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. If one root of the be double of the other, then

(A)  a=2b                                            (B)  b=2a

(C)   a=3b                                           (D)  b=3a

b.  is equal to

(A) tan A                                            (B)  tan 2A

(C)  cot A                                           (D)  cot 2A                                                               

             c.   The point (x, y) lies on the line joining (2, 1) and (–6, –3) if

(A) x = 2y                                          (B)  y = 2x

(C)  x = y                                            (D)  x+y = 0

             d.   The equation of the straight line which passes through (3, 5) and is parallel to 2x+3y = 7 is    

(A)  3x – 2y = 9                                 (B)  2x + 3y = 19

(C)  3x – 2y = –1                               (D)  2x + 3y = 21

             e.   The equation of the circle passing through the origin and making intercepts

                   –2 and 3 on x-axis and y-axis respectively is

(A) (B) 

(C)  (D)

             f.    If y = log(sec x + tan x), then   is equal to

(A) (B)  sec x

(C)  tan x                                             (D)  sec x – tan x

g.   The value of   is equal to

(A) 1                                                  (B)  2

(C)  0                                                  (D)  none of these

             h.   is equal to

(A) log (1 + sin x)                               (B)

(C) sec x – tan x                                  (D) tan x – sec x

             i.    The area bounded by the axis of x and the curve  is

(A) (B)

(C) (D) 1

             j.    The order and the degree of the differential equation  are

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

(C)  2, 4                                              (D) 3, 4

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

Q.2     a.   The coefficients of second, third and fourth terms in the expansion of  are in A.P.; find the value of n.                                                           (8)

             b.   If the sum of first n terms of an A.P. is zero, show that the sum of next m terms is , if a be the first term of the A.P.                            (8)

Q.3     a.   If , show that

(8)

             b.   In any triangle ABC, show that

                   a sin (B – C) + b sin (C – A) + c sin (A – B) = 0                                                      (8)

Q.4     a.   Determine the ratio in which 3x – 5y + 8 = 0 divides the join of (4, 3) and

                   (8, 7). Also find the coordinates of that point.                                                     (8)

b.   Find the equation of a straight line passing through the point of intersection of 5x – 3y = 1 and 2x + 3y = 23 and perpendicular to the line x – 2y = 3.       (8)

Q.5     a.   Find the equation of a circle whose centre is (3, – 4) and passes through the intersection of the straight lines 3x + 4y  = 0 and 4x +3y  = 0.              (8)

     b.   Find the vertex, focus, latus rectum and directrix of the parabola .                 (8)

  Q.6     a. Find the differential coefficient of tan x from first principle.                                   (8)

             b.   Find , if .                                                              (8)

Q.7     a.   Find the points at which the function y = (x–1) (x–2) (x–3) has maximum and minimum values.                                                                     (8)

b.   Evaluate.                                                                                      (8)

Q.8     a.   Evaluate .                                                                           (8)

             b.   Find the volume of the solid generated by the revolution of the semi-circle of radius a, about its bounding diameter.                                             (8)

Q.9           Solve the following differential equations:-

(i)    .

                   (ii)  .                                                              (2 x 8 = 16)