CUET Mathematics

Cuet (Common University Entrance Test) For Undergraduate Courses Mathematics (Code: 319) [Includes Sections A and B1 of the CUET Syllabus Only] Parmanand Gupta           B.Sc. (Hons.), M.Sc. (Delhi)              M.Phil (KU), Pre. Ph.D. (IIT Delhi) Associate Professor of Mathematics Former Head of Department of Mathematics Indira Gandhi National College, Ladwa Kurukshetra University, Haryana   LAXMI PUBLICATIONS (P) LTD (An ISO 9001:2015 Company) BEngalURU  • Chennai • Guwahati • Hyderabad • Jalandhar Kochi  • Kolkata • lucknow • mumbai • Ranchi New Delhi

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Introduction National Testing Agency has been entrusted with the responsibility of conducting the Undergraduate entrance tests for all the Central Universities (CUs) for the academic session 2022–2023. CUET will provide a single window opportunity to students to seek admission in any of the Central Universities (CUs) across the country. The CUET (UG)–2022 will be conducted in Computer Based Test (CBT) Mode. Examination Structure for CUET (UG)–2022: CUET (UG)–2022 will consist of the following four sections:  Section IA – 13 Languages  Section IB – 19 Languages  Section II – 27 Domain Specific Subjects  Section III – General Test Choosing options from each section is not mandatory. Choices should match the requirements of the desired University. Broad features of CUET (UG)–2022: Some features of CUET (UG)–2022 are as follows: Section Subjects/Tests Questions to Question Type Duration be Attempted Section IA– There are 13* different 40 questions to Language to be tested through 45 minutes Languages languages. Any of these be attempted for each languages may be chosen. out of 50 in each Reading Comprehension language language (based on different types of passages–Factual, Literary and Narrative, [Literary Aptitude and Vocabulary] Section IB– There are 19** languages. Languages Any other language apart from those offered in Section I A may be chosen. Section II– There are 27*** Domain 40 questions to • Input text can be used for 45 minutes Domain specific subjects being be attempted for each offered under this section. out of 50 MCQ Based Questions Domain Section III– A candidate may choose specific General a maximum of Six (06) 60 questions to • MCQs based on NCERT subjects Test Domains as desired by be attempted the applicable University/ out of 75 Class XII syllabus only 60 minutes Universities. For any such under- • Input text can be used for graduate programme/ programmes being offered MCQ Based Questions by Universities where a General Test is being used • General Knowledge, Current for admission. Affairs, General Mental Ability, Numerical Ability, Quantitative Reasoning (Simple application of basic mathematical concepts arithmetic/algebra geometry/ mensuration/stat taught till Grade 8), Logical and Analytical Reasoning

* Languages (13): Tamil, Telugu, Kannada, Malayalam, Marathi, Gujarati, Odia, Bengali, Assamese, Punjabi, English, Hindi and Urdu ** Languages (19): French, Spanish, German, Nepali, Persian, Italian, Arabic, Sindhi, Kashmiri, Konkani, Bodo, Dogri, Maithili, Manipuri, Santhali, Tibetan, Japanese, Russian and Chinese. *** DomainSpecificSubjects(27):1.Accountancy/BookKeeping;2.Biology/BiologicalStudies/Biotechnology/ Biochemistry; 3. Business Studies; 4. Chemistry; 5. Computer Science/Informatics Practices; 6. Economics/ Business Economics; 7. Engineering Graphics; 8. Entrepreneurship; 9. Geography/Geology; 10. History; 11. Home Science; 12. Knowledge Tradition and Practices of India; 13. Legal Studies; 14. Environmental Science; 15. Mathematics; 16. Physical Education/NCC/Yoga; 17. Physics; 18. Political Science; 19. Psychology; 20. Sociology; 21. Teaching Aptitude; 22. Agriculture; 23. Mass Media/Mass Communication; 24. Anthropology; 25. Fine Arts/Visual Arts (Sculpture/Painting)/Commercial Arts; 26. Performing Arts– (i) Dance (Kathak/Bharatnatyam/Oddisi/Kathakali/Kuchipudi/Manipuri (ii) Drama-Theatre (iii) Music General (Hindustani/Carnatic/Rabindra Sangeet/Percussion/Non-Percussion); 27. Sanskrit [For all Shastri (Shastri 3 years/4 years Honours) Equivalent to B.A./B.A. Honours courses i.e. Shastri in Veda, Paurohitya (Karmakand), Dharamshastra, Prachin Vyakarana, Navya Vyakarana, Phalit Jyotish, Siddhant Jyotish, Vastushastra, Sahitya, Puranetihas, Prakrit Bhasha,Prachin Nyaya Vaisheshik, Sankhya Yoga, Jain Darshan, Mimansa, AdvaitaVedanta, Vishihstadvaita Vedanta, Sarva Darshan, a candidate may choose Sanskrit as the Domain]. • A candidate can choose a maximum of any 3 languages from Section IA and Section IB taken together. (One of the languages chosen needs to be in lieu of Domain specific subjects). • Section II offers 27 subjects, out of which a candidate may choose a maximum of 6 subjects. • Section III comprises General Test. • For choosing Languages (up to 3) from Section IA and IB and a maximum of 6 subjects from Section II and General Test under Section III, the candidate must refer to the requirements of his/her intended University. Level of questions for CUET (UG)–2022: All questions in various testing areas will be benchmarked at the level of Class XII only. Marking Scheme of Examination For Multiple Choice Questions: To answer a question, the candidates need to choose one option corresponding to the correct answer or the most appropriate answer. However, if any anomaly or discrepancy is found after the process of challenges of the key verification, it shall be addressed in the following manner: (i) Correct answer or the most appropriate answer: Five marks (+ 5) (ii) Any incorrect option marked will be given minus one mark (– 1). (iii) Unanswered/Marked for Review will be given no mark (0). (iv) If more than one option is found to be correct then Five marks (+ 5) will be awarded to only those who have marked any of the correct options. (v) If all options are found to be correct then Five marks (+ 5) will be awarded to all those who have attempted the question. (vi) If none of the options is found correct or a Question is found to be wrong or a Question is dropped then all candidates who have appeared will be given five marks (+ 5). Note: C andidates are advised to visit the NTA CUET (UG)–2022 official website https://cuet.samarth.ac.in/, for latest updates regarding the Examination.

Contents ... 1 1. Relations and Functions ... 10 2. Inverse Trigonometric Functions 3. Matrices ... 15 4. Determinants 5. Continuity and Differentiability ... 27 6. Application of Derivatives 7. Integrals ... 34 8. Application of Integrals 9. Differential Equations ... 47 10. Vector Algebra 11. Three Dimensional Geometry ... 60 12. Linear Programming 13. Probability ... 77 Answers Mock Test Papers ... 81 ... 89 ... 101 ... 110 ... 113 ... 123 ... M-1–M-44

SYLLABUS (MATHEMATICS / APPLIED MATHEMATICS – 319) Note: There will be one Question Paper which will contain Two Sections i.e. Section A and Section B [B1 and B2]. Section A will have 15 questions covering both i.e. Mathematics/Applied Mathematics which will be compulsory for all candidates Section B1 will have 35 questions from Mathematics out of which 25 questions need to be attempted. Section B2 will have 35 questions purely from Applied Mathematics out of which 25 question will be attempted. SECTION A 1. Algebra (i) Matrices and types of Matrices (ii) Equality of Matrices, transpose of a Matrix, Symmetric and Skew Symmetric Matrix (iii) Algebra of Matrices (iv) Determinants (v) Inverse of a Matrix (vi) Solving of simultaneous equations using Matrix Method 2. Calculus (i) Higher order derivatives (ii) Tangents and Normals (iii) Increasing and Decreasing Functions (iv) Maxima and Minima 3. Integration and its Applications (i) Indefinite integrals of simple functions (ii) Evaluation of indefinite integrals (iii) Definite Integrals (iv) Application of Integration as area under the curve 4. Differential Equations (i) Order and degree of differential equations (ii) Formulating and solving of differential equations with variable separable 5. Probability Distributions (i) Random variables and its probability distribution (ii) Expected value of a random variable (iii) Variance and Standard Deviation of a random variable (iv) Binomial Distribution 6. Linear Programming (i) Mathematical formulation of Linear Programming Problem (ii) Graphical method of solution for problems in two variables (iii) Feasible and infeasible regions (iv) Optimal feasible solution









































































DIFFERENTIAL EQUATIONS 83 CASE BASED QUESTIONS Case Study 1 The degree of differential equation is the power of the highest order derivative, when differential If the equation is of the form dy  Py = Q, where P, coefficients are made free from radicals and fractions. dx Also, differential equation must be a polynomial equation in derivatives for the degree to be defined. Q are functions of x, then the solution of the differential equation is given by Based on the above information, answer the following questions: ye P dx = Qe P dx dx  c , 1. Find the degree of the differential equation where e P dx is called the integrating factor (I.F.) d2 y  dy  2 dx 2 dx 2 3 1  y 0 1. The integrating factor of the differential dy (a) 1 (b) 2 dx equation sin x  2y cos x = 1 is (sin x), (c) 3 (d) 4 where  is equal to 2. Order and degree of the differential equation (a) 1 (b) 2 y dy  x are respectively dx (c) 3 (d) 4 dy  dy  3 dx dx  2. Integrating factor of the differential equation (1  x2 ) dy (a) 1, 4 (b) 1, 3 dx  xy = 1 is: (c) 1, 2 (d) 1, 1 x 3. Find order and degree of the equation (b) 1  x2 (a) – x y  y2  ey  0 (c) 1  x2 1 (a) order = 3, degree = undefined 2 (d) log (1  x2 ) (b) order = 1, degree = 3 dy (c) order = 2, degree = undefined dx 3. The solution of  y = e–x, y(0) = 0 is: (d) order = 1, degree = 2 4. Find the degree of the differential equation (a) y = ex(x – 1) (b) y = xe–x ( a  x) .  dy   x  0 (c) y = xe–x + 1 (d) y = e–x(x + 1) dx dy (a) 1 (b) 2 dx 4. General solution of  tan x  y = sec x is (c) 3 (d) Not defined (a) y sec x = tan x + c (b) y tan x = sec x + c 5. Order and degree of the differential equation (c) tan x = y tan x + c (d) x sec x = tan y 5. The integrating factor of differential equation 1 dy 3 7 3 d2 y  dx  dx 2      7 are respectively   dy (a) 2, 1 (b) 2, 3 dx  3y = sin 2x is (c) 1, 3 (d) 1, 7 3 (a) e2x (b) 3x (c) e–3x (d) None of these Case Study 3 Case Study 2 In a college hostel accommodating 1000 students, one of the hostellers come in carrying corona virus, and The order of a differential equation is the order of the the hostel was isolated. The rate at which the virus highest order derivative appearing in the differential spreads is assumed to be proprotional to the product equation. of the number of infected students and remaining students. There are 40 infected students after 4 days.