Board Paper of Class 12Commerce 2005 Maths All India(SET 1)  Solutions
(i) The question paper consists of three sections A, B and C Section A is compulsory for all students. In addition to Section A, every student has to attempt either Section B OR Section C.
(ii) For Section A 
Question numbers 1 to 8 are of 3 marks each.
Question numbers 9 to 15 are of 4 marks each.
Question numbers 16 to 18 are of 6 marks each.<o:p></o:p></span></p>
(iii) For Section B/Section C
Question numbers 19 to 22 are of 3 marks each.
Question numbers 23 to 25 are of 4 marks each.
Question number 26 is of 6 marks.
(iv) All questions are compulsory.
(v) Internal choices have been provided in some questions. You have to attempt only one of the choices in such questions.
(vi) Use of calculator is not permitted. However, you may ask for logarithmic and statistical tables, if required.
 Question 1
If and f(x) = x^{2}− 2x− 3, show that f(A) = O.
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 Question 2
Using the properties of determinants, solve for x:
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 Question 3
An integer is chosen at random from the first 200 positive integers. Find the probability that it is divisible by 6 or 8.
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 Question 4
X is taking up subjectsMathematics, Physics and Chemistry in the examination, His probability of getting Grade A in these subjects are 0.2, 0.3, and 0.5 respectively. Find the probability that he gets.
(i) Grade A in all subjects
(ii) Grade A in no subject
(iii) Grade A in two subjects
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 Question 5
Evaluate:
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 Question 6
Evaluate:
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 Question 7
Solve the following differential equation:
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 Question 8
Solve the differential equation:
(x^{2}+ xy)dy= (x^{2}+ y^{2})dx
Or
Solve the following differential equation:
, given that, when x = 0
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 Question 9
Test the validity of the following argument:
S_{1}: p ∧ q; S_{2}: ∼p; S: ∼q
Or
If B is a Boolean algebra and x, y ∈ B, then show the following:
(x + y) + (x’ y’) = 1
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 Question 10
Evaluate: .
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 Question 11
Differentiatewith respect to xfrom the first principle.
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 Question 12
If, then prove that
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 Question 13
Find the intervals in which the function f(x) = 2x^{3}− 15x^{2}+ 36x + 1 is strictly increasing or decreasing. Also find the points on which the tangents are parallel to the xaxis.
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 Question 14
Evaluate:.
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 Question 15
Evaluate:, where
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 Question 16
Show that the height of the cone of maximum volume that can be inscribed in a sphere of radius 12 cm is 16 cm.
Or
Prove that the curves x = y^{2} and xy = k cut at right angle if 8k^{2} = 1.
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 Question 17
Using matrix method, solve the following system of linear equations:
x+ y − z = 1
x− y− z= −1
3x+ y − 2z = 3
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 Question 18
Find the area of the region bound by the curve y= x^{2}and the line y= x.
Or
Find the area enclosed by the parabola y^{2}= x and the line y+ x = 2 and the first quadrant.
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