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Board Paper of Class 12-Commerce 2011 Maths (SET 1) - 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

    State the reason for the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} not to be transitive.

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  • Question 3

    For a 2 × 2 matrix, A = [aij] whose elements are given by , write the value of a12.

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  • Question 4

    For what value of x, the matrix is singular?

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  • Question 8

    For what value of ‘a’ the vectors and are collinear?

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  • Question 9

    Write the direction cosines of the vector

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  • Question 10

    Write the intercept cut off by the plane 2x + y z = 5 on x-axis.

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  • Question 11

    Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by a * b = min. {a, b}. Write the operation table of the operation *.

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  • Question 12

    Prove the following:

    OR

    Find the value of

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  • Question 13

    Using properties of determinants, prove that

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  • Question 14

    Find the value of ‘a’ for which the function f defined as

    is continuous at x = 0

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  • Question 15

    Differentiate.

    OR

    If x = a (θ − sinθ), y = a (1 + cosθ), find .

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  • Question 16

    Sand is pouring from a pipe at the rate of 12 cm3/s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?

    OR

    Find the points on the curve x2 + y2 − 2x − 3 = 0 at which the tangents are parallel to x-axis.

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  • Question 18

    Solve the following differential equation:

    ex tan y dx + (1 − ex) sec2y dy = 0

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  • Question 19

    Solve the following differential equation:

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  • Question 20

    Find a unit vector perpendicular to each of the vector and where

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  • Question 21

    Find the angle between the following pair of lines:

    and check whether the lines are parallel or perpendicular.

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  • Question 22

    Probabilities of solving a specific problem independently by A and B are and respectively. If both try to solve the problem independently, find the probability that (i) the problem is solved (ii) exactly one of them solves the problem.

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  • Question 23

    Using matrix method, solve the following system of equations:

    OR

    Using elementary transformations, find the inverse of the matrix

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  • Question 24

    Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

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  • Question 25

    Using integration find the area of the triangular region whose sides have equations y = 2x + 1, y = 3x + 1 and x = 4.

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  • Question 27

    Find the equation of the plane which contains the line of intersection of the planes and which is perpendicular to the plane

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  • Question 28

    A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftsman’s time in its making while a cricket bat takes 3 hours of machine time and 1 hour of craftsman’s time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsman’s time.

    If the profit on a racket and on a bat is Rs 20 and Rs 10 respectively, find the number of tennis rackets and cricket bats that the factory must manufacture to earn the maximum profit. Make it as an L.P.P. and solve graphically.

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  • Question 29

    Suppose 5% of men and 0.25% of women have grey hair. A grey haired person is selected at random. What is the probability of this person being male? Assume that there are equal number of males and females.

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