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

Applied mathematics-1

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Question paper 1

  1. SECTION-A
    Q1. a) Choose the correct answer.

    i. If nP4 = 20 × nP2 then n is equal to
    a) 7   b) 8    c) 2      d) 4

    Answer:

     ,

    (a) option

  2. ii. Modulus of 3 – 4i is
    a) 6     b) 5      c) 4     d) 3

    Answer:

    b

  3. iii. Slope of line 3x + y – 2 = 0 is
    a) 3     b) 2     c) - 3    d) -2

    Answer:

    c

  4. iv. Mid points internally of (-a, b) and (a, -b) is
    a)
    b)
    c) a + b

    d) (0, 0)

    Answer:

    d

  5. v. The point (-4, -5) lies in quadrant
    a) 1st    b) 2nd    c) 3rd    d) 4th

    Answer:

    c

  6. b) State True or False.
    vi. Sin(A + B) = SinA CosB – CosA SinB

    Answer:

    False

  7. vii. Sum of first n natural numbers is

    Answer:

    False

  8. viii. If a, b, c are in A.P then 2b = 2a + c.

    Answer:

    False

  9. ix. Length of latus rectum of ellipse  +  = 1  is     .

    Answer:

    True

  10. x. The equation 3x + 4y + 5 = 0 and 4x – 3y + 7 = 0 represent perpendicular lines.

    Answer:

    True

  11. c) Fill in the blanks.
    xi. nth term of a G.P is ________

    Answer:

  12. xii. The eccentricity of parabola is _________

    Answer:

    1

  13. xiii. The equation x + 2y + 3 = 0 to the slope form is _________

    Answer:

  14. xiv. Two lines are ________if their slopes are equal.

    Answer:

    Parallel

  15. xv. Value of    = ________

    Answer:

    -1

  16. SECTION-B

    Q2. Attempt any six questions.

    a. If   Show that    .

    Answer:

  17. b) Given log2 = .30103, log5 = 0.69897 Solve the equations 2x. 5y = 1, 5x+1. 2y = 2.

    Answer:

    Equations

    Taking log on both sides

    Taking log on both sides

    Equation 1 multiplying by log 5 and equation 2 by log 2, then subtracting 1 and 2, we get

    ,   Ans

  18. c. Find the term independent of x in the expansion of .

    Answer:

    Find the term independent of x in expansion of

    Solution :

    Let  is the term independent of x.

             

            

    Term independent of x if

     Ans

  19. d. Resolve into partial fraction  .

    Answer:

    Resolve into partial fraction

    Sol:  

      

  20. e. Express the Complex number -3 + 3i in polar form.

    Answer:

    Express into    in polar form.

    From 1 and 2, we get

    From equation 3.

  21. f. Find the co-ordinates of foot of perpendicular from the point (2, 3) on the line y = 3x + 4.

    Answer:

    The slope of line perpendicular to

  22. g. Prove that   .

    Answer:

    Prove that  

    Solution :

  23. h. Sum the series 0.9 + 0.09 + 0.009 + ----- to nth term.

    Answer:

  24. i. The sum of two angles is and their difference is . Find the angles in degrees and radians.

    Answer:

  25. SECTION-C
    Q3. Attempt any three questions.

    i. Find the equation of circle which passes through the points (4, 1) and (6, 5) and has its centre lies on the line 4x + y = 16.
     

    Answer:

  26. (ii) If  

    Answer:

    --------1

    ---------2

    Squaring and adding  1 and 2

    2 sin

    2+2  

    Squaring 1 and 2 and subtranction 1 from 2

                                                                 Hence  Proved

  27. (iv) Find the co-ordinates of focus, vertex, the equation of directrix and axis of parabola y2 - 4y – 2x – 8 = 0

    Answer:

    Equation of parabola is  given by

                         ------------1

    Shift the origin to the point (-6,2) by putting   x=x1-6   and    y=y1+2

     vertex is (0,0)

    Focus  is   (12 , 0)  and  directrix   is ( 

    Vertex  is  (-6,2)

    Focus    is   (-6+

                (

    Directrix is     

                           X+6= 

                         X= -6- 

  28. (v) A boy observes the angle of elevation of a mountain top to be 600 and after walking directly away from it on level ground through 100 meters, the angle of elevation is 450. Find the height of the mountain and the distance between the mountain and first position of the boy.

    Answer:

    In right angle triangle ABD

    1=  

     h=x                                                                              -------1

    in right angle triangle ACD

                     --------2

    From eq 1 and 2               we get :

    =

    h (

     h = 

    h =