Solved question paper for MATH-2 Dec-2018 (DIPLOMA automobile engineering 1st-2nd)

Applied mathematics-2

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Question paper 1

  1. SECTION-A
    Q1. a) Choose the correct answer.
    i. If A is a non singular matrix, then A-1 is
    a) |A| adj A      b)      
    c) (adj A)T            d) 

    Answer:

    (b)

  2. ii.  
    a) π      b) 
    c)       d)  

    Answer:

    (c)

  3. (iii) limx->0   

    a) 1      b) π
    c)       d)  -π

    Answer:

    (a)

  4. iv. Order of differential equation (y''') + 2y'' + 3y = x is
    a) 3       b)         4       c) 1        d) 2
     

    Answer:

    (a)

  5. v. The differential coefficient of sinx2 w.r.t  cosx2 is
    a) -tan x2 b) -cot x2 c) 2x  d) -2x

    Answer:

    (b)

  6. b) State True or False.
    vi. The differential coefficient of a constant is one.

    Answer:

    False

  7. (vii)    if    is even.

    Answer:

    False

  8. viii. Mean Deviation =   Standard Deviation

    Answer:

    True

  9. ix. Volume of a sphere of radius 'a' is   π   

    Answer:

    True

  10. x. cos 2A = cos2A - sin2A

    Answer:

    True

  11. c) Fill in the blanks
    xi. The angles in trigonometric functions are supposed to be measured in ______.
     

    Answer:

    Degree or radian

  12. xii. A square matrix is said to be a diagonal matrix if all its non-diagonal elements are______.

    Answer:

    Zero

  13. xiii.  is equal to______.

    Answer:

    Log(g(x)) + constant

  14. xiv. Central value of the set of observation is called_______.

    Answer:

    Central Tendency

  15. xv. The derivative of ?ex is equal to________.

    Answer:

    ex

  16.                                            SECTION-B
    Q2. Attempt any six questions.
    a. In a class of 30 students with roll no. 1 to 30, a student is picked up at random to answer a question. Find the probability that the roll number of selected students is either a multiple of 4 or 7.

    Answer:

    = 4,8, 12,16,20,24, 28

      &    

  17. b. If ?y = ex+y prove that    = 

    Answer:

     

  18. c. Calculate by Simpson’s rule an approximate value of     by taking seven equidistant ordinates.

    Answer:

    7  equidistant co-ordinate                   

     

    Equation..  

     

     

  19. d. Find the equation of the tangent to the curve y = x2 whose slope is  . 

    Answer:

    Y =                  -------------  1

    The slope of tangent to 1 at point ( is

    Y = 

    Slope = 

    Also  lies on curve 1

    ∴Requrid equation of tangent is 

    Y =  

    16y= 8x-1

    8x -16y =1

  20. e. Evaluate  .

    Answer:

    I =  

    Put

    Dx =

     =  

    =

    = 2 

    = 2 tan 

  21. f. Find the area bounded by the curve y = log x between the x-axis and the ordinates x = 2 and x = 3.

    Answer:

    Y =logx

     = 

    = log   -1

  22. g. If y = tan-1x , prove that (1 +x2) y2 + 2xy1 = 0

    Answer:

    Y =

    = 

    = (1+    = -2x.    

  23. h. Solve the equations by Crammer’s rule.

    Answer:

    A=

    =   

    X =                     y =

    X= -1                                                     y = 4

  24. i. 5x + 2y = 3


     

    Answer:

  25. j. 3x + 2y = 5

    Answer:

  26. k. Evaluate  

    Answer:

    I =

      Put logx  = t

            =

     

     =

     =

  27. SECTION-C

    Q3. Attempt any three questions.
    i. Solve the following equations by matrix method
    3x + y + 2z = 3

    2x - 3y - z = -3

    x + 2y + z = 4

    Answer:

    3x+ y+2z=3

    2x-3y-z=-3

    X+2y+z=4

    Ax=B

    X = 

    Where

    A= 

          = -3-3+14= 8  0

    (adj.A)=  = 

    Ax = B

    X=

    X =  

      =

    X = 1       ,    y= -2 ,   x=1

  28. ii. Find the maximum and minimum values of the function x3 - 6x2 + 9x + 15

    Answer:

      

    = 6(3) -12>0

    ∴x=3 is Point of minima

    Maximum value is

        = 1-6 + 9 + 15 = 19

    Min.  value is =

                             = 27-54+27+15

                             =  15

  29. iii. Find the standard deviation from the following data

    Wages

    0-10

    10-20

    20-30

    30-40

    40-50

    50-60

    60-70

    70-80

    Frequency

    12

    18

    35

    42

    50

    45

    20

    8

    Answer:

    Wages

    0-10

    10-20

    20-30

    30-40

    40-50

    50-60

    60-70

    70-80

    F

    12

    18

    35

    42

    50

    45

    20

    8

    N= 230

    X

    5

    15

    25

    35=a

    45

    55

    65

    75

    D=x-35

    -30

    -20

    -10

     0

    10

    20

    30

    40

    Fd

    -360

    -360

    -350

    0

    500

    900

    600

    320

    10800

    7200

    3500

    0

    5000

    18000

    12800

     

     

    Standard  deviation =

    = =

    =  = 17.25

  30. iv. Solve the differential equation
    y2(x2 - 1)   - x2 (y2 - 1) = 0

    Answer:

    =

    =

  31. v. Integrate x2 sin2x dx

    Answer:

    I = 

    =

    =

    =

    =