Board Paper of Class 12Commerce 2004 Maths (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
 Question 1
If, show that
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 Question
 Question 2
Using properties of determinants, solve for x:
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 Question
 Question 3
An urn contains 7 white, 5 black and 3 red balls. Two balls are drawn at random. Find the probability that
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(i) both the balls are red
(ii) one ball is red, the other is black
(iii) one ball is white
 Question
 Question 4
A fair die is tossed twice. If the number appearing on the top is less than 3, it is a success. Find the probability distribution of X.
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 Question
 Question 5
Evaluate:
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 Question
 Question 6
Evaluate:
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 Question
 Question 7
Form the differential equation corresponding to where a andb are arbitrary constants.
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 Question
 Question 8
Solve the differential equation: given that y = 1, when x= 0.
OR
Solve the differential equation:
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 Question
 Question 9
Show that in a Boolean algebra, B:
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(i)
(ii)
Out of current syllabus
 Question
 Question 10
Evaluate:
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 Question
 Question 11
Differentiate w.r.t. x from first principles.
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 Question
 Question 12
Differentiate w.r.t. x.
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 Question
 Question 13
Find the equations of the tangent and the normal to the curve at.
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 Question
 Question 14
Evaluate:
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 Question
 Question 15
Evaluate:
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 Question
 Question 16
Using matrix method, solve the following system of linear equations:
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 Question
 Question 17
Show that a right circular cylinder, which is open at the top and has a given surface area, will have the greatest volume if its height is equal to the radius of its base.
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 Question
 Question 18
Using integration, find the area of the circle x^{2}+ y^{2}= 16, which is exterior to the parabola y^{2}=6x.
OR
Find the area of the smaller region bounded by the ellipse and the line.
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 Question
 Question 19
If , then show that the vectors and are orthogonal.
Or
Find x such that the four points A(3, 2, 1), B(4, x, 5), C(4, 2, −2), and D(6, 5, − 1) are coplanar.
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 Question
 Question 20
Prove that:
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 Question
 Question 21
Find the Cartesian and vector equations of a line, which passes through the point (1, 2, 3) and is parallel to the line
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 Question
 Question 22
Find the Cartesian equation of the sphere which has the points A (2, −3, 4) and B (−5, 6, −7) as the end points of one of its diameters. Also find its centre and radius.
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Out of current syllabus
 Question
 Question 23
Show that the lines and intersect. Find the point of intersection also.
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 Question
 Question 24
Three forces acting on a point are in equilibrium. If the angle between be double the angle between , prove that
Out of current syllabus
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 Question
 Question 25
Two unlike parallel forces act at two points units apart. Show that if the direction of be reversed, the resultant will be displaced by a distance units.
Out of current syllabus
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 Question
 Question 26
A particle starting with some initial velocity and moving with uniform acceleration acquires a velocity of 20 cm/sec after moving through 10 cm from a point P to Q and a velocity of 30 cm/sec after further moving 20 cm from Q to R in the same direction. Find
(i) its velocity at the point P
(ii) its acceleration
(iii) the time it will take and the distanceOut of current syllabus
OR
From a point on the ground at a distance from the foot of a vertical wall, a ball is thrown at an angle of 450 which just clears the top of the wall and afterwards strikes the ground at a distance of yon the other side of the wall. Find the height of the wall.
Out of current syllabus
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 Question
 Question 27
Find the banker's discount and true discount on a bill of Rs 22,800 due 4 months at 4% per annum.
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Out of current syllabus
 Question
 Question 28
A bill of exchange drawn on January 4, 2003 at 5 months date was discounted on march 26, 2003 at 3% per annum. If the banker's discount is Rs 1207.20, find the face value of the bill.
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Out of current syllabus
 Question
 Question 29
Three urns A, B, and C contain 6 red and 4 white, 2 red and 6 white, and 1 red and 5 white balls respectively. An urn is chosen at random and a ball is drawn. If the ball drawn is found to be red, find the probability that the ball was drawn from urn A.
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 Question
 Question 30
The mean and variance of a binomial distribution are 4 and respectively. Find P(X ≥1).
OR
If the sum of the mean and variance of a binomial distribution for 5 trials be 1.8, find the distribution.
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 Question
 Question 31
A, B, and C are partners in a business. A, being a working partner, receives 10% of the total profit as salary. The remaining profit is distributed among them in the ratio of 2 : 3 : 4. If A gets Rs 3,00,000 in all, find the shares of B and C.
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Out of current syllabus
 Question
 Question 32
Find the present value of an annuity of Rs 1,200 payable at the end of each of the 6 months for 3 years, when the interest is 8% per annum, compounded semiannually.
[Use (1.04)^{6}= 1.2653]
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 Question
 Question 33
The total cost and demand function of an item are given by
and p = 100 − xrespectively.
Write the total revenue function and the profit function. Find the number of items when the profit will be maximum. Find the maximum profit also.
 Question
 Question 34
An oil company requires 13,000, 20,000 and 15,000 barrels of high grade, medium grade and low grade oil respectively. Refinery A produces 100, 300 and 200 barrels per day of high, medium and low grade oil respectively whereas refinery B produces 200, 400 and 100 barrels per day respectively. If A costs Rs 400 per day and B costs Rs 300 per day to operate, how many days should each be run to minimize the cost of requirement?
OR
A firm makes items A and B and the total number of items it can make in a day is 24. It takes one hour to make an item of A and only half an hour to make an item of B. The maximum time available per day is 16 hours. The profit on an item of A is Rs 300 and on one item of B is Rs 160. How many items of each type should be produced to maximize the profit? Solve the problem graphically.

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