Board Paper of Class 12Humanities 2007 Maths Delhi(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 A=, show that A^{2}− 6A+ 17I = O. Hence find A^{−1}.
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 Question 2
An urn contains 7 red and 4 blue balls. Two balls are drawn at random with replacement. Find the probability of getting (a) 2 red balls (b) 2 blue balls (C) one red and one blue ball.
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 Question 3
Using properties of determinants prove the following:
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 Question 4
A card is drawn at random from a wellshuffled pack of 52 cards. Find the probability that it is neither an ace nor a king.
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 Question 5
Evaluate:
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 Question 6
 Question 7
Form the differential equation of the family of curves y = A cos 2x + B sin 2x, where A and B are constants.
Or
Solve the following differential equation:
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 Question 8
Evaluate:
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 Question 9
Using properties of definite integrals, prove the following:
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 Question 10
Evaluate:
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 Question 11
 Question 12
Differentiate sin (x^{2}+ 1) with respect to xfrom the first principle.
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 Question 13
Write the Boolean expression for the following circuit:
Simplify the Boolean expression.
Or
Show that the following argument is valid:
s_{1}: p ∧ q
s_{2}: ∼ q
s: p ∨ ∼ q
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 Question 14
If y= sin (log x), prove that
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 Question 15
Verify Rolle’s Theorem for the function f (x) = x^{2}− 5x+ 4 on [1, 4].
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 Question 16
Using matrices solve the following system of equations:
x+ 2y + 3z= 6
3x+ 2y − 2z = 3
2x− y+ z = 2
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 Question 17
Using integration, find the area of the region enclosed between the circles:
x^{2}+ y^{ 2}= 1 and (x− 1)^{2}+ y^{2}= 1
Or
Evaluate as limit of sums.
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 Question 18
Find the point on the curve x^{2}= 8y which is nearest to the point (2, 4).
Or
Show that the right circular cone of least curved surface and given volume has an altitude equal totimes the radius of the base.
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 Question 19
Find the projection of , where
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 Question 20
Find the value of λ which makes the vectors coplanar, where
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 Question 21
A particle starting with initial velocity of 30 m/sec, moves with a uniform acceleration of 9 m/sec^{2}Find:
(a) the velocity of the particle after 6 seconds
(b) how far will it go in 9 seconds.
(c) its velocity when it has travelled 150 m.
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 Question 22
Find the resultant of two velocities 4 m/sec and 6 m/sec inclined to one another at an angle of 120°.
Or
A ball projected with a velocity of 28 m/sec has a horizontal range 40 m. Find the two angles of projection.
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 Question 23
A body of weight 70 N is suspended by two strings of lengths 27 cm and 36 cm, fastened to two points in the same horizontal line 45 cm apart and is in equilibrium. Find the tensions in the strings.
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 Question 24
The resultant of two unlike parallel forces of 18 N and 10 N act along a line at a distance of 12 cm from the line of action of the smaller force. Find the distance between the lines of action of two forces.
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 Question 25
Find the equation of the plane which is perpendicular to the plane 5x + 3y + 6z + 8 = 0 and which contains the line of intersection of the planes x + 2y + 3z − 4 = 0 and 2x + y − z + 5 = 0.
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 Question 26
Find the equation of the sphere passing through the points (3, 0, 0), (0, −1, 0), (0, 0, −2) and having the centre on the plane 3x +2y + 4z = 1.
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 Question 27
Find the face value of a bill, discounted at 6% per annum 146 days before the due date, if the banker’s gain is Rs. 36.
Out of current syllabus
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 Question 28
A bill for Rs. 7650 was drawn on 8^{th} March, 2005 at 7 months. It was discounted on 18^{th} May, 2005 and the holder of the bill received Rs. 7497. What rate of interest did the banker charge?
Out of current syllabus
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 Question 29
There are two bags I and II. Bag I contains 2 white and 3 red balls and Bag II contains 4 white and 5 red balls. One ball is drawn at random from one of the bags and is found to be red. Find the probability that it was drawn from bag II.
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 Question 30
Find mean μ, variance σ^{2} for the following probability distribution:
X
0
1
2
3
P(X)
Or
Find the binomial distribution for which the mean is 4 and variance 3.
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 Question 31
A, B, C entered into a partnership investing Rs. 12000, Rs. 16000 and Rs. 20000 respectively. A as working partner gets 10% of the annual profit for the same. After 5 months, B invested Rs. 2000 more while C withdrew Rs. 2000 after 8 months from the start of the business. Find the share of each in an annual profit of Rs. 97000.
Out of current syllabus
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 Question 32
Find present value of an annuity due of Rs. 700 per annum payable at the beginning of each year for 2 years allowing interest 6% per annum, compounded annually. [Take (1.06)^{−1} = 0.943]
Out of current syllabus
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 Question 33
The total cost C(x), associated with the production and making x units of an item is given by
C(x) = 0.005 x^{3} − 0.02 x^{2} + 30 x + 5000
Find (i) the average cost function (ii) the average cost of output of 10 units (iii) the marginal cost function (iv) the marginal cost when 3 units are produced.
Out of current syllabus
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 Question 34
If a young man rides his motorcycle at 25 km/hour, he has to spend Rs. 2 per km on petrol. If he rides at a faster speed of 40 km/hour, the petrol cost increases at Rs. 5 per km. He has Rs. 100 to spend on petrol and wishes to find the maximum distance he can travel within one hour. Express this as an LPP and solve it graphically.
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