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# Board Paper of Class 12-Commerce 2013 Maths Delhi(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

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 2

Write unit vector in the direction of the sum of vectors and VIEW SOLUTION
• Question 3

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

x + y + z.

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• Question 4

Write the degree of the differential equation .

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• Question 5

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 6

If , then find the matrix A.

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• Question 7

If , then write the value of x.

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• Question 8

Find the value of a if VIEW SOLUTION
• Question 9

Write the value of .

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• Question 10

Write the principal value of .

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• Question 11

A speaks truth in 75% of the cases, while B in 90% of the cases. In what percent of cases are the likely to contradict each other in stating the same fact?

Do you think that statement of B is true?

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• Question 12

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 13

Using vectors, find the area of the triangle ABC with vertices A(1, 2, 3), B(2, −1, 4) and C(4, 5, −1).

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• Question 14

Evaluate: .

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• Question 15

Evaluate: .

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• Question 16

Evaluate: OR

Evaluate: VIEW SOLUTION
• Question 17

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

OR

If x = a sin t and y = a , find VIEW SOLUTION
• Question 18

If , show that .

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• Question 19

Differentiate the following function with respect to x: VIEW SOLUTION
• Question 20

Using properties of determinants prove the following: .

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• Question 21

Find the value of the following: .

OR

Prove that: VIEW SOLUTION
• Question 22

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

Hence find f −1.

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• Question 23

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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• Question 24

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 25

Find the coordinates of the point where the line through (3, −4, −5) and (2, −3, 1) crosses the plane, passing through the points (2, 2, 1), (3, 0, 1) and (4, −1, 0).

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• Question 26

Show that the differential equation is homogeneous. Find the particular solution of this differential equation, given that x = 1 when y = 1.

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• Question 27

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 28

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 29

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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