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Question-1

Which of the following sentences are statements? Give reasons for your answer?

(i) There are 35 days in a month.

(ii) Mathematics is difficult.

(iii) The sum of 5 and 7 is greater than 10.

(iv) The square of a number is an even number.

(v) The sides of a quadrilateral have equal length.

(vi) Answer this question.

(vii) The product of –1 and 8 is 8.

(viii) The sum of all interior angles of a triangle is 180° .

(ix) Today is a windy day.

(x) All real numbers are complex numbers.

Solution:
(i) The given sentence is always false because the maximum number of days in a month is 31. Therefore, it is a statement.

(ii) The given sentence is not a statement because for some people mathematics can be easy and for some it can be difficult.

(iii) The given sentence is always true because the sum is 12 and it is greater than 10. Therefore it is a statement.

(iv) The given sentence is sometimes true and sometimes is not true. For example the square of 2 is an even number and the square of 3 is an odd number. Therefore it is not a statement.

(v) The given sentence is sometimes true and sometimes false. For example, the squares and rhombus have equal length whereas rectangles and trapezium have unequal length. Therefore it is not a statement.

(vi) It is a command and therefore it is not a statement.

(vii) This sentence is false as the product is (-8). Therefore it is a statement.

(viii) This sentence is always true and therefore it is a statement.

(ix) It is not clear from the context which day is referred and therefore, it is not a statement.

(x) This is a true statement because all real numbers can be written in the form a + ix0.

Question-2

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

Solution:
The three examples are:

(i) "Everyone in this class is intelligent". This is not a statement because from the context it is not clear which class is referred here and the term intelligent is not precisely defined.

(ii) "He is a physics student". This is also not a statement because who ‘he’ is.

(iii) "log tanf is always less than 1". Since we do not know the value of f , we cannot say whether the sentence is true or not.

Question-3

Write the negation of the following statement:

(i) Chennai is the capital of Tamilnadu.

(ii) is not a complex number

(iii) All triangles are not equilateral triangle.

(iv) The number 2 is greater than 7.

(v) Every natural number is an integer.

Solution:
(i) Chennai is not the capital of Tamilnadu.

(ii) 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.

Question-4

Are the following pairs of statements negations of each other:

(a) The number x is not a rational number. The number x is not an irrational number.

(b) The number x is a rational number. The number x is an irrational number.

Solution:
(a) The negation of the first statement is "the number x is not rational number" which is same as the second statement. This is because if that number is not irrational then it is rational.

(b) The negation of the first statement is " x is an irrational number" which is same as the second statement. Therefore the pairs are negations of each other.

Question-5

Find the component statements of the following compound statements and check whether they are true or false.

(i) The sun shines or it rains.

(ii) All integers are positive or negative.

(iii) 100 is divisible by 3, 11 and 5.

Solution:
(i) The sun shines, it rains. It is a true compound statement.

(ii) All integers are positive , all integers are negative.
It is a false compound statement.

(iii) 100 is divisible by 3. False.

Question-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 3x2 – x – 10 = 0.

Solution:
(i) "AND"
The component statements are:
All rational numbers are real.
All rational numbers are complex.

(ii) "OR"
The component statements are:
Square of an integer is positive.
Square of an integer is negative.

(iii) "AND"
The component statements are:
The sand heats up quickly in the sun.
The sand does not cool down fast at night.

(iv) "AND"
The component statements are:
x = 2 is a root of the equation 3x2 – x – 10 = 0
x = 3 is a root of the equation 3x2 – x – 10 = 0

Question-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 then x + 1.

(iii) There exists a capital for every state in India.

Solution:
(i) Quantifier is "there exists". Negation is: There does not exist a number which is equal to its square.

(ii) Quantifier is "For every". The negation statement is: There exists a real number x such that x is not less than x + 1.

(iii) Quantifier is "There exists". The negation statement is: There exists a state in India which does not have a capital.

Question-8

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

(i) x + y = y + x is true for every real number x and y.

(ii) There exists a real number x and y for which x + y = y + x.

Solution:
The above are not negation to each other since the negation of the statement in
(i) is "There exists real number x and y for which x + y y + x instead of the statement given in (ii).

Question-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) A lady gives birth to a baby boy or baby girl.

(iv) All integers are positive or negative.

Solution:
(i) OR used in the given statement is exclusive.

(ii) OR used in the given statement is inclusive.

(iii) OR used in the given statement is exclusive.

(iv) OR used in the given statement is exclusive.

Question-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:
Given statement is: "If a natural number is odd, then its square is also odd"

Identical five statements are:

(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.

Question-11

Write the contra positive and converse of the following statements.

(i) If x is a prime number, then x is odd.

(ii) If 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 4

Solution:
(i) The contra positive statement for the statement "If x is a prime number, then x is odd" is "If a number is not odd, then x is not a prime number".

The converse statement for the statement "If x is a prime number, then x is odd" is "If a number x is odd, then it is a prime number".

(ii) The contra positive statement for the statement "If the two lines are parallel, then they do not intersect in the same plane" is "If two lines intersect in the same plane, then they are not parallel" and the converse statement is "If two lines do not intersect in the same plane, then they are parallel.

(iii) The contra positive statement for the statement "Something is cold implies that it has low temperature" is "If something is not at low temperature, then it is cold" and the converse statement is "If something is at low temperature, then it is cold.

(iv) The contra positive statement for the statement "You cannot comprehend geometry if you do not know how to reason deductively" is "If you know how to reason deductively, then you cannot comprehend geometry" and the converse statement is "If you do not know how to reason deductively, then you cannot comprehend geometry".

(v) The contra positive statement for the statement "x is an even number implies x is divisible by 4" is "If x is not divisible by 4, then x is not an even number" and the converse statement is "If x is divisible by 4, then x is an even number.

Question-12

Write each of the following statements in the form of "if-then"

(i) You get a job implies that your credentials are good.

(ii) The Banana tree will bloom if it stays warm for a month.

(iii) A quadrilateral is a parallelogram if its diagonals bisect each other.

(iv) To get an A+ in the class, it is necessary that you do all the exercises of the book.

Solution:
(i) If you get a job, then your credentials are good.

(ii) If the banana tree stays warn for a month, then it will bloom.

(iii) If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
(iv) If you want to get A+ in the class, then you do all the exercises of the book.

Question-13

Given statements in (a) and (b).
Identify the statements given below as contra positive or converse of each other.
(a)
(i) If you live in Delhi, then you have winter clothes.
(ii) If you do not have winter clothes, then you do not live in Delhi.

(b)
 (i) If a quadrilateral is a parallelogram, then its diagonals bisect each other.
(ii) If the diagonals of a quadrilateral do not bisect each other, then the
quadrilateral is not a parallelogram.

Solution:
(a) (i) Contra positive statement
(ii)Converse statement.

(b) (i) Contra positive statement
(ii) Converse statement.

Question-14

Show that the statement p: "If x is a real number such that x3 + 4x = 0, then x is 0" is true by

(i) Direct method.

(ii) Method of contradiction.

(iii) Method of contra positive.

Solution:
(i) Direct method: x3 + 4x = 0 or x(x2 + 4) = 0
Now x2 + 4 0 and x R and hence x = 0.

(ii) Method of Contradiction: Let x 0 and let x = p, p R is a root of x3 + 4x = 0. Therefore p3 + 4p = 0
(or) p(p2 + 4) = 0 as p = 0 Thus p2 + 4 = 0 which is not possible. Therefore, our supposition is wrong. Hence p = 0 or x = 0
 
(iii) Contra positive method: p is not true

Let x = 0 is not true Let x = p 0

Therefore p3 + 4p = 0

P being the root of x2 + 4 = 0

(or) p(p2 + 4) = 0

Now p = 0, also (p2 + 4) = 0

Which implies p(p2 + 4) = 0

Now p = 0, also p2 + 4 = 0

p(p2 + 4) 0 if p is not true.

Hence x = 0 is the root of x3 + 4x = 0

Question-15

Show that the statement "For any real number a and b, a2 = b2 implies that a = b" is not true by giving a counter example.

Solution:
Let a = 1, b = -1 and a2 = b2 but a b. Thus we observe that the given statement is not true.

Question-16

Show that the following statement is true by the method of contra positive.

Solution:
p: If x is an integer and x2 is even, then x is also even.
Let x be not even i.e. Let x = 2n + 1.
Therefore, x2 = (2n + 1)2 = 4n2 + 4n + 1 = 4(n2 + n) + 1.
Thus 4(n2 + n) + 1 is odd.
i.e. If q is not true, then p is not true, is proved.
Hence the given statement is not true.

Question-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 not obtuse angled triangle.

(ii) q: The equation x2 – 1 = 0 does not have root lying between 0 and 2.

Solution:
(i) p: If all angles of a triangle are equal then the triangle is an obtuse angled triangle. Let an angle of a triangle be 90° + θ (obtuse angle)

Now sum of the angles of a triangle is 3(90 + θ ) = 270 + 3θ which is greater than 180° .

Hence a triangle having equal angles cannot be obtuse angled triangle.

(ii) q: The equation x2 – 1 = 0 does not have a root lying between 0 and 2.

The equation x2 – 1 = 0 has the root x = 1 which lies between 0 and 2. Hence the given statement is not true.

Question-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: is a rational number.

Solution:
(i) p: Each radius of a circle is a chord of the circle. ‘p’ is false. By definition of the chord, it should intersect the circle in two points.

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

‘q’ is false. We can show this by giving a counter-example. A chord which is not a diameter does not pass through the centre.

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

‘r’ is true. In the question of an ellipse if we put a = b, then it is a circle using direct method.

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

By rule of inequality ‘s’ is true.

(v) t:is a rational number.

‘r’ is false. Since 11 is a prime number, therefore is irrational.





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