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NCERT Solutions for Exercise 1.2 Class 12 Maths Chapter 1 - Relations and Functions

NCERT Solutions for Exercise 1.2 Class 12 Maths Chapter 1 - Relations and Functions

Edited By Ramraj Saini | Updated on Dec 03, 2023 01:18 PM IST | #CBSE Class 12th

NCERT Solutions For Class 12 Maths Chapter 1 Exercise 1.2

NCERT Solutions for Exercise 1.2 Class 12 Maths Chapter 1 Relations and Functions are discussed here. These NCERT solutions are created by subject matter expert at Careers360 considering the latest syllabus and pattern of CBSE 2023-24. NCERT Solutions for Class 12 Maths chapter 1 exercise 1.2 is the most important exercise of chapter Relations and Functions and it includes topics like types of relations, functions, binary operations etc. Exercise 1.2 Class 12 Maths exposes students to questions like proving one to one functions etc. Such questions are generally asked in Board examinations. Solving NCERT syllabus for Class 12 Maths chapter 1 exercise 1.2 is recommended to students to score well in CBSE Class 12 board exam. The contribution of chapter Relations and Functions is high in competitive exams also like JEE main and NEET. There are many questions asked in previous years which are based on concepts of Class 12 Maths chapter 1 exercise 1.2.

This Story also Contains
  1. NCERT Solutions For Class 12 Maths Chapter 1 Exercise 1.2
  2. Assess NCERT Solutions for Class 12 Maths Chapter 1 Exercise 1.2
  3. NCERT Solutions For Class 12 Maths Chapter 1 Relations And Functions: Exercise 1.2
  4. More About NCERT Solutions for Class 12 Maths Chapter 1 Exercise 1.2
  5. Benefits of NCERT Solutions for Class 12 Maths Chapter 1 Exercise 1.2
  6. Key Features Of NCERT Solutions for Exercise 1.2 Class 12 Maths Chapter 1

12th class Maths exercise 1.2 answers are designed as per the students demand covering comprehensive, step by step solutions of every problem. Practice these questions and answers to command the concepts, boost confidence and in depth understanding of concepts. Students can find all NCERT exercise together using the link provided below.

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Assess NCERT Solutions for Class 12 Maths Chapter 1 Exercise 1.2

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NCERT Solutions For Class 12 Maths Chapter 1 Relations And Functions: Exercise 1.2

Question:1 Show that the function f: R_* \longrightarrow R_{*} defined by f(x) = \frac{1}{x} is one-one and onto,where R ∗ is the set of all non-zero real numbers. Is the result true, if the domain R ∗ is replaced by N with co-domain being same as R ∗ ?

Answer:

Given, f: R_* \longrightarrow R_{*} is defined by f(x) = \frac{1}{x} .

One - One :

f(x)=f(y)

\frac{1}{x}=\frac{1}{y}

x=y

\therefore f is one-one.

Onto:

We have y \in R_* , then there exists x=\frac{1}{y} \in R_* ( Here y\neq 0 ) such that

f(x)= \frac{1}{(\frac{1}{y})} = y

\therefore f is \, \, onto .

Hence, the function is one-one and onto.

If the domain R ∗ is replaced by N with co-domain being same as R ∗ i.e. g: N \longrightarrow R_{*} defined by

g(x)=\frac{1}{x}

g(x_1)=g(x_2)

\frac{1}{x_1}=\frac{1}{x_2}

x_1=x_2

\therefore g is one-one.

For 1.5 \in R_* ,

g(x) = \frac{1}{1.5} but there does not exists any x in N.

Hence, function g is one-one but not onto.

Question:2(i) Check the injectivity and surjectivity of the following functions:

(i) f : N\rightarrow N given by f(x) = x^2

Answer:

f : N\rightarrow N

f(x) = x^2

One- one:

x,y \in N then f(x)=f(y)

x^{2}=y^{2}

x=y

\therefore f is one- one i.e. injective.

For 3 \in N there is no x in N such that f(x)=x^{2}=3

\therefore f is not onto i.e. not surjective.

Hence, f is injective but not surjective.

Question:2(ii) Check the injectivity and surjectivity of the following functions:

(ii) f : Z \rightarrow Z given by f(x) = x^2

Answer:

f : Z \rightarrow Z

f(x) = x^2

One- one:

For -1,1 \in Z then f(x) = x^2

f(-1)= (-1)^{2}

f(-1)= 1 but -1 \neq 1

\therefore f is not one- one i.e. not injective.

For -3 \in Z there is no x in Z such that f(x)=x^{2}= -3

\therefore f is not onto i.e. not surjective.

Hence, f is neither injective nor surjective.

Question:2(iii) Check the injectivity and surjectivity of the following functions:

(iii) f: R \rightarrow R given by f(x) = x^2

Answer:

f: R \rightarrow R

f(x) = x^2

One- one:

For -1,1 \in R then f(x) = x^2

f(-1)= (-1)^{2}

f(-1)= 1 but -1 \neq 1

\therefore f is not one- one i.e. not injective.

For -3 \in R there is no x in R such that f(x)=x^{2}= -3

\therefore f is not onto i.e. not surjective.

Hence, f is not injective and not surjective.

Question:2(iv) Check the injectivity and surjectivity of the following functions:

(iv) f: N \rightarrow N given by f(x) = x^3

Answer:

f : N\rightarrow N

f(x) = x^3

One- one:

x,y \in N then f(x)=f(y)

x^{3}=y^{3}

x=y

\therefore f is one- one i.e. injective.

For 3 \in N there is no x in N such that f(x)=x^{3}=3

\therefore f is not onto i.e. not surjective.

Hence, f is injective but not surjective.

Question:2(v) Check the injectivity and surjectivity of the following functions:

(v) f : Z \rightarrow Z given by f(x) = x^3

Answer:

f : Z \rightarrow Z

f(x) = x^3

One- one:

For (x,y) \in Z then f(x) = f(y)

x^{3}=y^{3}

x=y

\therefore f is one- one i.e. injective.

For 3 \in Z there is no x in Z such that f(x)=x^{3}= 3

\therefore f is not onto i.e. not surjective.

Hence, f is injective but not surjective.

Question:3 Prove that the Greatest Integer Function f : R\longrightarrow R , given by f (x) = [x] , is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x .

Answer:

f : R\longrightarrow R

f (x) = [x]

One- one:

For 1.5,1.7 \in R then f(1.5)=\left [ 1.5 \right ] = 1 and f(1.7)=\left [ 1.7 \right ] = 1

but 1.5\neq 1.7

\therefore f is not one- one i.e. not injective.

For 0.6 \in R there is no x in R such that f(x)=\left [ 0.6 \right ]

\therefore f is not onto i.e. not surjective.

Hence, f is not injective but not surjective.

Question:4 Show that the Modulus Function f : R → R, given by f (x) = | x | , is neither one-one nor onto, where | x | is x, if x is positive or 0 and | x | is - x , if x is negative.

Answer:

f : R \rightarrow R

f (x) = | x |

f (x) = | x | = x \, if\, x\geq 0 \,\, and \, \, -x\, if\, x< 0

One- one:

For -1,1 \in R then f (-1) = | -1 |= 1

f (1) = | 1 |= 1

-1\neq 1

\therefore f is not one- one i.e. not injective.

For -2 \in R ,

We know f (x) = | x | is always positive there is no x in R such that f (x) = | x |=-2

\therefore f is not onto i.e. not surjective.

Hence, f (x) = | x | , is neither one-one nor onto.

Question:5 Show that the Signum Function f : R \rightarrow R , given by

f (x) = \left\{\begin{matrix} 1 & if\;x>0 \\ 0& if\;x=0 \\ -1& if\;x<0 \end{matrix}\right. is neither one-one nor onto.

Answer:

f : R \rightarrow R is given by

f (x) = \left\{\begin{matrix} 1 & if\;x>0 \\ 0& if\;x=0 \\ -1& if\;x<0 \end{matrix}\right.

As we can see f(1)=f(2)=1 , but 1\neq 2

So it is not one-one.

Now, f(x) takes only 3 values (1,0,-1) for the element -3 in codomain R ,there does not exists x in domain R such that f(x)= -3 .

So it is not onto.

Hence, signum function is neither one-one nor onto.

Question:6 Let A = \{1, 2, 3\} , B = \{4, 5, 6, 7\} and let f = \{(1, 4), (2, 5), (3, 6)\} be a function from A to B. Show that f is one-one.

Answer:

A = \{1, 2, 3\}

B = \{4, 5, 6, 7\}

f = \{(1, 4), (2, 5), (3, 6)\}

f : A \rightarrow B

\therefore f(1)=4,f(2)=5,f(3)=6

Every element of A has a distant value in f.

Hence, it is one-one.

Question:7(i) In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.

(i) f: R\rightarrow R defined by f(x) = 3 -4x

Answer:

f: R\rightarrow R

f(x) = 3 -4x

Let there be (a,b) \in R such that f(a)=f(b)

3-4a = 3 -4b

-4a = -4b

a = b

\therefore f is one-one.

Let there be y \in R , y = 3 -4x

x = \frac{(3-y)}{4}

f(x) = 3 -4x

Puting value of x, f(\frac{3-y}{4}) = 3 - 4(\frac{3-y}{4})

f(\frac{3-y}{4}) = y

\therefore f is onto.

f is both one-one and onto hence, f is bijective.

Question:7(ii) In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.

(ii) f : R\rightarrow R defined by f(x) = 1 + x^2

Answer:

f : R\rightarrow R

f(x) = 1 + x^2

Let there be (a,b) \in R such that f(a)=f(b)

1+a^{2} = 1 +b^{2}

a^{2}=b^{2}

a = \pm b

For f(1)=f(-1)=2 and 1\neq -1

\therefore f is not one-one.

Let there be -2 \in R (-2 in codomain of R)

f(x) = 1 + x^2 = -2

There does not exists any x in domain R such that f(x) = -2

\therefore f is not onto.

Hence, f is neither one-one nor onto.

Question:8 Let A and B be sets. Show that f : A \times B \rightarrow B \times A such that f (a, b) = (b, a) is
bijective function.

Answer:

f : A \times B \rightarrow B \times A

f (a, b) = (b, a)

Let (a_1,b_1),(a_2,b_2) \in A\times B

such that f (a_1, b_1) = f(a_2, b_2)

(b_1,a_1)=(b_2,a_2)

\Rightarrow b_1= b_2 and a_1= a_2

\Rightarrow (a_1,b_1) = (a_2,b_2)

\therefore f is one- one

Let, (b,a) \in B\times A

then there exists (a,b) \in A\times B such that f (a, b) = (b, a)

\therefore f is onto.

Hence, it is bijective.

Question:9 Let f : N \rightarrow N be defined by f(n) = \left\{\begin{matrix} \frac{n+1}{2} & if\;n\;is\;odd \\ \frac{n}{2} & if\;n\;is\;even \end{matrix}\right. for all n\in N . State whether the function f is bijective. Justify your answer.

Answer:

f : N \rightarrow N , n\in N

f(n) = \left\{\begin{matrix} \frac{n+1}{2} & if\;n\;is\;odd \\ \frac{n}{2} & if\;n\;is\;evem \end{matrix}\right.

Here we can observe,

f(2)=\frac{2}{2}=1 and f(1)=\frac{1+1}{2}=1

As we can see f(1)=f(2)=1 but 1\neq 2

\therefore f is not one-one.

Let, n\in N (N=co-domain)

case1 n be even

For r \in N , n=2r

then there is 4r \in N such that f(4r)=\frac{4r}{2}=2r

case2 n be odd

For r \in N , n=2r+1

then there is 4r+1 \in N such that f(4r+1)=\frac{4r+1+1}{2}=2r +1

\therefore f is onto.

f is not one-one but onto

hence, the function f is not bijective.

Question:10 Let A = R - \{3\} and B = R - \{1\} . Consider the function f : A\rightarrow B defined by f(x) = \left (\frac{x-2}{x-3} \right ) . Is f one-one and onto? Justify your answer.

Answer:

A = R - \{3\}

B = R - \{1\}

f : A\rightarrow B

f(x) = \left (\frac{x-2}{x-3} \right )

Let a,b \in A such that f(a)=f(b)

\left (\frac{a-2}{a-3} \right ) = \left ( \frac{b-2}{b-3} \right )

(a-2)(b-3)=(b-2)(a-3)

ab-3a-2b+6=ab-2a-3b+6

-3a-2b=-2a-3b

3a+2b= 2a+3b

3a-2a= 3b-2b

a=b

\therefore f is one-one.

Let, b \in B = R - \{1\} then b\neq 1

a \in A such that f(a)=b

\left (\frac{a-2}{a-3} \right ) =b

(a-2)=(a-3)b

a-2 = ab -3b

a-ab = 2 -3b

a(1-b) = 2 -3b

a= \frac{2-3b}{1-b}\, \, \, \, \in A

For any b \in B there exists a= \frac{2-3b}{1-b}\, \, \, \, \in A such that

f(\frac{2-3b}{1-b}) = \frac{\frac{2-3b}{1-b}-2}{\frac{2-3b}{1-b}-3}

=\frac{2-3b-2+2b}{2-3b-3+3b}

=\frac{-3b+2b}{2-3}

= b

\therefore f is onto

Hence, the function is one-one and onto.

Question:11 Let f : R \rightarrow R be defined as f(x) = x^4 . Choose the correct answer.

(A) f is one-one onto

(B) f is many-one onto

(C) f is one-one but not onto

(D) f is neither one-one nor onto.

Answer:

f : R \rightarrow R

f(x) = x^4

One- one:

For a,b \in R then f(a) = f(b)

a^{4}=b^{4}

a=\pm b

\therefore f(a)=f(b) does not imply that a=b

example: and 2\neq -2

\therefore f is not one- one

For 2\in R there is no x in R such that f(x)=x^{4}= 2

\therefore f is not onto.

Hence, f is neither one-one nor onto.

Option D is correct.

Question:12 Let f : R\rightarrow R be defined as f(x) = 3x . Choose the correct answer.

(A) f is one-one onto

(B) f is many-one onto

(C) f is one-one but not onto

(D) f is neither one-one nor onto.

Answer:

f : R\rightarrow R

f(x) = 3x

One - One :

Let \left ( x,y \right ) \in R

f(x)=f(y)

3x=3y

x=y

\therefore f is one-one.

Onto:

We have y \in R , then there exists x=\frac{y}{3} \in R such that

f(\frac{y}{3})= 3\times \frac{y}{3} = y

\therefore f is \, \, onto .

Hence, the function is one-one and onto.

The correct answer is A .

More About NCERT Solutions for Class 12 Maths Chapter 1 Exercise 1.2

The NCERT Class 12 Maths chapter Relations and Functions has a total of 5 exercises including miscellaneous. Exercise 1.2 Class 12 Maths covers solutions to 12 main questions and their sub-questions. Most of the questions are related to proving a function one to one. Hence NCERT Solutions for Class 12 Maths chapter 1 exercise 1.2 can be referred for learning the concepts related to proof etc.

Also Read| NCERT Notes For Class 12 Mathematics Chapter 1

Benefits of NCERT Solutions for Class 12 Maths Chapter 1 Exercise 1.2

  • The Class 12th Maths chapter 1 exercise provided here is solved by subject matter experts having rich experience in the domain of competitive exam preparation.
  • Students can practice Exercise 1.2 Class 12 Maths to prepare various concepts like signus functions, one to one functions etc, many direct questions are asked in Board exams from this chapter.
  • These Class 12 Maths chapter 1 exercise 1.2 solutions can be referred by students to revise before the exam and clarify any doubt regarding solution of exercise questions.
  • NCERT Solutions for Class 12 Maths chapter 1 exercise 1.2 provided here are most recommended solutions for students aspiring to score well in examinations.
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Key Features Of NCERT Solutions for Exercise 1.2 Class 12 Maths Chapter 1

  • Comprehensive Coverage: The solutions encompass all the topics covered in ex 1.2 class 12, ensuring a thorough understanding of the concepts.
  • Step-by-Step Solutions: In this class 12 maths ex 1.2, each problem is solved systematically, providing a stepwise approach to aid in better comprehension for students.
  • Accuracy and Clarity: Solutions for class 12 ex 1.2 are presented accurately and concisely, using simple language to help students grasp the concepts easily.
  • Conceptual Clarity: In this 12th class maths exercise 1.2 answers, emphasis is placed on conceptual clarity, providing explanations that assist students in understanding the underlying principles behind each problem.
  • Inclusive Approach: Solutions for ex 1.2 class 12 cater to different learning styles and abilities, ensuring that students of various levels can grasp the concepts effectively.
  • Relevance to Curriculum: The solutions for class 12 maths ex 1.2 align closely with the NCERT curriculum, ensuring that students are prepared in line with the prescribed syllabus.

Also see-

NCERT Solutions Subject Wise

Subject wise NCERT Exemplar solutions

Happy learning!!!

Frequently Asked Questions (FAQs)

1. Which concepts are covered in Exercise 1.2 Class 12 Maths?

Concepts related to one to one functions, reflexive functions etc, are discussed in the Exercise 1.2 Class 12 Maths

2. What is a set?

In Mathematics, A set is a collection of distinct or well-defined numbers or elements

3. How much weightage is given to chapter NCERT book chapter Relations and Functions for the CBSE board exam ?

Weightage of the chapters 'relation and function' is around 5 % weightage in the CBSE final board exam.

4. What are the various ways of representing a set?

There are 3 ways to represent a set:

a. Statement form.

b. Roaster form .

c. Set Builder form.

5. What is the meaning of the empty set in Class 12 Maths?

 A set with no elements is called an empty set. Also known by Null set or void set.

6. What is the meaning of Relation discussed in Exercise 1.2 Class 12 Maths

A relation is the set of ordered pair numbers.

7. How many questions are covered in Exercise 1.2 Class 12 Maths ?

12 questions  are there  in Exercise 1.2 Class 12 Maths

8. What is the total number of exercises there are in the NCERT Class 12 Maths chapter 1 Relations and Functions ?

5 exercises are there including a miscellaneous exercise in the NCERT class 12 maths chapter 1.

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quartely question paper for mathematics cbse

Hello there! Thanks for reaching out to us at Careers360.

Ah, you're looking for CBSE quarterly question papers for mathematics, right? Those can be super helpful for exam prep.

Unfortunately, CBSE doesn't officially release quarterly papers - they mainly put out sample papers and previous years' board exam papers. But don't worry, there are still some good options to help you practice!

Have you checked out the CBSE sample papers on their official website? Those are usually pretty close to the actual exam format. You could also look into previous years' board exam papers - they're great for getting a feel for the types of questions that might come up.

If you're after more practice material, some textbook publishers release their own mock papers which can be useful too.

Let me know if you need any other tips for your math prep. Good luck with your studies!

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Jefferson 25 September,2024
i have taken admission in clg after my campartment exam in chemistry along with the neet 2025 preperation but unfortunately i again got campartment after doing all this preparation i got 15 marks only now I can't able to think what should I do next.

It's understandable to feel disheartened after facing a compartment exam, especially when you've invested significant effort. However, it's important to remember that setbacks are a part of life, and they can be opportunities for growth.

Possible steps:

  1. Re-evaluate Your Study Strategies:

    • Identify Weak Areas: Pinpoint the specific topics or concepts that caused difficulties.
    • Seek Clarification: Reach out to teachers, tutors, or online resources for additional explanations.
    • Practice Regularly: Consistent practice is key to mastering chemistry.

  2. Consider Professional Help:

    • Tutoring: A tutor can provide personalized guidance and support.
    • Counseling: If you're feeling overwhelmed or unsure about your path, counseling can help.

  3. Explore Alternative Options:

    • Retake the Exam: If you're confident in your ability to improve, consider retaking the chemistry compartment exam.
    • Change Course: If you're not interested in pursuing chemistry further, explore other academic options that align with your interests.

  4. Focus on NEET 2025 Preparation:

    • Stay Dedicated: Continue your NEET preparation with renewed determination.
    • Utilize Resources: Make use of study materials, online courses, and mock tests.

  5. Seek Support:

    • Talk to Friends and Family: Sharing your feelings can provide comfort and encouragement.
    • Join Study Groups: Collaborating with peers can create a supportive learning environment.

Remember: This is a temporary setback. With the right approach and perseverance, you can overcome this challenge and achieve your goals.

I hope this information helps you.







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Kanishka kaushikii 17 September,2024
i scored 84 percent in cbse 2 years ago.Am i eligible for medhavi scholarship if give 12th again after 2 drop years and will got 75+ in mpboard

Hi,

Qualifications:
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Get Scholarship: Following a successful verification process, the scholarship will be given. You need to have at least passed the 10th grade/matriculation scholarship amount will be transferred directly to your bank account.

Scholarship Details:

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Scholarships can range from Rs. 2,000 to Rs. 18,000 per month, depending on the marks obtained and the type of scholarship exam (SAKSHAM, SWABHIMAN, SAMADHAN, etc.).

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Hope you find this useful!

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Yuvan 30 August,2024
i upload 12th board result screenshot in ncweb form from the cbse site so it is valid

hello mahima,

If you have uploaded screenshot of your 12th board result taken from CBSE official website,there won,t be a problem with that.If the screenshot that you have uploaded is clear and legible. It should display your name, roll number, marks obtained, and any other relevant details in a readable forma.ALSO, the screenshot clearly show it is from the official CBSE results portal.

hope this helps.

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Ashvadha 12 July,2024
Kya mujhe class 12 ka important question milega jo exam mai per year puche jate hai

Hello Akash,

If you are looking for important questions of class 12th then I would like to suggest you to go with previous year questions of that particular board. You can go with last 5-10 years of PYQs so and after going through all the questions you will have a clear idea about the type and level of questions that are being asked and it will help you to boost your class 12th board preparation.

You can get the Previous Year Questions (PYQs) on the official website of the respective board.

I hope this answer helps you. If you have more queries then feel free to share your questions with us we will be happy to assist you.

Thank you and wishing you all the best for your bright future.

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kumardurgesh1802 10 July,2024
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 Option 4)

20,000 \, \, J - 50,000 \, \, J
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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
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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 .
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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
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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.
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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.
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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
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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
Read More

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October 23, 2024 - 04:01 PM IST :
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