NIMCET 2019 Mathematics PYQ — A particle P starts from the point , where . It moves first horiz… | Mathem Solvex | Mathem Solvex
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NIMCET 2019 — Mathematics PYQ
NIMCET | Mathematics | 2019
A particle P starts from the point z0=1+2i, where i=−1. It moves first horizontally away from the 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 to reach z2, and then it moves through an angle 2π in an anti-clock-wise direction on a circle with center at origin, to reach a point z3. The point z3 is given by:
Choose the correct answer:
A.
6+7i
B.
−7+6i
C.
7+6i
D.
−6+7i
(Correct Answer)
Correct Answer:
−6+7i
Explanation
Concept:
The complex nunbera +ib is represented on a two dinnensional plane where a is represented on the x-axis (the real axis) and b is represented on the y-axis (the imaginary axis). The point on aline whakes an angle θ with the x- axis, is given by ( rcosθ, r sin of the point from the origin. If a point on the complex plane rotates by an angle θ in the anti-clock-wise direction, then its value becomes eiθ times of itself, or (cos θ+i sin θ) times. In case of a clock-wise rotation, the value become e−iθ times of itself, or (cosθ−isinθ)times.
Calculation:
When the point z0=1+2i moves horizontally, its real part will change and when it moves vertically, its
imaginary part will change.
∴z1=(1+5)+(2+3)i
⇒z1=6+5i
The vector î+j makes an angle of 45° with the x-axis. Moving a point by √2 units along this direction will
change its value by (2cos45∘,2sin45∘)=(1,1).
∴z2=(6+1)+(5+1)i
⇒z2=7+6i
Finally, rotating the point in the anti-clock-wise direction by an angle of 2π changes its value by e2πi times or (cos2π+isin2π)=0+i.1=i times. ∴z3=(7+6i˙).i˙=−6+7i˙.
Explanation
Concept:
The complex nunbera +ib is represented on a two dinnensional plane where a is represented on the x-axis (the real axis) and b is represented on the y-axis (the imaginary axis). The point on aline whakes an angle θ with the x- axis, is given by ( rcosθ, r sin of the point from the origin. If a point on the complex plane rotates by an angle θ in the anti-clock-wise direction, then its value becomes eiθ times of itself, or (cos θ+i sin θ) times. In case of a clock-wise rotation, the value become e−iθ times of itself, or (cosθ−isinθ)times.
Calculation:
When the point z0=1+2i moves horizontally, its real part will change and when it moves vertically, its
imaginary part will change.
∴z1=(1+5)+(2+3)i
⇒z1=6+5i
The vector î+j makes an angle of 45° with the x-axis. Moving a point by √2 units along this direction will
change its value by (2cos45∘,2sin45∘)=(1,1).
∴z2=(6+1)+(5+1)i
⇒z2=7+6i
Finally, rotating the point in the anti-clock-wise direction by an angle of 2π changes its value by e2πi times or (cos2π+isin2π)=0+i.1=i times. ∴z3=(7+6i˙).i˙=−6+7i˙.
