JEE Mains · Maths · STD 11 - 4.1 complex nubers
Let \(z \in C\) be such that \(\frac{z^2+3 i}{z-2+i}=2+3 i\). Then the sum of all possible values of \(z^2\) is
- A \(19-2 \mathrm{i}\)
- B \(-19-2 \mathrm{i}\)
- C \(19+2 i\)
- D \(-19+2 i\)
Answer & Solution
Correct Answer
(B) \(-19-2 \mathrm{i}\)
Step-by-step Solution
Detailed explanation
\(z^2+3 i=z(2+3 i)-7-4 i\) \(\mathrm{z}^2-\mathrm{z}(2+3 \mathrm{i})+7+7 \mathrm{i}=0\) \(\begin{aligned} & z_1^2+z_2^2=\left(z_1+z_2\right)^2-2 z_1 z_2 \\ & =4-9+12 i-14-14 i \\ & =-19-2 i \end{aligned}\)
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
- Let \(m, n \in N\) and \(\operatorname{gcd}(2, n)=1\). If \(30\left(\begin{array}{l}30 \\ 0\end{array}\right)+29\left(\begin{array}{l}30 \\ 1\end{array}\right)+\ldots+2\left(\begin{array}{l}30 \\ 28\end{array}\right)+1\left(\begin{array}{l}30 \\ 29\end{array}\right)= n .2^{ m }\) then \(n + m\) is equal to (Here \(\left.\left(\begin{array}{l} n \\ k \end{array}\right)={ }^{ n } C _{ k }\right)\)JEE Mains 2021 Hard
- Let the coefficients of three consecutive terms \(T_r, T_{r+1}\) and \(T_{r+2}\) in the binomial expansion of \((a+b)^{12}\) be in a G.P. and let \(p\) be the number of all possible values of \(r\). Let \(q\) be the sum of all rational terms in the binomial expansion of \((\sqrt[4]{3}+\sqrt[3]{4})^{12}\). Then \(\mathrm{p}+\mathrm{q}\) is equal to :JEE Mains 2025 Medium
- The two adjacent sides of a cyclic quadrilateral are \(2\) and \(5\) and the angle between them is \(60^o\). If the area of the quadrilateral is \(4\sqrt 3 \) , then the perimeter of the quadrilateral isJEE Mains 2017 Hard
- Let the volume of a parallelopiped whose coterminous edges are given by \(\overrightarrow{\mathrm{u}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\lambda \hat{\mathrm{k}}, \overrightarrow{\mathrm{v}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+3 \hat{\mathrm{k}} \) and \(\overrightarrow{\mathrm{w}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}\) be \(1\; cu.\) unit. If \(\theta\) be the angle between the edges \(\overrightarrow{\mathrm{u}}\) and \(\overrightarrow{\mathrm{w}},\) then \(\cos \theta\) can beJEE Mains 2020 Hard
- Let \(f(x)+2 f\left(\frac{1}{x}\right)=x^2+5\) and \(2 g(x)-3 g\left(\frac{1}{2}\right)=x, x \gt 0\). If \(\alpha=\int_1^2 f(x) d x\), and \(\beta=\int_1^2 g(x) d x\), then the value of \(9 \alpha+\beta\) is:JEE Mains 2025 Hard
- \(\lim _{n \rightarrow \infty} \frac{1}{2^{n}}\left(\frac{1}{\sqrt{1-\frac{1}{2^{a}}}}+\frac{1}{\sqrt{1-\frac{2}{2^{n}}}}+\frac{1}{\sqrt{1-\frac{3}{2^{a}}}}+\ldots \ldots+\frac{1}{\sqrt{1-\frac{2^{a}-1}{2^{n}}}}\right)\) is equal toJEE Mains 2022 Hard
More PYQs from JEE Mains
- The number of natural numbers lying between \(1012\) and \(23421\) that can be formed using the digits \(2,3,4,5,6\) (repetition of digits is not allowed) and divisible by \(55\) is \(....\)JEE Mains 2022 Hard
- Let \(\mathrm{A}(-2,-1), \mathrm{B}(1,0), \mathrm{C}(\alpha, \beta)\) and \(\mathrm{D}(\gamma, \delta)\) be the vertices of a parallelogram \(A B C D\). If the point \(C\) lies on \(2 x-y=5\) and the point \(D\) lies on \(3 x-2 y=6\), then the value of \(|\alpha+\beta+\gamma+\delta|\) is equal to ...........JEE Mains 2024 Hard
- A function \(y=f(x)\) satisfies \(f(x) \sin 2 x+\sin x-\left(1+\cos ^2 x\right) f^{\prime}(x)=0\) with condition \(f(0)=0\). Then \(f\left(\frac{\pi}{2}\right)\) is equal toJEE Mains 2024 Medium
- Let \(S =\{\sqrt{ n }: 1 \leq n \leq 50\) and \(n\) is odd \(\}\) Let \(a \in S\) and \(A =\left[\begin{array}{ccc}1 & 0 & a \\ -1 & 1 & 0 \\ - a & 0 & 1\end{array}\right]\) If \(\sum_{ a \in S } \operatorname{det}(\operatorname{adj} A )=100 \lambda\), then \(\lambda\) is equal toJEE Mains 2022 Medium
- \(\mathop {\lim }\limits_{n \to \infty } \left( {\frac{{{{\left( {n + 1} \right)}^{1/3}}}}{{{n^{4/3}}}} + \frac{{{{\left( {n + 2} \right)}^{1/3}}}}{{{n^{4/3}}}} + .... + \frac{{{{\left( {2n} \right)}^{1/3}}}}{{{n^{4/3}}}}} \right)\) is equal toJEE Mains 2019 Hard
- Let \(\mathrm{g}: \mathrm{N} \rightarrow \mathrm{N}\) be defined as \(g(3 n+1)=3 n+2\) \(g(3 n+2)=3 n+3\) \(g(3 n+3)=3 n+1, \text { for all } n \geq 0\) Then which of the following statements is true?JEE Mains 2021 Hard