1.  1. A particle P starts from the point z_{0} = 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 z_{1}. From z_{1} 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 z_{2}. The point z_{2} is given by 

Hint
Halfnhalf Clue
 

2.  2. Let the function g : (− ∞, ∞) →  (−  π 2  ,  π 2  ) be given by g(u) = 2 tan^{−1} (e^{u}) −  π 2  . Then, g is  
 Halfnhalf Clue
 

3.  3. Consider a branch of the hyperbola x^{2} – 2 y^{2} – 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 
 Halfnhalf Clue
 

4.  4. The area of the region between the curves and bounded by the lines x = 0 and x = π /4 is 
 √2 − 1 ∫ 0  t (1 + t^{2}) √(1 − t^{2})  dt   √2 − 1 ∫ 0  4t (1 + t^{2}) √(1 − t^{2})  dt   √2 + 1 ∫ 0  4t (1 + t^{2}) √(1 − t^{2})  dt   √2 + 1 ∫ 0  t (1 + t^{2}) √(1 − t^{2})  dt   Halfnhalf Clue
 

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

6.  6. An experiment has 10 equally likely outcomes. Let A and B be two nonempty 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 
 Halfnhalf Clue
 

7.  7. Let two noncollinear 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 
 Halfnhalf Clue
 
