 # Mathematical Reasoning

## Class 11 NCERT

### NCERT

(i) Chennai is not the capital of Tamil Nadu.$\\$ (ii) $\sqrt{2}$ is a complex number.$\\$ (iii) All triangles are equilateral triangles.$\\$ (iv) The number $2$ is not greater than $7.$$\\ (v) Every natural number is not an integer. 4 Are the following pairs of statements negations of each other?\\ (i) The number x is not a rational number.\\ The number x is not an irrational number.\\ (ii)The number x is a rational number.\\ The number x is an irrational number. ##### Solution : (i) The negation of the first statement is ‘the number x is a rational number’. This is same as the second statement. This is because if a number is not an irrational number, then it is a rational number.\\ Therefore, the given statements are negations of each other.\\ (ii) The negation of the first statement is ‘the number x is not a rational number’. This means that the number x is an irrational number, which is the same as the second statement. Therefore, the given statements are negations of each other. 5 Find the component statements of the following compound statements and check whether they are true or false.\\ (i) Number 3 is prime or it is odd.\\ (ii) All integers are positive or negative.\\ (iii) 100 is divisible by 3, 11 and 5. ##### Solution : (i) The component statements are as follows.\\ p: Number 3 is prime.\\ q: Number 3 is odd.\\ Both the statements are true.\\ (ii) The component statements are as follows.\\ p: All integers are positive.\\ q: All integers are negative.\\ Both the statements are false.\\ (iii) The component statements are as follows.\\ p: 100 is divisible by 3.$$\\$ q: $100$ is divisible by $11.$$\\ r: 100 is divisible by 5.$$\\$ Here, the statements, $p$ and $q,$ are false and statement $r$ is true.

6   For each of the following compound statements first identify the connecting words and then break it into component statements.$\\$ (i) All rational numbers are real and all real numbers are not complex.$\\$ (ii) Square of an integer is positive or negative.$\\$ (iii) The sand heats up quickly in the Sun and does not cool down fast at night.$\\$ (iv) $x = 2$ and $x = 3$ are the roots of the equation $3x 2 n - x n - 10 = 0.$

##### Solution :

(i) Here, the connecting word is ‘and’.$\\$ The component statements are as follows.$\\$ p: All rational numbers are real.$\\$ q: All real numbers are not complex.$\\$ (ii) Here, the connecting word is ‘or’.$\\$ The component statements are as follows.$\\$ p: Square of an integer is positive.$\\$ q: Square of an integer is negative.$\\$ (iii) Here, the connecting word is ‘and’.$\\$ The component statements are as follows.$\\$ p: The sand heats up quickly in the sun.$\\$ q: The sand does not cool down fast at night.$\\$ (iv) Here, the connecting word is ‘and’.$\\$ The component statements are as follows.$\\$ $p: x = 2$ is a root of the equation $3x ^2 n - x n - 10 = 0$$\\ q: x = 3 is a root of the equation 3x^ 2 n - x n - 10 = 0$$\\$

7   Identify the quantifier in the following statements and write the negation of the statements.$\\$ (i) There exists a number which is equal to its square.$\\$ (ii) For every real number $x, x$ is less than $x + 1.$$\\$ (iii) There exists a capital for every state in India.

##### Solution :

(i) The quantifier is ‘There exists’.$\\$ The negation of this statement is as follows.$\\$ There does not exist a number which is equal to its square.$\\$ (ii) The quantifier is ‘For every’.$\\$ The negation of this statement is as follows.$\\$ There exist a real number x such that x is not less than x + 1.$\\$ (iii) The quantifier is ‘There exists’.$\\$ The negation of this statement is as follows.$\\$ There exists a state in India which does not have a capital.$\\$

8   Check whether the following pair of statements is negation of each other. Give reasons for the answer.$\\$ (i) $x + y = y + x$ is true for every real numbers $x$ and $y$.$\\$ (ii) There exists real number$x$ and $y$ for which $x + y = y + x.$

##### Solution :

The negation of statement (i) is as follows.$\\$ There exists real number $x$ and $y$ for which $x + y = y + x.$ This is not the same as statement (ii) Thus, the given statements are not negation of each other.

9   State whether the ‘Or’ used in the following statements is exclusive ‘or’ inclusive. Give reasons for your answer.$\\$ (i) Sun rises or Moon sets.$\\$ (ii) To apply for a driving license, you should have a ration card or a passport.$\\$ (iii) All integers are positive or negative.

##### Solution :

(i) Here, ‘or’ is exclusive because it is not possible for the Sun to rise and the moon to set together.$\\$ (ii) Here, ‘or’ is inclusive since a person can have both a ration card and a passport to apply for a driving license.$\\$ (iii) Here, ‘or’ is exclusive because all integers cannot be both positive and negative.

10   Rewrite the following statement with ‘if-then’ in five different ways conveying the same meaning. If a natural number is odd, then its square is also odd.

##### Solution :

The given statements can be written in five different ways as follows.$\\$ (i) A natural number is odd implies that its square is odd.$\\$ (ii) A natural number is odd only if its square is odd.$\\$ (iii) For a natural number to be odd, it is necessary that its square is odd.$\\$ (iv) For the square of a natural number to be odd, it is sufficient that the number is odd.$\\$ (v) If the square of a natural number is not odd, then the natural number is not odd.

11   Write the contrapositive and converse of the following statements.$\\$ (i) If x is a prime number, then x is odd.$\\$ (ii) It the two lines are parallel, then they do not intersect in the same plane.$\\$ (iii) Something is cold implies that it has low temperature.$\\$ (iv) You cannot comprehend geometry if you do not know how to reason deductively.$\\$ (v) x is an even number implies that x is divisible by 4v

##### Solution :

(i) The contrapositive is as follows. $\\$ If a number x is not odd, then x is not a prime number. $\\$ The converse is as follows. $\\$ If a number x is odd, then it is a prime number. $\\$ (ii) The contrapositive is as follows. $\\$ If two lines intersect in the same plane, then they are not parallel. $\\$ The converse is as follows. $\\$ If two lines do not intersect in the same plane, then they are parallel. $\\$ (iii) The contrapositive is as follows. $\\$ If something does not have low temperature, then it is not cold. $\\$ The converse is as follows. $\\$ If something is at low temperature, then it is cold. $\\$ (iv) The contrapositive is as follows. $\\$ If you know how to reason deductively, then you can comprehend geometry. $\\$ The converse is as follows. $\\$ If you do not know how to reason deductively, then you cannot comprehend geometry. $\\$ (v) The given statement can be written as follows. $\\$ If x is an even number, then x is divisible by 4. $\\$ The contrapositive is as follows. $\\$ If x is not divisible by 4, then x is not an even number. $\\$ The converse is as follows. $\\$ If x is divisible by 4, then x is an even number. $\\$

12   Write each of the following statement in the form ‘if-then’. $\\$ (i) You get a job implies that your credentials are good. $\\$ (ii) The Banana trees will bloom if it stays warm for a month. $\\$ (iii) A quadrilateral is a parallelogram if its diagonals bisect each other. $\\$ (iv) To get A $^+$ in the class, it is necessary that you do the exercises of the book $\\$

##### Solution :

(i) If you get a job, then you credentials are good. $\\$ (ii) If the Banana tree stays warm for a month, then it will bloom. $\\$ (iii) If the diagonals of a quadrilateral bisect each other, then it is parallelogram. $\\$ (iv) If you want to get an A$^ +$ in the class, then you do all the exercise of the book. $\\$

13   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. $\\$ $\qquad$(i) If you do not have winter clothes, then you do not live in Delhi. $\\$ $\qquad$(ii) If you have winter clothes, then you live in Delhi. $\\$ (b) If a quadrilateral is a parallelogram, then its diagonals bisect each other. $\\$ $\qquad$(i) If the diagonals of a quadrilateral do not bisect each other, then the quadrilateral is not a parallelogram. $\\$ $\qquad$(ii) If the diagonals of a quadrilateral bisect each other, then it is a parallelogram. $\\$

##### Solution :

(a) (i) This is the contrapositive of the given statement (a).$\\$ (ii) This is the converse of the given statement (a).$\\$ (b) (i) This is the contrapositive of the given statement (b).$\\$ (ii) This is the converse of the given statement (b).

14   Show that the statement $\\$ p: ‘If x is a real number such that x$^ 3$ + 4x = 0, then x is 0’ is true by $\\$ (i) direct method $\\$ (ii) method of contradiction $\\$ (iii) method of contrapositive $\\$

##### Solution :

p: ‘If x is a real number such that x $^3$ + 4x = 0, then x is 0’. $\\$ Let q: x is a real number such that $^3$ + 4x = 0 $\\$ r: x is 0. $\\$ (i) To show that statement p is true, we assume that q, is true and then show that r is true. $\\$ Therefore, let statement q be true. $\\$ $^3$ + 4x = 0 $\\$ x ($^2$ + 4) = 0 $\\$ x = 0 or x 2 + 4 = 0 $\\$ However, since x is real, it is 0. $\\$ Thus, statement r is true. $\\$ Therefore, the given statement is true. $\\$ (ii) To show statement p to be true by contradiction, we assume that p is not true. $\\$ Let x be a real number such that $^3$ + 4x = 0 and let x is not 0. $\\$ Therefore, $^3$ + 4x = 0 $\\$ x ($^2$ + 4) = 0 $\\$ x = 0 or $^2$ + 4 = 0 $\\$ x = 0 or $^2$ = -4 $\\$ However, x is real. $\\$Therefore, x = 0, which is a contradiction since we have assumed that x is not 0.

Thus, the given statement p is true. $\\$ (iii)To prove statement p to be true by contrapositive method, we assume that r is false and prove that q must be false. $\\$ Here, r is false implies that it is required to consider the negation of statement r. This obtains the following statement. $\\$ $\cong$ r: x is not 0 $\\$ I can be seen that (x$^ 2$ + 4) will always be positive $\\$ x = 0 implies that the product of any positive real number with x is not zero. $\\$ Let us consider the product of x with ($^ 2$ + 4) $\\$ $\\$ x ($^ 2$ + 4) = 0 $\\$ $^ 3$ + 4x = 0 $\\$ This shows that statement q is not true. $\\$ Thus, it has been proved that $\\$ $\cong$ r $\cong$ q $\\$ Therefore, the given statement p is true.

15   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 counter-example.

##### Solution :

The given statement can be written in the form of ‘if-then’ as follows. $\\$ 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 $\\$ The given statement has to be proved false. For this purpose, it has to be proved that if p, then $\\$ $\sim$ q To sow this, two real numbers, a and b, with a$^ 2$ = b $^ 2$ $\\$are required such that a $\neq$ b. $\\$ Let a = 1 and b = -1 $\\$ a $^ 2$ = (1)$^ 2$ and b$^ 2$ = (-1) 2 = 1 $\\$ a $^ 2$ = b$^ 2$ $\\$ However, a = b $\\$ Thus, it can be concluded that the given statement is false.

16   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 :

p: If x is an integer and x$^ 2$ is even, then x is also even. $\\$ Let q: x is an integer and x $^ 2$ is even. $\\$ r: x is even. $\\$ To prove that p is true by contrapositive method, we assume that r is false, and prove that q is also false. $\\$ Let x is not even. $\\$ To prove that q is false, it has to be proved that x is not an integer or x $^ 2$ is not even. $\\$ x is not even implies that x $^ 2$ is also not even. $\\$ Therefore, statement q is false. $\\$ Thus, the given statement p is true. $\\$

17   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) 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. $\\$ The given statement p has to be proved false. For this purpose, it has to be proved that if q, $\\$ then $\\$ $\sim$ r. $\\$ To show this, angles of a triangle are required such that none of them is an obtuse angle. $\\$ It is known that the sum of all angles of a triangle is 180°. Therefore, if all the three angles $\\$ are equal, then each of them is of measure 60°, which is not an obtuse angle. $\\$ In an equilateral triangle, the measure of all angles is equal. However, the triangle is not an obtuse-angled triangle. $\\$ Thus, it can be concluded that the given statement p is false. $\\$ (ii) The given statement is as follows. $\\$ q: The equation x $^2$ - 1 = 0 does not have a root lying between 0 and 2. $\\$ This statement has to be proved false. To show this, a counter example is required. $\\$ Consider x $^2$ - 1 = 0 $\\$ x $^2$ = 1 $\\$ x = $\pm$1 $\\$ One root of the equation x $^2$ - 1 = 0, i.e. the root x = 1, lies between 0 and 2. $\\$ Thus, the given statement is false. $\\$

18   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. $\\$

##### Solution :

(i) The given statement p is false. $\\$ According to the definition of chord, it should intersect the circle at two distinct points. $\\$ (ii) The given statement q is false. $\\$ If the chord is not the diameter of the circle, then the centre will not bisect that chord. $\\$ In other words, the centre of a circle only bisects the diameter, which is the chord of the circle. $\\$ (iii) The equation of an ellipse is, $\\$ $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ $\\$ If we put a = b = 1, then we obtain $\\$ x 2 + y 2 =1,which is an equation of a circle $\\$ Therefore, circle is a particular case of an eclipse. $\\$ Thus, statement r is true. $\\$ (iv) x > y $\\$ $\implies$ x < - y (By a rule of inequality) $\\$ Thus, the given statement s is true. $\\$ (v) 11 is a prime number and we know that the square root of any prime number is an irrational number. $\\$ Therefore, 11 is an irrational number. $\\$ Thus, the given statement t is false. $\\$

19   Write the negation of the following statements: $\\$ (i) p: For every positive real number x, the number x - 1 is also positive. $\\$ (ii) q: All cats scratch. $\\$ (iii) r: For every real number x, either x > 1 or x < 1. $\\$ (iv) s: There exists a number x such that 0 < x < 1. $\\$

##### Solution :

(i) The negation of statement p is as follows. $\\$ There exists a positive real number x, such that x - 1 is not positive. $\\$ (ii) The negation of statement q is as follows. $\\$ There exist a cat that does not scratch. $\\$ (iii) The negation of statement r is as follows. $\\$ There exists a real number x, such that neither x > 1 nor x < 1. $\\$ (iv) The negation of statement s is as follows. $\\$ There does not exist a number x, such that 0 < x < 1.

20   State the converse and contrapositive of each of the following statements: $\\$ (i) p: A positive integer is prime only if it has no divisors other than 1 and itself. $\\$ (ii) q: I go to a beach whenever it is a sunny day. $\\$ (iii) r: If it is hot outside, then you feel thirsty. $\\$

##### Solution :

(i) Statement p can be written as follows. $\\$ If a positive integer is prime, then it has no divisors other than 1 and itself. $\\$ The converse of the statement is as follows. $\\$ If a positive integer has no divisors other than 1 and itself, then it is prime. $\\$ The contrapositive of the statement is as follows. $\\$ If positive integer has divisors other than 1 and itself, then it is not prime. $\\$ (ii) The given statement can be written as follows. $\\$ If it is a sunny day, then I go to a beach. $\\$ The converse of the statement is as follows. $\\$ If I go to a beach, then it is a sunny day. $\\$ The contrapositive of the statement is as follows. $\\$ If I go to a beach, then it is not a sunny day. $\\$ (iii) The converse of statement r is as follows. $\\$ If you feel thirsty, then it is hot outside. $\\$ The contrapositive of statement r is as follows. $\\$ If you do not feel thirsty, then it is not hot outside. $\\$

21   Write each of the statements in the form ‘if p, then q’. $\\$ (i) p: It is necessary to have a password to log on to the server. $\\$ (ii) q: There is traffic jam whenever it rains. $\\$ (iii) r: You can access the website only if you pay a subscription fee. $\\$

##### Solution :

(i) Statement p can be written as follows. $\\$ If you log on to the server, then you have a password. $\\$ (ii) Statement q can be written as follows. $\\$ If it rains, then there is a traffic jam. $\\$ (iii) Statement r can be written as follows. $\\$ If you can access the website, then you pay a subscription fee. $\\$

22   Re write 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: For you to get an A grade, it is necessary and sufficient that you do all the homework regularly. $\\$ (iii) r: If a quadrilateral is equiangular, then it is a rectangle and if a quadrilateral is a rectangle, then it is equiangular. . $\\$

##### 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 and only if it is a rectangle. $\\$

23   Given below are two statements $\\$ p: 25 is a multiple of 5. $\\$ q: 25 is a multiple of 8. $\\$ Write the compound statements connecting these two statements with ‘And’ $\\$ and ‘Or’. In both cases check the validity of the compound statement. $\\$

##### Solution :

The compound statement with ‘And’ is ‘25 is a multiple of 5 and 8’. $\\$ This is a false statement, since 25 is not a multiple of 8. $\\$ The compound statement with ‘Or’ is ‘25 is a multiple of 5 or 8’. $\\$ This is a true statement, since 25 is not a multiple of 8 but it is a multiple of 5. $\\$

24   Check the validity of the statements given below by the method given against it. $\\$ (i) p: The sum of an irrational number and a rational number is irrational (by contradiction method). $\\$ (ii) q: If n is a real number with n > 3, then n$^ 2$ > 9 (by contradiction method). $\\$

##### Solution :

(i) The given statement is as follows. p: the sum of an irrational number and a rational number is irrational. $\\$ Let us assume that the given statement, p, is false. That is, we assume that the sum of an irrational number and a rational number is rational. $\\$ Therefore, $\sqrt{a}+\dfrac{b}{c}=\dfrac{d}{e},$when $\sqrt{a}$ is irrational and b, c, d, e are integers. $\dfrac{b}{e}=\dfrac{d}{e}$is a rational number and is an irrational number. $\\$ This is a contradiction. Therefore, our assumption is wrong. $\\$ Therefore, the sum of an irrational number and a rational number is rational. $\\$ Thus, the given statement is true. $\\$ (ii) The given statement, q is as follows. $\\$ If n is a real number with n > 3, then n$^ 2$ > 9. $\\$ Let us assume that n is a real number with n > 3, but n $^ 2$ > 9 is not true. $\\$ That is, n $^ 2$ < 9 $\\$ Then, n > 3 and n is a real number. $\\$ Squaring both the sides, we obtain $\\$ n $^ 2$ > (3) 2 $\\$ $\implies$ n $^ 2$ > 9, which is a contradiction, since we have assumed that n $^ 2$ < 9. $\\$ Thus, the given statement is true. That is, if n is a real number with n > 3, then $\\$ n 2 > 9. $\\$

25   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 :

The given statement can be written in five different ways as follows. $\\$ (i) A triangle is equiangular implies that is an obtuse-angled triangle. $\\$ (ii) A triangle is equilateral only if it an obtuse-angled triangle. $\\$ (iii) For a triangle to be equiangular, it is necessary that the triangle is an obtuse-angled triangle. $\\$ (iv) For a triangle to be an obtuse-angled triangle, it is sufficient that the triangle is equiangular. $\\$ (v) If a triangle is not an obtuse-angled triangle, then the triangle is not equiangular. $\\$