JEE Advance - Mathematics (2008 - Paper 2 Offline - No. 2)
A particle P stats from the point $${z_0}$$ = 1 +2i, where $$i = \sqrt { - 1} $$. It moves horizontally away from origin by 5 unit and then vertically away from origin by 3 units to reach a point $${z_1}$$. From $${z_1}$$ the particle moves $$\sqrt 2 $$ units in the direction of the vector $$\hat i + \hat j$$ and then it moves through an angle $${\pi \over 2}$$ in anticlockwise direction on a circle with centre at origin, to reach a point $${z_2}$$. The point $${z_2}$$ is given by
6 + 7i
-7 + 6i
7 + 6i
- 6 + 7i
Explanation
given $$-$$ $$\alpha$$ particle
In the direction of, or, Now rotation about origin through angle of means multiply by
$${z_0} = 1 + 2i = (1,2) = ({x_0},{y_0})$$
$${z_1} = ({x_0} + 5,{y_0} + 3)$$
$$ = (6,5) = 6 + 5i$$
$$\Rightarrow 2$$ in the direction of $$i = j$$
$${x_1} = 2\cos 45^\circ $$
$${y_1} = 2\sin 45^\circ $$
$${z_2}(7 + 6i)$$
Now, rotation about origin through an angle of 2$$\pi$$ means multiply z$$_2$$ by $$ \to {e^{i\pi /2}}$$
$$\cos 90^\circ + i\sin 90^\circ = i$$
$${z_3} = i(7 + 6i)$$
$$ = - 6 + 7i$$
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