Solved question paper for MATH-1 May-2018 (DIPLOMA Mechanical Engineering (RAC) 1st-2nd)

Applied mathematics-1

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Question paper 1

  1. SECTION-A

    Q1. Choose the correct answer.

    (i) The modulus of 1 + Ý…√3 is a) √2 b) -1 c) 2 d) 0

    Answer:

    z = 1 +

    |z| =

    Option C

  2. (ii) The value of 3π/12 radians in degree is

    a) 60° b) 45° c) 90° d) 120°

    Answer:

    π radian = 1800

    Option B

  3. (iii) Characteristic of log 0.07426 is

    a) 1 b) 2 c) 0 d) 1

    Answer:

    log0.07426 = -2.6001 (Find with the help of log table)

    option B

  4. (iv) If Sin (A-B) = ½ and Cos (A+B) = ½ then value of A and B will be

    a) A=15° , B=45° b) A=45°, B=15° c) A=45°, B=45° d) A=30°, B=60°

    Answer:

    We know that 

    Option B

  5. (v) The centroid of a triangle with two vertices (3,4) (-1,-9) is (2, -4) then third vertex is a) (-4 , -7) b) (4, -7) c) (4,7) d) (-4,7)

    Answer:

    Vertices of triangle are (3,4),(-1,-9) and let third vertices is (x,y).

    Its centroid is (2,-4)

    Formula of centroid

     =

     

     

    Option B

  6. Q2. State True or False.

    a. The series of the R.H.S of the expansion (1 + x)n extends to infinity

    Answer:

    False

  7. b. If k, k+1, k+3 are in G.P, then k=2

    Answer:

    k, k+1,k+3, ------

        k = 2

    2,3,5,….. is not in G.P.

    False

  8. c. Value of tan 120° is √3

    Answer:

    False

  9. d. Sec(270 ÌŠ + θ) = Cosecθ

    Answer:

    True

  10. e. The point (3,4); (7,7); (x,4) are collinear, if x=3

    Answer:

    (3,4),(7,7) and (x,4) are collinear if x=3

    A (3,4), B (7,7), C(3,4)

    By distance formula

    True

  11. Q3. Fill in the blanks.

    i. Radius is a -------- angle.

    Answer:

    constant

  12. ii. The revolving line is always ---------

    Answer:

    rotate about fixed point.

  13. iii. If CosA = ½ then Cos3A =

    Answer:

    cos3A = -1

  14. iv. The conic is parabola if ---------

    Answer:

    e = 1

  15. v. Equation of line perpendicular to line ax+by+c=0 is -------

    Answer:

    -bx+ay+c = 0

  16. SECTION-B

    Q4. Attempt any six questions.

    a. In how many ways, 3 boys and 3 girls are seated at round table, so that no two girls sit together.

    Answer:

    3 boys can be seated at a round table in  i.e.  Ways

    When 3 boys have occupied their seats in any one of these  Ways, then 3 girls can occupy any 3 out of 3 seats between boys so that no two girls are sitting together. This can be done in  ways

    Hence required number of ways 

                                                                     

                                                                    = 12 Ans

  17. b. Find the co-ordinates of the incentre of the triangle whose vertices are (-36,7), (20,7) and (0,-8)

    Answer:

    Formula of the center of the triangle is

     

     , where a,b,c are the length of the sides of triangle.

    To find a,b and c by distance formula

     

     

      

    e

     

     

      

     

    Put all values in formula

  18. c. Resolve  \((3x+7) \over (x+3)(x^2 + 1)\)  into partial fractions

    Answer:

    Resolve     into partial fractions.

     =

     

     =

     

     

    Comparing coefficients of x, x2 and constant terms

            ----- 1

               ----- 2

              ----- 3

     

    From 2

    This is put in 1

        ------ 4

    Subtract 4 from 3

    As  

    From 3

     

     =  Ans

  19. d. A (10, 4); B (-4, 9); C (-2,-1) are the vertices of a triangle ABC, find the equation of the median through A.

    Answer:

    D is mid point of BC

    To find the co-ordinate of D is

    We write the equation of line AB by using Two-point form,

  20. e. Prove that Cos α + Cos(α+2π/3) + Cos(α+4π/3) = 0

    Answer:

    Prove that  

    L.H.S.

  21. f. If   l\({log x \over y-z} = {logy \over z-x} = {log z \over x-y} \)  then show that x2y2z2  = 1

    Answer:

    If   then  Show that 

    Sol : 

    And

    Take  

    Taking log on both sides

    Using 1 and 2

    Hence it is proved that

  22. g. Prove that  \(sinA Sin3A Sin5A Sin7A\over CosA Cos 3A Cos 5A cos 7A\) = tan4A

    Answer:

    Prove that  

    Sol. L.H.S.

    By using C,D formulae from trignomentary

  23. h. Prove that \( { cot\theta + Coses\theta - 1\over cot\theta + Coses\theta + 1} = {1+Cos \theta \over sin \theta}\)

    Answer:

    Prove that

    Sol:

    L.H.S.

  24. i. How many terms of the series 3+8+13+18+ ------ must be taken so that their sum is 1010?

    Answer:

    3+8+13+18+  ……………

    This is A.P

    Formula