DipIETE – ET / CS (OLD SCHEME)

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 the best alternative in the following:                                  (210)

             a.    If  , then the value of  is equal to

(A) (B)

(C) (D)

             b.   If   then the value of cosec is equal to

(A) (B)

(C) (D)

             c.    The value of definite integral  is equal to

(A) a                                                         (B) a²

(C) 0                                                        (D) 2a

             d.   If (3, –4) and  (–6, 5) are the extremities of the diagonal of a parallelogram and (–2, 1) is the third vertex, then the fourth vertex is

(A) (–1, 0)                                                (B) (0,–1)

(C) (–1, 1)                                                (D) None of  these.

             e.    If the circle cuts  in A and B, then the equation of the circle on AB as diameter is

(A)

                    (B)

(C)

                    (D) _{None of these.}

f.    If the r^th, (r+1)^th and (r+2)^th terms in the expansion of (1+x)¹⁴ are in A.P., then the value of r is given by

(A) 8                                                        (B) 6

(C) 7                                                        (D) 9

             g.    If   then equals to

(A) (B)

(C) (D)

             h.    The curve touches the line  at the points (a,b) for n =

(A) 1                                                        (B) 2

(C) 3                                                        (D) all non-zero values of n.

             i.     The value of is equal to

(A) (e–1)                                                  (B) 2(e–1)

(C) 3(e–1)                                                (D) 2(1–e)

             j. The solution of  is

(A) (B)

(C) (D) None of these.

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

Q.2   a.     In a triangle ABC,   a, b, c are    the   sides of   triangle and A, B, C are the angles, then find the value of  in terms of angles.                (5)

            b.     In a triangle, the lengths of the two larger sides are 10   and 9 respectively. If the angles are in A.P, then find  the length of the third side.                                                                                    (5)

            c.     If , then find the value of x.                             (6)

Q.3   a.     If the third term in the expansion of     is  then   find  the value of  x.           (8)

            b.     If      and   , then (8)

Q.4   a.     The  function

is continuous for  then find the most suitable values of a and b.              (8)

            b.     If  f(x) is twice  differentiable  such that  and

, then find the value of h(10) if h(5) = 11.                     (8)

Q.5   a.     A, B  are     two   points  (3, 4) and  (5, -2);  find  the   point  P such   that PA = PB and the area of triangle PAB = 10.                                                                                                          (8)

            b.     If   p and are   the   perpendicular   from the origin on the straight lines whose  equations   are prove that .              (8)

Q.6   a.     Let A be the centre of the circle x²+y²–2x–4y–20=0. Suppose the tangents   at the points B (1,7) and D (4, –2) on the circle meet at the point C. Find the area of quadrilateral ABCD.                              (8)

            b.     If the normal at the end of a latus rectum of an ellipse passes through one extremity of a minor axes, show that eccentricity of the curve is given by e⁴ + e² – 1 = 0.                                        (8)

Q.7   a.     Prove   that   the sum of intercepts on the coordinate axes of any tangent to the curve is constant.                                                                                                              (8)

            b.     Show that the semi-vertical angle of the cone of maximum volume of given slant height is .                   (8)

Q.8   a.     Prove that .                                              (8)

            b.     If  then prove that .  Deduce that (8)

Q.9   a.     Find   the   volume   formed   by   the   revolution   of the loop of the curve

 about x-axis.                                                                     (8)

            b.     Solve (8)