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   Hint 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 (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,   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+ a^. b^)1/2   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+ 2a^. b^)1/2   Half-n-half Clue