MATHS QUESTION PAPER 2006

SECTION – A

Q. 1. Express the matrix as the sum of a symmetric and a skew-symmetric matrix.

Q. 2. Using properties of determinants, prove the following:

Q. 3. Solve the following differentia equation:

Q. 4. Verify that is solution of the differential equation

Or

Solve the following differential equation:

Q. 5.

Or

Q. 6.

Q. 7. Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the probability distribution of number of jacks.

Q. 8. A and B toss a coin alternately till one of them gets a head and wins the game. If A starts first, find the probability that B will win the game.

Q. 9. Shove that the following argument is in valid:

Or

Construct an input/output table for the following circuit:

Q. 10.

Q. 11. Differentiate w. r. t. x from first principle.

Q. 12.

Or

Q. 13.

Q. 14.

Q. 15. Find the intervals in which the function f (x) = x³ - 12x² + 36x + 17 is (a) increasing, (b) decreasing.

Q. 16. Using matrices, solve the following system of equations:

Q. 17. An open box, with a square base, is to be made out of a given quantity of metal sheet of area C². Show that the maximum volume of the box is

Or

A window is in the form of a rectangle surmounted by a semi-circle. If the total perimeter of the window is3O m, find the dimensions of the window so that maximum light is admitted.

Q. 18. Find the area of the region bounded by the parabola x² = 4y and the line x = 4y – 2.

Or

SECTION – B

Q. 19. Find the angle between the vectors

Q. 20. Using vectors, prove that in a

where a, b and c are lengths of the sides opposite, respectively, to the angles A, B and C of

Q. 21. Find the coordinates of the point where the line

meets the plane x + y + 4z = 6.

Q. 22. Find the image of the point (1, 2, 3) in the plane x + 2y + 4z = 38.

Q. 23. Find the radius of the circular section of sphere by the plane .

Q. 24. The resultant of two forces acting at an angle is equal to and when they act an angle , the resultant is equal to, Show that

Q. 25. A body of mass 10 kg hags by a string from a fixed point. The string is drawn out of the vertical by applying a force 49 N to the body. In which direction should this force be applied in order that, in equilibrium, the deflection of the string from the vertical may be of 300? Also find the tension in the string.

Q. 26. A particle is projected so as to graze the tops of two walls, each of height 10m, at distances 15 m and 45 m respectively from the point of projection. Find the angle of projection.

SECTION – C

Q. 19. In a bolt factory, machines A, B, C manufacture 25%, 35% and 40% respectively of the total bolts. Of their output 5%, 4% and 2% respectively are defective bolts. A bolt is drawn at random and is found to be defective. Find the probability that it is manufactured by machine B.

Q. 20. The mean and variance of the binomial distribution are 4 and 4/3 respectively. Find the distribution and

Q. 21. The banker’s discount and banker’s gain on a certain bill of exchange, due after a certain time, are respectively Rs.1,250 and Rs.50. Find the face value of the bill.

Q. 22. A bill for Rs.5,050 is drawn on April 13, 2005. It is discounted on July 4,2005 at 5% per annum. If the banker’s gain in the transaction is Rs. 0.50 find the nominal date of maturity of the bill.

Q. 23. A machine costs a company Rs 5,75,000 and its effective life is estimated to be 20 years. A sinking fund is created for replacing the machine at the end of its life-time when its scrap realises a sum of Rs.75,000 only. Calculate what amount should be provided every year out of profits, for the sinking fund if it accumulates an interest of 5% per annum, compounded annually. [Use(1.05)20 = 2.6551]

Q. 24. A and B are partners sharing profits and losses in the ratio 5 : 3. C pays Rs.96,000 as premium for the goodwill of the business and is admitted for 1/5th share. Show how this amount will be shared by A and B, assuming that the share of C is contributed by A and B in their profit sharing ratio. Find also the new profit sharing ratio.

Q. 25. The manufacturing cost of an item consists of Rs. 900 as overheads, the material costs Rs.3 per item and Iabour cost is x²/100 for x items produced. How many items must be produced to have minimum average cost?

Q. 26. David wants to invest at most Rs. 12,000 in bonds A and B. According to the rule, he has to invest at least Rs. 2,000 in Bond A and at least Rs.4,000 in Bond B. If the rates of interest on Bonds A and B respectively are 8% and 10% per annum, formulate the problem as L.P.P. and solve it graphically for maximum interest. Also determine the maximum interest received in a year.

CHEMISTRY BOARD QUESTION PAPER 2006

Q. 1. Name the non-stoichiometric point defect responsible for colour in alkali halides. (1)

Q. 2. Define ‘mole fraction’ of a substance in a solution. (1)

Q. 3. A reaction is 50% complete in 2 hours and 75% complete in 4 hours. What is the order of the reaction? (1)

Q. 4. Write the IUPAC name of CH₃COCH ₂ COCH ₃. (1)

Q. 5. Give a chemical test to distinguish between a primary and a secondary amine. (1)

Q. 6. Account for the following:

1. N₂ has higher bond dissociation energy than NO. (1)
2. N₂ and CO both have same bond order but CO is more reactive than N₂ (1)

Q. 7. At absolute zero, an exothermic reaction is always spontaneous but at temperatures above absolute zero, we have to consider both enthalpy and entropy before we can predict spontaneity. Why? (2)

Q. 8. Write the chemical equations involved in the preparation of the following: (2)

1. XeF₄
2. H₃PO₃

Q 9 Why is the +2 oxidation state of manganese quite stable, while the same is not true for iron? [Mn = 25, Fe = 26] (2)

Q. 10. Differentiate between conformation and configuration in open chain molecules by giving one example each. (2)

Q. 11. Give reasons for the following:

1. Ortho-nitrophenol is more acidic than ortho-methoxyphenol. (1)
2. Glycerol is used in cosmetics. (1)

Q. 12. Write the structures of monomoers used and one use of each of the following polymers: (1)

1. Teflon
2. Buna-N

Or

What are biodegradale polymers? Give two examples.

Q. 13. What is meant by dual nature of electrons? Calculate the energy and wavelength of the photon emitted by hydrogen atom when the electron makes a transition from n =2 to n = 1. Given that the ionization potential is 13.6 eV.
[1 eV =1.6 x 10^—19 J]

Q. 14. Calculate the distance between Na⁺ and Cl^— ions in NaCl crystal if its density is 2.165 g cm^-3. [ Molar mass of NaCl= 58.5 g mole^-1; N_A = 6.02 x 10²³ moI^-1] (3)

Q. 15.

1. Urea forms an ideal solution in water. Determine the vapour pressure of an aqueous solution containing 10% by mass of urea at 40° C.
   (Vapour pressure of water at 40° C = 55.3 mm of Hg) (2)
2. Why is freezing point depression of 0.1 M sodium chloride solution nearly twice that of 0.1 M glucose solution? (1)

Q. 16. How is the concept of coupling reactions useful in explaining the occurrence of non-spontaneous thermochemical reactions? Explain giving an example. (3)

Q. 17. A certain reaction is 50% complete in 20 minutes at 300 K and the same reaction is again 50% complete in 5 minutes at 350 K. Calculate the activation energy. if it is a first order reaction. [R = 8.314 JK ^-1 mol ^-1; log 4 = 0.602] (3)

Q. 18.

1. In which of the following does adsorption take place and why? (1)
1. Silica gel placed in the atmosphere saturated with water.
2. Anhydrous CaCl₂ placed in the atmosphere saturated with water.
2. How does BF₃ act as a catalyst in industrial process? (1)
3. Give an example of shape-selective catalysis. (1)

Or

1. What are micelles? How do they differ from ordinary colloidal particles? Give two examples of micelles forming substances. (2)
2. State Hardy-Schulze rule. (1)

Q. 19.

1. Write the electronic configuration of the element with atomic number 102. (1)
2. What is lanthanoid contraction? What is its effect on the chemistry of the elements which follow the lanthanoids? (2)

Q. 20.

1. Using valence bond theory, predict the shape and magnetic character of [Ni (CO)₄]. [Ni = 28] (2)
2. Give one example of application of coordination compounds in medicine. (1)

Q. 21.

1. State Group Displacement Law. Calculate the number of ά- particles and ß-- particles emitted when U(238) changes to Pb(208) (2)
2. What is meant by K-capture in nuclear chemistry? (1)

Q. 22.

1. Write the steps and conditions involved in the following conversions:
1. Acetophenone to 2-phenyl-2butanol (1)
2. Propene to acetone. (1)
2. Give a chemical test to distinguish between Methyl acetate and Ethyl acetate.

Q. 23.

1. Explain the following giving suitable examples: (2)
1. Sandmeyer’s reaction
2. Coupling reaction of a diazonium salt
2. Explain the observed K _b order:
   Et₂NH > ET₃N > EtNH₂ in aqueous solution (1)

Q. 24. Define the following and give one example of each: (3)

1. Antipyretics
2. Vat dyes
3. Antibiotics

Q. 25.

1. State two advantages of H₂ – O₂ fuel cell over ordinary cell. (2)
2. Silver is electrodeposited on a metallic vessel of total surface area 900 cm² by passing a current of 0.5 amp for two hours. Calculate the thickness of silver deposited.
   [Given: Density of silver = 10.5 g cm^-3,
   Atomic mass of silver = 108 amu, F= 96,500 C mol^—1] (3)

Or

1. Give reasons for the following: (2)
1. Rusting of iron is quicker in saline water than in ordinary water.
2. Aluminium metal cannot be produced by the electrolysis of aqueous solution of aluminium salt.
2. Resistance of a conductivity cell filled with 0.1 M KCI solution is 100 ohm. If the resistance of the same cell when filled with 0.02 M K Cl solution is 520 ohms, calculate the conductivity and molar conductivity of 0.02 M K Cl solution. Conductivity of 0.1 KCI solution is 1.29 S m^--1 (3)

Q. 26. Give reasons for each of the following:

1. SiF₆^2-- is known but SiCl₆^2-- is not known. (1)
2. Sulphur in vapour state exhibits paramagnetic behaviour. (1)
3. PbO₂ is a stronger oxidizing agent than Sn0₂ (1)
4. H₃PO₂ acts as a monobasic acid. (1)
5. Bond dissociation energy of F₂ is less than that of Cl₂ (1)

Or

1. Account for the following:
1. Thermal stability of water is much higher than that of H₂S (1)
2. Anhydrous aluminium chloride acts as a catalyst. (1)
3. White phosphorus is more reactive than red phosphorus. (1)
2. Draw the structures of
1. H₃PO₃ and
2. XeOF₄ (2)

Q. 27.

1. What are essential and non-essential amino acids? Give two examples of each. (2)
2. What are the two types of photosynthesis in green plants? Give the basic equations of photosynthesis. (2)
3. Mention the two products of glycolysis. (1)

Or

1. Define the following terms: (3)
1. Co-enzymes
2. Mutation in biomolecules
3. Nucleotides
2. List four main functions of carbohydrates in organisms. (2)

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Making the correct choice

Multiple choice questions can be tough nuts to crack. Breathe easy, for here are ways in which the formidable task of choosing the right aswers can be made somewhat easy.

STRATEGIC CHOICE: Apply your mind and try to maximise your chances of getting the answers right.

A typical multiple choice question has two parts: the stem and a set of alternatives containing the key (the correct or "best" answer) and distracters (incorrect answers). Your job is to select the best alternative(s).

Multiple-choice questions are basically true/false questions arranged in groups and are usually not "tricky." Questions are designed to challenge and encourage you to think, not lead you to the wrong answer. The test taker is expected to select the best or most appropriate one among four or five given answer choices. The challenge is not to be distracted by the distracters.

Read the question carefully: Should you select one correct response or select several correct responses? Always check the context. All choices may be correct, but which fits the question best?

Example:

In a two-hour exam of multiple-choice questions, the best way to organise time is to:

A: Calculate the time available for each question, answer the easy ones, spend more time on tough ones till you run out of time, then quickly guess the answers to the rest.

B: Read the entire stem and every alternative, eliminating obviously incorrect answers.

C: Mark questions with negatives, superlatives, and qualifiers, then grammatically break down difficult questions.

D: Take a few minutes to relax by taking a few deep breaths and get focused.

Choices B, C and D are correct, but they have nothing to do with managing your time.

They do not fit the context of the question. The only choice that is correct and fits the context is A.

Three distinct methods have been identified for answering multiple choice questions

The answer-search method, using recognition memory: read the stem, derive the answer in your mind, and then find the answer within the alternatives.

Tension, however, can interfere with recognition. Being relaxed and confident helps recognition memory. Keep relaxed and "distant" from the items - wait for the correct answer to "jump" at you.

Come up with the answer in your head before looking at the alternatives - they won't trick you.

Read all the choices before choosing your answer. If a clearly correct answer doesn't emerge easily, move on to the next question.

Example: Divide x⁵ by x²

(a) x^2.5 (b) x³ (c) x⁷ (d) x^-3

You immediately realise that x³ should be the correct answer. When you look for x³ among the answer choices you find that it is choice (b) and indicate your choice as (b).

The elimination method involves ignoring the clearly incorrect alternatives and then making a selection from the remaining possibilities.

Example:

Which of the following numbers could be described as natural, rational, an integer and a whole number?

(a) 0 (b) 1 (c) 2.33 (d) -3

A quick look tells us that: (a) 0 is not a rational number (c) 2.33 is not a natural number and is also not an integer (d) -3 is not a natural number

That leaves only answer choice (b) 1 which must be the correct answer. Verify that it is natural, rational, an integer and also a whole number, and indicate your choice as (b).

Use logic and common sense for elimination. Even if you are unfamiliar with the subject, it is possible to reason out the correct answer.

Example:

Prejudice and discrimination are:

(a) Harmful to society because they waste the country's economic and social resources.

(b) Helpful because they ensure against attack from within.

(c) Harmful because they create negative images in foreign countries.

(d) Helpful because they keep the majority pure and united against minorities.

Through logic and common sense, we can eliminate choices (b) and (d). Prejudice and discrimination are not positive, desirable, or helpful. We are left with two choices, (a) or (c). The attitude of other countries is not as serious as a waste of economic and social resources, eliminating (c). So the best answer choice is (a), which offers a stronger reason why prejudice and discrimination are harmful. Notice that it is also the longest answer and a positive one (actually a double negative)

Try to identify the approach that works best for you by analysing previous tests.

Use a combination of the answer-search and elimination methods within each group of questions. Apply the answer-search strategy first to all questions within a group of questions, and then apply the elimination method to the questions you skipped in the same group before going on to the next group of questions.

The answer - search method will be more successful in groups of "easy" questions than in relatively tougher groups of questions, and takes up comparatively less time. Apply these two methods to all groups of questions, including the ones you are "doubtful" about.

Use the answer-search method in the shortest time possible, and spend more time using the elimination method. Don't waste too much time with these methods on the "doubtful" list.

You are now left with the questions you could not answer so far using both methods. Use the final strategy now!

The method of guessing intelligently: This final strategy can be applied after the other methods have failed to lead you to the correct answer.

If there are four alternatives, the probability of guessing the right answer is one in four (25 per cent).

If you can eliminate one alternative, your chances will rise to one in three (33 per cent). You can eliminate two alternatives, your chances will rise to one in two (50 per cent). So if you must guess, do it, but from among the smallest number of alternatives as possible.

Never guess blindly, unless there is no negative marking - use strategies to make an educated guess.

Multiple-answer multiple-choice questions are those that ask you to select between possible combinations of responses, and the responses form a part of the stem itself.

A statement and several conclusions that can be drawn from it are given. You have to select the correct conclusion(s).

Example:

Statement: Few shops on this road have neon lights, but they all have signboards.

Conclusions:

(a). Some shops have either signboards or neon lights

(b). Some shops have both signboards and neon lights

(c). Some shops have no neon lights

(d). Some shops have no signboards

The correct conclusion(s) are: (You have to choose one of 1,2,3 or 4)

(1) a (2) a & c (3) b (4) b & c

Here, you are not selecting the best answer; you must select the choice that contains all the correct conclusions (4. b & c in this question).

The strategy to use with such questions is to treat each statement as a true/false question.

First try to prove that (a) is false. If you cannot prove it false, accept it as true and place a mark alongside (a). Continue this way with (b), (c) and (d) statements, and then mark off the appropriate choices.

THE WORLD IS NOT ENOUGH!

High standards of education, affordable living costs and scholarships were the major attractions for Indian students who thronged the two-day European higher education fair in New Delhi.

The United Kingdom and the United States might still be the favourite destination for most students wanting to apply abroad for higher studies, but increasingly other countries around the world are vying with one another to attract the attention of Indian students. With a whole range of options in new upcoming sectors like biotechnology and evergreen sectors like engineering, students can now hope to look beyond hitherto popular destinations.

The two-day European Higher Education Fair that was held in New Delhi on November 24 and 25, probably best illustrated the extent of their interest. With over a hundred recognised European institutions from 25 countries at the fair, it was a chance for students to explore and get information about the university of their choice under one roof.

``This year, Germany, France and the Netherlands had the biggest contingents. Italy also brought in a large number of universities,'' said an organiser. Apart from the bigger education destinations, smaller countries such as Cyprus and Ireland were present to woo Indian students to their shores. While there were some students who wanted to go abroad for undergraduate degrees, most of the visitors to the fair wanted to go to Europe for pursuing their master's. ``I think opportunities in America are really hyped. It is also much more expensive to go to America I believe. Europe offers a high level of quality in education,'' said Ranjit Singh, who intends to do his master's and then Ph.D. in Europe. With high-standards of education, cheaper living costs and scholarships, Europe is becoming an attractive destination for Indian students, especially those looking for opportunities in the fields of engineering, biotechnology and sciences.

Here is a sample of what was on offer:

Sweden

KTH, Royal Institute

Main field of study: First-class education in a wide range of fields, including natural sciences, all branches of engineering, architecture, industrial management, urban planning and environmental technology. Tuition fees: There is no fee for registration, tuition or language courses. However, membership to student union is compulsory and costs about SEK 400 per year. Website: www.kth.se/eng.

France

France has about 2,56,000 students from around the world studying there. French higher education is made up, in large part, of public institutions, financed and controlled by the Government. Compared to other countries, education is affordable, but the cost varies from course to course. ``Most of the enquiries we got from students were at the master's level. We had expected that there would be enquiries in biotechnology. But we also got enquiries from people wanting to pursue forensic sciences too. There were queries about sustainable development and gender issues,'' said EduFrance India coordinator Archana Chaudhary.

Ireland

One of the biggest advantages Ireland has is that it is an English-speaking country. While the Emerald Isle is small, it is hoping to be able to have great connections with India and has recently set-up an Enterprise Ireland office in India``Our thrust areas are information technology, management and hospital management. Engineering and MBA are other fields that students can pursue in the country,'' said Gabriel McCarrick of Enterprise Ireland.

The Netherlands

The first non-English-speaking country to offer courses taught in English, The Netherlands is renowned worldwide for offering high quality education. Institutions in Holland offer about 1,150 international study programmes and courses that are taught in English. ``This was the first time that The Netherlands brought so many institutes to the fair. The country is looking seriously at attracting Indian students and setting up an education office here. The main areas that Indian students will find interesting are information technology, science and technology. We are mainly hoping to attract students at the master's level,'' said an official from The Netherlands Embassy.