Board Paper of Class 12Commerce 2010 Maths All India(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
Find λ if
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 Question 2
Write the value of the following integral:
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 Question 3
Evaluate:
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 Question 4
Write the adjoint of the following matrix:
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 Question 5
If f: R → R be defined by, then find of fof(x).
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 Question 6
Write the principal value of.
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 Question 7
What positive values of x makes the following pair of determinant equal?
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 Question 8
A is a square matrix of order 3 and. Write the value of .
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 Question 9
 Question 10
If and are two vectors such that, then what is the angle between and?
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 Question 11
 Question 12
A family has 2 children. Find the probability that both are boys, if it is known that
(i) at least one of the children is a boy,
(ii) the elder child is a boy.
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 Question 13
Show that the relation S in the set A = {x ∈ Z: 0 ≤ x ≤ 12} given by S = {(a, b): a, b ∈ Z, a − b is divisible by 4} is an equivalence relation. Find the set of all elements related to 1.
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 Question 14
If , then find the value of A^{2} − 3A + 2I.
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 Question 15
Find the points on the line at a distance of 5 units from the point P (1, 3, 3).
OR
Find the distance of the point P (6, 5, 9) from the place determined by the points A (3, −1, 2), B (5, 2, 4) and C (−1, −1, 6).
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 Question 16
If and , find a vector of magnitude 6 units which is parallel to the vector
OR
Let and Find a vector which is perpendicular to both and and = 18.
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 Question 17
Solve the following differential equation:
x ≠ 1
OR
Solve the following differential equation:
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 Question 18
Evaluate:
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 Question 19
If , −1 ≤ x ≤1, then show that
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 Question 20
Show that the following differential equation is homogeneous, and then solve it:
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 Question 21
If find
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 Question 22
Evaluate the following:
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 Question 23
Find the equations of the tangent and the normal to the curve at
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 Question 24
Find the equation of the plane passing through the point P (1, 1, 1) and containing the line. Also, show that the plane contains the line
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 Question 25
Using properties of determinants, prove the following
OR
Find the inverse of the following matrix using elementary operations
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 Question 26
A bag contains 4 balls. Two balls are drawn at random, and are found to be white. What is the probability that all balls are white?
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 Question 27
Find the area of the circle 4x^{2} + 4y^{2} = 9 which is interior to the parabola x^{2} = 4y.
OR
Using integration, find the area of the triangle ABC, coordinates of whose vertices are A (4, 1), B (6, 6) and C (8, 4)
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 Question 28
One kind of cake required 300g of flour and 15g of fat, another kind of cake requires 150g of flour and 30g of fat. Find the maximum number of cakes which can be made from 7.5 kg of flour and 600g of fat, assuming that there is no shortage of the other ingredients used in making the cakes. Make it as an L.P.P and solve it graphically.
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 Question 29
If the length of three sides of a trapezium other than the base is 10 cm each, find the area of the trapezium, when it is maximum.
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