Board Paper of Class 12Commerce 2012 Maths (SET 3)  Solutions
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
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 Question 3
 Question 4
 Question 5
 Question 6
 Question 7
Find ‘λ’when the projection of
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 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:
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 Question 10
If , write the cofactor of the element a_{22}.
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 Question 11
 Question 12
Let A = R − {3} and B = R − {1}. Consider the function f : A → B defined by
Show that is oneone and onto and hence find f^{−1}.
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 Question 13
Find the point on the curve y = x^{3} − 11x + 5 at which the equation of tangent is y = x − 11.
OR
Using differentials, find the approximated value of
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 Question 14
 Question 15
If are three vectors such that Find the value of
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 Question 16
Solve the following differential equation:
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 Question 17
 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:
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 Question 20
If y = sin^{−1}x, show that
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 Question 21
Find the particular solution of the following differential equation:
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 Question 22
Find the equation of a line passing through the point P(2, −1, 3) and perpendicular to the lines
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 Question 23
Using matrices, solve the following system of linear equations:
OR
Using elementary operations, find the inverse of the following matrix:
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 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
 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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