Board Paper of Class 12Humanities 2004 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, prove that, n ∈ N.
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 Question 2
Using the properties of determinants, prove that
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 Question 3
From a bag containing 20 tickets, numbered from 1 to 20, two tickets are drawn at random. Find the probability that
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(i) Both the tickets have prime numbers on them
(ii) On one there is prime number and on the other there is a multiple of 4
 Question 4
Two dice are tossed once. Find the probability of getting an even number on the first die or a total of 8.
Or
From a lot of 30 bulbs, which includes 6 defective bulbs, a sample of 3 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.
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 Question 5
Evaluate:
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 Question 6
Evaluate:
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 Question 7
Solve the differential equation:
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 Question 8
Solve the differential equation:, given that , when x = 0.
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 Question 9
For each x in a Boolean Algebra B, prove that
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 Question 10
 Question 11
Differentiate with respect to x:
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 Question 12
Differentiate with respect to xfrom first principles.
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 Question 13
The volume of a spherical balloon is increasing at the rate of 25 cm^{3}/sec. Find the rate of change of its surface area at the instant when its radius is 5 cm.
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 Question 14
Evaluate:
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 Question 15
 Question 16
Using matrix method, solve the following system of linear equations:
x+ y − z = 1
3x+ y − 2z = 3
x− y− z= −1
Or
If, prove that A^{2}− 4A− 5I= O. Hence, find A^{−1}.
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 Question 17
Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius cm is 500πcm^{3}.
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 Question 18
Find the area bounded by the circle x^{2}+ y^{2}= 16 and the line y= x in the first quadrant.
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 Question 19
If the vectors are coplanar, then find the value of λ.
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 Question 20
Prove that:
Or
The volume of the parallelopiped whose edges are is 546 cubic units. Find the value of λ.
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 Question 21
The Cartesian equations of a line are 3x+ 1 = 6y − 2 = 1 − z. Find the fixed point through which it passes, its direction ratios, and also its vector equation.
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 Question 22
Find the equation of the plane passing through the points (0, −1, 0), (1, 1, 1) and (3, 3, 0).
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 Question 23
Prove that the plane x + 2y − z = 4 intersects the sphere x^{2} + y^{2} + z^{2} − x + z − 2 = 0 in a circle of radius unity. Also, find the centre of the circle.
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 Question 24
Three forces and act along OA, OB and OC, where O is the orthocenter of ΔABC. If the forces be in equilibrium, prove that P: Q: R= a: b: c.
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Out of current syllabus
 Question 25
The resultant of two like parallel forces and acting at A and B, 8 cm apart, is 40 N. If the resultant passes through C, where AC = 3 cm, find the magnitude of the forces.
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 Question 26
A particle moving in a straight line with uniform acceleration describes successive equal distance in times t_{1}, t_{2}and t_{3}. Prove that.
Or
A stone is projected at an angle αwith the horizontal. Given its velocity when it attains half of the maximum height (it can attain) is times of the velocity at the maximum height. Prove that
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Out of current syllabus
 Question 27
The true discount and the banker’s gain on a certain bill of exchange due after a certain period of time are respectively Rs 700 and Rs 17.50. Find the face of the bill.
Out of current syllabus
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 Question 28
A bill of Rs 1,000 drawn on May 7, 2003 for six months was discounted on August 29, 2003 for a cash payment of Rs 988. Find the rate of interest charged by the bank.
Out of current syllabus
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 Question 29
A company has two plants to manufacture T.V.s. The first plant manufactures 70% of the T.V.s. and the rest are manufactured by the other plant. 80% of the T.V.s. manufactured by the first plant is rated of standard quality while that of the second plant, only 70% are of standard quality. If a T.V. chosen at random is found to be of standard quality, find the probability that it was produced by the first plant.
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 Question 30
A pair of dice is thrown 7 times. If getting the total 7 is considered a success, find the probability of (i) no success (ii) at least 6 successes.
Or
If the probability that an individual suffers from reaction by an injection is, find the probability that out of 5000 individuals given that injection (i) exactly 5 will suffer from reaction (ii) no one will suffer from reaction.
Given that
Out of current syllabus
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 Question 31
X and Y entered into a joint business with their capitals in the ratio 3 : 2. At the end of 3 months, X took out onethird of his capital, but after another 3 months Y put in a sum equal to what X had taken out. If at the end of the year Y gets Rs 11,000 more than what X got as profit, find
(i) the total profit in the business
(ii) the profit share of X in the businessOut of current syllabus
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 Question 32
A television set is available for Rs 20, 000 cash or Rs 5,000 as cash down payment followed by 6 equal annual instalments, the first to be paid one year after the date of purchase. If the rate of interest under the instalment plan is 10% per annum, determine the amount of instalment. [Given that (1.1)^{−6} = 0.5644]
Out of current syllabus
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 Question 33
If the cost function of an article manufactured by a company is given by
C(x) =, where x stands for the output.
Find the output at which:
(i) The marginal cost is minimum
(ii) The average cost is minimumOut of current syllabus
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 Question 34
A diet for a sick person must contain at least 4000 units of vitamins, 50 units of minerals and 1400 calories. Two foods X and Y are available at a cost of Rs 4 and Rs 3 per unit respectively. One unit of the food X contains 200 units of vitamins, 1 unit of minerals and 40 calories, whereas one unit of food Y contains 100 units of vitamins, 2 units of minerals and 40 calories. Find what combination of X and Y should be used to have least cost, satisfying the requirements.
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