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IIT JEE 2008
Paper II


Formats Worksheet / Test Paper Quiz Review
: Multiple choice

Your Performance  

1.

1. A particle P starts from the point z0 = 1 + 2i, where i = √–1. 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 √2 units in the direction of the vector i^+ j^and then it moves through an angle π /2 in anticlockwise direction on a circle with centre at origin, to reach a point z2. The point z2 is given by

  6 + 7i

  – 7 + 6i

  7 + 6i

  – 6 + 7i

  Half-n-half Clue
 

2.

2.
Let the function g : (− ∞, ∞) → ( π

2
, π

2
) be given by g(u) = 2 tan−1 (eu) − π

2
. Then, g is

  even and is strictly increasing in (0, ∞)

  odd and is strictly decreasing in (– ∞, ∞)

  odd and is strictly increasing in (– ∞, ∞)

  neither even nor odd, but is strictly increasing in (– ∞, ∞)

  Half-n-half Clue
 

3.

3. Consider a branch of the hyperbola

x2 – 2 y2 – 2 √2 x – 4 √2 y – 6 = 0


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

 
1 – 2

3

 
3

2
− 1

 
1 + 2

3

 
3

2
+ 1

  Half-n-half Clue
 

4.

4. The area of the region between the curves
y = 1 + sin x

cos x
and
y = 1 − sin x

cos x
bounded by the lines x = 0 and x = π /4 is

 
2 − 1

0
t

(1 + t2) √(1 − t2)
dt

 
2 − 1

0
4t

(1 + t2) √(1 − t2)
dt

 
2 + 1

0
4t

(1 + t2) √(1 − t2)
dt

 
2 + 1

0
t

(1 + t2) √(1 − t2)
dt

  Half-n-half Clue
 

5.

5. Consider three points P = (– sin (βα), – cos β), Q = (cos (βα), sin β) and R = (cos (βα + θ), sin (βθ)), where 0 < α , β, θ < π /4. Then,

  P lies on the line segment RQ

  Q lies on the line segment PR

  R lies on the line segment QP

  P, Q, R are non-collinear

  Half-n-half Clue
 

6.

6. 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

  2, 4 or 8

  3, 6 or 9

  4 or 8

  5 or 10

  Half-n-half Clue
 

7.

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

 
u^ =
a^b^

| a^ b^ |
and M = ( 1+ 2a^. b^)1/2

 
u^ =
a^b^

| a^b^ |
and M = ( 1+a^. b ^)1/2

 
u^ =
a^ + b^

| a^+ b^ |
and M = ( 1+ a^. b^)1/2

 
u^ =
a^ + b^

| a^+ b^ |
and M = ( 1+ 2a^. b ^)1/2

  Half-n-half Clue
 
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