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

Time: 3 Hours                                                          June 2006                                            Max. Marks: 100

NOTE: There are 9 Questions in all.

· Question 1 is compulsory and carries 20 marks. Answer to Q. 1. must be written in the space provided for it in the answer book supplied and nowhere else.

· Out of the remaining EIGHT Questions answer any FIVE Questions. Each question carries 16 marks.

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

Q.1       Choose the correct or best alternative in the following:                                         (2x10)

a. Sum of the series  is equal to

(A)  348551                                        (B)  -1000

(C)   5151                                           (D)  None of the above

b. The value of tan  is

(A) (B)

(C) (D)

             c.   In a triangle ABC, let a = BC, b = CA and c = AB.  If , then

(A) (B)

(C) (D)  None of the above

             d.   The circles  and  cut orthogonally if the value of p is

(A)  3                                                      (B) -2

(C)  -3                                                (D)  1       

             e.   The eccentricity of the ellipse  is

(A) 3                                                  (B) 

(C)  5                                                  (D)

             f.    The derivative of   – cos (log x) is 

(A) sin (log x)                                     (B)

(C)  – sin (log x)                                  (D)

             g.   The value of the  is

(A) 0                                                  (B)  1

(C)  e                                                  (D)  Does not exist

             h.   The integral  is equal to

(A) (B)

(C) 0                                                   (D) 1

             i.    The area under the curve  between x = 0 and x = 1 is

(A) 1                                                  (B)

(C) (D)

             j.    The solution of  is

(A)  (B)

(C) (D)

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

Q.2     a.   Show that the sum to n terms of the series 1.3.5 + 3.5.7 + 5.7.9 + … is .                                                              (8)

             b.   If  are the roots of the quadratic equation  and   are the roots of the quadratic equation  then show that . (8)

Q.3     a.   If A + B + C = , prove that .                                                                    (8)

             b.   Show that sin  is a root of the equation .                   (8)

Q.4     a.   Find the value of  such that the circles  and  touch each other.                                 (8)

b.   For what values of k the points  and  are collinear?             (8)

Q.5     a.   Find the equation of the circle for which  is a tangent and  are normals.                                              (8)

     b.   Find the values of a, b such that the line ax + by + 1 = 0 is tangent to the hyperbola  and is parallel to the line y = 2x  + 4.              (8)

  Q.6     a. Evaluate the limit.                                                          (8)

             b.   Consider the function .  Find  using first principle.  Is  continuous at x = 0?                                              (8)

Q.7     a.   Find the local maximum and minimum values of in .              (8)

b.   Find the area of the region bounded by ,  and             x = 1.                   (8)

Q.8     a.   Evaluate the following integral .                                            (8)

             b.   Evaluate  the following definite integral .                                     (8)

Q.9     a.   Solve the differential equation

.                                                                                  (8)

             b.   Solve the differential equation

.                                                                       (8)