TS EAMCET · Maths · Complex Number
\(\text { If } z_1=x_1+i y_1, z_2=x_2+i y_2, z_3=x_1+\frac{i x_2}{2}\) \(z_4=2 y_1+i y_2\) are complex numbers such that \(\left|z_1\right|=1,\left|z_2\right|=2\) and \(\operatorname{Re}\left(z_1 z_2\right)=0\), then
- A \(\left|z_3\right|=1,\left|z_4\right|=2, \operatorname{Im}\left(z_3 z_4\right)=0\)
- B \(\left|z_3\right|=2,\left|z_4\right|=1, \operatorname{Re}\left(z_3 z_4\right)=0\)
- C \(\left|z_3\right|=1,\left|z_4\right|=2, \operatorname{Re}\left(z_3 z_4\right)=0\)
- D \(\left|z_3\right|=2,\left|z_4\right|=1, \operatorname{Re}\left(z_1 z_3\right)=\operatorname{Im}\left(z_2 z_4\right)=0\)
Answer & Solution
Correct Answer
(C) \(\left|z_3\right|=1,\left|z_4\right|=2, \operatorname{Re}\left(z_3 z_4\right)=0\)
Step-by-step Solution
Detailed explanation
We have, \(z_1=x_1+i y_1\) \(z_2=x_2+i y_2\) \(z_3=x_1+\frac{i x_2}{2}\) \(z_4=2 y_1+i y_2\) \(\left|z_1\right|=1,\left|z_2\right|=2\) and \(\operatorname{Re}\left(z_1 z_2\right)=0\) Let \(z_1=e^{i \alpha}\) and \(z_2=2 e^{i \beta}\) \(z_1 z_2=2 e^{i(\alpha+\beta)}\)…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- If \([x]\) is the greatest integer function then \(\lim _{x \rightarrow 3^{-}} \frac{(3-|x|+\sin |3-x|) \cos [9-3 x]}{|3-x|[3 x-9]}=\)TS EAMCET 2025 Hard
- If \(x=\frac{1-\sqrt{y}}{1+\sqrt{y}}\), then \((x+1) \frac{d^2 y}{d x^2}+\left(\frac{3 \sqrt{y}+1}{\sqrt{y}}\right) \frac{d y}{d x}\) equalsTS EAMCET 2015 Medium
- If the volume of a tetrahedron having \(\bar{i}+2 \bar{j}-3 \bar{k}, 2 \bar{i}+\bar{j}-3 \bar{k}\) and \(3 \bar{i}-\bar{j}+\mathrm{p} \bar{k}\) as its coterminous edges is 2, then the values of \(p\) are the roots of the equationTS EAMCET 2025 Medium
- Two ships leave a port from a point at the same time. One goes with a velocity of \(3 \mathrm{~km} / \mathrm{h}\) along North-East maling an angle of \(45^{\circ}\) with East direction and the other travels with a velocity of \(4 \mathrm{~km} / \mathrm{h}\) along South-East maling an angle of \(15^{\circ}\) with East direction. Then, the distance between the ships at the end of two hours isTS EAMCET 2018 Medium
- If the equation of a tangent drawn to the curve \(y=\cos (x+y),-1 \leq x \leq 1+\pi\) is \(x+2 y=k\), then \(k=\)TS EAMCET 2023 Hard
- 68. If \(f(x)=\frac{e^{-x} \sin x}{\log _e x}\) and \(f^{\prime}(x)=f(x) . g(x)\), then \(g^{\prime}(\mathrm{e})=\)
(a) \(e^{-2}-\operatorname{cosec}^2(e)\)
(b) \(2 e^{-2}-\operatorname{cosec}^2(e)\)
(c) \(2 e^{-2}-\operatorname{cosec}^2(e)\)
(d) \(2 e^{-2}+\operatorname{cosec}^2(e)\)TS EAMCET 2022 Easy
More PYQs from TS EAMCET
- A ball \(A\) of mass \(m\) moving along positive \(x\)-direction with kinetic energy \(K\) and momentum \(p\) undergoes elastic head on collision with a stationary ball \(B\) of mass \(M\) after collision the ball \(A\) moves along negative \(x\)-direction with kinetic energy \(\frac{K}{9}\), final momentum of \(B\) isTS EAMCET 2012 Hard
- Two tangents are drawn from the point \((-1,-2)\) to the parabola \(\mathrm{y}^2=4 \mathrm{x}\). If \(\theta\) is the angle between these tangents, then \(\tan \theta=\)TS EAMCET 2023 Medium
- An object of mass \(15 \mathrm{~kg}\) is attached to the end of a metal wire of unstretched length \(1.0 \mathrm{~m}\). The object is then whirled in a vertical circle with an angular velocity of 4 \(\mathrm{rad} / \mathrm{s}\) at the bottom of the circle. If the cross sectional area of the wire is \(0.05 \mathrm{~cm}^2\) and Young's modulus of metal is \(2 \times 10^{11} \mathrm{~N} / \mathrm{m}^2\), then the elongation of the wire when the mass is at the lowest point of its path (Take \(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2\) )TS EAMCET 2022 Easy
- If and are the greatest divisors of and respectively for all , thenTS EAMCET 2018 Medium
- If \(\sqrt{5}-i \sqrt{15}=r(\cos \theta+i \sin \theta),-\pi \lt \theta \lt \pi\), then \(r^2\left(\sec \theta+3 \operatorname{cosec}^2 \theta\right)=\)TS EAMCET 2024 Medium
- A ball of mass \(1 \mathrm{~kg}\) moves in a straight line with velocity \(v=c x^a\), where \(c=1\) (SI unit) and \(\alpha\) is a constant. If the work done by the net force during its displacement from \(\mathrm{x}=0\) to \(\mathrm{x}=4 \mathrm{~m}\) is \(128 \mathrm{~J}\), then the \(\alpha\) isTS EAMCET 2022 Medium