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Board Paper of Class 12-Commerce 2013 Maths (SET 1) - 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 6

    Write the degree of the differential equation.

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  • Question 7

    If and are two equal vectors, then write the value of

    x + y + z.

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  • Question 8

    If a unit vector makes angles with, with and acute angles θ with, then find the value of θ.

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  • Question 9

    Find the Cartesian equation of the line which passes through the point (−2, 4, −5) and is parallel to the line.

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  • Question 10

    The amount of pollution content added in air in a city due to x-diesel vehicles is given by P(x) = 0.005x3 + 0.02x2 + 30x. Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above question.

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  • Question 11

    Show that the function f in defined as is one-one and onto.

    Hence find f −1.

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  • Question 12

    Find the value of the following:

    .

    OR

    Prove that:

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  • Question 13

    Using properties of determinants prove the following:

    .

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  • Question 14

    Differentiate the following function with respect to x:

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  • Question 16

    Show that the function is continuous but not differentiable at x = 3.

    OR

    If x = a sin t and y = a, find

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  • Question 20

    If and are two vectors such that, then prove that vector is perpendicular to vector.

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  • Question 21

    Find the coordinates of the point, where the line intersects the plane x − y + z − 5 = 0. Also find the angle between the line and the plane.

    OR

    Find the vector equation of the plane which contains the line of intersection of the planes and which is perpendicular to the plane.

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  • Question 22

    A speaks truth in 60% of the cases, while B in 90% of the cases. In what percent of cases are they likely to contradict each other in stating the same fact? In the cases of contradiction do you think, the statement of B will carry more weight as he speaks truth in more number of cases than A?

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  • Question 23

    A school wants to award its students for the values of Honesty, Regularity and Hard work with a total cash award of Rs 6,000. Three times the award money for Hard work added to that given for honesty amounts to Rs 11,000. The award money given for Honesty and Hard work together is double the one given for Regularity. Represent the above situation algebraically and find the award money for each value, using matrix method. Apart from these values, namely, Honesty, Regularity and Hard work, suggest one more value which the school must include for awards.

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  • Question 24

    Show that the height of the cylinder of maximum volume, which can be inscribed in a sphere of radius R is. Also find the maximum volume.

    OR

    Find the equation of the normal at a point on the curve x2 = 4y which passes through the point (1, 2). Also find the equation of the corresponding tangent.

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  • Question 25

    Using integration, find the area bounded by the curve x2 = 4y and the line x = 4y − 2.

    OR

    Using integration, find the area of the region enclosed between the two circles x2 + y2 = 4 and (x − 2)2 + y2 = 4.

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  • Question 26

    Show that the differential equation 2yx/y dx + (y − 2x ex/y) dy = 0 is homogeneous. Find the particular solution of this differential equation, given that x = 0 when y = 1.

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  • Question 27

    Find the vector equation of the plane passing through three points with position vectors. Also find the coordinates of the point of intersection of this plane and the line.

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  • Question 28

    A cooperative society of farmers has 50 hectares of land to grow two crops A and B. The profits from crops A and B per hectare are estimated as Rs 10,500 and Rs 9,000 respectively. To control weeds, a liquid herbicide has to be used for crops A and B at the rate of 20 litres and 10 litres per hectare, respectively. Further not more than 800 litres of herbicide should be used in order to protect fish and wildlife using a pond which collects drainage from this land. Keeping in mind that the protection of fish and other wildlife is more important than earning profit, how much land should be allocated to each crop so as to maximize the total profit? Form an LPP from the above and solve it graphically. Do you agree with the message that the protection of wildlife is utmost necessary to preserve the balance in environment?

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  • Question 29

    Assume that the chances of a patient having a heart attack is 40%. Assuming that a meditation and yoga course reduces the risk of heart attack by 30% and prescription of certain drug reduces its chance by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options, the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga. Interpret the result and state which of the above stated methods is more beneficial for the patient.

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