{"id":292680,"date":"2025-11-17T14:08:25","date_gmt":"2025-11-17T08:38:25","guid":{"rendered":"https:\/\/www.aakash.ac.in\/blog\/?p=292680"},"modified":"2025-11-17T14:12:16","modified_gmt":"2025-11-17T08:42:16","slug":"rmo-2025-answer-key-out-by-aakash-download-pdf-exam-analysis-result-updates","status":"publish","type":"post","link":"https:\/\/www.aakash.ac.in\/blog\/rmo-2025-answer-key-out-by-aakash-download-pdf-exam-analysis-result-updates\/","title":{"rendered":"RMO 2025 Answer Key OUT by Aakash: Download PDF, Exam Analysis, Result Updates"},"content":{"rendered":"<p style=\"text-align: justify;\" data-start=\"390\" data-end=\"877\">The Regional Mathematical Olympiad (RMO) stands among India\u2019s most prestigious school-level mathematics competitions. Conducted by the Homi Bhabha Centre for Science Education (HBCSE), it is the second stage of the Mathematical Olympiad Programme and serves as the gateway to the Indian National Mathematical Olympiad (INMO). With the <strong data-start=\"725\" data-end=\"786\">RMO 2025 exam successfully conducted on November 16, 2025<\/strong>, thousands of students nationwide have been eagerly waiting to evaluate their performance.<\/p>\n<p style=\"text-align: justify;\" data-start=\"879\" data-end=\"1276\">To help students assess their responses and understand the right approach to solving Olympiad-level problems, <strong data-start=\"989\" data-end=\"1068\">Aakash Institute has officially released the Unofficial RMO Answer Key 2025<\/strong>. Curated by India\u2019s top mathematics experts, the answer key provides accurate and detailed solutions to all RMO problems\u2014enabling students to estimate their scores and prepare effectively for the next stage.<\/p>\n<p style=\"text-align: justify;\" data-start=\"1278\" data-end=\"1441\">This article provides complete information on the <strong data-start=\"1328\" data-end=\"1351\">RMO 2025 Answer Key<\/strong>, Aakash\u2019s expert analysis, result date, selection process, and INMO preparation guidance.<\/p>\n<h2 style=\"text-align: justify;\" data-start=\"1448\" data-end=\"1477\"><strong data-start=\"1451\" data-end=\"1477\">RMO 2025 Exam Overview<\/strong><\/h2>\n<p style=\"text-align: justify;\" data-start=\"1479\" data-end=\"1722\">The RMO 2025 exam was conducted nationwide on <strong data-start=\"1525\" data-end=\"1546\">November 16, 2025<\/strong>, from <strong data-start=\"1553\" data-end=\"1575\">13:00 to 16:00 hrs<\/strong>, following the rescheduling of IOQM in certain regions. The three-hour examination tested students on advanced concepts and mathematical thinking.<\/p>\n<h3 style=\"text-align: justify;\" data-start=\"1724\" data-end=\"1751\"><strong data-start=\"1728\" data-end=\"1751\">Key Exam Highlights<\/strong><\/h3>\n<ul style=\"text-align: justify;\" data-start=\"1752\" data-end=\"2038\">\n<li data-start=\"1752\" data-end=\"1812\">\n<p data-start=\"1754\" data-end=\"1812\"><strong data-start=\"1754\" data-end=\"1768\">Exam Name:<\/strong> Regional Mathematical Olympiad (RMO) 2025<\/p>\n<\/li>\n<li data-start=\"1813\" data-end=\"1843\">\n<p data-start=\"1815\" data-end=\"1843\"><strong data-start=\"1815\" data-end=\"1835\">Conducting Body:<\/strong> HBCSE<\/p>\n<\/li>\n<li data-start=\"1844\" data-end=\"1880\">\n<p data-start=\"1846\" data-end=\"1880\"><strong data-start=\"1846\" data-end=\"1860\">Exam Date:<\/strong> November 16, 2025<\/p>\n<\/li>\n<li data-start=\"1881\" data-end=\"1906\">\n<p data-start=\"1883\" data-end=\"1906\"><strong data-start=\"1883\" data-end=\"1896\">Duration:<\/strong> 3 hours<\/p>\n<\/li>\n<li data-start=\"1907\" data-end=\"1933\">\n<p data-start=\"1909\" data-end=\"1933\"><strong data-start=\"1909\" data-end=\"1929\">Total Questions:<\/strong> 6<\/p>\n<\/li>\n<li data-start=\"1934\" data-end=\"1989\">\n<p data-start=\"1936\" data-end=\"1989\"><strong data-start=\"1936\" data-end=\"1954\">Question Type:<\/strong> Proof-based subjective questions<\/p>\n<\/li>\n<li data-start=\"1990\" data-end=\"2038\">\n<p data-start=\"1992\" data-end=\"2038\"><strong data-start=\"1992\" data-end=\"2007\">Next Stage:<\/strong> INMO 2026 (January 18, 2026)<\/p>\n<\/li>\n<\/ul>\n<p style=\"text-align: justify;\" data-start=\"2040\" data-end=\"2214\">The paper covered topics from <strong data-start=\"2070\" data-end=\"2125\">number theory, geometry, algebra, and combinatorics<\/strong>, requiring students to justify steps clearly and present mathematically rigorous proofs.<\/p>\n<h2 style=\"text-align: justify;\" data-start=\"2221\" data-end=\"2274\"><strong data-start=\"2224\" data-end=\"2272\">RMO Answer Key 2025 by Aakash (Now Released)<\/strong><\/h2>\n<p style=\"text-align: justify;\" data-start=\"2275\" data-end=\"2512\">Aakash Institute has released the <strong data-start=\"2309\" data-end=\"2343\">Unofficial RMO 2025 Answer Key<\/strong>, prepared soon after the exam concluded. The answer key has been developed by Aakash\u2019s specialised team of Olympiad faculty who analysed the paper question-by-question.<\/p>\n<h3 style=\"text-align: center;\" data-start=\"2221\" data-end=\"2274\"><strong data-start=\"2224\" data-end=\"2272\"><a href=\"https:\/\/dcx0p3on5z8dw.cloudfront.net\/Aakash\/s3fs-public\/pdf_management_files\/target_solutions\/RMO-2025-26_%28Answers%20%26%20Solutions%29.pdf\" target=\"_blank\" rel=\"noopener\">Click To Download &#8211; RMO Answer Key 2025 by Aakash<\/a> <\/strong><\/h3>\n<h2 style=\"text-align: justify;\" data-start=\"2514\" data-end=\"2573\"><strong data-start=\"2518\" data-end=\"2573\">Why Aakash\u2019s RMO Answer Key 2025 is Highly Reliable?<\/strong><\/h2>\n<ol style=\"text-align: justify;\" data-start=\"2574\" data-end=\"3228\">\n<li data-start=\"2574\" data-end=\"2746\">\n<p data-start=\"2577\" data-end=\"2746\"><strong data-start=\"2577\" data-end=\"2627\">Prepared by India\u2019s top Maths Olympiad faculty<\/strong><br data-start=\"2627\" data-end=\"2630\" \/>Aakash\u2019s expert educators have years of experience mentoring RMO and INMO achievers, ensuring accuracy and depth.<\/p>\n<\/li>\n<li data-start=\"2748\" data-end=\"2922\">\n<p data-start=\"2751\" data-end=\"2922\"><strong data-start=\"2751\" data-end=\"2789\">Step-by-step proof-based solutions<\/strong><br data-start=\"2789\" data-end=\"2792\" \/>Every question includes detailed reasoning, structured proofs, diagrams (if required), and alternative problem-solving methods.<\/p>\n<\/li>\n<li data-start=\"2924\" data-end=\"3077\">\n<p data-start=\"2927\" data-end=\"3077\"><strong data-start=\"2927\" data-end=\"2979\">Faster availability compared to the official key<\/strong><br data-start=\"2979\" data-end=\"2982\" \/>Students can analyse their performance immediately without waiting for the official release.<\/p>\n<\/li>\n<li data-start=\"3079\" data-end=\"3228\">\n<p data-start=\"3082\" data-end=\"3228\"><strong data-start=\"3082\" data-end=\"3124\">Improves INMO-level conceptual clarity<\/strong><br data-start=\"3124\" data-end=\"3127\" \/>The answer key helps students understand how to write complete proofs\u2014an essential skill for INMO.<\/p>\n<\/li>\n<\/ol>\n<h2 style=\"text-align: justify;\" data-start=\"3230\" data-end=\"3282\"><strong data-start=\"3234\" data-end=\"3282\">What the Aakash RMO 2025 Answer Key Contains?<\/strong><\/h2>\n<ul style=\"text-align: justify;\" data-start=\"3283\" data-end=\"3488\">\n<li data-start=\"3283\" data-end=\"3319\">\n<p data-start=\"3285\" data-end=\"3319\">Question-wise complete solutions<\/p>\n<\/li>\n<li data-start=\"3320\" data-end=\"3350\">\n<p data-start=\"3322\" data-end=\"3350\">Region-wise paper insights<\/p>\n<\/li>\n<li data-start=\"3351\" data-end=\"3381\">\n<p data-start=\"3353\" data-end=\"3381\">Answer presentation format<\/p>\n<\/li>\n<li data-start=\"3382\" data-end=\"3411\">\n<p data-start=\"3384\" data-end=\"3411\">Expected difficulty level<\/p>\n<\/li>\n<li data-start=\"3412\" data-end=\"3449\">\n<p data-start=\"3414\" data-end=\"3449\">Suggested marking-style breakdown<\/p>\n<\/li>\n<li data-start=\"3450\" data-end=\"3488\">\n<p data-start=\"3452\" data-end=\"3488\">Expert comments on common mistakes<\/p>\n<\/li>\n<\/ul>\n<p style=\"text-align: justify;\" data-start=\"3490\" data-end=\"3598\">Students can now <strong data-start=\"3507\" data-end=\"3564\">download the Aakash RMO Answer Key 2025 in PDF format<\/strong> directly from the Aakash website.<\/p>\n<h2 style=\"text-align: justify;\" data-start=\"3605\" data-end=\"3658\"><strong data-start=\"3608\" data-end=\"3658\">How to Download the Aakash RMO Answer Key 2025<\/strong><\/h2>\n<p style=\"text-align: justify;\" data-start=\"3659\" data-end=\"3719\">Students can access the answer key by following these steps:<\/p>\n<ol style=\"text-align: justify;\" data-start=\"3721\" data-end=\"3941\">\n<li data-start=\"3721\" data-end=\"3774\">\n<p data-start=\"3724\" data-end=\"3774\">Visit the official <strong data-start=\"3743\" data-end=\"3763\">Aakash Institute<\/strong> website.<\/p>\n<\/li>\n<li data-start=\"3775\" data-end=\"3812\">\n<p data-start=\"3778\" data-end=\"3812\">Go to the <strong data-start=\"3788\" data-end=\"3801\">Olympiads<\/strong> section.<\/p>\n<\/li>\n<li data-start=\"3813\" data-end=\"3866\">\n<p data-start=\"3816\" data-end=\"3866\">Click on <strong data-start=\"3825\" data-end=\"3863\">\u201cRMO Answer Key 2025 (Unofficial)\u201d<\/strong>.<\/p>\n<\/li>\n<li data-start=\"3867\" data-end=\"3895\">\n<p data-start=\"3870\" data-end=\"3895\">Select your <strong data-start=\"3882\" data-end=\"3892\">region<\/strong>.<\/p>\n<\/li>\n<li data-start=\"3896\" data-end=\"3941\">\n<p data-start=\"3899\" data-end=\"3941\">Download the PDF and match your solutions.<\/p>\n<\/li>\n<\/ol>\n<p style=\"text-align: justify;\" data-start=\"3943\" data-end=\"4046\">The answer key serves as a reliable tool for score estimation and strategy building for the next stage.<\/p>\n<h2 style=\"text-align: justify;\" data-start=\"4053\" data-end=\"4105\"><strong data-start=\"4056\" data-end=\"4105\">RMO Result 2025 \u2014 Expected Date &amp; Direct Link<\/strong><\/h2>\n<p style=\"text-align: justify;\" data-start=\"4107\" data-end=\"4164\">HBCSE is expected to announce the <strong data-start=\"4141\" data-end=\"4160\">RMO Result 2025<\/strong> in:<\/p>\n<ul style=\"text-align: justify;\" data-start=\"4166\" data-end=\"4237\">\n<li data-start=\"4166\" data-end=\"4204\">\n<p data-start=\"4168\" data-end=\"4204\"><strong data-start=\"4168\" data-end=\"4198\">Last week of December 2025<\/strong>, or<\/p>\n<\/li>\n<li data-start=\"4205\" data-end=\"4237\">\n<p data-start=\"4207\" data-end=\"4237\"><strong data-start=\"4207\" data-end=\"4237\">First week of January 2026<\/strong><\/p>\n<\/li>\n<\/ul>\n<p style=\"text-align: justify;\" data-start=\"4239\" data-end=\"4350\">The result will be available region-wise on the official HBCSE Olympiad portal. Students will be able to check:<\/p>\n<ul style=\"text-align: justify;\" data-start=\"4352\" data-end=\"4466\">\n<li data-start=\"4352\" data-end=\"4385\">\n<p data-start=\"4354\" data-end=\"4385\">Names of qualified candidates<\/p>\n<\/li>\n<li data-start=\"4386\" data-end=\"4409\">\n<p data-start=\"4388\" data-end=\"4409\">Region-wise cutoffs<\/p>\n<\/li>\n<li data-start=\"4410\" data-end=\"4425\">\n<p data-start=\"4412\" data-end=\"4425\">Merit lists<\/p>\n<\/li>\n<li data-start=\"4426\" data-end=\"4466\">\n<p data-start=\"4428\" data-end=\"4466\">Eligibility status for <strong data-start=\"4451\" data-end=\"4464\">INMO 2026<\/strong><\/p>\n<\/li>\n<\/ul>\n<p style=\"text-align: justify;\" data-start=\"4468\" data-end=\"4545\">A direct link to download the RMO Result 2025 will be provided once released.<\/p>\n<h2 style=\"text-align: justify;\" data-start=\"4552\" data-end=\"4606\"><strong data-start=\"4555\" data-end=\"4606\">What Happens After RMO 2025? Focus on INMO 2026<\/strong><\/h2>\n<p style=\"text-align: justify;\" data-start=\"4608\" data-end=\"4862\">Students who qualify for RMO automatically progress to <strong data-start=\"4663\" data-end=\"4676\">INMO 2026<\/strong>, which will be held on <strong data-start=\"4700\" data-end=\"4720\">January 18, 2026<\/strong>. INMO features highly challenging six-question proof-based problems that test creativity, deep thinking, and strong mathematical foundations.<\/p>\n<h2 style=\"text-align: justify;\" data-start=\"4869\" data-end=\"4929\"><strong data-start=\"4872\" data-end=\"4929\">How Aakash Supports Students in INMO 2026 Preparation?<\/strong><\/h2>\n<p style=\"text-align: justify;\" data-start=\"4931\" data-end=\"5109\">Aakash has been a trusted name in Olympiad training, helping students build strong conceptual understanding and proof-writing skills. For INMO-level preparation, Aakash provides:<\/p>\n<h3 style=\"text-align: justify;\" data-start=\"5111\" data-end=\"5156\"><strong data-start=\"5115\" data-end=\"5154\">1. Expert-curated INMO Problem Sets<\/strong><\/h3>\n<p style=\"text-align: justify;\" data-start=\"5157\" data-end=\"5247\">Designed by experienced Olympiad teachers who understand the style and difficulty of INMO.<\/p>\n<h3 style=\"text-align: justify;\" data-start=\"5249\" data-end=\"5296\"><strong data-start=\"5253\" data-end=\"5294\">2. Dedicated Olympiad Faculty Support<\/strong><\/h3>\n<p style=\"text-align: justify;\" data-start=\"5297\" data-end=\"5372\">Personalised mentorship ensures individual guidance and concept refinement.<\/p>\n<h3 style=\"text-align: justify;\" data-start=\"5374\" data-end=\"5417\"><strong data-start=\"5378\" data-end=\"5415\">3. Past Year RMO &amp; INMO Solutions<\/strong><\/h3>\n<p style=\"text-align: justify;\" data-start=\"5418\" data-end=\"5472\">Helps students analyse patterns and improve reasoning.<\/p>\n<h3 style=\"text-align: justify;\" data-start=\"5474\" data-end=\"5528\"><strong data-start=\"5478\" data-end=\"5526\">4. Special Sessions for Proof-Writing Skills<\/strong><\/h3>\n<p style=\"text-align: justify;\" data-start=\"5529\" data-end=\"5629\">Since INMO requires well-structured proofs, Aakash trains students on clarity, structure, and logic.<\/p>\n<h3 style=\"text-align: justify;\" data-start=\"5631\" data-end=\"5685\"><strong data-start=\"5635\" data-end=\"5683\">5. Regular Doubt Sessions &amp; Strategy Classes<\/strong><\/h3>\n<p style=\"text-align: justify;\" data-start=\"5686\" data-end=\"5743\">Ensures students stay focused and confident for the exam.<\/p>\n<p style=\"text-align: justify;\" data-start=\"5745\" data-end=\"5909\">Many RMO and INMO rankers over the years have been trained by Aakash faculty, showcasing the institute\u2019s expertise in preparing students for higher-stage Olympiads.<\/p>\n<h2 style=\"text-align: justify;\" data-start=\"5916\" data-end=\"5971\"><strong data-start=\"5919\" data-end=\"5971\">Why Students Prefer Aakash for Olympiad Training?<\/strong><\/h2>\n<p style=\"text-align: justify;\" data-start=\"5973\" data-end=\"6032\">Aakash has built a strong reputation for excellence due to:<\/p>\n<ul style=\"text-align: justify;\" data-start=\"6034\" data-end=\"6254\">\n<li data-start=\"6034\" data-end=\"6076\">\n<p data-start=\"6036\" data-end=\"6076\">Highly skilled and experienced faculty<\/p>\n<\/li>\n<li data-start=\"6077\" data-end=\"6111\">\n<p data-start=\"6079\" data-end=\"6111\">Research-backed study material<\/p>\n<\/li>\n<li data-start=\"6112\" data-end=\"6160\">\n<p data-start=\"6114\" data-end=\"6160\">Proven and systematic teaching methodologies<\/p>\n<\/li>\n<li data-start=\"6161\" data-end=\"6207\">\n<p data-start=\"6163\" data-end=\"6207\">Mock tests based on real Olympiad patterns<\/p>\n<\/li>\n<li data-start=\"6208\" data-end=\"6254\">\n<p data-start=\"6210\" data-end=\"6254\">Strong track record of Olympiad selections<\/p>\n<\/li>\n<\/ul>\n<p style=\"text-align: justify;\" data-start=\"6256\" data-end=\"6432\">With structured preparation and expert-led guidance, Aakash helps students gain the confidence and skills needed to excel at every stage of the Mathematical Olympiad Programme.<\/p>\n<p style=\"text-align: justify;\" data-start=\"6458\" data-end=\"6890\">The RMO 2025 exam marks a significant milestone for thousands of young math enthusiasts across India. With the <strong data-start=\"6569\" data-end=\"6612\">Aakash RMO Answer Key 2025 now released<\/strong>, students can evaluate their responses, identify mistakes, and understand how to present solutions for Olympiad-level problems. As the RMO result approaches, students aiming for INMO 2026 should begin refining their proof-writing skills and practising advanced-level questions.<\/p>\n<p style=\"text-align: justify;\" data-start=\"6892\" data-end=\"7137\">Aakash Institute continues to support students through expert-reviewed solutions, detailed exam analysis, and structured Olympiad guidance\u2014helping them move confidently toward their dream of excelling in India\u2019s premier mathematics competitions.<\/p>\n<h2 style=\"text-align: justify;\" data-start=\"7144\" data-end=\"7182\"><strong data-start=\"7147\" data-end=\"7182\">RMO 2025 Aakash Answer Key FAQs<\/strong><\/h2>\n<p style=\"text-align: justify;\" data-start=\"7184\" data-end=\"7238\"><strong data-start=\"7188\" data-end=\"7238\">1. Is the Aakash RMO 2025 Answer Key released?<\/strong><\/p>\n<p style=\"text-align: justify;\" data-start=\"7239\" data-end=\"7366\">Yes, Aakash has released the <strong data-start=\"7268\" data-end=\"7302\">Unofficial RMO 2025 Answer Key<\/strong> with complete, expert-verified solutions for all six questions.<\/p>\n<p style=\"text-align: justify;\" data-start=\"7368\" data-end=\"7431\"><strong data-start=\"7372\" data-end=\"7431\">2. Where can I download the Aakash RMO Answer Key 2025?<\/strong><\/p>\n<p style=\"text-align: justify;\" data-start=\"7432\" data-end=\"7539\">You can download the answer key from the <strong data-start=\"7473\" data-end=\"7510\">official Aakash Institute website<\/strong> under the Olympiads section.<\/p>\n<p style=\"text-align: justify;\" data-start=\"7541\" data-end=\"7607\"><strong data-start=\"7545\" data-end=\"7607\">3. What does the RMO 2025 exam analysis by Aakash include?<\/strong><\/p>\n<p style=\"text-align: justify;\" data-start=\"7608\" data-end=\"7637\">The exam analysis provides:<\/p>\n<ul style=\"text-align: justify;\" data-start=\"7638\" data-end=\"7788\">\n<li data-start=\"7638\" data-end=\"7658\">\n<p data-start=\"7640\" data-end=\"7658\">Difficulty level<\/p>\n<\/li>\n<li data-start=\"7659\" data-end=\"7685\">\n<p data-start=\"7661\" data-end=\"7685\">Question-wise insights<\/p>\n<\/li>\n<li data-start=\"7686\" data-end=\"7705\">\n<p data-start=\"7688\" data-end=\"7705\">Expected cutoff<\/p>\n<\/li>\n<li data-start=\"7706\" data-end=\"7725\">\n<p data-start=\"7708\" data-end=\"7725\">Common mistakes<\/p>\n<\/li>\n<li data-start=\"7726\" data-end=\"7754\">\n<p data-start=\"7728\" data-end=\"7754\">Region-wise observations<\/p>\n<\/li>\n<li data-start=\"7755\" data-end=\"7788\">\n<p data-start=\"7757\" data-end=\"7788\">INMO qualification expectations<\/p>\n<\/li>\n<\/ul>\n<p style=\"text-align: justify;\" data-start=\"7790\" data-end=\"7843\"><strong data-start=\"7794\" data-end=\"7843\">4. When will the RMO Result 2025 be declared?<\/strong><\/p>\n<p style=\"text-align: justify;\" data-start=\"7844\" data-end=\"7935\">HBCSE is likely to release the RMO results in <strong data-start=\"7890\" data-end=\"7934\">late December 2025 or early January 2026<\/strong>.<\/p>\n<p style=\"text-align: justify;\" data-start=\"7937\" data-end=\"7988\"><strong data-start=\"7941\" data-end=\"7988\">5. How many students qualify for INMO 2026?<\/strong><\/p>\n<p style=\"text-align: justify;\" data-start=\"7989\" data-end=\"8091\">Typically, up to <strong data-start=\"8006\" data-end=\"8032\">30 students per region<\/strong> are selected based on RMO performance and cutoff criteria.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The Regional Mathematical Olympiad (RMO) stands among India\u2019s most prestigious school-level mathematics competitions. Conducted by the Homi Bhabha Centre for Science Education (HBCSE), it is the second stage of the Mathematical Olympiad Programme and serves as the gateway to the Indian National Mathematical Olympiad (INMO). With the RMO 2025 exam successfully conducted on November 16, [&hellip;]<\/p>\n","protected":false},"author":55,"featured_media":292682,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6120],"tags":[14010,14019,14015,14018,13606,14009,14014,14012,14017,14013,14016,14011],"class_list":["post-292680","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-rmo","tag-aakash-rmo-answer-key","tag-hbcse-rmo-answer-key","tag-inmo-2025","tag-regional-maths-olympiad-2025","tag-rmo-2025","tag-rmo-2025-answer-key","tag-rmo-cutoff-2025","tag-rmo-exam-analysis-2025","tag-rmo-paper-analysis","tag-rmo-result-2025","tag-rmo-solutions-pdf","tag-rmo-unofficial-answer-key"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.0 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>RMO 2025 Answer Key OUT by Aakash: Download PDF, Exam Analysis, Result Updates<\/title>\n<meta name=\"description\" content=\"Get RMO 2025 exam analysis, difficulty level, and the unofficial answer key prepared by Aakash faculty. 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