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Board Paper of Class 12-Science 2015 Maths Abroad(SET 2) - Solutions

General Instructions :
(i) All questions are compulsory.
(ii) Please check that this Question Paper contains 26 Questions.
(iii) Marks for each question are indicated against it.
(iv) Questions 1 to 6 in Section-A are Very Short Answer Type Questions carrying one mark each.
(v) Questions 7 to 19 in Section-B are Long Answer I Type Questions carrying 4 marks each.
(vi) Questions 20 to 26 in Section-C are Long Answer II Type Questions carrying 6 marks each.
(vii) Please write down the serial number of the Question before attempting it.
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  • Question 1
    If A =56-3-432-4-73, then write the cofactor of the element a21 of its 2nd row. VIEW SOLUTION
  • Question 2
    Write the sum of the order and degree of the differential equation

    d2ydx22 + dydx3 + x4 = 0. VIEW SOLUTION
  • Question 3
    Write the solution of the differential equation

    dydx= 2-y VIEW SOLUTION
  • Question 4
    Find the unit vector in the direction of the sum of the vectors 2i^+3j^-k^ and 4i^-3j^+2k^. VIEW SOLUTION
  • Question 5
    Find the area of a parallelogram whose adjacent sides are represented by the vectors 2i^-3k^ and 4j^+2k^. VIEW SOLUTION
  • Question 6
    Find the sum of the intercepts cut off by the plane 2x+y-z=5, on the coordinate axes. VIEW SOLUTION
  • Question 8
    Three machines E1, E2 and E3 in a certain factory producing electric bulbs, produce 50%, 25% and 25% respectively, of the total daily output of electric bulbs. It is known that 4% of the bulbs produced by each of machines E1 and E2 are defective and that 5% of those produced by machine E3 are defective. If one bulb is picked up at random from a day's production, calculate the probability that it is defective.
     
    OR

    Two numbers are selected at random (without replacement) from positive integers 2, 3, 4, 5, 6 and 7. Let X denote the larger of the two numbers obtained. Find the mean and variance of the probability distribution of X. VIEW SOLUTION
  • Question 9
    The two vectors j^+k^ and 3i^-j^+4k^ represent the two sides vectors AB and AC respectively of triangle ABC. Find the length of the median through A. VIEW SOLUTION
  • Question 10
    Find the equation of a plane which passes through the point (3, 2, 0) and contains the line x-31=y-65=z-44. VIEW SOLUTION
  • Question 11
    If 2 tan−1 (cos θ) = tan−1 (2 cosec θ), (θ ≠ 0), then find the value of θ.

    OR

    If tan-111+1.2+tan-111+2.3+...+tan-111+n.n+1=tan-1 θ, then find the value of θ. VIEW SOLUTION
  • Question 12
    If A=   2-1-1   2 and I is the identity matrix of order 2, then show that A2= 4 A − 3 I. Hence find A−1.
    OR

    If A=1-12-1 and B=a   1b-1 and A+B2=A2+B2, then find the values of a and b. VIEW SOLUTION
  • Question 13
    Using properties of determinants, prove the following :
    1aa2a21aaa21=1-a32 VIEW SOLUTION
  • Question 14

    Evaluate :

    sin x-asin x+adx

    OR

    Evaluate :

    x2x2+4x2+9dx VIEW SOLUTION
  • Question 15
    Find whether the following function is differentiable at x = 1 and x = 2 or not :
    fx=x,x < 12-x,1x2-2+3x-x2,  x>2  VIEW SOLUTION
  • Question 16
    In a parliament election, a political party hired a public relations firm to promote its candidates in three ways − telephone, house calls and letters. The cost per contact (in paise) is given in matrix A as

    A = 140200150TelephoneHouse CallLetters     

    The number of contacts of each type made in two cities X and Y is given in the matrix B as

            TelephoneHouse CallLettersB = 1000     500    50003000      1000      10000City XCity Y

    Find the total amount spent by the party in the two cities.

    What should one consider before casting his/her vote − party's promotional activity or their social activities ? VIEW SOLUTION
  • Question 18
    Find the point on the curve 9y2 = x3, where the normal to the curve makes equal intercepts on the axes. VIEW SOLUTION
  • Question 19
    If y=x+1+x2n, then show that1+x2d2ydx2+xdydx=n2y. VIEW SOLUTION
  • Question 20
    Find the minimum value of (ax + by), where xy = c2.

    OR

    Find the coordinates of a point of the parabola y = x2 + 7x + 2 which is closest to the straight line y = 3x − 3. VIEW SOLUTION
  • Question 21
    Maximise z = 8x + 9y subject to the constraints given below :
    2x + 3y ≤ 6
    3x − 2y ≤6
    y ≤ 1
    x, y ≥ 0 VIEW SOLUTION
  • Question 22
    Find the distance of the point (1, −2, 3) from the plane x − y + z = 5 measured parallel to the line whose direction cosines are proportional to 2, 3, −6. VIEW SOLUTION
  • Question 23
    Let f : N → ℝ be a function defined as f(x) = 4x2 + 12x + 15. Show that f : N → S, where S is the range of f, is invertible. Also find the inverse of f. VIEW SOLUTION
  • Question 24
    Using integration, find the area of the region bounded by the line x y + 2 = 0, the curve x = y and y-axis. VIEW SOLUTION
  • Question 25
    Find the probability distribution of the number of doublets in four throws of a pair of dice. Also find the mean and variance of this distribution. VIEW SOLUTION
  • Question 26
    Solve the following differential equation :

    y-x cosyxdy+y cosyx-2x sinyxdx=0
    OR
     
    Solve the following differential equation :

    1+x2+y2+x2 y2 dx+xy dy = 0 VIEW SOLUTION
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