Solved question paper for BM May-2018 (BCA 2nd)

Question paper 1

1. BASIC MATHEMATICS

   (SECTION-A)

   1. (a) prove that sum and production of two complex numbers are real if they are a conjugate of each other.

   Answer:

   

    

   

   

   

   

   

   a

   

   

   

   

    

    

    

   

    

   

   

   

2. (b) find the square root of -15-8i

   Answer:

   

   Squaring

   

   ;

   

   

   Now

   

   

   

   

   

   

   

   

3. 2.  (a) The sum of roots of a quadratic equation is 5 and the sum of squares is 13 find the equation.

   Answer:

   Let α,b be roots of qn.

   

   

   

   

   

   

   The reqd quadratic q.is:

   

   

4. (b) find the distance between the parallel lines

   

   Answer:

   

    

   

   

6. 3 (a) Find the equations of the median of the triangle ABC whose vertices are A(2,5), B(-4,9) and  C(-2,1).

   Answer:

   

    

   Mid points of AB,BC&CA are resp.

   

   

   Eq of CD:

   

   

   

   Eq.of Bf:

   

   

   

   

   Eq of AE:

   

   

   

   

   

7. (b) Find the equation of the circle whose Center is (-4,2) and the line x-y=3 touches are a circle.

   Answer:

   The distance from a point (x,y) to the line  Ax+by+c=0 is given by:

   

   The centre of the circle is A(-4,2)

   Also, the line touches the circle is: X-y=3----(1)

   The distance of point A(-4,2) to the line   (1)

   Is the radius of the circle

   But distance

   

   

   

8. 4. (a) Solve the equation  the squared difference of its roots is equal to 144.

   Answer:

   

   

   

   

   

   

   

   

   

   

   

   

   

   

   

   

   

9. (b) Find the real values of x and y for which the complex number   are conjugate of each other.

   Answer:

   Are conjugate

   

   

   

   

   

   

   

   

   

   

11. SECTION - B

    (5) (a) Find the order of convergence of Regula-Falsi method.

    Answer:

    Order of convergence of Regula Falsi method :

    Let us assume that

    The general iterative formula for Regula falsi method is

    

    are the errors in the approximations

    

    

    

    

    

    

    

    

    

    

    

    

     

    

    

    [Neglecting higher order terms]

    (3) in terms of errors is called errors q.n.

    Let the straight line joining the pts lies  above the cure y=f(x)

    One of the pts xo or x1 is always  &  other varies with k it the it  is fixed then in each iteration f(x) is  by the straight line joining the pts 

    So the error q. (3) becomes

    

    

     

    Asymptotic erros constant

    From qn (4) it is clear that Regula Falsi method is lines order convergent i.e order of convergence of this method is 1.

12. (b) Find the inverse of    

    Answer:

     let

     

     

    Co- factor matrix

    

    

13. (6) (a) Use Newton-Rapson’s method to find a root of the equation  which is nearer to x =3

    Answer:

    Let

    

     

    

    So real root of given q lies b/w 2.5 & 3 also (3) is neares  to 0 than f(2.5) so take initial approx.   x to tj=he root as 3.

     Iteration-1 the first approx.  to the root is given by

        

    

    Iteration -2 The 2^nd approx.  to the  root is given by

    

    

    

    

    Iteration-3  The 3^rd approx. to the root is given by

    

    From 2^nd  & 3^rd  we see that there is no change in the successive approx. to the root

    So a root of given qn is given by

    [x= 2.7984]

14. (b)  Solve by using Gauss elimination method:

     ,  

    Answer:

    

    

    

    

     

    

    

    

    

    

    

    

    ; [z=5]; putin(8)

    

    

    

    

16. (7). (a) If A =  , show that . Hence find

    Answer:

    

    

    

    

    

    

     

    

    

    

17. (b) Verify the following metrics that  .

    Answer:

     

    

    Co factor matric of

    

    

    

    

     

    

18.        (8) (a)  use matrices, slove :    ,  ,

    Answer:

     

     

    

    

    

    

     

    

    Co-factor matrix of  

    

    

     

    

    

    

    

    

19.  (b).  Find a real root  between 2 and 3 correct up to four decimal places using the bisection method.

    Answer:

     

    

    

    So a real root of given qn. Lies in interval (2,3) & neares to 2

    Iteration (1) Taking a=2 & b=3

    The first approximation to the root id=s given by

    Now

    

    

    So a real root of the given q lies in the interval (2,2.5) & neares to 2.5

    Iteration (2) Taking

    

    

    

    

    Iteration(3) Taking

    

    

    

     

    

    Iteration(4)  

     

    

     

    

     

    Iteration (5)  

    

    

     

    

    

    Iteration (5)  

    

    

    

    

21. SECTION - C

    (9). Attempt all the following:

    (i) Express  in the standard form a+ib.

    Answer:

    

     

    

    

     

     

    .

22. (ii) Find the multiplicative inverse of 3+2i

    Answer:

     

    Multiplicative inverse of  

    

    

    

    

    

23. (iii) If  , find 2A + 3B.

    Answer:

     

    

     

24.       (iv) If the roots of the equations  are real and distinct , the find all the possible values of a.

    Answer:

    

    All roots of (1) are real  & distinct

    

     

     

    

      

     

    

26. (v) Find the equation of the line passing through (1,1) and parallel to line 4x + 4y = 7.

    Answer:

    Let the qn of the reqd line be

    y=mx+c---1

    Since (1) passes thro (1,1)

    

    Is parallel to

    

    

    

    

    

    

    

27. (vi) Give the advantages of Regula-Falsi method.

    Answer:

    Advantages of Regula Falsi method :

    1. It doesn’t require the derivative calculation
    2. It is linearly convergent
    3. It is a quick method .