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Board Paper of Class 12-Commerce 2012 Maths Delhi(SET 2) - 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 5

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

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  • Question 6

    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 9

    Find the sum of the following vectors:

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  • Question 10

    If , write the cofactor of the element a32.

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  • Question 11

    If are three vectors such that Find the value of

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  • Question 12

    Solve the following differential equation:

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  • Question 13

    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 14

    OR

    If sin y = x sin(a + y), prove that

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  • Question 15

    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 17

    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

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  • Question 19

    Using properties of determinants, prove the following:

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  • Question 20

    If y = 3 cos (log x) + 4 sin(log x), show that

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  • Question 21

    Find the equation of the line passing through the point (− 1, 3, − 2) and perpendicular to the lines

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  • Question 22

    Find the particular solution of the following differential equation:

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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

    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 25

    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

    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 28

    A girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes the number of heads. If she gets 1, 2, 3 or 4, she tosses a coin two times and notes the number of heads obtained. If she obtained exactly two heads, what is the probability that she threw 1, 2, 3 or 4 with the die?

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  • Question 29

    Using the method of integration, find the area of the region bounded by the following lines: 3x y − 3 = 0, 2x + y − 12 = 0, x − 2y − 1 = 0

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