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NCERT Exemplar Class 11 Maths Solutions Chapter 14 Mathematical Reasoning

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NCERT Exemplar Class 11 Maths Solutions Chapter 14 Mathematical Reasoning

Edited By Ravindra Pindel | Updated on Sep 12, 2022 05:57 PM IST

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.

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.

NCERT Exemplar Class 11 Maths Solutions Chapter 14: Exercise: 1.3

Question:1

Which of the following sentences are statements? Justify
i) A triangle has three sides.
ii) 0 is a complex number.
iii) Sky is red
iv) Every set is an infinite set.
v) 15+8>23
vi) y+9=7
vii) Where is your bag?
viii) Every square is a rectangle.
ix) Sum of opposite angles of a cyclic quadrilateral is 180°.
x) \sin ^{2}x+\cos ^{2}x=0

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 15 + 8 > 23 because LHS\neq RHS.
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 \sin ^{2}x+\cos^{2}x=1 is the correct expression according to the laws of trigonometry.

Question:2

Find the component statements of the following compound statements.
i) Number 7 is prime and odd.
ii) Chennai is in India and is the capital of Tamil Nadu.
iii) The number 100 is divisible by 3,11 and 5.
iv) Chandigarh is the capital of Haryana and U.P.
v) \sqrt{7} is a rational number or an irrational number.
vi) 0 is less than every positive integer and every negative integer.
vii) Plants use sunlight, water and carbon dioxide for photosynthesis.
viii) Two lines in a plane either intersect at one point or they are parallel.
ix) A rectangle is a quadrilateral or a 5 - sided polygon.

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 \sqrt{7} is a rational number or an irrational number”, will be –
p: \sqrt{7} is a rational number &
q: \sqrt{7} 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

Write the component statements of the following compound statements and check whether the compound statement is true or false.
57 is divisible by 2 or 3.
ii) 57 is divisible by 2 or 3.
24 is a multiple of 4 and 6.
iii) 57 is divisible by 2 or 3.
All living things have two eyes and two legs.
iv) 57 is divisible by 2 or 3.
2 is an even number and a prime number.

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 P\wedge Q, 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

Write the negation of the following simple statements
i) The number 17 is prime.
ii) 2 + 7 = 6.
iii) Violets are blue.
iv) \sqrt{5} is a rational number.
v) 2 is not a prime number.
vi) Every real number is an irrational number.
vii) Cow has four legs.
viii) A leap year has 366 days.
ix) All similar triangles are congruent.
x) Area of a circle is same as the perimeter of the circle.

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 – 2+7\neq 6
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 –
\sqrt{5} 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

Translate the following statements into symbolic form
(i) Rahul passed in Hindi and English.
(ii) x and y are even integers.
(iii) 2, 3 and 6 are factors of 12.
(iv) Either x or x+1 is an odd integer.
(v) A number is either divisible by 2 or 3.
(vi) Either x=2 or x=3 is a root of 3x^{2}-x-10=0
(vii) Students can take Hindi or English as an optional paper.

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 –
p \wedge q – 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 –
p \wedge q -x \; and\; y 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 –
p \wedge q \wedge r: 2, 3 \; and\; 6 are factors of 12.
(iv) It is a compound statement whose components are –
p: x is an odd integer &
q:x+1 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 –
p: x = 2 is a root of 3x^2 - x -10 = 0 &
q: x = 3 is a root of 3x^2 - x -10 = 0
Symbolically the function is represented as –
p v q – Either x = 2 or x = 3 is a root of 3x^2 - x -10 = 0
(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

Write down the negation of following compound statements
(i) All rational numbers are real and complex.
(ii) All real numbers are rationals or irrationals.
(iii) x=2 and x=3 are roots of the Quadratic equation x^{2} - 5x + 6 = 0
(iv) A triangle has either 3-sides or 4-sides.
(v) 35 is a prime number or a composite number.
(vi) All prime integers are either even or odd.
(vii) \left | x \right | is equal to either x or – x.
(viii) 6 is divisible by 2 and 3.

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,(p \wedge q) =All rational numbers are real and complex.
& -(p \wedge q) = -p V -q =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.
& -(p \wedge q) = -p V -q= 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 x^{2} - 5x + 6 = 0.
-p: x = 2 is not a root of Quadratic equation x^{2} - 5x + 6 = 0.
q: x = 3 is a root of Quadratic equation x^{2} - 5x + 6 = 0.
-q: x = 3 is not a root of Quadratic equation x^{2} - 5x + 6 = 0.
Thus, (p \wedge q) = x = 2 \; and\; x = 3 are roots of Quadratic equation x^{2} - 5x + 6 = 0.
& -(p \wedge q) = -p V -q
Neither x = 2 nor x = 3 are roots of Quadratic equation x^{2} - 5x + 6 = 0.
(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 &-(p V q) = -p \wedge -q = 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: \left | x \right | is equal to x.
-p: \left | x \right | is not equal to x.
q: \left | x \right | is equal to-x.
-q: \left | x \right | is not equal to-x.
Thus, (p V q) = x is either equal to x or-x & -(p V q) = -p ? – = \left | x \right | 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

Rewrite each of the following statements in the form of conditional statements
(i) The square of an odd number is odd.
(ii) You will get a sweet dish after the dinner.
(iii) You will fail, if you will not study.
(iv) The unit digit of an integer is 0 or 5 if it is divisible by 5.
(v) The square of a prime number is not prime.
(vi) 2b = a + c, if a, b and c are in A.P.

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

Form the biconditional statement P\leftrightarrow q, where
i) p : The unit digit of an integer is zero.
q : It is divisible by 5.
ii) p : A natural number n is odd.
q : Natural number n is not divisible by 2.
iii) p : A triangle is an equilateral triangle.
q : All three sides of a triangle are equal.

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

Write down the contrapositive of the following statements:
(i) If x = y \; and \; y = 3, then \; x = 3.
(ii) If n is a natural number, then n is an integer.
(iii) If all three sides of a triangle are equal, then the triangle is equilateral.
(iv) If x and y are negative integers, then xy is positive.
(v) If natural number n is divisible by 6, then n is divisible by 2 and 3.
(vi) If it snows, then the weather will be cold.
(vii) If x is a real number such that 0 < x < 1, then\; x^{2} < 1.

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 x^{2} > 1 then, x is not a real number such that 0 < x < 1.

Question:10

Write down the converse of following statements :
(i) If a rectangle ‘R’ is a square, then R is a rhombus.
(ii) If today is Monday, then tomorrow is Tuesday.
(iii) If you go to Agra, then you must visit Taj Mahal.
(iv) If the sum of squares of two sides of a triangle is equal to the square of third side of a triangle, then the triangle is right angled.
(v) If all three angles of a triangle are equal, then the triangle is equilateral.
(vi) If x : y = 3 : 2, then 2x = 3y.
(vii) If S is a cyclic quadrilateral, then the opposite angles of S are supplementary.
(viii) If x is zero, then x is neither positive nor negative.
(ix) If two triangles are similar, then the ratio of their corresponding sides are equal.

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

Identify the Quantifiers in the following statements.
(i) There exists a triangle which is not equilateral.
(ii) For all real numbers x and y, xy = yx.
(iii) There exists a real number which is not a rational number.
(iv) For every natural number x, x + 1 is also a natural number.
(v) For all real numbers x with x > 3, x2 is greater than 9.
(vi) There exists a triangle which is not an isosceles triangle
(vii) For all negative integers x, x3 is also a negative integers.
(viii) There exists a statement in above statements which is not true.
(ix) There exists a even prime number other than 2.
(x) There exists a real number x such that x^{2} + 1 = 0.

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 n, n^{3} - n is always even.

Answer:

Given : n^{3}-n
Let us consider that, n is even
Thus, n=2k .......(k-natural\; no)
\\n^{3} - n = (2k)^{3} - (2k) \\n^{3} - n = 2k (4k^{2} - 1)
[Let us consider that k(4k^{2}-1)=m]
Thus, n^{3}-n=2m
Thus,
n^{3}-n is even
Now, we will assume that n is an odd no.
Thus, n = (2k + 1) ....... (k - natural \; no.)
\\n^{3} - n = (2k + 1)^{3} - (2k + 1)\\ n^{3} - n = (2k + 1) [(4k2 + 4k + 1 - 1)]
\\n^{3} - n = (2k + 1) [(4k2 + 4k)]\\ n^{3} - n = 4k (2k + 1) (k + 1)
\\n^{2} - n = 2.2k (2k + 1) (k + 1)\\ n^{3} - n = 2\lambda
Thus, from this we can say that n^{3}-n is always even.

Question:13

Check the validity of the following statement.
i) p : 125 is divisible by 5 and 7.
ii) q : 131 is a multiple of 3 or 11.

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

Prove the following statement by contradiction method.
p: The sum of an irrational number and a rational number is irrational.

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, n\rightarrow rational \; no.
& \sqrt{\lambda }\rightarrow irrational\; no.
Now,
\sqrt{\lambda }+n=r
\sqrt{\lambda }=r-n
But since, \sqrt{\lambda } 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 x,y if x=y, then x^{2}=y^{2}.

Answer:

Given: For any real no. x, y, x = y.
Let us consider that,
p: x = y, where x & y are real no.
If we square both sides,
q: x^{2} = y^{2}
……. (assumption)
Thus, p=q.

Question:16

Using the contra positive method prove that if n2 is an even integer, then n is also an even integer.

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, -q\rightarrow -p=If n is not an even integer, then n^{2} is not an even integer.
Thus, -q\; and\; \rightarrow -p 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. 2+2=4
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 2 + 7 > 9 \; or\; 2 + 7 < 9 is
A. and
B. or
C. >
D. <

Answer:

Since the two statements 2 + 7 > 9\; \; 2 + 7 < 9 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
-(q \wedge r) = -q V -r
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 CO_{2} and give out O_{2}” is

A. Plants do not take in CO_{2} and do not give out O_{2}.
B. Plants do not take in CO_{2} or do not give out O_{2}.
C. Plants take in CO_{2} and do not give out O_{2}.
D. Plants take in CO_{2} or do not give out O_{2}.

Answer:

In the given statement-
p: Plants take in CO_{2} and give out O_{2}
q: Plants take in CO_{2} & -q: Plants do not take in CO_{2}
r: Plants give out O_{2} & -r: Plants do not give out O_{2}
-(q \wedge r) = -q V -r
Thus, Plants do not take in CO_{2} or do not give out O_{2}.
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
-(q \; \vee r) = -q \wedge -r
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, p: x > y
& q: x + a > y + a
Thus, p\rightarrow q, whose converse will be –
q\rightarrow p viz., if x + a> y + a, then x + a > y + a.
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, p \rightarrow q, whose converse will be –
q\rightarrow p 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 \sim q.
C. If \sim q , then \sim p.
D. If \sim p, then \sim q.

Answer:

Here the statement is in the form – “If p, then q” viz. p\rightarrow q
whose converse will be -q\rightarrow p
Thus, if –q, then –p.
Therefore, option (C) is the correct answer.

Question:31

The statement

“If x^{2} is not even, then x is not even” is converse of the statement

A. If x^{2} is odd, then x is even.
B. If x is not even, then x^{2} is not even.
C. If x is even, then x^{2} is even.
D. If x is odd, then x^{2} is even.

Answer:

Here, let p: x2 is not even & q: x is not even
Thus, p \rightarrow q, whose converse will be –
q \rightarrow p viz., If x is not even, then x^{2} 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 p\rightarrow q?
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 p\rightarrow q.
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

State whether the following sentences are statements or not:
(i) The angles opposite to equal sides of a triangle are equal.
(ii) The moon is a satellite of earth.
(iii) May God bless you!
(vi) Asia is a continent.
(v) How are you?

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.

More About NCERT Exemplar Class 11 Maths Chapter 14

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.

Main Subtopics in NCERT Exemplar Class 11 Maths Solutions Chapter 14 Mathematical Reasoning

· 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

What students will learn from NCERT Exemplar Class 11 Maths Solutions Chapter 14?

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.

NCERT Solutions for Class 11 Maths Chapters

Important Topics to Cover from NCERT Exemplar Class 11 Maths Solutions Chapter 14 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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Frequently Asked Question (FAQs)

1. Does the NCERT Exemplar Solutions for Class 11 Maths Chapter 14 include textbook-based exercises?

Yes, most of the questions solved in NCERT Exemplar Class 11 Maths Chapter 14 Solutions include textbook-based exercises

2. Can the format of the solutions be followed for board examination?

Yes, the format of the solutions can be followed for board examinations.

3. Where can I download the NCERT Exemplar Class 11 Maths Chapter 14 Solutions?

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A block of mass 0.50 kg is moving with a speed of 2.00 ms-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is

Option 1)

0.34\; J

Option 2)

0.16\; J

Option 3)

1.00\; J

Option 4)

0.67\; J

A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times.  Assume that the potential energy lost each time he lowers the mass is dissipated.  How much fat will he use up considering the work done only when the weight is lifted up ?  Fat supplies 3.8×107 J of energy per kg which is converted to mechanical energy with a 20% efficiency rate.  Take g = 9.8 ms−2 :

Option 1)

2.45×10−3 kg

Option 2)

 6.45×10−3 kg

Option 3)

 9.89×10−3 kg

Option 4)

12.89×10−3 kg

 

An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range

Option 1)

2,000 \; J - 5,000\; J

Option 2)

200 \, \, J - 500 \, \, J

Option 3)

2\times 10^{5}J-3\times 10^{5}J

Option 4)

20,000 \, \, J - 50,000 \, \, J

A particle is projected at 600   to the horizontal with a kinetic energy K. The kinetic energy at the highest point

Option 1)

K/2\,

Option 2)

\; K\;

Option 3)

zero\;

Option 4)

K/4

In the reaction,

2Al_{(s)}+6HCL_{(aq)}\rightarrow 2Al^{3+}\, _{(aq)}+6Cl^{-}\, _{(aq)}+3H_{2(g)}

Option 1)

11.2\, L\, H_{2(g)}  at STP  is produced for every mole HCL_{(aq)}  consumed

Option 2)

6L\, HCl_{(aq)}  is consumed for ever 3L\, H_{2(g)}      produced

Option 3)

33.6 L\, H_{2(g)} is produced regardless of temperature and pressure for every mole Al that reacts

Option 4)

67.2\, L\, H_{2(g)} at STP is produced for every mole Al that reacts .

How many moles of magnesium phosphate, Mg_{3}(PO_{4})_{2} will contain 0.25 mole of oxygen atoms?

Option 1)

0.02

Option 2)

3.125 × 10-2

Option 3)

1.25 × 10-2

Option 4)

2.5 × 10-2

If we consider that 1/6, in place of 1/12, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

Option 1)

decrease twice

Option 2)

increase two fold

Option 3)

remain unchanged

Option 4)

be a function of the molecular mass of the substance.

With increase of temperature, which of these changes?

Option 1)

Molality

Option 2)

Weight fraction of solute

Option 3)

Fraction of solute present in water

Option 4)

Mole fraction.

Number of atoms in 558.5 gram Fe (at. wt.of Fe = 55.85 g mol-1) is

Option 1)

twice that in 60 g carbon

Option 2)

6.023 × 1022

Option 3)

half that in 8 g He

Option 4)

558.5 × 6.023 × 1023

A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t2) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m2 , the number of rotations made by the pulley before its direction of motion if reversed, is

Option 1)

less than 3

Option 2)

more than 3 but less than 6

Option 3)

more than 6 but less than 9

Option 4)

more than 9

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Pathologist

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.

5 Jobs Available
Speech Therapist
4 Jobs Available
Gynaecologist

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. 

4 Jobs Available
Oncologist

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.

3 Jobs Available
Audiologist

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.

3 Jobs Available
Healthcare Social Worker

Healthcare social workers help patients to access services and information about health-related issues. He or she assists people with everything from locating medical treatment to assisting with the cost of care to recover from an illness or injury. A career as Healthcare Social Worker requires working with groups of people, individuals, and families in various healthcare settings such as hospitals, mental health clinics, child welfare, schools, human service agencies, nursing homes, private practices, and other healthcare settings.  

2 Jobs Available
Actor

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. 

4 Jobs Available
Acrobat

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.

3 Jobs Available
Video Game Designer

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.

3 Jobs Available
Radio Jockey

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.

3 Jobs Available
Multimedia Specialist

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. 

2 Jobs Available
Producer

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.

2 Jobs Available
Fashion Blogger

Fashion bloggers use multiple social media platforms to recommend or share ideas related to fashion. A fashion blogger is a person who writes about fashion, publishes pictures of outfits, jewellery, accessories. Fashion blogger works as a model, journalist, and a stylist in the fashion industry. In current fashion times, these bloggers have crossed into becoming a star in fashion magazines, commercials, or campaigns. 

2 Jobs Available
Photographer

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.

2 Jobs Available
Copy Writer

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. 

5 Jobs Available
Journalist

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.

3 Jobs Available
Publisher

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.

3 Jobs Available
Vlogger

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. 

3 Jobs Available
Editor

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.

3 Jobs Available
Advertising Manager

Advertising managers consult with the financial department to plan a marketing strategy schedule and cost estimates. We often see advertisements that attract us a lot, not every advertisement is just to promote a business but some of them provide a social message as well. There was an advertisement for a washing machine brand that implies a story that even a man can do household activities. And of course, how could we even forget those jingles which we often sing while working?

2 Jobs Available
Photographer

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.

2 Jobs Available
Social Media Manager

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.

2 Jobs Available
Welding Engineer

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. 

5 Jobs Available
QA Manager
4 Jobs Available
Product Manager

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.  

3 Jobs Available
Quality Controller

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.

3 Jobs Available
Production Manager
3 Jobs Available
Procurement Manager

The procurement Manager is also known as  Purchasing Manager. The role of the Procurement Manager is to source products and services for a company. A Procurement Manager is involved in developing a purchasing strategy, including the company's budget and the supplies as well as the vendors who can provide goods and services to the company. His or her ultimate goal is to bring the right products or services at the right time with cost-effectiveness. 

2 Jobs Available
Process Development Engineer

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.

2 Jobs Available
Merchandiser
2 Jobs Available
AWS Solution Architect

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. 

4 Jobs Available
QA Manager
4 Jobs Available
Azure Administrator

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. 

4 Jobs Available
Information Security Manager

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 

3 Jobs Available
Computer Programmer

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.

3 Jobs Available
Product Manager

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.  

3 Jobs Available
ITSM Manager
3 Jobs Available
.NET Developer

.NET Developer Job Description: A .NET Developer is a professional responsible for producing code using .NET languages. He or she is a software developer who uses the .NET technologies platform to create various applications. Dot NET Developer job comes with the responsibility of  creating, designing and developing applications using .NET languages such as VB and C#. 

2 Jobs Available
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