Launch Your Career : CBSE,IIT-JEE,Pre medical ,AIEEE,UPSC,MBA and Entrance and Carrer Exam

to get solved paper visit http://jee.iitm.ac.in/ PATTERN OF JEE-2008 Question Papers There will be two question papers, each of three hours duration. Both the question papers would consist of three separate sections on Physics, Chemistry, and Mathematics. Questions in these papers will be of objective type, which are to be answered on a specially designed machine-gradable sheet (ORS – Optical Response Sheet) using HB pencils only. Incorrect answers will be awarded negative marks. Language and Font of Question Papers Candidates can opt for Question Papers either in English or in Hindi. This option should be exercised while filling the application form, and it cannot be changed at any later stage. Visually impaired candidates will be provided with question papers with 20% enlarged font. However, to avail this facility, candidates should make a request along with the application form. Calculators and Log Tables Use of calculators and log tables is NOT permitted in JEE-2008. Aptitude Test for B.Arch. and B.Des. Candidates called for counselling and desirous of joining the B.Arch. and B.Des. courses will be required to qualify in an Aptitude Test to be conducted at each counselling institute on June 21, 2008. The test will consist of two papers, each of two hours duration – from 10.00 a.m. to 12.00 noon, and from 2.00 pm to 4:00 pm. Candidates who fail to qualify in the Aptitude Test will not be eligible for admission to either B.Arch. or B.Des. courses. Question papers for aptitude test for B.Arch and B.Des will be in English only.

The Student-Tutor Relationship

The student-tutor relationship is a very important one. Nowhere is this more true than in e-learning. Yet students are often unsure as to how best to take fullest advantage of the unique access an e-learning course can provide them to a qualified expert/tutor.

The best tutors will guarantee a full response to a student's questions or queries within 24-48 hours. This is good, because it means that the student can feel the tutor is really there for them. Make sure that any e-mail is addressed directly to the tutor or expert concerned. Though length of query is not an issue, clarity is. Ensure that you use punctuation, and paragraphs. If you are asking multiple questions, consider numbering them or using headers. Spell-check before you send, to avoid potential misunderstandings that might delay a response.

Don't feel shy about sending a number of e-mails to your tutor. This is common practice, and it demonstrates your proactive engagement in the course. The more you ask, the more guidance you will receive back in return. For example, lessons with assessment tasks or questions can be answered and sent back for marking. Don't be afraid to make mistakes - the tutor will not only correct them, but show you how to get it right in future.

To save time, start each e-mail to your tutor with the number of the lesson (and, where applicable, the task or sentence) in question.

E-learning tutors are experts in their respective field, and often qualified and experienced teachers. The opportunity to have one-on-one, direct access to them and their expertise and knowledge is an exciting and extremely empowering one for the student.

Medicine is one of the most sort out and rewarding career for those interested in Science and dealing with sick people. Over the years, the field of medical studies have undergone various stages of development, it has become so vast that specialisations within are increasing day by day. There is great scope for medicine as a professional career.
A doctor's profession involves a lot of hard work and at the same time, it gives the satisfaction of curing patients at times even saving lives. It is a very demanding profession. To be in this profession is more a responsibility than a privilege. It is more of what you can give the community than what you can get from it.

The increasing complicated lifestyles giving birth to a variety of ailments have made it impossible for General Physicians with an MBBS degree to handle all ailments. It is here that specialisation in a particular branch of medicine becomes a necessity. The major Specialisations include General Medicine, General Surgery, Paediatrics, Obstetrics & Gynaecology, Dermatology, Ophthalmology, Orthopedics, ENT (Ear, Nose and Throat), Psychiatry, Anesthesiology etc. There is also Super Specialisations which require a further 3-5 years of study in areas such as Plastic Surgery, Neurosurgery, Cardio-thoracic surgery, Conito-urinary surgery, Paediatric Surgery, Gastroenterology, Endocrinology and Clinical Haematology.

The demand for medical professionals are tremendously increasing with the unfortunate upsurge of diseases and ailments day by day. At the same time super specialty hospitals are mushrooming both within the country and abroad offering employment opportunities. These along with liberalization of economy could bring better opportunities for these professionals in terms of remuneration, research and working facilities.

Other than Allopathy, medicine also covers different systems like Homeopathy, Ayurveda, Siddha, Unani etc.

Eligibility : The basic or degree level qualification for a medical profession is MBBS (Bachelor of Medicine And Bachelor Of Surgery) which is of 5 1/2 years duration (including 1 year internship). Click here for more

Job Prospects and Career Options : Some of the common areas of specialization in medicine and surgery are General Practitioner or Physician, General Surgeon, Anaesthetist or Anaesthesiologists, Psychiatrist, Neurologist etc. Click here for more

Remuneration : The earnings of a doctor through private practice depends on his/her popularity. The doctors who are working in government hospitalrs are well paid. The pay varies for those working with private hospitals. It may vary from Rs 10,000 to Rs 15,000.

Dentists beginning their careers in government hospitals can expect their salaries to be around Rs 7,000, and those who hold post graduate degrees can begin at salaries of Rs. 8,000. A dentist setting up a private practice can earn, on an average, Rs 6,000 a month.

Ayurveda : Ayurveda is a very ancient system which evolved around 600 BC in India. Ayurvedic treatments are person specific rather than disorder specific.

Dentistry : Dentistry deals with medical or surgical treatment of diseases and disorders related to teeth, gums and soft tissues of mouth.

Homeopathy : Homeopathy is a system of medicine which is practiced on the principle, that a drug and a disease which produce similar symptoms cancel each other out in some way thereby restoring the patients to health.

Engineering is a vast field that offers infinite specialisation. All fields in the modern economy has been invigorated by engineering technology. This field deals with designing and its application. The three traditional branches of engineering are civil, mechanical and electrical. Apart form these, there are various other branches like Aeronautical engineering, Ceramic engineering, chemical engineering, computer engineering, automobile engineering, industrial engineering, environmental engineering, marine engineering, textile engineering etc.

B.E. (Bachelor of Engineering)

Civil, Mechanical and Electrical Engineering: Civil engineering deals with construction activities including building roads, bridges, tunnels etc. Mechanical engineering deals with design and production of tools, machines etc. to be used in industries. Electrical engineering deals with production of electrical equipments.

Eligibility : 10 + 2 Science with high percentage of marks in Science subjects
Duration : 4 years
Fees: In private colleges fees range from Rs.1 lakh to Rs. 2 lakh annually. In IIT's it varies between Rs. 15,000/- to Rs. 20,000/- per year
Job Prospects: They can find job in Government departments, private and public sector industries, research and teaching institutions etc.

Aeronautical/ Aerospace Engineering

This field deals with development of new technology in the field of aviation, space exploration and defence systems. It specialises in the development and research of commercial and military aircraft, missiles, spaceships etc.

Eligibility : 10 + 2 Science with high percentage of marks in Science subjects
Duration : 4 years
Job Prospects: Jobs are available with the airline services and aircraft manufacturing units. The defence service and Indian Space Research Organisation (ISRO) are two other important employers.

Ceramic Engineering

They specialises in production of ceramic products like glass, electric power line insulators, semi conductors etc.

Eligibility : 10 + 2 Science with high percentage of marks in Science subjects
Fees: Around Rs. 1 lakh per year and in IIT's fees range from Rs. 15,000/- to Rs. 20,000/- per year.
Job Prospects: They can find employment in ceramic industry.

Computer Engineering

Deals with computer hardware and software. Manufacture and research, development of new design, technical improvements, alterations etc.

Eligibility : 10 + 2 Science with high percentage of marks in Science subjects
Duration : 5 years
Job Prospects: The scope of the job is directly related to the growth and development of computer industry. They can get employment in computer fields and also work as maintenance consultants.

Chemical Engineering

It gives a vast knowledge of production of chemicals and related products.

Eligibility : 10 + 2 Science with high percentage of marks in Science subjects
Duration: 4 years
Job Prospects: They can find employment opportunities with chemical industries, refineries, manufacturers of acids, medicines, varnishes, paints, fertilizers etc. They can also find employment in research laboratories.

Automobile Engineering

This field deals with design, development and maintenance of automobile and its spare parts.

Eligibility : 10 + 2 Science with Physics, Chemistry and Mathematics.
Duration : 4 years
Job prospects : He can find employment in automobile industries, service stations, transport companies etc.

Industrial Engineering

The industrial engineers aim is to increase productivity through management of people and methods of business organisation. They are the bridges between management and operations.

Eligibility : 10 + 2 Science with Physics, Chemistry and Mathematics plus qualifying exam
Duration : 5 years after 10 +2, 1 year after 4 years of mechanical or electrical engineering degree course.
Job Prospects : They can find the jobs in R & D establishments , Self employment opportunities are also possible as consultants.

Environmental Engineering

It is concerned with the conservation of environment.

Eligibility : 10 + 2 Science with Physics, Chemistry and Mathematics plus qualifying exam
Duration : 4 years
Job Prospects : In this polluted environment they can find jobs in Government as well as private sectors, particularly in chemical manufacturing units, mines etc.

Marine Engineering

Marine Engineers have the complete responsibility of the ship's engine room. Their responsibilities involve development and designing of the engines related to ships and propulsion system. Ministry of Surface Transport, Government of India, takes care of the training needs and conducts competency exam through the Directorate General of Shipping (DGS).

Eligibility : 10 + 2 Science (Physics, Chemistry & Maths)
Duration : 1 year for Navigation course, 3 years for B.Sc Nautical science, 4 years for (Marine Engineering Research Institute) MERI course
Selection: Written test / interview (Joint Entrance Examination { JEE})
Job Prospects : This is a highly paid job with lots of overseas travel. Jobs are available with Shipping companies in India and other countries.

Textile Engineering

They deals with the development, design, manufacturing and quality control of the fibers and fiber products.

Eligibility : 10 + 2 Science (Physics, Chemistry & Maths)
Duration : 4 yrs for Engineering degree, 3 years for B.Sc (Tech) degree
Fees: Around Rs. 1 lakh per year and in IIT's fees range from Rs. 15,000/- to Rs. 20,000/- per year
Job prospectus : They can find employment in Textile industry.

B.Arch (Bachelor of Architecture)

Architects design and supervise construction of varies types of buildings keeping in view the primary consideration of stability, utility and beauty. An ideal architect needs skill in visualising, designing, engineering and communicating your ideas to the client. In short Architecture is a unique blend of art and science.
In Architecture there are several areas of specialisation like industrial design, interior design, town planning, regional planning, housing, transport planning, landscape planning, environmental planning etc.

Eligibility : 10 + 2 with high percentage of marks + qualifying examination and interview
Duration : 5 years/4 years
Fees: Around Rs. 1 lakh per year. In IIT's fees are lower.
Prospects : Job opportunities are available in construction firms, government departments, town planning offices, private architectural firms, landscaping consultants etc.

1. Make a time table for a regular study or revision of minimum 6-8 hours daily. Do not study at a stretch, take a few minutes break.
2. Time management is very important. Learn to time yourself simulate examination situations while practicing.
3. Writing practice should be done by solving different questions.
4. Relaxation-practice deep breathing , yoga or any other relaxation technique to improve concentration.
5. Quickest and most effective way of eliminating stress is to shut down your eyes and take deep breaths.
6. As far as possible continue with the normal routine of sleeping and eating .
7. A balanced diet will boost energy.
8. Regular and moderate exercises reduce stress by relaxing tensed muscles.
9. Take help of the teachers and parents from time to time.
10. Believe in yourself and prepare well.
11. Never fear exams, avoid panic.and most important, do not worry about your results. Give your best shots and move over.
12. Students may also practice from the Sample Question Papers brought out by the CBSE, Marking Schemes performance analysis which analysis common mistakes committed by the students and provides remedial measures.

DESIGN OF THE QUESTION PAPER
CHEMISTRY CLASS - XII
Time : Three Hours Max. Marks : 70
The weightage of the distribution of marks over different dimensions of the question paper shall be as follows:
A. Weightage to content/subject units
Unit Title Marks
1. Solid state 4
2. Solutions 5
3. Electrochemistry 5
4. Chemical Kinetics 5
5. Surface Chemistry 4
6. General principles and process of Isolation of elements 3
7. p-Block Elements 8
8. d-and f-Block Elements 5
9. Coordination Compounds 3
10. Haloalkanes and Haloarenes 4
11. Alcohols, Phenols and Ethers 4
12. Aldehydes, Ketones and Carboxylic acids 6
13. Organic Compounds containing Nitrogen 4
14. Biomolecules 4
15. Polymers 3
16. Chemistry in Everyday life 3
Total 70
B. Weightage to form of questions
S.No. Form of Questions Marks for each No. of Total Marks
question questions
1. Long Anwer Type (LA) 5 3 15
2. Short Answer (SAI) 3 9 27
3. Short Answer (SAII) 2 10 20
4. Very Short Answer (VSA) 1 08 08
Total - 30 70
(2)
C. Scheme of Options
1. There will be no overall option.
2. Internal choices (either/or type) in five questions has been given in questions testing higher mental abilities in
the following types of questions :-
(i) One in two marks questions.
(ii) One in three marks questions.
(iii) All the three in five marks questions.
D. Guidelines for Units 10-13 of syllabus.
These units include questions on:

Nomenclature : 2 marks

Reasoning : 6 marks

Distinguishing between compounds : 2 marks

Name reactions : 2 marks

Reaction Mechanism : 2 marks

Word problems (conversions) covering
Properties and reactions of functional groups : 5 marks
E. Numericals :
Weightage of 8 -10 marks in total has been assigned to numericals.
F. Weightage to difficulty level of questions
S.No. Estimated difficulty level Percentage
1. Easy 15
2. Average 70
3. Difficult 15
A weightage of 20% has been assigned to questions which test higher order thinking skills of students.
(3) BLUE-PRINT I
Class XII
CHEMISTRY SAMPLE PAPER
S.NO. UNIT VSA SA I SAII LA TOTAL
(1 Mark) (2 Marks) (3 Marks) (5 Marks)
1. Soild State - 4 (2) - - 4 (2)
2. Solutions - 2(1) 3(1) - 5(2)
3. Electrochemistry - 2(1) 3 (1) - 5(2)
4. Chemical Kinetics 5 (1) 5(1)
5. Surface Chemistry 1(1) 3 (1) - 4(2)
6. General principles and processes - - 3(1) 3(1)
of Isolation of Elements
7. p -Block Elements - - 3 (1) 5 (1) 8 (2)
8. d- and f-Block Elements - 2(1) 3(1) - 5(2)
9. Coordination Compounds 1(1) 2 (1) - - 3(2)
10. Haloalkanes and Haloarenes - 4(2) - - 4(2)
11. Alcohols, Phenols and Ethers 1 (1) - 3 (1) - 4 (2)
12. Aldehydes, Ketones 1 (1) - - 5 (1) 6 (2)
and Carboxylic Acids
13. Organic Compounds Containing 1 (1) - 3 (1) - 4 (2)
Nitrogen
14. Biomolecules 1 (1) - 3 (1) - 4 (2)
15. Polymers 1 (1) 2 (1) - - 3 (2)
16. Chemistry in Everyday Life 1 (1) 2 (1) - - 3 (2)
Total 8(8) 20(10) 27(9) 15(3) 70(30)
(4)
CHEMISTRY SAMPLE PAPER - I
CLASS - XII
Time : Three Hours Max. Marks : 70
General Instructions
1. All questions are compulsory.
2. Question nos. 1 to 8 are very short answer questions and carry 1 mark each.
3. Question nos. 9 to 18 are short answer questions and carry 2 marks each.
4. Question nos. 19 to 27 are also short answer questions and carry 3 marks each
5. Question nos. 28 to 30 are long answer questions and carry 5 marks each
6. Use log tables if necessary, use of calculators is not allowed.
(1) Why is ferric chloride preferred over potassium chloride in case of a cut leading to bleeding? 1
(2) Why does a tetrahedral complex of the type [MA2 B2] not show geometrical isomerism? 1
(3) How do you account for the miscibility of ethoxyethane with water. 1
(4) Give the IUPAC name of the organic compound 1
O
II
(CH3)2 C CH C CH3
= − −
(5) Name the monomers of nylon 2 or nylon 6 ploymer. 1
(6) Give one example of an artificial sweetener used by the diabetic patients. 1
(7) Direct nitration of aniline is not carried out. Explain why? 1
(8) What type of linkage holds together the monomers of D.N.A.? 1
(9) Examine the illustration of a portion of the defective crystal given below and answer the following questions.
(5)
(i) What are these type of vacancy defects called?
(ii) How is the density of a crystal affected by these defects?
(iii) Name one ionic compound which can show this type of defect in the crystalline state
(iv) How is the stoichiometry of the compound affected? 2
10. Analysis shows that a metal oxide has the empirical formula M0.96 O1.00. Calculate the percentage of M2+ and M3+
ions in this crystal? 2
OR
In an ionic compound the anion (N¯) form cubic close type of packing. While the cation (M+) ions occupy one
third of the tetrahedral voids. Deduce the empirical formula of the compound and the coordination number
of (M+) ions. 2
11. Given below is the sketch of a plant for carrying out a process.
(i) Name the process occurring in the above plant.
(ii) To which container does the net flow of solvent take place?
(iii) Name one SPM which can be used in this plant.
(iv) Give one practical use of the plant. 2
12. Write the chemical equations for all the steps involved in the rusting of iron. Give any one method to prevent rusting
of iron. 2
13. A metal ion Mn+ having d4 valence electronic configuration combines with three didentate ligands to form a complex
compound. Assuming
(i) draw the diagram showing d orbital splitting during this complex formation.
(ii) write the electronic configuration of the valence electrons of the metal Mn+ ion in terms of t2g and eg.
(iii) what type of hybridisation will Mn+ ion have?
(iv) name the type of isomerism exhibited by this complex. 2
14. A mixed oxide of iron and chromium FeOCr2O3 is fused with sodium carbonate in the presence of air to form a
yellow coloured compound (A). On acidification the compound (A) forms an orange coloured compound (B),
which is a strong oxidising agent. Identify
(i) the compounds (A) and (B)
(ii) write balanced chemical equation for each step 2
15. An optically active compound having molecular formula C7H15Br reacts with aqueous KOH to give a racemic
mixture of products. Write the mechanism involved for this reaction. 2
(6)
16. Write the formula of main product formed in the following chemical reactions.
(i) (CH3)2 CH-C1
(ii) CH3Br + AgF
(iii) CH3CH2Br + Nal
(iv) 2
17. Differentiate the following pair of polymers based on the property mentioned against each.
(i) Novolac and Bakelite (structure)
(ii) Buna-s and Terylene (intermolecular forces of attraction) 2
18. In order to wash clothes with water containing dissolved calcium hydrogencarbonate, which cleaning agent will you
prefer and why: soaps or synthetic detergents? Give one advantage of soaps over synthetic detergents. 2
19. Heptance and octane form an ideal solution at 373 K, The vapour pressures of the pure liquids at this terperature
are 105.2 KPa and 46.8 KPa respectively. If the solution contains 25g of heptance and 28.5g of octane, calculate
(i) vapour pressure exerted by heptane
(ii) vapour pressure exerted by solution
(iii) mole fraction of octane in the vapour phase. 3
20. The following chemical reaction is occurring in an electrochemical cell.
Mg(s) + 2 Ag+ (0.0001 M) Mg2+ (0.10M) + 2 Ag(s)
The electrode values are
Mg2+ / Mg = – 2. 36 V
Ag+ / Ag = 0.81 V
For this cell calculate / write
(a) (i) EO value for the electrode 2Ag+ / 2Ag
(ii) Standard cell potential EO
cell.
(b) Cell potential (E)cell
(c) (i) Symbolic representation of the above cell.
(ii) Will the above cell reaction be spontaneous? 3
21. Consider the adsorption isotherms given below and interpret the variation in the extent of adsorption (x/m) when
(7)
(a) (i) temperature increases at constant pressure
(ii) pressure increases at constant temperature
(b) Name the catalyst and the promoter used in Haber’s process for manufacture of ammonia. 3
22. Account for the following facts
(a) the reduction of a metal oxide is easier if the metal formed is in liquid state at the temperature of reducation.
(b) the reduction of Cr2O3 with AI is thermodynamically feasible, yet it does not occur at room temperature.
(c) pine oil is used in froth floatation method. 3
23. Explain the following facts
(a) transition metals act as catalysts.
(b) chromium group elements have the highest melting points in their respective series.
(c) transition metals form coloured complexes. 3
24. (a) Give a chemical test to distinguish between the following pairs of compounds.
(i)
(ii) and
(b) Why is phenol more acidic than ethanol? 3
25. Account for the following observations
(i) among the halogens F2 is the strongest oxidising agent?
(ii) fluorine exhibits only – 1 oxidation state whereas other halogens exhibit higher positive oxidation states also.
(iii) acidity of oxo acid of chlorine is
HOCl < HOClO < HOClO2 < HOClO3 3
26. (a) Give plausible explanation for each of the following.
(i) The presence of a base is needed in the ammonolysis of alkyl halides.
(ii) Aromatic primary amines cannot be prepared by Gabriel phthaliminde syntheses.
(b) Write the IUPAC name of
3
27. An optically active compound having molecular formula C6H12O6 is found in two isomeric forms (A) and (B) in
nature. When (A) and (B) are dissolved in water they show the following equilibrium.
(A) Equilibrium mixture (B)
D = 1110 52.20 19.20
(i) What are such isomers called?
(ii) Can they be called enantiomers? Justify your answer.
(iii) Draw the cyclic structure of isomer (A) 3
(8)
OR
An optically active amino acid (A) can exist in three forms depending on the pH of the medium. If the molecular
formula of (A) is C3H7NO2 write
(i) structure of compound (A) in aqueous medium. What are such ions called?
(ii) In which medium will the cationic form of compound (A) exist?
(iii) In alkaline medium, towards which electrode will the compound (A) migrate in electric field? 3
28. For a certain chemical reaction variation in the concentration in [R] vs. time (s) plot is given below.
For this reaction write / draw
(i) what is the order of the reactions?
(ii) what are the units of rate constant k?
(iii) give the relationship between k and t ½ (half life period)
(iv) what does the slope of the above line indicate?
(v) draw the plot log [R]0 / [R] vs time t(s) 5
OR
For a certain chemical reaction
A + 2B 2C + D
The experimentally obtained information is tabulated below.
Experiment [A]0 [B]0 Initial rate
of reaction
1 0.30 0.30 0.096
2 0.60 0.30 0.384
3 0.30 0.60 0.192
4 0.60 0.60 0.768
For this reaction
(i) derive the order of reaction w.r.t. both the reactants A and B.
(ii) write the rate law.
(iii) calculate the value of rate constant k
(iv) write the expression for the rate of reaction in terms of A and C. 5
(9)
29. A translucent white waxy solid (A) on heating in an inert atmosphere is converted to its allotropic form (B).
Allotrope (A) on reaction with very dilute aqueous KOH liberates a highly poisonous gas (C) having rotten fish
smell. With excess of chlorine forms (D) which hydrolyses to compound (E). Identify compounds (A) to (E). 5
OR
Concentrated sulphuric acid is added followed by heating to each of the following test tubes labelled (i) to (v)
(i) (ii) (iii) (iv) (v)
Identify in which of the above test tube the following change will be observed. Support your answer with the help
of a chemical equation.
(a) formation of black substance
(b) evolution of brown gas
(c) evolution of colour less gas
(d) formation of brown substance which on dilution becomes blue.
(e) disappearance of yellow powder along with evolution of colourless gas. 5
30. Identify the unknown organic compounds (A) to (E) in the following series of chemical reactions.
(i) (A) + (B)
(ii) (A) + (B) (C) + H2O
(iii) (C) (A) + (D)
(iv) (D) (E) 5
OR
An organic compound (A) having molecular formula C9H10O forms an orange red precipitate (B) with 2, 4 - DNP
reagent. Compound (A) gives a yellow precipitate (C) when heated in the presence of iodine and NaOH along
with a colourless compound (D). (A) does not reduce Tollen’s reagent or Fehling’s solution nor does it decolorise
bromine water. On drastic oxidation of (A) with chromic acid, a corboxylic acid (E) of molecular formula C7H6O2
is formed. Deduce the structures of the organic compounds (A) to (E).

Sample Paper – 2008

Class – XII

Subject - Mathematics

____________________________________________________________

Application of Derivatives

Q1 The volume of a cube is increasing at a constant rate. Prove that the increase in surface area varies inversely as the length of the edge of the cube.

Q2 Use differentials to find the approximate value of

Q3 It is given that for the function f(x) = x³ – 6x² + ax + b on [1, 3], Rolle’s theorem holds with

c = 2+ . Find the values of a and b if f(1)= f(3) = 0

Q4 Find a point on the curve y = (x – 3)², where the tangent is parallel to the line joining (4, 1)

and (3, 0).

Q5 Find the intervals in which the function f(x) = x⁴ – 8x³ + 22x² – 24x + 21 is decreasing or increasing.

Q6 Find the local maximum or local minimum of the function.

f(x) = sin⁴x + cos⁴x, 0p/2.

Q7 Find the point on the curve y² = 4x which is nearest to the point (2, 1).

Q8 A figure consists of a semi-circle with a rectangle on its diameter. Given the perimeter of the figure, find its dimensions in order that the area may be maximum.

Q9 A balloon which always remain spherical has a variable diameter . Find the rate of change of its volume with respect to x.

Q10 Find the intervals in which f(x) = (x+1)³ (x – 3)³ is strictly increasing or decreasing.

Q11 Prove that the curves x = y² and xy = k cut at right angles if 8k² = 1

Q12 Using differentials, find the approximate value of (26.57)^1/3

Q13 Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

Q14 Find the equation of the tangent and normal to the hyperbola at the point (x₀,y₀)

Q15 Find the intervals of the function f(x) = sinx + cosx, 0£x£2p is strictly increasing or strictly decreasing.

Q16 An open topped box is to be constructed by removing equal squares from each corner of a 3 metre by 8 metre rectangular sheet of aluminium and folding up the sides. Find the volume of the largest such box.

Inverse Trigonometric Functions

Class – XII

Q.1. Find the value of : tan^-1 (1) + cos ^-1 (-1/2) + sin^-1 (-1/2).

Q.2. Prove : tan^-1x + tan^-1 = tan^-1 , çxê<

Q.3. If tan^-1 then find the value of x.

Q.4. Find the value of sin .

Q.5. Prove : sin^-1

Q.6. Solve : tan^-12x + tan^-13x =

Q.7. Prove :

Q.8. Solve : sin^-1 ( 1 –x) – 2sin^-1x = .

Q.9. Evaluate: tan^-1 - sec^-1 (-2) + cosec^-1 .

Q.10. Prove : tan^-1 =

Q.11. Simplify: sin^{-1 ,}

Q.12. Prove: sec² (tan^-12) + cosec² ( cot^-13) = 15.

Q.13. Simplify : tan^-1

Q.14. Prove : tan^-1 =

Q.15. If sin(sin^-1 , then find the value of x.

Q16.. Prove that :

2tan^-1 = cos^-1

MATRICES

Class - XII

Q.1. Construct a 3´4 matrix, whose elements are given by a_ij=

Q.2. If A = and B = , then find the matrix X, such that 2A + 3X = 5B.

Q.3. If A = and I = , then show that I+A= (I – A)

Q.4. Express the matrix A= as the sum of symmetric and skew-symmetric matrix

Q.5. Obtain the inverse of the matrix A = using elementary transformations.

Q.6. If f(x)= Prove that f(x). f(y) = f(x + y)

Q.7. Show that the matrix B^¢AB is symmetric or skew-symmetric according as A is symmetric or skew symmetric.

Q.8. If A and B are invertible matrices of the same order, then prove that (AB)^-1 = B^-1A^-1

Q.9. Let f(x) = x² – 5x + 6. Find f(A) If A =

Q.10. If A = Show that A² -5A + 7I = 0, Use this to find A⁴.

Q.11. Express the matrix A = as the sum of a symmetric and a skew-symmetric matrix.

Q.12. Find the values of x, y, z if the matrix A = satisfy the equation A¢A = I₃.

Q.13. Show that : =

Q.14. Show that the following system of equations is consistent 2x – y + 3z = 5, 3x + 2y – z = 7, 4x + 5y – 5z = 9, Also, find the solution.

DETERMINANTS

Class - XII

Q.1. Prove that : = (1- x³)²

Q.2. Find the equation of the line joining A(1,3) and B(0,0) using determinants and find if D (K, 0) is a point such that area of a triangle ABD is 3 square units.

Q.3. If A = Verify that A³ – 6A² + 9A – 4I = 0 and hence find A^-1

Q.4. Prove that : = (1 – x²)

Q.5. Solve by matrix method:

2x + y + z = 1

x - 2y – z = 3/2

3y - 5z = 9

Q.6. Prove that :

= a³

Q.7. Prove that : = abc + bc + ca + ab.

Q.8. Solve : = 0

Q.9. Using determinants, find the area of the triangle whose vertices are (1, 4), (2, 3), (-5, 3). Are the given points collinear.

Q.10. If the points (a₁, b₁), (a₂, b₂) and (a₁ + a₂, b₁ + b₂) are collinear, Show that a₁b₂ = a₂b₁.

Q.11. If a, b, c are all positive and are p^th , q^th and r^th terms of G.P., then show that

D = = 0

Q.12. If = 0, then Prove that

a, b, c are in G.P or x, y, z are in G.P

Integrals

Class – XII

Q.1.

Q.2. dx

Q.3.

Q.4.

Q.5.

Q.6.

Q.7.

Q.8.

Q.9.

Q.10. Evaluate: .

Q.11.

Q12.

Q.13.

Q.14

Q.15.

Q.16.

Q.17.

Q.18. Evaluate

Differential Equations

Class – XII

Q.1. sec²x. tany dx + sec²y. tanx dy = 0.

Q.2. xdy – ydx =

Q.3. 2xy + y² – 2x² dy/dx = 0, y = 2 when x = 1.

Q.4. x logx. dy/dx + y = (2/x) logx

Q.5. y dx + (x – y²) dy = 0

Q.6. (tan^-1y - x)dy = (1 + y²) dx

Q.7. (1 + x²)dy/dx + 2xy = ; y = 0 when x = 1

Q.8. x²dy + (xy + y²)dx = 0, y = 1 when x = 1

Q.9. dx/dy + ycotx = 2x + x²cotx, y = 0 when x =

Q.10. (x + y)dy = dx

Q.11. x (xdy – ydx) = ydx, y(1) = 1.

Q.12. dy/dx = cos(x + y) + sin(x + y)

Q.13. x dy/dx = y – x tan(y/x).

Q.14. (x² + 1) dy/dx + 2xy =

Q.15. (x² + y²)dx + xy.dy = 0, y(1) = 1

Q.16. (x + y + 1)² dy = dx, y(-1) = 0

Q.17. (xy² + 2x)dx + (x²y + 2y)dy = 0

Q.18. dy/dx + = cosx +

Q.19. Find the differential equation of all circles in the first quadrant which touch the co-ordinate axis.

Q.20. Form the differential equation corresponding to y² = m(a² – x²) by eliminating parameters m and a.

Application of Integrals

Class – XII

Q.1. Find the area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle

Q.2. Find the area of the region bounded by the ellipse .

Q.3. Find the area of the region bounded by the parabola y = x² and y = .

Q.4. Find the area of the smaller part of the circle x² + y² = a² cut off by the linex=^.

Q.5. Using integration, find the area of the region bounded by the triangle whose vertices are (1, 0), (2,2) and (3, 1).

Q.6. Prove that the curves y² = 4x and x² = 4y divide the area of the square bounded by x=0, x=4, y=4 and y=0 into three equal parts.

Q.7. Sketch the graph of y=

Q.8. Using the method of integration, find the area bounded by the curve .

Q.9. Find the area of the smaller region bounded by the ellipse .

Q.10. Using integration, find the area of the triangular region, the equations of whose sides are y=2x + 1, y=3x +1 and x = 4.

Q.11. Find the area of the region

Q.12. Find the area of the region between the circles x² + y² = 4 and (x – 2)² + y² = 4.

Q.13. Find the area bounded by the ellipse and the co-ordinates x = ae and x = 0, where b²=a²(1 – e²) and e<1.

Q.14. Find the area bounded by the curve y² = 4a²(x – 1) and the lines x = 1and y = 4a.

Q.15. Using integration, find the area of the region bounded by the following curves, after making a rough sketch:

y = 1 +

Q16. Draw a rough sketch of the curves y = sinx and y = cosx as x varies from o to and find the area of the region enclosed by them and x-axis.

Continuity & Differentiation

Class – XII

Q.1. Find the values of a and b such that the function defined by

f(x) = ( 5, if x £ 2

ax + b if 2

21, if x 10 ) is a continuous function

Q.2. Find of sin²y + cos (xy) = p

Q.3. Differentiate w.r.t. x (x cosx)^x + (x sinx)^1/x

Q.4. If x = , y = , show that

Q.5. If y = (tan^-1x)², show that

(x² + 1)² y₂ + 2x (x² + 1) y₁ = 2.

Q.6. Differentiate sin^-1 w.r.t. x

Q.7. If x for -1

Q.8. Find if y = a ^{t + 1/t} , x = ( t + 1/t)^a

Q9. Discuss the continuity of the function given by:-

f(x) = êx-1ç + çx-2ê at x = 1, and x = 2.

Q10 If the function f(x) is given by f(x) = { 3ax + b) if x>1

11 if x = 1

(5ax – 2b if x<1}

is continuous at x = 1, find the values of a and b.

Q11 If y = [x + ]ⁿ, then prove that

Q12 Prove :

Q13 Find when y = sec^-1

Q14 If e^x + e^y = e^x+y, prove that

Q15 Given that cos-------------------= prove that

Q16 If x=a(q + sinq), y= a(1+ cosq), prove that

VECTOR ALGEBRA

Class – XII

Q.1. Find a vector in the direction of vector that has magnitude 7 units.

Q.2. Show that the points A, B and C with position vectors, respectively, form the vertices of a right angled triangle.

Q.3. Find , if two vectors are such that .

Q.4. Find the area of the parallelogram whose adjacent sides are determined by the vectors

Q.5. If a unit vector makes angles with , and acute angle θ with , then find θ and hence

the components of .

Q.6. Let,and be three vectors such that and each one of them being perpendicular to the sum of other two, Find

Q.7. Find the value of

Q.8. The scalar product of the vector with a unit vector along the sum of vectors Find the value of l.

Q.9. If the sum of two unit vectors is a unit vector, Prove that the magnitude of their difference is .

Q.10. If are position vectors of points A and B respectively, then find the position vector of points of trisection of AB.

Q.11. Prove that the line segment joining the mid-points of two sides of a triangle is parallel to the third side and equal to half of it.

Q.12. ABCD is a parallelogram. If the coordinates of A, B, C are (-2, -1), (3, 0) and (1, -2) respectively, Find the co-ordinate of D.

Q.13. Show that the points A, B, C with position vectors

Q.14. If a vector makes a, b, with OX, OY and OZ respectively, prove that sin²a+sin²b+sin²γ=2.

Q.15. If inclined at an angle , then prove that sin = .

Q.16 If .

Q.17. If .

Three Dimensional Geometry

Class – XII

Q.1. Find the direction cosines of X, Y and Z-axis.

Q.2. Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector .

Q.3. Find the value of p so that the lines,

Q.4. Find the shortest distance between the lines whose vector equations are:-

Q.5. Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 2x – 3y + 4z -6 = 0.

Q.6. Find the vector equation of the plane passing through the intersection of planes

Q.7. Find the angle between the line

Q.8. Prove that if a plane has the intercepts a, b, c and is at a distance of p units from the origin, then

Q.9. Show that the angles between the diagonals of a cube is cos^-1.

Q.10. Find the equation of the line passing through the point (-1, 3, -2) and perpendicular to the lines

Q.11. Find the foot of the perpendicular drawn from the point (0, 2, 3) on the line Also, find the length of the perpendicular.

Q.12. Find the shortest distance between the following pairs of lines whose cartesian equations are :

Q.13. A plane meets the coordinate axis in A, B, C such that the centroid of triangle ABC is the point (p,q,r). Show that the equation of the plane is .

Q.14. Find the equation of the plane passing through the point (1, 1, -1) and perpendicular to the planes x + 2y + 3z – 7=0 and 2x – 3y + 4z = 0.

Q.15. Find the distance between parallel planes,

Q.16. Show that the lines :

are coplanar.

Also, find the plane containing these two lines.

Class-XII

PROBABILITY

Q.1. A die is thrown twice and the sum of the numbers appearing is observed to be 6. What is the conditional probability that the number 4 has appeared at least once.

Q.2. Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that at least one is a girl.

Q.3. If A and B are two independent events, show that the probability of occurrence of at least one of A and B is given by :

1 – P(A').P(B')

Q.4. Probability of solving specific problem independently by A and B are ½ and 1/3 respectively. If both try to solve the problem independently, find the probability that the problem is solved.

Q.5. An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour is noted and is returned to the urn. Moreover, 2 additional balls of the colour drawn are put in the urn and then a ball is drawn at random. What is the probability that the second ball is red.

Q.6. Two cards are drawn simultaneously (or successively without replacement) from a well shuffled pack of 52 cards. Find the mean, variance and standard deviation of the number of kings.

Q.7. In an examination, 20 questions of true-false type are asked. Suppose a student tosses a fair coin to determine his answer to each question. If the coin falls heads, he answers ‘true’, if it falls tails, he answers ‘false’. Find the probability that he answers at least 12 questions correctly.

Q.8. A die is thrown again and again until three sixes are obtained. Find the probability of obtaining the third six in the sixth throw of the die.

Q.9. If P(A) = 3/8, P(B) = ½ and P (AÇB) = ¼, find P and P.

Q.10. The probability that a certain person will buy a shirt is 0.2, the probability that he will buy a trouser is 0.3 and the probability that he will buy a shirt given that he buys a trouser is 0.4. Find the probability that he will buy both a shirt and a trouser. Find also the probability that he will buy a trouser given that he buys a shirt.

Q.11. A can solve 90% of the problem given in a book and B can solve 70%. What is the probability that at least one of them will solve the problem, selected at random from the book.

Q.12. Three persons A, B, C throw a die in succession till one gets a ‘six’ and wins the game. Find their respective probabilities of winning, if A begins.

Q.13. A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is a six. Find the probability that it is actually a six.

Q.14. An urn contains 4 white and 3 red balls. Find the probability distribution of the number of red balls in a random draw of three balls.

Q.15. In a meeting 70% of the members favour a certain proposal,30% being opposed. A member is selected at random and let X=0 if he opposed and X =1 if he is in favour. Find E(x) and Var(x).

Q.16. Find the probability distribution of the number of doublets in 4 throws of a pair of dice.

LINEAR PROGRAMMING

Q.1. One kind of cake requires 200gm of flour and 25gm of fat, and another kind of cake requires 100gm of flour and 50gm of fat. Find the maximum number of cakes which can be made from 5 kg of flour and 1kg of fat assuming that there is no shortage of the other ingredients used in making the cakes.

Q.2. A dietician wishes to mix together two kinds of food X and Y in such a way that the mixture contains at least 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C. The Vitamin contents of one Kg food is given below :-

Food	Vitamin A	Vitamin B	Vitamin C
X	1	2	3
Y	2	2	1

One Kg of food X costs Rs. 16 and one Kg of food Y costs Rs. 20. Find the least cost of the mixture which will produce the required diet.

Q.3. Maximise and Minimise:

Z = x + 2y

Subject to constraints x + 2y ³100, 2x – y £ 0, 2x + y £ 200, x, y ³ 0

Q.4. An aeroplane can carry a maximum of 200 passengers. A profit of Rs. 1000 is made on each executive class ticket and a profit of Rs. 600 is made on each economy class ticket. The airline reserves at least 20 seats for executive class. However, at least 4 times as many passengers prefer to travel by economy class than by the executive class. Determine how many tickets of each type must be sold in order to maximise the profit for the airline. What is the maximum profit.

Q.5. Two godowns A and B have grain capacity of 100 quintals and 50 quintals respectively. They supply to 3 ration shops, D, E and F whose requirements are 60, 50 and 40 quintals respectively. The cost of transportation per quintal from the godowns to the shops are given in following table :

Transportation Cost Per Quintal (in Rs.)

From / To	A	B
D	6	4
E	3	2
F	2.50	3

How should the supplies be transported in order that the transportation cost is minimum. What is the minimum cost.

Q.6. Two tailors A and B earn Rs. 150 and Rs. 200 per day respectively. A can stitch 6 shirts and 4 pants per day, while B can stitch 10 shirts and 4 pants per day. Form a linear programming problem to minimize the labour cost to produce at least 60 shirts and 32 pants.

Q.7. Solve the following L.P.P graphically:

Maximise: Z = 60x + 15y

Subject to constraints

x + y £ 50

3x + y £ 90, x, y ³ 0

Q.8. A dealer wishes to purchase a number of fans and sewing machines. He has only Rs 5,760 to invest and has space for at most 20 items. A fan costs him Rs. 360 and a sewing machine Rs 240. His expectation is that he can sell a fan at a profit of Rs. 22 and a sewing machine at a profit of Rs. 18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximise his profit.

Q.9. If a young man drives his vehicle at 25 km/hr, he has to spend Rs. 2/km on petrol. If he drives it at a faster speed of 40km/hr, the petrol cost increases to Rs. 5/km. He has Rs. 100 to spend on petrol and travel within one hour. Express this as an L.P.P. and solve.