Code: D-01 / DC-01                                                                        Subject: MATHEMATICS - I

Time: 3 Hours                                                                                                     Max. Marks: 100

 

NOTE: There are 11 Questions in all.

 

·      Question 1 is compulsory and carries 16 marks. Answer to Q. 1. must be written in the space provided for it in the answer book supplied and nowhere else.

·      Answer any THREE Questions each from Part I and Part II. Each of these questions carries 14 marks.

·      Any required data not explicitly given, may be suitably assumed and stated.

 

 

 

Q.1       Choose the correct or best alternative in the following:                                           (2x8)

 

a.       The co-ordinates of the middle points of the sides of a triangle are (4, 2)    (3, 3) and (2, 2).  Then the co-ordinates of the centroid are

                  

                   (A)  .                                      (B)  (3, 3).

                   (C)  (4, 3).                                          (D) (4, 7).

 

b.      If x, 2x +2, 3x + 3 are first three terms of a G.P. then its 4th term is

 

(A)    27.                                               (B)  -27.

(C)  13.5.                                            (D)  -13.5.

 

             c.   The angle made by any diameter of a circle at any point on the circumference is

                  

(A)     90°                                              (B)  180°

                   (C)  45°                                              (D)  60°

 

             d.   If  then the value of r is

 

                   (A) 6.                                                  (B) 5.

(C)    4.                                                 (D) 7.

 

             e.   is equal to

 

(A)     .                          (B)  .

(C)  .                          (D)  .

 

                                                                                                                     

             f.    If  then is equal to 

 

(A)     4896.                                           (B)  816.

(C)  1632.                                           (D)  408.

 

             g.     is

                  

                   (A) .                                              (B)5.

(C) Not defined.                                  (D) .

            

             h.   is equal to

 

                   (A)  .                                               (B)  .

                   (C)  .                                            (D)  .

 

PART I

Answer any THREE Questions. Each question carries 14 marks.

 

  Q.2     a.   If  are the roots of the equation .  Find the equation whose roots are  and .                                            (7)

 

             b.   If the roots of the equation  are equal, show that .                                                                       (7)

 

  Q.3     a.   In a  show that .                      (7)

 

             b.   If  then show that .              (7)

 

  Q.4     a.   Evaluate .                                                              (7)

       

             b.   Differentiate  by the first principle.                                                (7)

 

  Q.5     a.   Find the area bounded by the curve  and the straight line .       

                                                                                                                                             (7)

       

             b.   Find the equation of tangent to  at , where  and .                                                                      (7)

  Q.6     a.   Find the equation of a line passing through  and perpendicular to the line 3x –y +5 = 0.                  (7)

 

             b.   Find the equation of the circle whose centre lies on the line x – 4y =1 and which passes through the points (3, 7) and (5, 5).                                       (7)

                                                                             

PART II

Answer any THREE Questions. Each question carries 14 marks.

 

  Q.7     a.   Find the term independent of x in the expansion of .                         (7)

 

             b.  Find the equation of the parabola whose focus is (1, -1) and whose vertex is (2,1).              (7)

                  

  Q.8     a.   Show that the height of a cylinder of maximum volume that can be inscribed in a sphere of radius R is .  Also find the maximum value.                       (7)

 

             b.   Evaluate .                                                                   (7)

                  

  Q.9     a.   Using induction, prove that  for all n.                                                       (7)

       

             b.   Solve .                                                                             (7)

 

Q.10           a.                                                        Evaluate .                                 (7)

                    

             b.   Evaluate .                                                                                  (7)     

 

Q.11           a.                                                        Solve , given y =1 when x = 1. (7)

 

             b.   Find the differential equation of which  is a solution.     (7)