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 |
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Hint
Half-n-half Clue
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2. | 2. Let the function g : (− ∞, ∞) → | (− | π 2 | , | π 2 | ) be given by g(u) = 2 tan−1 (eu) − | π 2 | . Then, g is | |
| Half-n-half Clue
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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 |
| Half-n-half Clue
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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 + 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
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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, |
| Half-n-half Clue
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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 |
| Half-n-half Clue
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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 |
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