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Board Paper of Class 12-Science 2012 Maths (SET 3) - Solutions

General Instructions:
i. All questions are compulsory.
ii. The question paper consists of 29 questions divided into three sections A, B and C. Section A comprises of 10 questions of one mark each, Section B comprises of 12 questions of four marks each, and Section C comprises of 7 questions of six marks each.
iii. All questions in section A are to be answered in one word, one sentence or as per the exact requirements of the question.
iv. There is no overall choice. However, internal choice has been provided in 4 questions of four marks each and 2 questions of six marks each. You have to attempt only one of the alternatives in all such questions.
v. Use of calculators is not permitted.
• Question 1

Let * be a ‘binary’ operation on N given by a * b = LCM (a, b) for all a, b ∈ N. Find 5 * 7.

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• Question 2

Write the principal value of VIEW SOLUTION
• Question 3 VIEW SOLUTION
• Question 4 VIEW SOLUTION
• Question 5 VIEW SOLUTION
• Question 6 VIEW SOLUTION
• Question 7

Find ‘λ’when the projection of VIEW SOLUTION
• Question 8

If a line has direction ratios 2, − 1, − 2, then what are its direction cosines?

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• Question 9

Find the sum of the following vectors: VIEW SOLUTION
• Question 10

If , write the cofactor of the element a22.

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• Question 11 OR VIEW SOLUTION
• Question 12

Let A = R − {3} and B = R − {1}. Consider the function f : A → B defined by Show that is one-one and onto and hence find f−1.

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• Question 13

Find the point on the curve y = x3 − 11x + 5 at which the equation of tangent is y = x − 11.

OR

Using differentials, find the approximated value of VIEW SOLUTION
• Question 14 OR VIEW SOLUTION
• Question 15

If are three vectors such that Find the value of VIEW SOLUTION
• Question 16

Solve the following differential equation: VIEW SOLUTION
• Question 17 OR

If sin y = x sin(a + y), prove that VIEW SOLUTION
• Question 18

How many times must a man toss a fair coin, so that the probability of having at least one head is more than 80%?

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• Question 19

Using properties of determinants, prove the following: VIEW SOLUTION
• Question 20

If y = sin−1x, show that VIEW SOLUTION
• Question 21

Find the particular solution of the following differential equation: VIEW SOLUTION
• Question 22

Find the equation of a line passing through the point P(2, −1, 3) and perpendicular to the lines VIEW SOLUTION
• Question 23

Using matrices, solve the following system of linear equations: OR

Using elementary operations, find the inverse of the following matrix: VIEW SOLUTION
• Question 24

Show that the height of a closed right circular cylinder of given surface and maximum volume, is equal to the diameter of its base.

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• Question 25 OR VIEW SOLUTION
• Question 26

Find the equation of the plane determined by the points A (3, −1, 2), B (5, 2, 4) and C (− 1, −1, 6) and hence find the distance between the plane and the point P (6, 5, 9)

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• Question 27

A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of Rs 17.50 per package on nuts and Rs 7 per package of bolts. How many packages of each should be produced each day so as to maximize his profits if he operates his machines for at the most 12 hours a day? From the above as a linear programming problem and solve it graphically.

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• Question 28

Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5 black balls. Two balls are transferred at random from Bag I to Bag II and then a ball is drawn from Bag II. The ball so drawn is found to be red in colour. Find the probability that the transferred ball were both black.

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• Question 29

Using the method of integration, find the area of the region bounded by the following lines: 5x − 2y − 10 = 0, x + y − 9 = 0, 2x − 5y − 4 = 0

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