 x = 0 or x^{2} + 4 = 0 x = 0 OR x^{2} = 4 ⇒ x = √4 Since, x is real. Therefore, x^{2} + 4 ≠ 0. x = 0 Hence, the statement r is true. (ii) Let statement p be false. Then, x is a real number such that x^{3} + 4x = 0 and x ≠ 0. Therefore, x^{3} + 4x = 0 x(x^{2} + 4) = 0 x = 0 or x^{2} + 4 = 0 x = 0 OR x^{2} = 4 ⇒ x = √4 But, x is real. So, x = 0 which contradicts our assumption that x ≠ 0 and statement q is false. Therefore, statement p is true. (iii) Let the statement r be false. Then, x \ne0 x^{2} + 4 is positive The product of a positive term with x cannot be zero as x ≠ 0. Then, product of x and x^{2} + 4 will be x(x^{2} + 4) ≠ 0 x^{3} + 4x ≠ 0 Therefore, statement q is false. ∼r ⇒ ∼q Hence, the given statement p is true. 2. Show that the statement "For any real numbers a and b, a^{2} = b^{2} implies that a = b" is not true by giving a counterexample. SOLUTION The given statement can be rewritten as: If a and b are real numbers such that a^{2} = b^{2}, then a = b. Let p: a and b are real numbers such that a^{2} = b^{2} q: a = b Let us consider that a = 1 and b = 1. Then, a^{2} = 1^{2} = 1 AND b^{2} = (1)^{2} = 1 So, a^{2} = b^{2}. But a ≠ b Hence, the given statement is false. 3. Show that the following statement is true by the method of contrapositive. p: If x is an integer and x^{2} is even, then x is also even. SOLUTION Let q: x be an integer and x^{2} be even r: x is even Let the statement r be false. Then, x is not even So, x^{2} is also not even. Therefore, statement q is false. Hence, the give statement p is true. 4. By giving a counter example, show that the following statements are not true. (i) p: If all the angles of a triangle are equal, then the triangle is an obtuse angled triangle. (ii) q: The equation x^{2}  1 = 0 does not have a root lying between 0 and 2. SOLUTION (i) Let q: All the angles of a triangle are equal r: The triangle is an obtuse angled triangle We know that the sum of all interior angles of a triangle is 180°. If all the interior angles of the triangle are equal, then each angle must be 60°, which is not an obtuse angle. So, the triangle is not an obtuse angled triangle when all angles are equal. Hence, the give statement p is false. (ii) Consider the give equation x^{2}  1 = 0 x^{2} = 1 x = ±1 x = 1 and x = 1 The root x = 1 lies between 0 and 2. Hence, the given statement q is false. 5. Which of the following statements are true and which are false? In each case, give a valid reason for saying so.
SOLUTION (i) The given statement p is false. A chord must intersect two distinct points on a circle. A radius, only intersects one point. (ii) The given statement q is false. A chord does not need to pass through the centre and is therefore not bisected by it. The diameter of a circle is the only chord that gets bisected hy the centre. (iii) The equation of an ellipse is x^{2}/a^{2} + y^{2}/b^{2} = 1 Let us consider that a = 1 and b = 1 x^{2}/12 + y^{2}/12 = 1 x^{2} + y^{2} = 1, which is the equation of a circle. Therefore, circle is a particular case of an ellipse. Hence the give statement r is true. (iv) Consider x > y. x  y > 0 y > x Or x < y Hence, the give statement s is true. (v) The given statement t is true. The square root of a prime number is irrational. We know that the number 11 is a prime number. Therefore, √11 will be irrational. Miscellaneous Exercise1. Write the negation of the following statements:
SOLUTION (i) There exists a positive real number x, such that x  1 is not positive. (ii) There exists a cat which does not scratch. (iii) There exists a real number x, such that neither x > 1 nor x < 1. (iv) There does not exist a number x, such that 0 < x < 1. 2. State the converse and contrapositive of each of the following statements:
SOLUTION (i) The statement p can be rewritten as: If a positive integer is prime, then it has no divisors other than 1 and itself. Converse: If a positive integer has no divisors other than 1 and itself, then it is prime. Contrapositive: If a positive integer has divisors other than 1 and itself, then it is not prime. (ii) The statement q can be rewritten as: If it is a sunny day, then I go to a beach. Converse: If I go to a beach, then it is a sunny day. Contrapositive: If I do not go to a beach, then it is not a sunny day. (iii) Converse: If you feel thirsty, then it is hot outside. Contrapositive: If you do not feel thirsty, then it is not hot outside. 3. Write each of the statements in the form "if p, then q"
SOLUTION (i) If you log on to the server, then you have a password. (ii) If it rains, then there is a traffic jam. (iii) If you can access the website, then you pay a subscription fee. 4. Rewrite each of the following statements in the form "p if and only if q"
SOLUTION (i) You watch television if and only if your mind is free. (ii) You get an A grade if and only if you do all the homework regularly. (iii) A quadrilateral is equiangular if only if it is a rectangle. 5. Given below are two statements p : 25 is a multiple of 5. Write the compound statements connecting these two statements with "And" and "Or". In both cases check the validity of the compound statement. SOLUTION Compound statement using "And": 25 is a multiple of 5 and 8. This compound statement is false as 25 is not a multiple of 8. Compound statement using "Or": 25 is a multiple of 5 and 8. This compound statement is true as 25 is not a multiple of 8 but it is a multiple of 5. 6. Check the validity of the statements given below by the method given against it.
SOLUTION (i) Let statement p be false. Then, √a + b/c = d/e where √a is irrational and b, c, d, and e are integers d/e  b/c = √a But this is not possible as d/e  b/c is a rational number and √a is irrational. Thus, our assumption was wrong and statement p is true. (ii) Let us assume that n is a real number with n > 3, but n^{2}> 9 is false. Then, n^{2} < 9 We have n > 3. Square both sides n^{2} > 9 This contradicts our assumption that n^{2}> 9 is false. Hence, the statement q is true. 7. Write the following statement in five different ways, conveying the same meaning. p: If a triangle is equiangular, then it is an obtuse angled triangle. SOLUTION
