Board Paper of Class 12Commerce 2013 Maths (SET 2)  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
Write the differential equation representing the family of curves y = mx, where m is an arbitrary constant.
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 Question 2
If A_{ij} is the cofactor of the element a_{ij} of the determinant, then write the value of a_{32} · A_{32}.
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 Question 3
Write the value of.
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 Question 4
For what value of x, is the matrix a skewsymmetric matrix?
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 Question 5
Write the principal value of.
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 Question 6
The money to be spent for the welfare of the employees of a firm is proportional to the rate of change of its total revenue (marginal revenue). If the total revenue (in rupees) received from the sale of x units of a product is given by R(x) = 3x^{2} + 36x + 5, find the marginal revenue, when x = 5, and write which value does the question indicate.
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 Question 7
Find the length of the perpendicular drawn from the origin to the plane 2x − 3y + 6z + 21 = 0.
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 Question 8
Find, if for a unit vector .
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 Question 9
If matrix and A^{2} = pA, then write the value of p.
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 Question 10
A and B are two points with position vectors respectively. Write the position vector of a point P which divides the line segment AB internally in the ratio 1 : 2.
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 Question 11
Differentiate the following with respect to x :
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 Question 12
 Question 13
Using properties of determinants, prove the following:
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 Question 14
 Question 15
Consider f : R_{+}→ [4, ∞) given by f(x) = x^{2} + 4. Show that f is invertible with the inverse f^{−1} of f given by, where R_{+} is the set of all nonnegative real numbers.
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 Question 16
Find the value of k, for which
is continuous at x = 0.
OR
If x = a cos^{3}θ and y = a sin^{3}θ, then find the value of at .
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 Question 17
Show that the lines
are intersecting. Hence find their point of intersection.
OR
Find the vector equation of the plane through the points (2, 1, −1) and (−1, 3, 4) and perpendicular to the plane x − 2y + 4z = 10.
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 Question 18
The probabilities of two students A and B coming to the school in time are respectively. Assuming that the events, ‘A coming in time’ and ‘B coming in time’ are independent, find the probability of only one of them coming to the school in time.
Write at least one advantage of coming to school in time.
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 Question 19
If x^{y} = e^{x − y}, prove that
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 Question 20
Evaluate:
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 Question 21
Evaluate:
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 Question 22
If , then find the value of λ, so that are perpendicular vectors.
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 Question 23
Find the equation of the plane passing through the line of intersection of the planes and , whose perpendicular distance from origin is unity.
OR
Find the vector equation of the line passing through the point (1, 2, 3) and parallel to the planes.
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 Question 24
In a hockey match, both teams A and B scored same number of goals up to the end of the game, so to decide the winner, the referee asked both the captains to throw a die alternately and decided that the team, whose captain gets a six first, will be declared the winner. If the captain of team A was asked to start, find their respective probabilities of winning the match and state whether the decision of the referee was fair or not.
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 Question 25
A manufacturer considers that men and women workers are equally efficient and so he pays them at the same rate. He has 30 and 17 units of workers (male and female) and capital respectively, which he uses to produce two types of goods A and B. To produce one unit of A, 2 workers and 3 units of capital are required while 3 workers and 1 unit of capital is required to produce one unit of B. If A and B are priced at Rs 100 and Rs 120 per unit respectively, how should he use his resources to maximise the total revenue? Form the above as an LPP and solve graphically.
Do you agree with this view of the manufacturer that men and women workers are equally efficient and so should be paid at the same rate?
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 Question 26
Find the area of the greatest rectangle that can be inscribed in an ellipse.
OR
Find the equations of tangents to the curve 3x^{2} − y^{2} = 8, which pass through the point.
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 Question 27
The management committee of a residential colony decided to award some of its members (say x) for honesty, some (say y) for helping others and some others (say z) for supervising the workers to keep the colony neat and clean. The sum of all the awardees is 12. Three times the sum of awardees for cooperation and supervision added to two times the number of awardees for honesty is 33. If the sum of the number of awardees for honesty and supervision is twice the number of awardees for helping others, using matrix method, find the number of awardees of each category. Apart from these values, namely, honesty, cooperation and supervision, suggest one more value which the management of the colony must include for awards.
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 Question 28
Find the area of the region {(x, y): y^{2}≤ 6ax and x^{2} + y^{2}≤ 16a^{2}} using method of integration.
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 Question 29
Show that the differential equation is homogeneous. Find the particular solution of this differential equation, given that when x = 1.
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