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.  is equal to

(A) .                                     (B) .

(C) .                         (D) .

b. If  then x is equal to

(A) 3                                                  (B)  27

(C)  9                                                  (D)  15

             c.   The value of  is equal to

(A) 1.                                                 (B)  0.

(C)  -1.                                               (D) ½

             d.   If  are the roots of  then  is

(A) .                                            (B) .

(C) .                                    (D) .

             e.   is equal to

(A) 1.                                                 (B)  0.

(C)  e.                                                 (D)  .

             f.    is equal to

(A) .                           (B) .

(C)  .                                          (D)  .

             g.   The maximum value of y = 2 cos 2x – cos 4x, is

(A) -1.                                                (B) .

(C) .                                              (D) 1.

             h.   The equation of the line which is perpendicular to the line 3x – 4y +7=0 and passes through the point (-3, 2) is

(A)  4x + 3y + 5 = 0.                           (B)  4x + 3y –3 = 0.

(C)  4x + 3y + 6 = 0.                           (D)  3x – 4y + 6 = 0.

PART I

Answer any THREE Questions. Each question carries 14 marks.

Q.2     a.   Prove that 7 divides  for all positive integers n.                                        (7)

             b.   Find the condition that the roots of equation  are equal.            (7)

Q.3     a.   Evaluate .                                                                                           (6)

             b.   If  prove that .                                          (8)

Q.4     a.   If a, b, c are lengths of sides opposite to angles A, B, C in a triangle ABC, then show that .               (7)

             b.   Show that in a triangle ABC,

                   a Sin (B – C) + b Sin (C – A) + c Sin (A – B) = 0,                                                           

                   where a , b, c are lengths of sides opposite to angles A, B, C.                             (7)

  Q.5     a.   Find the condition that the points (1, 1), (3, 5) and (a, b) are collinear.         (7)

             b.   Find equations of lines which pass through the point (4, 5) and make an angle  with the line 2x + y +1 = 0.                                                        (7)

  Q.6     a. Find the equation of the circle concentric with the circle                                          

 and which passes through (-4, 5).                            (7)

     b.   Show that represents a parabola.  Find its focus, vertex and directrix.                                                                      (7)

PART II

Answer any THREE Questions. Each question carries 14 marks.

Q.7     a.   Find .                                                                                             (6)

             b.  Examine the continuity of the function f(x) = [x],

                   where [x] is greatest integer , x being any real number.                          (8)

Q.8     a.   Show that the semi verticle angle of a cone of maximum volume and a given slant height is .                                                                    (7)

b.   Find the equation of tangent and normal to the curve  at the point where it intersects the positive x-axis.                                                 (7)

Q.9     a.   Find a reduction formula for the integral .                                       (7)

             b.   Evaluate    .                                                                (7)

Q.10           a.                                                        Find the area bounded by  and its latus rectum.   (7)

             b.   Find the volume of the solid obtained by revolving the ellipse , about its major axis.                                                               (7)     

Q.11           a.                                                        Solve the equation .                             (6)

b.   Solve the equation .                                                                  (8)