RD Sharma Solutions For Class 10 Math Chapter 1 Real Numbers Exercise 1.1
Q1. If a and b are two odd positive integers such that a > b, then prove that one of the two numbers \(\frac{a+b}{2} \) and \(\frac{a-b}{2} \) is odd and the other is even.
Solution: We know that any odd positive integer is of the form 4q+1 or, 4q+3 for some whole number q. Now that it’s given a > b
So, we can choose a= 4q+3 and b= 4q+1.
∴ \( \frac{a+b}{2}=\frac{\left[ (4q+3)+(4q+1) \right]}{2} \)
⇒ \(\Rightarrow \frac{a+b}{2}=\frac{8q+4}{2} \)(a+b)/2 = (8q+4)/2 ⇒ (a+b)/2 = 4q+2 = 2(2q+1) which is clearly an even number.
Now, doing (a-b)/2
⇒ \(\Rightarrow \frac{a-b}{2}=\frac{\left[ (4q+3)-(4q+1) \right]}{2} \)
⇒ (a-b)/2 = (4q+3-4q-1)/2
⇒ (a-b)/2 = (2)/2
⇒ (a-b)/2 = 1 which is an odd number.
Hence, one of the two numbers (a+b)/2 and (a-b)/2 is odd and the other is even.
2
CBSE Date Sheet 2025 PDF: Steps to Download Time Table Online @cbse.gov.in
CBSE Date Sheet 2025 Class 10th
Check out the schedule for all subjects below.
| Date and Time |
Subject Code |
Subject Name |
| Saturday, 15th February, 2025 |
||
| 10:30 AM - 01:30 PM |
101 |
English (Communicative) |
| 184 |
English (Language and Literature) |
|
| Monday, 17th February, 2025 |
||
| 10:30 AM - 12:30 PM |
036 |
Hindustani Music (Per Ins) |
| 131 |
Rai |
|
| 132 |
Gurung |
|
| 133 |
Tamang |
|
| 134 |
Sherpa |
|
| 254 |
Elements of Book Keeping & Accountancy |
|
| 418 |
Physical Activity Trainer |
|
| Tuesday, 18th February, 2025 |
||
| 403 |
Security |
|
| 404 |
Automotive |
|
| 405 |
Introduction to Financial Markets |
|
| 406 |
Introduction to Tourism |
|
| 407 |
Beauty & Wellness |
|
| 408 |
Agriculture |
|
| 409 |
Food Production |
|
| 410 |
Front Office Operations |
|
| 411 |
Banking & Insurance |
|
| 412 |
Marketing & Sales |
|
| 414 |
Apparel |
|
| 415 |
Multi-Media |
|
| 416 |
Multi Skill Foundation Course |
|
| 419 |
Data Science |
|
| 420 |
Electronics & Hardware |
|
| 421 |
Foundation Skills for Sciences |
|
| 422 |
Design Thinking & Innovation |
|
| Thursday, 20th February, 2025 |
||
| 10:30 AM - 01:30 PM |
086 |
Science |
| Saturday, 22nd February, 2025 |
||
| 10:30 AM - 01:30 PM |
018 |
French |
| 119 |
Sanskrit (Communicative) |
|
| 122 |
Sanskrit |
|
| Tuesday, 25th February, 2025 |
||
| 10:30 AM - 01:30 PM |
087 |
Social Science |
| Thursday, 27th February, 2025 |
||
| 10:30 AM - 01:30 PM |
003 |
Urdu Course-A |
| 005 |
Bengali |
|
| 006 |
Tamil |
|
| 009 |
Marathi |
|
| 010 |
Gujarati |
|
| 011 |
Manipuri |
|
| 308 |
Urdu Course-B |
|
| Friday, 28th February, 2025 |
||
| 10:30 AM - 01:30 PM |
002 |
Hindi Course-A |
| 008 |
Hindi Course-B |
|
| Saturday, 1st March, 2025 |
||
| 10:30 AM - 12:30 PM |
049 |
Painting |
| Monday, 3rd March, 2025 |
||
| 10:30 AM - 12:30 PM |
413 |
Health Care |
| Wednesday, 5th March, 2025 |
||
| 10:30 AM - 01:30 PM |
154 |
Elements of Business |
| 10:30 AM - 12:30 PM |
401 |
Retail |
| Thursday, 6th March, 2025 |
||
| 10:30 AM - 01:30 PM |
017 |
Tibetan |
| 020 |
German |
|
| 076 |
National Cadet Corps |
|
| 088 |
Bhoti |
|
| 089 |
Telugu - Telangana |
|
| 092 |
Bodo |
|
| 093 |
Tangkhul |
|
| 094 |
Japanese |
|
| 095 |
Bhutia |
|
| 096 |
Spanish |
|
| 097 |
Kashmiri |
|
| 098 |
Mizo |
|
| 099 |
Bahasa Melayu |
|
| Monday, 10th March, 2025 |
||
| 10:30 AM - 01:30 PM |
041 |
Mathematics Standard |
| 241 |
Mathematics Basic |
|
| Wednesday, 12th March, 2025 |
||
| 10:30 AM - 01:30 PM |
007 |
Telugu |
| 016 |
Arabic |
|
| 021 |
Russian |
|
| 023 |
Persian |
|
| 024 |
Nepali |
|
| 025 |
Limboo |
|
| 026 |
Lepcha |
|
| 031 |
Carnatic Music (Vocal) |
|
| 032 |
Carnatic Music (Melodic Instruments) |
|
| 10:30 AM - 12:30 PM |
033 |
Carnatic Music (Percussion Instruments) |
| 034 |
Hindustani Music (Vocal) |
|
| 035 |
Hindustani Music (Melodic Instruments) |
|
| 136 |
Thai |
|
| Thursday, 13th March, 2025 |
||
| 10:30 AM - 01:30 PM |
064 |
Home Science |
| Monday, 17th March, 2025 |
||
| 10:30 AM - 01:30 PM |
012 |
Punjabi |
| 013 |
Sindhi |
|
| 017 |
Malayalam |
|
| 018 |
Odia |
|
| 014 |
Assamese |
|
| 015 |
Kannada |
|
| 091 |
Kokborok |
|
| Tuesday, 18th March, 2025 |
||
| 10:30 AM - 12:30 PM |
165 |
Computer Applications |
| 402 |
Information Technology |
|
| 417 |
Artificial Intelligence | |
Also Read: CBSE Date Sheet 2025: Class XII Time Table PDF Out at cbse.gov.in
CBSE Class 12 Date Sheet 2025
Have a look at the following table for the schedule of all subjects:
| Date and Time |
Subject Code |
Subject Name |
| Saturday, 15th February, 2025 |
||
| 10:30 AM - 01:30 PM |
66 |
Entrepreneurship |
| Monday, 17th February, 2025 |
||
| 10:30 AM - 01:30 PM |
48 |
Physical Education |
| Tuesday, 18th February, 2025 |
||
| 10:30 AM - 12:30 PM |
35 |
Hindustani Music (Melodic Instruments) |
| 36 |
Hindustani Music (Percussion Instruments) |
|
| 821 |
Multi-Media |
|
| 804 |
Automotive |
|
| 813 |
Health Care |
|
| 844 |
Data Science |
|
| 847 |
Electronics & Hardware |
|
| Wednesday, 19th February, 2025 |
||
| 10:30 AM - 01:30 PM |
809 |
Food Production |
| 824 |
Office Procedures & Practices |
|
| 830 |
Design |
|
| 10:30 AM - 12:30 PM |
342 |
Early Childhood Care & Education |
| Thursday, 20th February, 2025 |
||
| 10:30 AM - 01:30 PM |
817 |
Typography & Computer Applications |
| Friday, 21st February, 2025 |
||
| 10:30 AM - 01:30 PM |
42 |
Physics |
| Saturday, 22nd February, 2025 |
||
| 10:30 AM - 01:30 PM |
54 |
Business Studies |
| 833 |
Business Administration |
|
| Monday, 24th February, 2025 |
||
| 10:30 AM - 01:30 PM |
29 |
Geography |
| Tuesday, 25th February, 2025 |
||
| 10:30 AM - 01:30 PM |
118 |
French |
| 822 |
Taxation |
|
| 829 |
Textile Design |
|
| 10:30 AM - 12:30 PM |
843 |
Artificial Intelligence |
| Thursday, 27th February, 2025 |
||
| 10:30 AM - 01:30 PM |
43 |
Chemistry |
| Friday, 28th February, 2025 |
||
| 10:30 AM - 01:30 PM |
805 |
Financial Markets Management |
| 807 |
Beauty & Wellness |
|
| 828 |
Medical Diagnostics |
|
| Saturday, 1st March, 2025 |
||
| 10:30 AM - 01:30 PM |
46 |
Engineering Graphics |
| 10:30 AM - 12:30 PM |
57 |
Bharatanatyam - Dance |
| 58 |
Kuchipudi - Dance |
|
| 59 |
Odissi - Dance |
|
| 60 |
Manipuri - Dance |
|
| 61 |
Kathakali - Dance |
|
| 10:30 AM - 01:30 PM |
816 |
Horticulture |
| 823 |
Cost Accounting |
|
| 836 |
Library & Information Science |
|
| Monday, 3rd March, 2025 |
||
| 10:30 AM - 01:30 PM |
74, 801 |
Legal Studies; Retail |
| Tuesday, 4th March, 2025 |
||
| 10:30 AM - 01:30 PM |
76, 811 |
National Cadet Corps; Banking |
| Wednesday, 5th March, 2025 |
||
| 10:30 AM - 01:30 PM |
808, 812 |
Agriculture; Marketing |
| Thursday, 6th March, 2025 |
||
| 10:30 AM - 01:30 PM |
837 |
Fashion Studies |
| Friday, 7th March, 2025 |
||
| 10:30 AM - 01:30 PM |
835 |
Mass Media Studies |
| 848 |
Design Thinking & Innovation |
|
| Saturday, 8th March, 2025 |
||
| 10:30 AM - 01:30 PM |
41 |
Mathematics |
| 241 |
Applied Mathematics |
|
| Monday, 10th March, 2025 |
||
| 10:30 AM - 12:30 PM |
806 |
Tourism |
| 10:30 AM - 01:30 PM |
827 |
Air-Conditioning & Refrigeration |
| 831 |
Salesmanship |
|
| Tuesday, 11th March, 2025 |
||
| 10:30 AM - 01:30 PM |
1 |
English Elective |
| 301 |
English Core |
|
| Wednesday, 12th March, 2025 |
||
| 10:30 AM - 01:30 PM |
841 |
Yoga |
| 845 |
Physical Activity Trainer |
|
| Thursday, 13th March, 2025 |
||
| 10:30 AM - 01:30 PM |
803 |
Web Application |
| Saturday, 15th March, 2025 |
||
| 10:30 AM - 01:30 PM |
002 |
Hindi Elective |
| 302 |
Hindi Core |
|
| Monday, 17th March, 2025 |
||
| 10:30 AM - 01:30 PM |
003 |
Urdu Elective |
| 022 |
Sanskrit Elective |
|
| 10:30 AM - 12:30 PM |
031 |
Carnatic Music Vocal |
| 032 |
Carnatic Music Melodic Instruments |
|
| 033 |
Carnatic Music Percussion Instruments |
|
| 056 |
Kathak - Dance |
|
| 10:30 AM - 01:30 PM |
303 |
Urdu Core |
| 810 |
Front Office Operations |
|
| 814 |
Insurance |
|
| 818 |
Geospatial Technology |
|
| 819 |
Electrical Technology |
|
| Tuesday, 18th March, 2025 |
||
| 10:30 AM - 12:30 PM |
049 |
Painting |
| 050 |
Graphics |
|
| 051 |
Sculpture |
|
| 052 |
Applied Art (Commercial Art) |
|
| Wednesday, 19th March, 2025 |
||
| 10:30 AM - 01:30 PM |
30 |
Economics |
| Thursday, 20th March, 2025 |
||
| 10:30 AM - 12:30 PM |
034 |
Hindustani Music (Vocal) |
| Friday, 21st March, 2025 |
||
| 10:30 AM - 01:30 PM |
045 |
Biotechnology |
| 073 |
Knowledge Tradition & Practices of India |
|
| 188 |
Bhoti |
|
| 191 |
Kokborok |
|
| 820 |
Electronic Technology |
|
| 825 |
Shorthand (English) |
|
| 826 |
Shorthand (Hindi) |
|
| 834 |
Food Nutrition & Dietetics |
|
| Saturday, 22nd March, 2025 |
||
| 10:30 AM - 01:30 PM |
028 |
Political Science |
| Monday, 24th March, 2025 |
||
| 10:30 AM - 01:30 PM |
322 |
Sanskrit Core |
| Tuesday, 25th March, 2025 |
||
| 10:30 AM - 01:30 PM |
44 |
Biology |
| Wednesday, 26th March, 2025 |
||
| 10:30 AM - 01:30 PM |
55 |
Accountancy |
| Thursday, 27th March, 2025 |
||
| 10:30 AM - 01:30 PM |
39 |
Sociology |
| Saturday, 29th March, 2025 |
||
| 10:30 AM - 01:30 PM |
65 |
Informatics Practices |
| 83 |
Computer Science |
|
| 802 |
Information Technology |
|
| Tuesday, 1st April, 2025 |
||
| 10:30 AM - 01:30 PM |
27 |
History |
| Wednesday, 2nd April, 2025 |
||
| 10:30 AM - 01:30 PM |
104 |
Punjabi |
| 105 |
Bengali |
|
| 106 |
Tamil |
|
| 107 |
Telugu |
|
| 108 |
Sindhi |
|
| 109 |
Marathi |
|
| 110 |
Gujarati |
|
| 111 |
Manipuri |
|
| 112 |
Malayalam |
|
| 113 |
Odia |
|
| 114 |
Assamese |
|
| 115 |
Kannada |
|
| 116 |
Arabic |
|
| 117 |
Tibetan |
|
| 120 |
German |
|
| 121 |
Russian |
|
| 123 |
Persian |
|
| 124 |
Nepali |
|
| 125 |
Limboo |
|
| 126 |
Lepcha |
|
| 189 |
Telugu Telangana |
|
| 192 |
Bodo |
|
| 193 |
Tangkhul |
|
| 194 |
Japanese |
|
| 195 |
Bhutia |
|
| 196 |
Spanish |
|
| 197 |
Kashmiri |
|
| 198 |
Mizo |
|
| Thursday, 3rd April, 2025 |
||
| 10:30 AM - 01:30 PM |
064 |
Home Science |
| Friday, 4th April, 2025 |
||
| 10:30 AM - 01:30 PM |
037 |
Psychology |
Keep checking this page for a detailed subject-wise schedule!
CBSE Exam Date Sheet news
CBSE Date Sheet 2025 Live: The Central Board of Secondary Education, CBSE, on November 20 released the timetable for classes 10 and 12 board exams. Students appearing for the examinations can download the subject-wise date and time for both classes on the official website at cbse.gov.in. Direct link to check CBSE Class 10th, 12th exam 2025 date sheet
The CBSE 10th and 12th exams are scheduled to begin on February 15. The class 10th exam will begin with Communicative English/English Language and Literature. The exam will be held between 10.30 am to 1.30pm. The Class 12th exams will begin with Entrepreneurship between 10.30 am to 1.30pm.
Furthermore, in order to be eligible for the board examinations 2025, it is essential for students to maintain an attendance of 75 per cent or more.
Around 44 lakh students from 8,000 schools in India and abroad are eligible to appear for the Class 10 and 12 board exams this year.
Meanwhile, the Uttar Pradesh Madhyamik Shiksha Parishad (UPMSP) announced the board exam dates 2025.
Both examinations will be held from February 24 to March 12.
CBSE date sheet 2025: Here's how to download
- Visit the official website at cbse.gov.in.
- Open the main website link
- Open the Class 10 or 12 time table PDF, as required
- Download the time table and check exam dates.
Follow this blog for the live updates on CBSE board exam date sheet.
