Code: D-01 / DC-01                                                                        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 equation of the straight line which makes equal intercepts on the axes and passes through the point (1, 2) is 

(A)  x + y = 3                                      (B)  x + 2y = 5

(C)   x – y = 1                                     (D)  2x + y = 4

b. Area of the triangle whose vertices are (a, b) (a, a + b), (-a, -a + b) is

(A) a²b² (B)  a² + b²

(C)  a² (D)  b²

             c.  

(A) 1                                                  (B)

(C) (D)  Zero

             d.   The point on the curve y² = 4x at which the tangent to the curve is parallel to y = x is

(A)  (0, 0)                                          (B) (2,

(C)  (4, 4)                                           (D)  (1, 2) 

             e.   is equal to

(A) tan x – cot x                                 (B)  tan x + cot x

(C)  sec x + cosec x                            (D)  sec x - cosec x

f.    Sin³x dx is equal to

(A) (B)

(C) (D)

             g.   Solution of differential equation is

(A) e^x + e^y = const                             (B)  e^x – e^y = const

(C)  e^x . e^y   = const                             (D)  e^x / e^y  = const

             h.   Period of Sin (2x + 3) is

(A) 2π                                                (B)

(C) π                                                   (D)

             i.    The value of Sin 105⁰ + Cos 105⁰ is

(A) (B)

(C) (D)

             j.    If pth, (2p)^th and (3p)^th terms of a G.P. are x, y, z respectively, then x, y, z are in

(A)  A.P.                                             (B)  H.P.

(C)  G.P.                                             (D) None of these

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

Q.2     a.   Find the term independent of x in the expansion of .                            (8)

             b.   If the p^th, q^th and r^th terms of an A.P. are x, y, z respectively, show that        x (q – r) + y (r – p) + z (p – q) = 0.                                                                                                                   (8)

Q.3     a.   If A + B + C = π, show that

(8)

             b.   In any triangle ABC, show that

(8)

Q.4     a.   Find the equation of a straight line when p is the length of perpendicular on it from the origin and the inclination of this perpendicular to the x – axis is .                                                            (8)

b.   Find the equation of the straight line which passes through the intersection of the straight lines 2x – 3y + 4 = 0 and 3x + 4y + 5 = 0 and is perpendicular to the straight line 6x – 7y + 8 = 0.      (8)

Q.5     a.   Show that x² + y² + 2gx + 2fy + c = 0 represents a circle. Find its centre and radius.                      (6)

     b.   Find the vertex, focus, latus rectum and directrix of the parabola.              x² = 4x – y.                    (10)

  Q.6     a. Evaluate , by using the fact that .              (8)

             b.   Differentiate with respect to x.                                                (8)

Q.7     a.   Find the points at which the function

y = 3 Sin²x + 4 Cos²x

                   has maximum and minimum values in the interval (8)

b.   Evaluate , where a, b are not both zero.                                 (8)

Q.8     a.   Find the area common to the circles x² + y² – 2ax = 0 and

x² + y² – 2ay = 0.                                                                                            (10)

             b.   Evaluate  .                                                                                 (6)

Q.9           Solve following the differential equations

(i)        ydx – xdy = .                                                                  (8)

(ii) Cos²x .                                                                       (8)