Board Paper of Class 12Humanities 2012 Maths (SET 3)  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 the distance of the plane 3x − 4y + 12z = 3 from the origin.
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 Question 2
Find the scalar components of the vector with initial point A (2, 1) and terminal point B (−5, 7).
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 Question 3
Find the principal value of .
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 Question 4
Write f(x) satisfying the above.
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 Question 5
Evaluate:
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 Question 6
Let A be a square matrix of order 3 × 3. Write the value of 2A, where A = 4.
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 Question 7
 Question 8
The binary operation *: R × R → R is defined as a * b = 2a + b. Find (2 * 3) * 4.
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 Question 9
Write the value of .
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 Question 10
Find the value of x + y from the following equation:
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 Question 11
Using properties of determinants, show that
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 Question 12
 Question 13
A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall?
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 Question 14
Let . Find a vector which is perpendicular to both and and
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 Question 15
 Question 16
Show that f: N → N, given by is both oneone and onto.
OR
Consider the binary operations *: R × R → R and o: R × R → R defined as and a * b = a − b and a o b = a for all a, b ∈ R. Show that ‘*’ is commutative but not associative, ‘o’ is associative but not commutative.
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 Question 17
Two cards are drawn simultaneously (without replacement) from a wellshuffled pack of 52 cards. Find the mean and variance of the number of red cards.
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 Question 18
Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.
OR
Find the particular solution of the differential equation
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 Question 19
 Question 20
Find the coordinates of the point where the line through the points (3, −4, −5) and (2, −3, 1) crosses the plane 3x + 2y + z + 14 = 0
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 Question 21
Find the particular solution of the following differential equation :
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 Question 22
Prove the following:
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 Question 23
 Question 24
Prove that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half of that of the cone.
OR
An open box with a square base is to be made out of a given quantity of cardboard of area c^{2 }square units. Show that the maximum volume of the box is cubic units.
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 Question 25
Find the area of the region. .
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 Question 26
A dietician wishes to mix two types of foods in such a way that the vitamin contents of the mixture contains at last 8 units of vitamin A and 10 units of vitamin C. Food I contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C while Food II contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. It costs Rs 5 per kg to purchase Food I and Rs 7 per kg to purchase Food II. Determine the minimum cost of such a mixture. Formulate the above as a LPP and solve it graphically.
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 Question 27
Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin times and notes the number of heads. If she gets 1, 2, 3, or 4 she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is the probability that she threw 1, 2, 3, or 4 with the die?
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 Question 28
Find the coordinates of the foot of perpendicular and the length of the perpendicular drawn from the point P (5, 4, 2) to the line . Also find the image of P in this line.
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 Question 29
Using matrices, solve the following system of equations:
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