यूपी बोर्ड कक्षा 12 गणित की बोर्ड परीक्षाएं 3 मार्च, 2025 को आयोजित की जाएंगी। बोर्ड परीक्षा के लिए बस कुछ ही दिन बचे हैं, छात्रों को इस बचे हुए समय का कुशलतापूर्वक उपयोग करना चाहिए। यूपी बोर्ड कक्षा 12 गणित महत्वपूर्ण विषयों 2025 पर ध्यान केंद्रित करने से छात्रों को अच्छे अंक प्राप्त करने में मदद मिलेगी।
UP Board Class 12 Maths Important Topics 2025
UP Board Class 12 Maths Important Topics
Unit 1 – 22 marks
Unit 2 – 08 marks
Unit 3 – 20 marks
Unit 4 – 12 marks
Unit 5 – 12 marks
Unit 6 – 06 marks
Unit 7 – 05 marks
Unit 8 – 15 marks
UPMSP 12th Maths Important Topics Unit 1
The first unit of the UP Board class 12 mathematics syllabus is Algebra. Check the important topics from this unit below. UP Board Class 12 Maths Important Topics 2025 ; 3 March
1 – आव्यूह
संकल्पना, संकेतन, क्रम, समानता, आव्युहों के प्रकार, शून्य आव्यूह, एक आव्यूह का परिवर्त
सममित तथा विषम सममित आव्यूह, आव्यूहों का योग तथा अदिश गुणन योग गुणन तथा अदिश गुणन के सरल गुणधर्म.
व्युत्क्रमणीय आव्यूह तथा व्युत्क्रम की अद्वितीयता, यदि उसका अस्तित्व है (यहाँ सभी आव्युहों के अवयव वास्तविक संख्याएं हैं).
2 – सारणिक
एक वर्ग आव्यूह का सारणिक (3 x 3 के वर्ग आव्यूह तक), सारणिको के गुणधर्म, उपसारणिक तथा सहखण्ड, सारणिकों का अनुप्रयोग त्रिभुज का क्षेत्रफल ज्ञात करने में सहखण्ड आव्यूह का व्युत्क्रम.
दो अथवा तीन चरों में रैखिक समीकरण निकाय को (जिनका अद्वितीय हल हो) के प्रतिलोम का प्रयोग कर हल करना. क्रैमर का नियम तथा इसके अनुप्रयोग.
UPMSP 12th Maths Important Topics Unit 2
प्रतिलोम त्रिकोणमितीय फलनों के आलेख
प्रतिलोम त्रिकोणमितीय फलनों के प्रारम्भिक गुणधर्म
UPMSP 12th Maths Important Topics Unit 3
1 – शंकु परिच्छेद –
शंकु परिच्छेद, दीर्घवृत्त, परिवलय एवं अनिपरवलय.
एक बिंदु, एक सरल रेखा तथा प्रतिच्छेदी रेखाओं का एक युग्म, शंकु परिच्छेद के अपभ्रष्ट रूप में.
वृत्त का मानक समीकरण वृत्त का सामान्य समीकरण
सरल रेखा Y=mx+c के वृत्त, परवलय, दीर्घवृत्त तथा अनिपरवलय की स्पर्श रेखा होने का प्रतिबन्ध.
2 – त्रिकोणमीय ज्यामिती का परिचय –
त्रिविसिय अंतरिक्ष में निर्देश तथा निर्देशांक तल, एक बिंदु के निर्देशांक, दो बिन्दुओं के बीच दुरी तथा खण्ड सूत्र
UPMSP 12th Maths Important Topics Unit 4
1 – अवकलन तथा अवकलनियता–
फलनों के योग, अन्तर, गुणन तथा भाग द्वारा बने फलनों का अवकलन करना. बहुपद फलनों तथा त्रिकोणमितिय फलनों का अवकलन ज्ञात करना .
अवकलनियता , संयुक्त फलनों का अवकलन, श्रृंखला नियम, प्रतिलोम त्रिकोणमितिय फलनों का अवकलन, अस्पष्ट फलनों का अवकलन, चर घातांकी तथा लघुगुणकीय फलनों की संकल्पना तथा उनका अवकलन
2 – अवकलनों के अनुप्रयोग–
अवकलनों के अनुप्रोयग, परिवर्तन की दर वृद्धि / हार्श मान फलन,
अभिलम्ब तथा स्पर्श रेखायें, सन्निकटन, उच्चतम तथा निम्नतम (प्रथम अवकलन परिक्षण की ज्यामितीय प्रेरणा तथा द्वितीय अवकलन परिक्षण उपपति लायक टूल).
UPMSP 12th Maths Important Topics Unit 5
(1) समाकलन
समाकलन, अवकलन के व्युत्क्रम प्रक्रम के रूप में, कई प्रकार के फलनों का समाकलन-प्रतिस्थापन द्वारा, आंशिक भिन्नो द्वारा
योगफल की सीमा के रूप में निश्चित समाकलन, कलन का आधारभुत प्रमेय (बिना उपपति के), निश्चित समाकलनों के मूल गुणधर्म, तथा उनके मान ज्ञात करना .
(2) – समाकलनों के अनुप्रयोग
अनुप्रयोग : साधारण वक्रों के अन्तर्गत क्षेत्रफल ज्ञात करना, विशेषतया रेखायें
वृत/परवलय/दीर्घवृत (केवल मानक रूप में) का क्षेत्रफल.
UPMSP 12th Maths Important Topics Unit 6
अवकल समीकरण
अवकल समीकरण का व्यापक एवं विशिष्ट हल
पृथक्करणीय चर के तरीके द्वारा अवकल समीकरणों का हल
प्रथम कोटि एवं प्रथम घात वाले समघातीय अवकलन समीकरणों का हल
निम्न प्रकार के रैखिक अवकल समीकरणों का हल जहाँ p व q, x के फलन हैं
UPMSP 12th Maths Important Topics Unit 7
सदिश तथा त्रिविमीय ज्यामिति
सदिश तथा अदिश, एक सदिश का परिमाण व दिशा, सदिशों के दिककोसाइन/अनुपात, सदिशों के प्रकार (समान, मात्रक, शून्य, समांतर तथा संरेख सदिश)
किसी बिन्दु का स्थिति, ऋणात्मक सदिश, एक सदिश के घटक, सदिशों का योगफल
एक सदिश का अदिश से गुणन, दो बिन्दुओं को मिलाने वाले रेखाखण्ड को किसी अनुपात में बाँटने वाले बिंदु का स्थिति सदिश, सदिशो, का अदिश गुणनफल.
दो बिन्दुओं को मिलाने वाली रेखा के दिक् कोसाइन/अनुपात, एक रेखा का कार्तीय तथा सदिश समीकरण, समतलीय तथा विषमतलीय रेखायें, दो रेखाओं के बीच की न्यूनतम दुरी. एक तल के कार्तीय तथा सदिश समीकरण,
अगर आपकी वेबसाइट के लिए AdSense का वेरिफिकेशन नहीं हो रहा है, तो यह कुछ सामान्य कारणों में से एक हो सकता है। Google AdSense की वेरिफिकेशन प्रक्रिया में यह सुनिश्चित किया जाता है कि आपकी वेबसाइट उनके दिशानिर्देशों और नीतियों के अनुरूप हो। नीचे संभावित समस्याओं और उनके समाधान का विवरण दिया गया है:
1. वेबसाइट की सामग्री (Content Issues):
Low-Quality Content: अगर वेबसाइट पर कम गुणवत्ता वाली या कॉपी-पेस्ट की गई सामग्री है, तो AdSense इसे अस्वीकार कर सकता है।
Insufficient Content: वेबसाइट पर पर्याप्त मात्रा में सामग्री (जैसे लेख, ब्लॉग) का होना ज़रूरी है। हर पेज पर कम से कम 500-700 शब्द होने चाहिए।
Unoriginal Content: अगर सामग्री कॉपीराइटेड है या किसी अन्य साइट से ली गई है, तो यह AdSense वेरिफिकेशन में समस्या पैदा कर सकता है।
समाधान:मूल, उच्च गुणवत्ता और उपयोगकर्ता के लिए उपयोगी सामग्री तैयार करें।
2. वेबसाइट की डिज़ाइन और नेविगेशन:
Poor Design: वेबसाइट अगर उपयोगकर्ता के लिए समझने में कठिन है या प्रोफेशनल डिज़ाइन नहीं है, तो इसे रिजेक्ट किया जा सकता है।
Bad Navigation: अगर आपकी साइट का नेविगेशन (मेनू, लिंक) सही ढंग से काम नहीं कर रहा है, तो यह समस्या बन सकता है।
समाधान:
वेबसाइट का डिज़ाइन सिंपल और प्रोफेशनल बनाएं।
नेविगेशन को उपयोगकर्ता-अनुकूल बनाएं।
Broken Links को ठीक करें।
3. नीतियों का पालन न करना (Policy Issues):
Privacy Policy और About Us का अभाव: Google AdSense आपकी साइट पर Privacy Policy, About Us, और Contact Us पेज की मांग करता है।
Illegal Content: साइट पर अश्लील, हिंसक, या अवैध सामग्री नहीं होनी चाहिए।
समाधान:
Privacy Policy, Disclaimer, और Terms of Service पेज जोड़ें।
"About Us" और "Contact Us" पेज अनिवार्य रूप से जोड़ें।
4. डोमेन और होस्टिंग की समस्या:
Unverified Domain: अगर आपका डोमेन नया है या Google द्वारा ठीक से वेरिफाई नहीं किया गया है, तो यह अस्वीकार हो सकता है।
Website Loading Issues: वेबसाइट का लोडिंग टाइम बहुत धीमा है तो यह समस्या पैदा कर सकता है।
समाधान:
सुनिश्चित करें कि आपका डोमेन कम से कम 6 महीने पुराना है।
साइट की स्पीड बढ़ाने के लिए CDN (जैसे Cloudflare) का उपयोग करें।
5. ट्रैफ़िक और यूजर इंगेजमेंट:
Low Traffic: अगर आपकी साइट पर पर्याप्त ट्रैफ़िक नहीं है, तो AdSense अप्रूव नहीं होगा।
Spam Traffic: अगर साइट पर नकली या बॉट ट्रैफ़िक है, तो इसे रिजेक्ट किया जा सकता है।
समाधान:
ऑर्गेनिक ट्रैफ़िक बढ़ाने के लिए SEO और सोशल मीडिया का उपयोग करें।
बॉट ट्रैफ़िक से बचने के लिए Analytics में फिल्टर सेट करें।
6. AdSense Code Implementation:
अगर आपने AdSense का कोड ठीक से नहीं लगाया है, तो Google आपकी साइट को वेरिफाई नहीं कर पाएगा।
समाधान:AdSense का कोड अपनी साइट के <head> या <body> टैग में सही से लगाएं।
Google AdSense वेरिफिकेशन में सफलता के लिए सुझाव:
वेबसाइट को मोबाइल-फ्रेंडली बनाएं।
सभी पेज SSL (https://) से सुरक्षित होने चाहिए।
साइट पर कम से कम 20-30 उच्च गुणवत्ता वाले आर्टिकल पोस्ट करें।
नियमित रूप से Content अपडेट करते रहें।
Google Search Console और Analytics से साइट को कनेक्ट करें।
अगर आप इन सभी सुझावों को लागू करते हैं, तो AdSense वेरिफिकेशन मिलने की संभावना बढ़ जाएगी।
Philosophy is one of the most interesting optional subjects, with easy conceptual understanding gaining importance in the UPSC exam. Philosophy is the most chosen subject for optional as it has a minimal syllabus to grasp, and no prior information in this subject is required. The content in this optional is mostly subjective-based. Philosophy is a budding subject that talks about the meaning of life.
WHY TO CHOOSE PHILOSOPHY IN UPSC
It is not only helpful in the UPSC exam; rather, it develops an understanding of life, i.e., life existence, problems, solutions, etc. Philosophy deals with various ideas and beliefs of the people followed to lead a better life. This optional subject is open to aspirants of any background without any fear of understanding the subject. Scoring marks in philosophy is comparatively easier because it does not involve any kind of mugging up, nor does it have too many technical concepts from science.
There is a term known as spirituality, which may be uncommon to many, but that has a base in philosophy, as philosophy helps to understand the worth or meaning of life or one's existence beyond gods and evils. It helps to enhance one’s thinking ability, as aspirants who opt for philosophy can actually relate every subject to the real world. It will also help them in the administrative working of the country by setting their morals and values, which may exist much beyond good or bad things.
WHY TO CHOOSE PHILOSOPHY IN UPSC ?
Philosophy is a subject with primarily practical knowledge. The concepts in philosophy are easy to understand and can be implemented in real life. Philosophy helps to increase the thinking capability of an aspirant, and they have a bent towards practical life more.
They can understand the existence of everything around them. They can relate to different beliefs and ideologies prevailing in the world. The syllabus for this subject is very small and can be easily covered in a time span of a month or two.
An aspirant with a good understanding of the subject can score really well in the exam. The success rate of an aspirant achieving a good rank in the exam also increases, i.e., 8 out of 10 aspirants tend to improve their merit with the help of this optional subject. No previous knowledge is required in philosophy, and aspirants from any background can choose philosophy as their optional. It also has a base in the interview round and GS paper IV, i.e., ethics, integrity, and aptitude.
Philosophy also helps in easy writing and provides a broader perspective on many issues taking place in society. It helps to understand the function of human ideas, countries, and societies of various groups, which helps an aspirant in fieldwork after the clearance of the exam.
PROS AND CONS OF PHILOSOPHY OPTIONAL
Pros for philosophy optional
Philosophy optional has a minimal syllabus to prepare compared to other optional subjects in UPSC, which helps to devote extra time to other subjects in which one may require extra effort or time to prepare.
It can be easily prepared in the time span of a month or two on average, and even if one’s grasping or understanding skill is weak, this subject will take a maximum of 3 months for preparation with enough knowledge required for achieving excellent results in this subject.
The syllabus is also well-defined, meaning one doesn't need to go beyond the given syllabus to score better in exams.
Since philosophy generally deals with the meaning of life, ideas, and beliefs, which are expected to be the same throughout the years, hardly any news article will be found in the newspaper on the topic of philosophy. This simply means philosophy is a static subject with hardly any dynamic part to prepare, reducing the burden on an aspirant for creating a notes bank of current affairs.
In case an aspirant plans to take a second or third attempt in philosophy, then he may just require to revise what he has already learned and save his time for other subjects and revisions because the syllabus or concepts of philosophy do not change for years.
No prior information about the subject is required. An aspirant from any background can choose philosophy as an optional subject and can easily increase their chance of scoring better for getting selected in the UPSC exam compared to aspirants from other optional subjects.
Philosophy is also based in GS paper IV, i.e., ethics, integrity, and aptitude. Questions on philosophy can also be asked in the interview round, as interviewers try to check an aspirant's ideas and beliefs on various issues taking place in society.
It can also be useful in essay writing. One can write quotations, case studies, ideas, etc., with the real world in their essay to give strength to their essay writing, which helps to score better marks in essays that hold 250 marks.
It also helps in answer writing for other subjects as well, such as history, social issues, Indian polity, etc., as an aspirant develops rational thinking about each topic in detail. It also helps to understand the working of the country and social issues in depth, which helps in the answer formation of such subjects as well. This means philosophy will be helpful throughout the journey of the UPSC exam.
Cons for philosophy optional
Since the syllabus for philosophy optional is very limited, in order to maintain a standard for the UPSC exam, the level of questions being asked for this subject has been increased over the years to maintain the difficulty level of the exam.
It may be difficult to gain marks in this optional subject if one’s skills are not good enough. To score each mark in the exam, one has to write extra but relatable content for the question being asked. As philosophy is quite a deep subject, there is not a specific amount of words given for writing an answer.
Philosophy involves good understanding skills and interest in the subject because philosophy is completely concept-based and can't be memorized like any other subject. One needs to have an in-depth knowledge of each topic given in the syllabus for philosophy subjects to excel in this exam.
Marks in philosophy subjects are mostly in the hands of the examiner, as it is a subjective-based subject, and the ideas or beliefs of an aspirant may vary from those of an examiner. Then he may deduct marks based on his personal understanding of the answer of an aspirant.
EXPERT STRATEGIES FOR PHILOSOPHY OPTIONAL
Though philosophy optional has less syllabus to prepare compared to other optional subjects, it involves a lot of strategic study as the level of the exam does not go down because of the lesser syllabus. Rather, the level is maintained by increasing the complexity of questions asked in the UPSC exam. For such strategies, one doesn't need to refer to many videos on the web; rather, they can simply refer to the content here as all the web content is compressed in an easy and understandable way for the aspirants. This way, they can save time watching various videos before planning their studies for a good score in their exam or to excel in UPSC. Some of the strategies are given below:
Give one or two readings to each general book and start preparing handwritten notes from those books because philosophy as a UPSC optional has less syllabus, and books referred for the subject generally have extra content. It is suggested to write down notes that are part of the given syllabus to avoid wasting time on content irrelevant for philosophy optional. This allows focus on the required content to strengthen their base in the subject.
It is suggested to finish note-making for optional before prelims because the time between prelims and mains is very limited and the most crucial part of the preparation. The time available for preparations for mains is less than the required time. So, time management is very important, and if the handwritten notes are available, one can save and invest the entire time in revision for better results.
After note-making, an aspirant is suggested to attempt questions from various previous year question papers as in philosophy quite a few questions can be repeated in the exam due to the type of syllabus, which is more of a static part. Though a few questions can be repeated, the major portion of the question paper is tricky and unexpected in optional exams to maintain the level of UPSC exams. Attempting previous years' question papers also helps in understanding the twist that UPSC tries to give in the questions and also helps in understanding the time required for attempting each question.
Plus, in optional papers, there are choices available in various questions to attempt, and most aspirants waste their time in making the choice of the question. So, here attempting previous year papers will be of great help to the aspirants as they will be able to understand their strong and weak parts in the syllabus and can also reduce their time in making choices of the question to be attempted.
Since the optional exam carries maximum marks in the UPSC board, i.e., 500 marks in total, with proper strategy and planning, one can easily increase their chance of being selected in the exam by performing better in the optional subject. Since the optional subject is chosen by the aspirant based on their interest, the level is also increased, keeping that factor in mind. Hence, while preparing for optional, one must always keep notes of every possible case study that can be used in answer writing to give strength to an answer for scoring better in the exam.
In philosophy, there are an infinite number of philosophers who have given their study, and the books recommended in philosophy optional are generally used for graduation and post-graduation purposes in philosophy. So, these books carry a maximum number of philosophies given by various philosophers, which may not be required for the UPSC exam as well. Hence, it is suggested that one must develop mind maps, tables, quick notes, etc., of all the philosophies by philosophers required for the UPSC exam alongside mentioning all the years in which they provided philosophical studies, and these must mainly include Indian philosophers.
After preparing all the handwritten notes, mind maps, case study bank, etc., one must focus and work on the structuring of an answer. This will help in the timely completion of the paper with proper presentation of answers without missing even a single point in the answer. It also helps to increase the speed of recalling the body of the answer without wasting any time.
BOOKS TO BE REFERRED
Western Philosophy
A History of Philosophy by Frank Thilly
A Critical History of Western Philosophy by Y Masih
Indian Philosophy
An Introduction to Indian Philosophy by S Chatterjee
A Critical Survey of Indian Philosophy by C Sharma
Social Political Philosophy
An Introduction to Political Theory by O P Gauba
Philosophy of Religion
Introduction to Religious Philosophy by Y Masih
Philosophy of Religion by John H. Hick
ANSWER WRITING SKILLS
Answer writing plays an important role in the mains exam. Since the syllabus in philosophy is less, an aspirant is suggested to give even more extra focus on answer writing because an examiner can check your skills and knowledge through your answer sheet. To present an ideal answer in your exam, here we will discuss some tips and tricks for answer writing that an aspirant must inherit to score better marks in the exam and stand apart from other aspirants. As it is considered that scoring marks in philosophy optional is a bit difficult due to the increased level of questions being asked in the exam, one must not give any chance to an examiner for deducting the marks. Hence, for better performance in the mains optional subject, follow the given tips and excel in your exam. The tips to be followed are:
Layout to be followed for an answer in the exam must be decided during preparations itself. In the exam, each answer must be written in a proper format, i.e., flowchart > introduction > body > conclusion. Or there can be any other layout in which an aspirant would like to answer during the exam. It helps to present an answer sheet in an organized manner, enhancing the quality of answers, scoring better marks, etc. But a layout must be decided during the time of answer writing practice because while writing answers in the exam, an aspirant may not have enough time to focus on answer layout.
Make choices of questions wisely. There are enough choices available in the philosophy optional paper for aspirants. So choosing the appropriate question for which they must have enough content to write is a real task. It can also affect an aspirant’s result as well. If the choice of the question is correct, then an aspirant can write an appreciable answer for the same and can come out with flying colors. For that, it is suggested that an aspirant must have attempted enough previous year papers during the time of preparations so that they know their strong and weak points in the syllabus. And they can choose questions from their strong part of the syllabus.
Writing each answer in sequence and with proper numbering. It is generally suggested to attempt all the questions in a given sequence so that when an examiner is checking your answer sheet, he may not miss any of the answers because of lack of continuity in the answer, which may increase his frustration level, and he may not be willing to check your answer sheet happily and may deduct marks at various points. Or he may miss some questions unknowingly, leading to a complete loss of marks there.
Try to write answers in pointers more than paragraphs. Many times aspirants tend to write big paragraphs for an answer, which can be a big reason for scoring fewer marks in the exam. Because an examiner has thousands of answer sheets to check, hence having very little time to devote to each answer sheet. But paragraph reading requires enough time, due to which an examiner might lose interest in reading one's answer and can give fewer marks for the same. So it is suggested to write an answer in brief pointers with specified headings for each pointer, which will help in the presentation of the answer sheet. Writing main headings for each pointer will help to score more and more questions because sometimes if an aspirant lacks an explanation of the pointer but their heading is to the point, an examiner can avoid reading explanations and provide marks on the basis of the heading itself, assuming that the aspirant might be well aware of the content.
Use flow charts or mind maps. Using flow charts and mind maps helps both aspirant and examiner. An aspirant can easily recall content for the answer with the help of flow charts and mind maps, which helps to save a lot of time during paper writing and make an answer presentable. Flow charts also help to highlight the key concepts tried to explain in the answer with detailed information. It also helps the examiner while checking an answer sheet. Since they have to check a huge bundle of answer sheets in a limited time span, they can't read the answer word for word. Hence they can refer to the flow chart to know if an aspirant is clear in his concept. It also makes the checking process easier for an examiner, which can lighten their mood while checking answers, and they can give more marks to an aspirant.
Try to quote the exact statement given by the philosopher. Philosophy contains a lot of definitions and statements made by various philosophers in the field. And they should be exact; one cannot use their own language in such exams. Technical language given in the syllabus plays a very important role in answer writing. It's said that the answer will be checked on the basis of quantity in such competitive exams rather than its quality, which matters more. So to score more marks, be accurate in the statements used in the exam.
Keep your language simple and understandable. It is not important to use high-level English or Hindi while writing an exam; rather, it is always suggested to use basic vocabulary or phrases when writing an answer. Using extraordinary words or statements in an answer can irritate the examiner while they are checking the answer sheet, as they have to check a huge bundle of answer sheets. Also, UPSC is available for aspirants from various backgrounds, so they check your knowledge in the field and not in complicated words or statements. Additionally, it is not necessary that the examiner knows those high-level vocabulary words, which may seem challenging to them and can bother their self-respect if they fail to understand them. They may deduct your marks for challenging their knowledge.
Differentiate headers with the explanations by using black and blue pens. It is suggested that the main headers used in answer writing must be differentiated from the explanations for those headers. Because while checking an answer sheet, the examiner might be in a hurry to complete the bundle of answer sheets available to them, so they might not go into the details of an answer and might simply refer to the main header with one or two explanatory lines for the same. So, by using black and blue pens in the exam, the header becomes bold and can easily catch the attention of an examiner while checking the answer sheet. Scoring good marks in an exam is a need of an aspirant, and it's not the need of the examiner to give good marks to the aspirant; hence, the aspirants have to fetch marks by keeping this point in mind and presenting their answer with both quality and quantity, leaving no scope for an examiner to deduct marks.
