Code: DE01 / DC01                                                                        Subject: MATHEMATICS - I

Time: 3 Hours                                                                                                     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. The number of terms in the sequence  are

(A)  8                                                  (B)  9

(C)   10                                               (D)  6

b. First three terms in the expansion of  are

(A)    (B)

(C) (D)

             c.   Value of  is

(A) (B)

(C) (D)

             d.   If , then the value of cos 2A is

(A) (B)

(C) (D)

             e.   The value of ‘x’ such that PQ = QR, where P, Q and R are (6, -1), (1, 3) and (x, 8) respectively is given by   

(A) 5, –3                                            (B)  3, 5

(C)  2, 5                                              (D)  2, 3

f.    Slope of the line passing through the points  & is

(A) (B)

(C) (D)

             g.   is equal to

(A) (B)

(C)  (D)

             h.   If  then  is equal to

(A) (B)

(C) (D)

             i.    is equal to

(A)  (B)

                   (C) (D)

             j.    Order and degree of the differential equation  is given by

(A) 3, 2                                              (B) 2, 3

(C) 1, 3                                               (D) 3, 1

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

Q.2     a.   If 5 times the 5^th term of an A.P. is equal to the 10 times the 10^th term, find the 15^th term of the A.P.                                                                      (8)

             b.   If  denotes the sum of n terms of a G.P., prove that .                (8)

Q.3     a.   Show that .                                  (8)

             b.   If in the triangle ABC, A = , prove that .              (8)

Q.4     a.   Find the equation of the straight line which passes through the intersection of the lines x + y – 3 = 0 and 2x – y = 0 and is inclined at an angle of  with x-axis.                                                            (8)

b.   Show that  represents an ellipse.  Find its centre, vertices, foci, eccentricity, directrices, latusrectum and equations of major and minor axis.                        (8)

Q.5     a.   Find the equation of the circle which passes (4, 1) & (6, 5) and having centre on the line 4x+y =16.                                                                       (8)

     b.   Find the value of (8)

  Q.6     a. Differentiate y = tan x  w.r.t. ‘x’ from first principle.                                            (6)

             b.   Differentiate y = w.r.t ‘x’.                                                       (10)

Q.7     a.   Prove that straight line  touches the curve  at the point where the curve crosses the axis of y.                                              (8)

b.   Find the volume generated by revolving the ellipse  about      x-axis.               (8)

Q.8     a.   Prove that .                                                           (10)

             b.   Solve .                                                                                    (6)

Q.9     a.   Solve .                                                        (8)

b. Solve  subject to the initial condition y(0) = 0.        (8)