Solutions to IIT–JEE 2008 (PAPER-2)

 

MATHEMATICS

PART  I

 

 

SECTION – I

 

Straight Objective Type

 

This section contains 9 multiple choice questions numbered 1 to 9. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct.

 

1.         Consider a branch of the hyperbola

           

with vertex at the point A. Let B be one of the end points of its latus rectum. If C is the focus of the hyperbola nearest to the point A, then the area of the triangle ABC is

            (A)                                                       (B)   

            (C)                                                       (D)   

1.         (B)

            x2 - 2y2 - 2x - 4y - 6 = 0

            x2 - 2y2 - 2 - 4y - 6 - 2 - 4 + 4 = 0

            (x - )2 - 2(y + )2 = 4

                    centre (h, k) = (, -)

           

                    b2 = a2 (e2 - 1)

                       

            l (latus rectum) end point

                       

            Focus (ae + h, K) =

            Vertex (2 + , -)

            \         AC =  - 2 unit and BC = 1 unit

                        DABC =  =

            ή       

2.         A particle P starts from the point z0= 1 + 2i, where . It moves first horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point z1. From z1 the particle moves  units in the direction of the vector  and then it moves through an angle  in anticlockwise direction on a circle with centre at origin, to reach a point z2. The point z2 is given by

            (A)    6 + 7i                                                      (B)    – 7 + 6i

            (C)    7 + 6i                                                      (D)    – 6 + 7i

2.         (D)

            z1 = zo + 5 + 3i

                        = (1 + 2i) + 5 + 3i

                        = 6 + 5i

            After moving distance , along (i  + j) it will reach at
(6 + 5i) +

                        = 7 + 6i

            The slope

            m’ =

            x < 0 hence z2 = -6 + 7i

 

3.         Let the function  be given by g(u) = 2 tan–1 (eu) . Then, g is

            (A)    even and is strictly increasing in (0,  ₯)

            (B)    odd and is strictly decreasing (– ₯,  ₯)

            (C)    odd and is strictly increasing in (– ₯,  ₯)

            (D)    neither even nor odd, but is strictly increasing (– ₯,  ₯)

3.        

           

           

           

            hence g(u) is odd function

           

            If since (eu > 0) " u Ξ R.

 

4.         An experiment has 10 equally likely outcomes. Let A and B be two non-empty events of the experiment. If A consists of 4 outcomes, the number of outcomes that B must have so that A and B are independent, is

            (A)    2, 4  or  8                                                (B)    3, 6  or  9

            (C)    4  or  8                                                    (D)    5  or  10

4.         (D)

 

5.         The area of the region between the curves  and  bounded by the lines x = 0 and  is

            (A)                                 (B)   

            (C)                                 (D)   

5.         (B)

           

            =

            Putting

                         

           

 

6.         Consider three points P = (– sin(b–a),  – cos b),  Q = (cos (b–a), sin b) and

            R=(cos (b–a+q),  sin (b–q)), where 0 < a < b , q < . Then,

            (A)    P lies on the line segment RQ                   (B)    Q lies on the line segment PR      

            (C)    R lies on the line segment QP                   (D)    P, Q, R are non – collinear

6.         (D)

 

7.         Let ,  .

            Then, for an arbitrary constant C, the value of  J – I equals

            (A)                           (B)   

            (C)                             (D)   

7.         (C)

 

8.         Let two non-collinear unit vectors  and   from an acute angle. A point P moves so that at any time t the position vector  (where O is the origin) is given by . When P is farthest from origin O, let M be the length of  and  be the unit vector along . Then,

            (A)    and               (B)    and

            (C)    and             (D)    and

8.         (A)

             = (cosq   + sinq)

             = (cosa + sina)

            (cosq.cost  + sinq cost

                        + (cosa . sint  + sina sint )

            (cosq.cost + cosasint)  + (sinqcost + sinasint)

            || = cos2q.cos2t + cos2a.sin2t + 2cosq.cosa cost.sint + sin2q . cos2t + sin2asin2t + 2sinq.sina. sint.cost

            ||2 = cos2t + sin2t + 2 cos(q - a) sint cost

                        = 1 + 2 cos(q - a)sin2t

            ||max = (1 +  . )1/2

                \         =  cos

                        =

 

9.         Let g(x) = log f(x) where f(x) is a twice differentiable positive function on (0,  ₯) such that f(x + 1) = x f(x). Then, for N =1, 2, 3,……,

                    

            (A)             (B)   

            (C)             (D)   

9.         (A)

                         and