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Board Paper of Class 12-Humanities 2012 Maths All India(SET 3) - Solutions

General Instructions:
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 6

    Let A be a square matrix of order 3 × 3. Write the value of |2A|, where |A| = 4.

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  • Question 8

    The binary operation *: R × R R is defined as a * b = 2a + b. Find (2 * 3) * 4.

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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 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 16

    Show that f: N N, given by is both one-one 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 well-shuffled 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 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 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 c2 square units. Show that the maximum volume of the box is cubic units.

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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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