Board Paper of Class 12Commerce 2010 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
Evaluate:
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 Question 2
If, then for what value of α is A an identity matrix?
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 Question 3
What is the principal value of?
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 Question 4
What is the cosine of the angle which the vector makes with yaxis?
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 Question 5
Write a vector of magnitude 15 units in the direction of vector
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 Question 6
What is the range of the function?
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 Question 7
Find the minor of the element of second row and third column (a_{23}) in the following determinant:
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 Question 8
Write the vector equation of the following line:
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 Question 9
What is the degree of the following differential equation?
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 Question 10
If, then write the value of k.
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 Question 11
 Question 12
 Question 13
On a multiple choice examination with three possible answers (out of which only one is correct) for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing?
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 Question 14
Let * be a binary operation on Q defined by
Show that * is commutative as well as associative. Also find its identify element, if it exists.
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 Question 15
Using elementary row operations, find the inverse of the following matrix:
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 Question 16
Find the Cartesian equation of the plane passing through the points A(0, 0, 0) and B(3, − 1, 2) and parallel to the line.
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 Question 17
Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are respectively, externally in the ratio 1:2. Also, show that P is the midpoint of the line segment RQ.
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 Question 18
Evaluate:
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 Question 19
 Question 20
Find the equations of the normals to the curve y = x^{3} + 2x + 6 which are parallel to the line x + 14y + 4 = 0
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 Question 21
Find the particular solution of the differential equation satisfying the given conditions:
x^{2}dy + (xy + y^{2}) dx = 0; y = 1 when x = 1.
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 Question 22
Find the general solution of the differential equation
OR
Find the particular solution of the differential equation satisfying the given conditions:
, given that y = 1 when x = 0
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 Question 23
Evaluateas limit of sums.
OR
Using integration, find the area of the following region:
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 Question 24
A small firm manufactures gold rings and chains. The total number of rings and chains manufactured per day is atmost 24. It takes 1 hour to make a ring the 30 minutes to make a chain. The maximum number of hours available per day is 16. If the profit on a ring is Rs. 300 and that on a chain is Rs. 90, find the number of rings and chains that should be manufactured per day, so as to earn the maximum profit. Make it as an L.P.P. and solve it graphically.
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 Question 25
A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn at random and are found to both clubs. Find the probability of the lost card being of clubs.
OR
From a lot of 10 bulbs, which includes 3 defectives, a sample of 2 bulbs is drawn at random. Find the probability distribution of the number of defective bulbs.
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 Question 26
Using properties of determinants, show the following:
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 Question 27
Find the values of xfor which f(x) = [x(x− 2)]^{2}is an increasing function. Also, find the points on the curve, where the tangent is parallel to xaxis.
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 Question 28
Show that the right circular cylinder, open at the top, and of given surface area and maximum volume is such that its height is equal to the radius of the base.
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 Question 29
Write the vector equations of the following lines and hence determine the distance between them:
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