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NCERT Exemplar Class 11 Maths solutions Chapter 14 Mathematical Reasoning covers the concept of Mathematical reasoning. In the chapter, the students will learn the fundamentals of Mathematical Reasoning and their application. Through the use of Class 11 Maths NCERT Exemplar Chapter 14 solutions, the students will also comprehensively understand the different types of reasoning with the help of different mathematically acceptable statements like negation and compound statements. The Class 11 MathsNCERT Exemplar Chapter 14 solutions act as a guide for students and can be referred to at any point in time.
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The NCERT Exemplar Class 11 Maths solutions Chapter 14 are given below and the pdf for the same can be downloaded. The solutions are based on the exercises given in the textbook and are solved by experts. The NCERT Exemplar solutions for Class 11 Maths Chapter 14 proves to be a great help to students who find the chapter Mathematical Reasoning difficult. The solutions are in a stepwise format which allows the students to understand the problem easily and excel in their exams.
Question:1
Answer:
i) Concept:
A statement is considered as an assertive sentence if it is either true or false, but it shouldn’t be both.The given sentence is a true statement, since we know that it is true that “a triangle has three sides.”
ii) Concept:
A statement is considered as an assertive sentence if it is either true or false, but it should not be both.The given sentence is a true statement, since we know that it is true that “0 is a complex number”, because we can also write it as a + ib, where the imaginary part is 0 as, a + 0i.
iii) Concept:
A statement is considered as an assertive sentence if it is either true or false, but it should not be both. The given sentence is a false statement, since we know that it is not true that “sky is red.”
iv) Concept:
A statement is considered as an assertive sentence if it is either true or false, but it should not be both.The given sentence is a false statement, since we know that it is not true that “every set is an infinite set.”
v) Concept:
A statement is considered as an assertive sentence if it is either true or false, but it should not be both. The given sentence is a false statement, since we know that it is not true that because .
vi) Concept:
A statement is considered as an assertive sentence if it is either true or false, but it should not be both.The given sentence is not a statement, since we know that “y + 9 = 7” will be true for some value & false for some. For example, at y = -2 is true & y = 1 or any other value is false.
vii) Concept:
A statement is considered as an assertive sentence if it is either true or false, but it should not be both. The given sentence is not a statement since we know that “where is your bag” is a question.
viii) Concept:
A statement is considered as an assertive sentence if it is either true or false, but it shouldn’t be both.Theve given sentence is a true statement, since we know that it is true that “ every square is a rectangle.”
ix) Concept:
A statement is considered as an assertive sentence if it is either true or false, but it should not be both. The given sentence is a true statement, since we know that “Sum of opposite angles of a cyclic quadrilateral is 180?” is true by the properties f quadrilateral.
x) Concept:
A statement is considered as an assertive sentence if it is either true or false, but it should not be both. The given sentence is a false statement since we know that is the correct expression according to the laws of trigonometry.
Question:2
Answer:
i) Concept:
A compound statement is defined as a combination of two statements or components. Here the components of the given statement “number 7 is prime and odd, will be –
p: Number 7 is prime &
q: Number 7 is odd
ii) Concept:
A compound statement is defined as a combination of two statements or components. Here the components of the given statement “Chennai is in India and is the capital of Tamil Nadu”, will be –
p: Chennai is in India
q: Chennai is the capital of Tamil Nadu
iii) Concept:
A compound statement is defined as a combination of two statements or components. Here the components of the given statement “the number 100 is divisible by 3, 11 & 5”, will be –
p: 100 is divisible by 3
q: 100 is divisible by 11 &
r: 100 is divisible by 5
iv) Concept:
A compound statement is defined as a combination of two statements or components. Here the components of the given statement “Chandigarh is the capital of Haryana and U.P.”, will be –
p: Chandigarh is the capital of Haryana &
q: Chandigarh is the capital of U.P.
v) Concept:
A compound statement is defined as a combination of two statements or components. Here the components of the given statement is a rational number or an irrational number”, will be –
p: is a rational number &
q: is an irrational number
vi) Concept:
A compound statement is defined as a combination of two statements or components.Here the components of the given statement “0 is less than every positive integer and every negative integer”, will be –
p: 0 is less than every positive integer &
q: 0 is less than every negative integer
vii) Concept:
A compound statement is defined as a combination of two statements or components. Here the components of the given statement “Plants use sunlight, water and carbon dioxide for photosynthesis”, will be –
p: Plants use sunlight for photosynthesis
q: Plants use water for photosynthesis &
r: Plants use carbon dioxide for photosynthesis
viii) Concept:
A compound statement is defined as a combination of two statements or components. Here the components of the given statement “two lines in a plane either intersect at one point or they are parallel”, will be –
p: Two lines in a plane intersect at one point &
q: Two lines in a plane are parallel
ix) Concept:
A compound statement is defined as a combination of two statements or components. Here the components of the given statement “A rectangle is a quadrilateral or a 5 – sided polygon”, will be –
p: A rectangle is a quadrilateral
q: A rectangle is a 5 – sided polygon.
Question:3
Answer:
i) Concept:
A compound statement is defined as a combination of two statements or components. Here the components of the given statement “57 is divisible by 2 or 3”, will be –
p: 57 is divisible by 2 &
q: 57 is divisible by 3
Here, the given statement is true. Compound statement is in the form P V Q, which has truth value T if either P or Q or both will be true.
ii) Concept:
A compound statement is defined as a combination of two statements or components. Here the components of the given statement “24 is a multiple of 4 & 6”, will be –
p: 24 is a multiple of 4 &
q: 24 is a multiple of 6
Here, the given statement is true as both p & q are true since 24 is a multiple of both 4 & 6.
iii) Concept
A statement is considered as an assertive sentence if it is either true or false, but it shouldn’t be both.The given sentence is a true statement, since we know that “All living things have two eyes and two legs”, will be –
p: All living things have two eyes &
q: All living things have two legs.
Compound statement is in the form , which has truth value T which will be true only if both the components will be true.
Here p is false & q is true. Hence, the given statement is false.
iv) Concept
A compound statement is defined as a combination of two statements or components. Here the components of the given statement “2 is an even number and a prime number”, will be –
p: 2 is an even number &
q: 2 is a prime number.
Compound statement is in the form P ? Q, which has truth value T which will be true only if both the components will be true. Here p & q both are true.
Hence, the given statement is true.
Question:4
Answer:
i) Concept
We know that negation of p is “not p”, viz. symbolized as –p, also its truth value of –p is the opposite of the truth value of p. Thus, the negation of this statement will be – “The number 17 is not prime.”
ii) Concept
We know that negation of p is “not p”, viz. symbolized as –p, also its truth value of –p is the opposite of the truth value of p. Thus, the negation of this statement will be –
iii) Concept
We know that negation of p is “not p”, viz. symbolized as –p, also its truth value of –p is the opposite of the truth value of p. Thus, the negation of this statement will be –
“Violets are not blue”
iv) Concept
We know that negation of p is “not p”, viz. symbolized as –p, also its truth value of –p is the opposite of the truth value of p. Thus, the negation of this statement will be –
“ is not a rational number."
v) Concept
We know that negation of p is “not p”, viz. symbolized as –p, also its truth value of –p is the opposite of the truth value of p. Thus, the negation of this statement will be –
“2 is a prime number.”
vi) Concept
We know that negation of p is “not p”, viz. symbolized as –p, also its truth value of –p is the opposite of the truth value of p. Thus, the negation of this statement will be –
“Every real number is not an irrational number.”
vii) Concept
We know that negation of p is “not p”, viz. symbolized as –p, also its truth value of –p is the opposite of the truth value of p. Thus, the negation of this statement will be –
“Cow does not have four legs.”
viii) Concept
We know that negation of p is “not p”, viz. symbolized as –p, also its truth value of –p is the opposite of the truth value of p. Thus, the negation of this statement will be –
“A leap year does not have 366 days.”
ix) Concept
We know that negation of p is “not p”, viz. symbolized as –p, also its truth value of –p is the opposite of the truth value of p. Thus, the negation of this statement will be –
“All similar triangles are not congruent.”
x) Concept
We know that negation of p is “not p”, viz. symbolized as –p, also its truth value of –p is the opposite of the truth value of p.
Thus, the negation of this statement will be – “Area of a circle is not same as the perimeter of the circle.”
Question:5
Answer:
(i) It is a compound statement whose components are –
p: Rahul passed in Hindi &
q: Rahul passed in English
Symbolically the function is represented as –
– Rahul passed in Hindi & English
(ii) It is a compound statement whose components are –
p: x is an even integer &
q: y is an even integer
Symbolically the function is represented as –
are even integers.
(iii) It is a compound statement whose components are –
p: 2 is a factor of 12
q: 3 is a factor of 12 &
r: 6 is a factor of 12
Symbolically the function is represented as –
are factors of 12.
(iv) It is a compound statement whose components are –
p: x is an odd integer &
is an odd integer
Symbolically the function is represented as –
p V q – Either x or x + 1 is an odd integer.
(v) It is a compound statement whose components are –
p: A number is divisible by 2 &
q: A number is divisible by 3
Symbolically the function is represented as –
p V q – A number is either divisble by 2 or 3.
(vi) It is a compound statement whose components are –
is a root of &
is a root of
Symbolically the function is represented as –
p v q – Either x = 2 or x = 3 is a root of
(vii) It is a compound statement whose components are –
p: Hindi is the optional paper &
q: English is the optional paper
Symbolically the function is represented as –
p v q – Either Hindi or English is optional paper.
Question:6
Answer:
(i) It is a compound statement whose components are –
p: all rational numbers are real
-p: all rational numbers are not real
q: All rational numbers are complex
-q: All rational numbers are not complex
Thus,All rational numbers are real and complex.
& All rational nos. are neither complex nor real
(ii) It is a compound statement whose components are –
p: all real numbers are rational
-p: all real numbers are not rational
q: All real numbers are not irrational &
-q: All real numbers are not irrational
Thus, (p v q) = All real nos. are either rational or irrational.
& = All real numbers are neither rational or irrational.
(iii) It is a compound statement whose components are –
p: x = 2 is a root of Quadratic equation .
-p: x = 2 is not a root of Quadratic equation .
q: x = 3 is a root of Quadratic equation .
-q: x = 3 is not a root of Quadratic equation .
Thus, are roots of Quadratic equation .
&
Neither x = 2 nor x = 3 are roots of Quadratic equation .
(iv) It is a compound statement whose components are –
p: A triangle has 3 sides
-p: A triangle does not have 3 sides
q: A triangle has 4 sides
-q: A triangle does not have 4 sides
Thus, (p V q) = A triangle has either 3 or 4 sides & = A triangle has neither 3 nor 4 sides
(v) It is a compound statement whose components are –
p: 35 is a prime no.
-p: 35 is not a prime no.
q: 35 is a composite no.
-q: 35 is not a composite no.
Thus, (p V q) = 35 is either a prime no. or a composite no. & -(p V q) = -p ? –q = 35 is neither a prime no. nor a composite no.
(vi) It is a compound statement whose components are –
p: All prime integers are even
-p: All prime integers are not even
q: All integers are odd
-q: All integers are not odd
Thus, (p V q) = All prime integers are either even or odd & -(p V q) = -p ? –q = All prime integers are neither even nor odd
(vii) It is a compound statement whose components are –
p: is equal to x.
-p: is not equal to x.
q: is equal to-x.
-q: is not equal to-x.
Thus, (p V q) = x is either equal to x or-x & -(p V q) = -p ? – = is neither equal to x nor-x.
(viii) ) It is a compound statement whose components are –
p: 6 is divisible by 2
-p: 6 is not divisible by 2
q: 6 is divisible by 3
-q: 6 is not divisible by 3
Thus, (p ^ q) = 6 is divisible by 2 & 3
& -(p v q) = -p V –q = 6 is neither divisible by 2 nor 3.
Question:7
Answer:
(i) The expression in the given conditional statement is- if p, then q.
p: The no. is odd
q: The square of the no. is odd
Thus, “if the no. is odd, then its square is even.
(ii) The expression in the given conditional statement is- if p, then q.
p: Take the dinner
q: you will get sweet dish
Thus, “If you take the dinner, then you will get sweet dish.”
(iii) The expression in the given conditional statement is- if p, then q.
p: You do not study
q: you will fail
Thus, “If you do not study, you will fail.”
(iv) The expression in the given conditional statement is- if p, then q.
p: An integer is divisible by 5
q: Unit digits of an integer are 0 or 5.
Thus, “If an integer is divisible by 5, then its unit digits are 0.”
(v) The expression in the given conditional statement is- if p, then q.
p: Any no. is prime
q: square of no. is not prime
Thus, “If any no. is prime, then its square is not prime.”
(vi) The expression in the given conditional statement is- if p, then q.
p: a, b & c are in AP
q: 2b = a + c
Thus, “If a, b & c are in AP, then 2b = a + c.”
Question:8
Answer:
i) We use only & only if in biconditional statements, here,
p: The unit digit of an integer is zero.
q: It is divisible by 5.
Thus, p ↔ q = Unit digit of an integer is zero if and only if it is divisible by 5.
ii) We use only & only if in biconditional statements, here,
p: A natural no. n is odd
q: Natural no. n is not divisible by 2.
Thus, p ↔ q = A natural no. is odd if and only if it is not divisible by 2.
iii) We use only & only if in biconditional statements, here,
p: A triangle is an equilateral triangle.
q: All three sides of a triangle are equal.
p ↔ q = A triangle is an equilateral triangle if and only if all three sides of triangle are equal.
Question:9
Answer:
(i) Contrapositive definition: A conditional statement is said to be logically equivalent to its contrapositive.
Thus, Contrapositive: If x ≠ 3, then y ≠x or y ≠3.
(ii) Contrapositive definition: A conditional statement is said to be logically equivalent to its contrapositive.
Thus, Contrapositive: If n is not an integer, then it is not a natural no.
(iii) Contrapositive definition: A conditional statement is said to be logically equivalent to its contrapositive.
Thus, Contrapositive: If the triangle is not equilateral, then all three sides of the triangle are not equal.
(iv) Contrapositive definition: A conditional statement is said to be logically equivalent to its contrapositive.
Thus, Contrapositive: If xy is not a positive integer, then either x or y is not a negative integer.
(v) Contrapositive definition: A conditional statement is said to be logically equivalent to its contrapositive.
Thus, Contrapositive: If natural no. ‘n’ is not divisible by 2 or 3, then n is not divisible by 6.
(vi) Contrapositive definition: A conditional statement is said to be logically equivalent to its contrapositive.
Thus, Contrapositive: The weather will not be cold if it doesn’t snow.
(vii) Contrapositive definition: A conditional statement is said to be logically equivalent to its contrapositive.
Thus, Contrapositive: If then, x is not a real number such that 0 < x < 1.
Question:10
Answer:
(i) Converse definition: A conditional statement is said to be not logically equivalent to its converse.Thus, Converse: If the rectangle R is rhombus, then it is square.
(ii) Converse definition: A conditional statement is said to be not logically equivalent to its converse. Thus, Converse: If tomorrow is Tuesday, then today is Monday.
(iii) Converse definition: A conditional statement is said to be not logically equivalent to its converse.Thus, Converse: If you must visit Taj Mahal, then you go to Agra.
(iv) Converse definition: A conditional statement is said to be not logically equivalent to its converse. Thus, Converse: If the triangle is right triangle, then the sum of the squares of a triangle is equal to the square of the third side.
(v) Converse definition: A conditional statement is said to be not logically equivalent to its converse. Thus, Converse: If the triangle is equilateral, then all three angles of the triangle are equal.
(vi) Converse definition: A conditional statement is said to be not logically equivalent to its converse. Thus, Converse: If 2x = 3y, then x:y = 3:2.
(vii) Converse definition: A conditional statement is said to be not logically equivalent to its converse. Thus, Converse: If the opposite angles of a quadrilateral are supplementary, then S is cyclic.
(viii) Converse definition: A conditional statement is said to be not logically equivalent to its converse. Thus, Converse: If x is neither positive nor negative, then x = 0.
(ix) Converse definition: A conditional statement is said to be not logically equivalent to its converse. Thus, Converse:If the ratio of the corresponding sides of two triangles are equal, then trianles are similar.
Question:11
Answer:
(i) Quantifiers: It is a phrase viz. used to make the prepositional statement, example – ‘there exist’, ‘for all’, ‘for every’, etc.
In the given statement, quantifier is – “There exist” Thus, there is a quantifier.
(ii) Quantifiers: It is a phrase viz. used to make the prepositional statement, example – ‘there exist’, ‘for all’, ‘for every’, etc.
In the given statement, quantifier is – “For all” Thus, there is a quantifier.
(iii) Quantifiers: It is a phrase viz. used to make the prepositional statement, example – ‘there exist’, ‘for all’, ‘for every’, etc.
In the given statement, quantifier is – “There exist” Thus, there is a quantifier.
(iv) Quantifiers: It is a phrase viz. used to make the prepositional statement, example – ‘there exist’, ‘for all’, ‘for every’, etc.
In the given statement, quantifier is – “For every” Thus, there is a quantifier.
(v) Quantifiers: It is a phrase viz. used to make the prepositional statement, example – ‘there exist’, ‘for all’, ‘for every’, etc.
In the given statement, quantifier is – “For all” Thus, there is a quantifier.
(vi) Quantifiers: It is a phrase viz. used to make the prepositional statement, example – ‘there exist’, ‘for all’, ‘for every’, etc.
In the given statement, quantifier is – “There exist” Thus, there is a quantifier.
(vii) Quantifiers: It is a phrase viz. used to make the prepositional statement, example – ‘there exist’, ‘for all’, ‘for every’, etc.
In the given statement, quantifier is – “For all” Thus, there is a quantifier.
(viii) Quantifiers: It is a phrase viz. used to make the prepositional statement, example – ‘there exist’, ‘for all’, ‘for every’, etc.
In the given statement, quantifier is – “There exist” Thus, there is a quantifier.
(ix) Quantifiers: It is a phrase viz. used to make the prepositional statement, example – ‘there exist’, ‘for all’, ‘for every’, etc.
In the given statement, quantifier is – “There exist” Thus, there is a quantifier.
(x) Quantifiers: It is a phrase viz. used to make the prepositional statement, example – ‘there exist’, ‘for all’, ‘for every’, etc.
In the given statement, quantifier is – “There exist” Thus, there is a quantifier.
Question:12
Prove by direct method that for any integer is always even.
Answer:
Given :
Let us consider that, n is even
Thus,
[Let us consider that ]
Thus,
Thus,
is even
Now, we will assume that n is an odd no.
Thus,
Thus, from this we can say that is always even.
Question:13
Answer:
i) Here, p: 125 is divisible by 5 & 7
Now let, q: 125 is divisible by 5 &
r: 125 is divisible by 7.
We know that q is true & r is false.
Thus, q ^ r is false..
Therefore, p is no valid.
ii) Here, q: 131 is a multiple of 3 or 11.
Now let, P: 131 is a multiple of 3
& Q: 131 is a multiple of 11.
We know that both P & Q are false.
Thus, P V Q is False & q is not valid.
Question:14
Answer:
p: The sum of an irrational number and a rational number is irrational.
We know that the sum of a rational no. & a irrational no. is irrational, p is false.
Let,
&
Now,
But since, is irrational and n is rational viz.contradictiona, our assumption will be false.
Thus, P is true.
Question:15
Prove by direct method that for any real numbers if , then
Answer:
Given: For any real no. .
Let us consider that,
, where x & y are real no.
If we square both sides,
……. (assumption)
Thus, .
Question:16
Answer:
Let us consider that,
p: n2 is an even integer
-p: n is not an even integer
Q: n is also an even integer
-q: n is not an even integer.
A conditional statement is said to be logically equivalent to its contrapositive.
Thus, If n is not an even integer, then is not an even integer.
Thus, are true.
Question:17
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 statement is considered as an assertive sentence if it is either true or false, but it should not be both.
Therefore, (C) 6 is a natural no. is true.
Question:18
Which of the following is not a statement
A. Smoking is injurious to health.
B.
C. 2 is the only even prime number.
D. Come here.
Answer:
Option D
‘Go there’ is not a statement to a given order like ‘Come here’.
Question:19
The connective in the statement is
A. and
B. or
C. >
D. <
Answer:
Since the two statements are connected by ‘or’ –
Option (B) is the correct answer.
Question:20
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:
The connective in the given statement is ‘and’
Thus, option (D) is the correct answer.
Question:21
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:
The negation of the given statement will be (C) A circle is not an ellipse
Question:22
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:
The negation of the given statement will be –
(B) 7 is not greater than 8
Question:23
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:
In the given statement-
p: 72 is divisible by 2 & 3
q: 72 is divisible by 2 & -q: 72 is not divisible by 2
r: 72 is divisible by 3 & -r: 72 is not divisible by 3
Thus, 72 is not divisible by 2 or 72 is not divisible by 3.
Thus, opt A is the correct answer.
Question:24
The negation of the statement
“Plants take in and give out ” is
A. Plants do not take in and do not give out .
B. Plants do not take in or do not give out .
C. Plants take in and do not give out .
D. Plants take in or do not give out .
Answer:
In the given statement-
p: Plants take in and give out
q: Plants take in & -q: Plants do not take in
r: Plants give out & -r: Plants do not give out
Thus, Plants do not take in or do not give out .
Thus, option B is the correct answer.
Question:25
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:
In the given statement-
p: Rajesh or Rajini lived in Bangalore
q: Rajesh lived in Bangalore & -q: Rajesh dis not lived in Bangalore
r: Rajini lived in Bandalore & -r: Rajini did not lived in Bangalore
Thus, Rajesh did not live in Bangalore and Rajini did not live in Bangalore.
Thus, Option C is the correct answer.
Question:26
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:
The negation of the given statement is –
101 is a multiple of 3.
Thus, option A is the correct answer.
Question:27
The contrapositive of the statement
“If 7 is greater than 5, then 8 is greater than 6” is
A. If 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:
Here, p: 7 is greater than 5 & -p: 7 is not greater than 5
& q: 8 is greater than 6 & -q: 8 is not greater than 6.
Now,
A conditional statement is said to be logically equivalent to its contrapositive.
Thus, -p → -q = If 8 is not greater than 6, then 7 is not greater than 5.
Thus, option (C) is the correct answer.
Question:28
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:
Here,
& q:
Thus, , whose converse will be –
viz., if , then .
Thus, option (B) is the correct answer.
Question:29
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 the 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:
Here, p: Sun is not shining
& q: Sky is filled with clouds.
Thus, whose converse will be –
viz., If the sky is filled with clouds, then the sun is not shining.
Thus, option (A) is the correct answer.
Question:30
The contrapositive of the statement
“If p, then q”, is
A. If q, then p.
B. If p, then .
C. If , then .
D. If , then .
Answer:
Here the statement is in the form – “If p, then q” viz.
whose converse will be
Thus, if –q, then –p.
Therefore, option (C) is the correct answer.
Question:31
The statement
“If is not even, then x is not even” is converse of the statement
A. If is odd, then x is even.
B. If x is not even, then is not even.
C. If x is even, then is even.
D. If x is odd, then is even.
Answer:
Here, let p: x2 is not even & q: x is not even
Thus, , whose converse will be –
viz., If x is not even, then is not even.
Thus, option B is the correct answer.
Question:32
The contrapositive of 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 Capital of Punjab.
C. If Chandigarh is not capital of Punjab, then Chandigarh is not capital of India.
D. If Chandigarh is capital of Punjab, then Chandigarh is not in India.
Answer:
Here, let us take,
p: Chandigarh is the capital of Punjab, thus –p: Chandigarh is not the capital of Punjab
& q: Chandigarh is in India, thus –q: Chandigarh is not in India.
Now, If (-q), then (-p),
Thus, If Chandigarh is not in India, then Chandigarh is not the capital of Punjab.
Thus, option A is the correct answer.
Question:33
Which of the following is the conditional ?
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 the same as .
Thus, option C is the correct answer.
Question:34
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:
The negation of the given statement is –
“It is false that the product of 3 & 4 is 9.”
Thus, option A is the correct answer.
Question:35
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 the above
Answer:
We know that the negation of the given statement is false, viz.
“It is false that a natural no. is not greater than zero.”
Thus, option C is the correct answer.
Question:36
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 given statements is separated by ‘and’, thus, option D is the correct answer.
Question:37
Answer:
(i) “The angles opposite to equal sides of a triangle are equal” is true, thus, it is clear that it is a statement.
(ii) “The moon is the satellite of the earth” is true, thus, it is clear that it is a statement.
(iii) “May God bless you!” is an exclamation sentence, thus, it is clear that it is not a statement.
(iv) “Asia is a continent” is true, thus, it is clear that it is a statement.
(v) “How are you?” is a question, thus, it is clear that it is not a statement.
Students can download the notes by clicking on the NCERT Exemplar Class 11 Maths solutions Chapter 14 PDF Download. The PDF consists of solutions to those questions which are important from the examination point of view and is very convenient for the students to refer to.
· Mathematical Statements
· The negation of a Statement
· Compound Statements
· Special Words/Phrases
· The word ‘And’
· The word ‘Or’
· Quantifiers
· Implications
· Contrapositive
· Converse
· Validating Statements
· Direct Method
· Contrapositive Method
· Contradiction Method
· Using a Counterexample Method
The experts have given the solutions in a simple manner which enables the students to grasp the concept of Mathematical Reasoning. The Class 11 Maths NCERT Exemplar Solutions Chapter 14 also includes the answers to the questions related to different sub-topics like contrapositive and converse implications and different types of methods.
With the help of illustrations and stepwise format of the solutions, the students can easily ace the chapter in their examination. As NCERT Exemplar Class 11 Maths solutions Chapter 14 are important, it cannot be skipped by the students. It is the most crucial chapter from the entire syllabus and many questions which are asked in the exam are based on Mathematical Reasoning.
Mathematical Statements: Mathematical Statements are divided into two sub-topics namely: Negation statements and Compound statements. A mathematically acceptable statement is a statement which can either be termed as true or false. Negation statements in mathematics are used to determine the opposite of a given statement, for example, the statement given is A, then the negation of the statement will be denoted by ∼A. Compound Statements in mathematics in simple terms consist of two smaller statements.
Validating Statements: NCERT Exemplar Class 11 Maths solutions Chapter 14 suggests that there are four ways in mathematics to validate statements by using the Direct Method, Contrapositive Method, Contradiction Method, and by using a Counterexample Method. The students should first understand the basic concept behind all these methods and then apply them to validate the given statement.
Implications: Implication is another important topic which must be studied thoroughly by the students. The students should learn what are the different types of implications used in mathematical reasoning and the precise way to apply them to statements. NCERT Exemplar solutions for Class 11 Maths Chapter 14 proves to be the perfect guide for learning Mathematical Reasoning.
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Yes, most of the questions solved in NCERT Exemplar Class 11 Maths Chapter 14 Solutions include textbook-based exercises
Yes, the format of the solutions can be followed for board examinations.
Students can use the NCERT Exemplar Class 11 Maths Solutions Chapter 14 PDF Download function to download the PDF for this chapter.
Database professionals use software to store and organise data such as financial information, and customer shipping records. Individuals who opt for a career as data administrators ensure that data is available for users and secured from unauthorised sales. DB administrators may work in various types of industries. It may involve computer systems design, service firms, insurance companies, banks and hospitals.
The field of biomedical engineering opens up a universe of expert chances. An Individual in the biomedical engineering career path work in the field of engineering as well as medicine, in order to find out solutions to common problems of the two fields. The biomedical engineering job opportunities are to collaborate with doctors and researchers to develop medical systems, equipment, or devices that can solve clinical problems. Here we will be discussing jobs after biomedical engineering, how to get a job in biomedical engineering, biomedical engineering scope, and salary.
A career as ethical hacker involves various challenges and provides lucrative opportunities in the digital era where every giant business and startup owns its cyberspace on the world wide web. Individuals in the ethical hacker career path try to find the vulnerabilities in the cyber system to get its authority. If he or she succeeds in it then he or she gets its illegal authority. Individuals in the ethical hacker career path then steal information or delete the file that could affect the business, functioning, or services of the organization.
GIS officer work on various GIS software to conduct a study and gather spatial and non-spatial information. GIS experts update the GIS data and maintain it. The databases include aerial or satellite imagery, latitudinal and longitudinal coordinates, and manually digitized images of maps. In a career as GIS expert, one is responsible for creating online and mobile maps.
The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.
Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.
Individuals who opt for a career as geothermal engineers are the professionals involved in the processing of geothermal energy. The responsibilities of geothermal engineers may vary depending on the workplace location. Those who work in fields design facilities to process and distribute geothermal energy. They oversee the functioning of machinery used in the field.
If you are intrigued by the programming world and are interested in developing communications networks then a career as database architect may be a good option for you. Data architect roles and responsibilities include building design models for data communication networks. Wide Area Networks (WANs), local area networks (LANs), and intranets are included in the database networks. It is expected that database architects will have in-depth knowledge of a company's business to develop a network to fulfil the requirements of the organisation. Stay tuned as we look at the larger picture and give you more information on what is db architecture, why you should pursue database architecture, what to expect from such a degree and what your job opportunities will be after graduation. Here, we will be discussing how to become a data architect. Students can visit NIT Trichy, IIT Kharagpur, JMI New Delhi.
Individuals who opt for a career as a remote sensing technician possess unique personalities. Remote sensing analysts seem to be rational human beings, they are strong, independent, persistent, sincere, realistic and resourceful. Some of them are analytical as well, which means they are intelligent, introspective and inquisitive.
Remote sensing scientists use remote sensing technology to support scientists in fields such as community planning, flight planning or the management of natural resources. Analysing data collected from aircraft, satellites or ground-based platforms using statistical analysis software, image analysis software or Geographic Information Systems (GIS) is a significant part of their work. Do you want to learn how to become remote sensing technician? There's no need to be concerned; we've devised a simple remote sensing technician career path for you. Scroll through the pages and read.
Budget analysis, in a nutshell, entails thoroughly analyzing the details of a financial budget. The budget analysis aims to better understand and manage revenue. Budget analysts assist in the achievement of financial targets, the preservation of profitability, and the pursuit of long-term growth for a business. Budget analysts generally have a bachelor's degree in accounting, finance, economics, or a closely related field. Knowledge of Financial Management is of prime importance in this career.
The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.
Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.
An underwriter is a person who assesses and evaluates the risk of insurance in his or her field like mortgage, loan, health policy, investment, and so on and so forth. The underwriter career path does involve risks as analysing the risks means finding out if there is a way for the insurance underwriter jobs to recover the money from its clients. If the risk turns out to be too much for the company then in the future it is an underwriter who will be held accountable for it. Therefore, one must carry out his or her job with a lot of attention and diligence.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
Individuals in the operations manager jobs are responsible for ensuring the efficiency of each department to acquire its optimal goal. They plan the use of resources and distribution of materials. The operations manager's job description includes managing budgets, negotiating contracts, and performing administrative tasks.
Individuals who opt for a career as a stock analyst examine the company's investments makes decisions and keep track of financial securities. The nature of such investments will differ from one business to the next. Individuals in the stock analyst career use data mining to forecast a company's profits and revenues, advise clients on whether to buy or sell, participate in seminars, and discussing financial matters with executives and evaluate annual reports.
A Researcher is a professional who is responsible for collecting data and information by reviewing the literature and conducting experiments and surveys. He or she uses various methodological processes to provide accurate data and information that is utilised by academicians and other industry professionals. Here, we will discuss what is a researcher, the researcher's salary, types of researchers.
Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues.
A career as Transportation Planner requires technical application of science and technology in engineering, particularly the concepts, equipment and technologies involved in the production of products and services. In fields like land use, infrastructure review, ecological standards and street design, he or she considers issues of health, environment and performance. A Transportation Planner assigns resources for implementing and designing programmes. He or she is responsible for assessing needs, preparing plans and forecasts and compliance with regulations.
Individuals who opt for a career as an environmental engineer are construction professionals who utilise the skills and knowledge of biology, soil science, chemistry and the concept of engineering to design and develop projects that serve as solutions to various environmental problems.
A Safety Manager is a professional responsible for employee’s safety at work. He or she plans, implements and oversees the company’s employee safety. A Safety Manager ensures compliance and adherence to Occupational Health and Safety (OHS) guidelines.
A Conservation Architect is a professional responsible for conserving and restoring buildings or monuments having a historic value. He or she applies techniques to document and stabilise the object’s state without any further damage. A Conservation Architect restores the monuments and heritage buildings to bring them back to their original state.
A Structural Engineer designs buildings, bridges, and other related structures. He or she analyzes the structures and makes sure the structures are strong enough to be used by the people. A career as a Structural Engineer requires working in the construction process. It comes under the civil engineering discipline. A Structure Engineer creates structural models with the help of computer-aided design software.
Highway Engineer Job Description: A Highway Engineer is a civil engineer who specialises in planning and building thousands of miles of roads that support connectivity and allow transportation across the country. He or she ensures that traffic management schemes are effectively planned concerning economic sustainability and successful implementation.
Are you searching for a Field Surveyor Job Description? A Field Surveyor is a professional responsible for conducting field surveys for various places or geographical conditions. He or she collects the required data and information as per the instructions given by senior officials.
Orthotists and Prosthetists are professionals who provide aid to patients with disabilities. They fix them to artificial limbs (prosthetics) and help them to regain stability. There are times when people lose their limbs in an accident. In some other occasions, they are born without a limb or orthopaedic impairment. Orthotists and prosthetists play a crucial role in their lives with fixing them to assistive devices and provide mobility.
A career in pathology in India is filled with several responsibilities as it is a medical branch and affects human lives. The demand for pathologists has been increasing over the past few years as people are getting more aware of different diseases. Not only that, but an increase in population and lifestyle changes have also contributed to the increase in a pathologist’s demand. The pathology careers provide an extremely huge number of opportunities and if you want to be a part of the medical field you can consider being a pathologist. If you want to know more about a career in pathology in India then continue reading this article.
Gynaecology can be defined as the study of the female body. The job outlook for gynaecology is excellent since there is evergreen demand for one because of their responsibility of dealing with not only women’s health but also fertility and pregnancy issues. Although most women prefer to have a women obstetrician gynaecologist as their doctor, men also explore a career as a gynaecologist and there are ample amounts of male doctors in the field who are gynaecologists and aid women during delivery and childbirth.
The audiologist career involves audiology professionals who are responsible to treat hearing loss and proactively preventing the relevant damage. Individuals who opt for a career as an audiologist use various testing strategies with the aim to determine if someone has a normal sensitivity to sounds or not. After the identification of hearing loss, a hearing doctor is required to determine which sections of the hearing are affected, to what extent they are affected, and where the wound causing the hearing loss is found. As soon as the hearing loss is identified, the patients are provided with recommendations for interventions and rehabilitation such as hearing aids, cochlear implants, and appropriate medical referrals. While audiology is a branch of science that studies and researches hearing, balance, and related disorders.
An oncologist is a specialised doctor responsible for providing medical care to patients diagnosed with cancer. He or she uses several therapies to control the cancer and its effect on the human body such as chemotherapy, immunotherapy, radiation therapy and biopsy. An oncologist designs a treatment plan based on a pathology report after diagnosing the type of cancer and where it is spreading inside the body.
Are you searching for an ‘Anatomist job description’? An Anatomist is a research professional who applies the laws of biological science to determine the ability of bodies of various living organisms including animals and humans to regenerate the damaged or destroyed organs. If you want to know what does an anatomist do, then read the entire article, where we will answer all your questions.
For an individual who opts for a career as an actor, the primary responsibility is to completely speak to the character he or she is playing and to persuade the crowd that the character is genuine by connecting with them and bringing them into the story. This applies to significant roles and littler parts, as all roles join to make an effective creation. Here in this article, we will discuss how to become an actor in India, actor exams, actor salary in India, and actor jobs.
Individuals who opt for a career as acrobats create and direct original routines for themselves, in addition to developing interpretations of existing routines. The work of circus acrobats can be seen in a variety of performance settings, including circus, reality shows, sports events like the Olympics, movies and commercials. Individuals who opt for a career as acrobats must be prepared to face rejections and intermittent periods of work. The creativity of acrobats may extend to other aspects of the performance. For example, acrobats in the circus may work with gym trainers, celebrities or collaborate with other professionals to enhance such performance elements as costume and or maybe at the teaching end of the career.
Career as a video game designer is filled with excitement as well as responsibilities. A video game designer is someone who is involved in the process of creating a game from day one. He or she is responsible for fulfilling duties like designing the character of the game, the several levels involved, plot, art and similar other elements. Individuals who opt for a career as a video game designer may also write the codes for the game using different programming languages.
Depending on the video game designer job description and experience they may also have to lead a team and do the early testing of the game in order to suggest changes and find loopholes.
Radio Jockey is an exciting, promising career and a great challenge for music lovers. If you are really interested in a career as radio jockey, then it is very important for an RJ to have an automatic, fun, and friendly personality. If you want to get a job done in this field, a strong command of the language and a good voice are always good things. Apart from this, in order to be a good radio jockey, you will also listen to good radio jockeys so that you can understand their style and later make your own by practicing.
A career as radio jockey has a lot to offer to deserving candidates. If you want to know more about a career as radio jockey, and how to become a radio jockey then continue reading the article.
The word “choreography" actually comes from Greek words that mean “dance writing." Individuals who opt for a career as a choreographer create and direct original dances, in addition to developing interpretations of existing dances. A Choreographer dances and utilises his or her creativity in other aspects of dance performance. For example, he or she may work with the music director to select music or collaborate with other famous choreographers to enhance such performance elements as lighting, costume and set design.
A career as social media manager involves implementing the company’s or brand’s marketing plan across all social media channels. Social media managers help in building or improving a brand’s or a company’s website traffic, build brand awareness, create and implement marketing and brand strategy. Social media managers are key to important social communication as well.
Photography is considered both a science and an art, an artistic means of expression in which the camera replaces the pen. In a career as a photographer, an individual is hired to capture the moments of public and private events, such as press conferences or weddings, or may also work inside a studio, where people go to get their picture clicked. Photography is divided into many streams each generating numerous career opportunities in photography. With the boom in advertising, media, and the fashion industry, photography has emerged as a lucrative and thrilling career option for many Indian youths.
An individual who is pursuing a career as a producer is responsible for managing the business aspects of production. They are involved in each aspect of production from its inception to deception. Famous movie producers review the script, recommend changes and visualise the story.
They are responsible for overseeing the finance involved in the project and distributing the film for broadcasting on various platforms. A career as a producer is quite fulfilling as well as exhaustive in terms of playing different roles in order for a production to be successful. Famous movie producers are responsible for hiring creative and technical personnel on contract basis.
In a career as a copywriter, one has to consult with the client and understand the brief well. A career as a copywriter has a lot to offer to deserving candidates. Several new mediums of advertising are opening therefore making it a lucrative career choice. Students can pursue various copywriter courses such as Journalism, Advertising, Marketing Management. Here, we have discussed how to become a freelance copywriter, copywriter career path, how to become a copywriter in India, and copywriting career outlook.
In a career as a vlogger, one generally works for himself or herself. However, once an individual has gained viewership there are several brands and companies that approach them for paid collaboration. It is one of those fields where an individual can earn well while following his or her passion.
Ever since internet costs got reduced the viewership for these types of content has increased on a large scale. Therefore, a career as a vlogger has a lot to offer. If you want to know more about the Vlogger eligibility, roles and responsibilities then continue reading the article.
For publishing books, newspapers, magazines and digital material, editorial and commercial strategies are set by publishers. Individuals in publishing career paths make choices about the markets their businesses will reach and the type of content that their audience will be served. Individuals in book publisher careers collaborate with editorial staff, designers, authors, and freelance contributors who develop and manage the creation of content.
Careers in journalism are filled with excitement as well as responsibilities. One cannot afford to miss out on the details. As it is the small details that provide insights into a story. Depending on those insights a journalist goes about writing a news article. A journalism career can be stressful at times but if you are someone who is passionate about it then it is the right choice for you. If you want to know more about the media field and journalist career then continue reading this article.
Individuals in the editor career path is an unsung hero of the news industry who polishes the language of the news stories provided by stringers, reporters, copywriters and content writers and also news agencies. Individuals who opt for a career as an editor make it more persuasive, concise and clear for readers. In this article, we will discuss the details of the editor's career path such as how to become an editor in India, editor salary in India and editor skills and qualities.
Individuals who opt for a career as a reporter may often be at work on national holidays and festivities. He or she pitches various story ideas and covers news stories in risky situations. Students can pursue a BMC (Bachelor of Mass Communication), B.M.M. (Bachelor of Mass Media), or MAJMC (MA in Journalism and Mass Communication) to become a reporter. While we sit at home reporters travel to locations to collect information that carries a news value.
Are you searching for a Corporate Executive job description? A Corporate Executive role comes with administrative duties. He or she provides support to the leadership of the organisation. A Corporate Executive fulfils the business purpose and ensures its financial stability. In this article, we are going to discuss how to become corporate executive.
A multimedia specialist is a media professional who creates, audio, videos, graphic image files, computer animations for multimedia applications. He or she is responsible for planning, producing, and maintaining websites and applications.
Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues.
A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product.
A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
A QA Lead is in charge of the QA Team. The role of QA Lead comes with the responsibility of assessing services and products in order to determine that he or she meets the quality standards. He or she develops, implements and manages test plans.
A Structural Engineer designs buildings, bridges, and other related structures. He or she analyzes the structures and makes sure the structures are strong enough to be used by the people. A career as a Structural Engineer requires working in the construction process. It comes under the civil engineering discipline. A Structure Engineer creates structural models with the help of computer-aided design software.
The Process Development Engineers design, implement, manufacture, mine, and other production systems using technical knowledge and expertise in the industry. They use computer modeling software to test technologies and machinery. An individual who is opting career as Process Development Engineer is responsible for developing cost-effective and efficient processes. They also monitor the production process and ensure it functions smoothly and efficiently.
An AWS Solution Architect is someone who specializes in developing and implementing cloud computing systems. He or she has a good understanding of the various aspects of cloud computing and can confidently deploy and manage their systems. He or she troubleshoots the issues and evaluates the risk from the third party.
An Azure Administrator is a professional responsible for implementing, monitoring, and maintaining Azure Solutions. He or she manages cloud infrastructure service instances and various cloud servers as well as sets up public and private cloud systems.
Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack
An Automation Test Engineer job involves executing automated test scripts. He or she identifies the project’s problems and troubleshoots them. The role involves documenting the defect using management tools. He or she works with the application team in order to resolve any issues arising during the testing process.