Complex Numbers and Quadratic Equations

JEE Advanced · Mathematics · 46 past questions

1. Q1 2026 Paper 1

   Match each entry in List-I to the correct entry in List-II and choose the correct option.

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2. Q2 2026 Paper 2

   Let $a,\ b,\ c$ be positive integers in arithmetic progression such that the equation…

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3. Q3 2025 Paper 1

   Let $\mathbb{R}$ denote the set of all real numbers. Let $z_1 = 1+2i$ and $z_2 = 3i$ be two complex numbers…

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4. Q4 2025 Paper 2

   For a non-zero complex number $z$, let $\arg(z)$ denote the principal argument of $z$, with…

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5. Q5 2024 Paper 1

   Let $f(x) = x^4 + ax^3 + bx^2 + c$ be a polynomial with real coefficients such that $f(1) = -9$. Suppose that…

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6. Q6 2023 Paper 1

   Let $A=\left\{\frac{1967+1686i\sin\theta}{7-3i\cos\theta}:\theta\in\mathbb{R}\right\}$. If $A$ contains…

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7. Q7 2023 Paper 1

   Let $z$ be a complex number satisfying $|z|^3+2z^2+4\bar z-8=0$, where $\bar z$ denotes the complex conjugate…

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8. Q8 2022 Paper 1

   Let $z$ be a complex number with non-zero imaginary part. If…

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9. Q9 2022 Paper 1

   Let $\bar z$ denote the complex conjugate of a complex number $z$ and let $i=\sqrt{-1}$. In the set of…

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10. Q10 2022 Paper 2

    Let $\bar{z}$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for…

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11. Q11 2021 Paper 1

    Let $\theta_1, \theta_2, \ldots, \theta_{10}$ be positive valued angles (in radian) such that…

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12. Q12 2021 Paper 1

    For any complex number $w=c+id$, let $\arg(w)\in(-\pi,\pi]$, where $i=\sqrt{-1}$. Let $\alpha$ and $\beta$ be…

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13. Q13 2021 Paper 1

    For $x \in \mathbb{R}$, the number of real roots of the equation…

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14. Q14 2020 Paper 1

    Suppose $a, b$ denote the distinct real roots of the quadratic polynomial $x^2+20x-2020$ and suppose $c, d$…

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15. Q15 2020 Paper 1

    Let $S$ be the set of all complex numbers $z$ satisfying $|z^2+z+1|=1$. Then which of the following…

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16. Q16 2020 Paper 2

    For a complex number $z$, let Re($z$) denote the real part of $z$. Let $S$ be the set of all complex numbers…

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17. Q17 2019 Paper 1

    Let $S$ be the set of all complex numbers $z$ satisfying $|z-2+i| \ge \sqrt{5}$. If the complex number $z_0$…

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18. Q18 2019 Paper 1

    Let $\omega \neq 1$ be a cube root of unity. Then the minimum of the set…

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19. Q19 2018 Paper 1

    For a non-zero complex number $z$, let $\arg(z)$ denote the principal argument with $-\pi < \arg(z) \le \pi$.…

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20. Q20 2018 Paper 2

    Let $s,t,r$ be non-zero complex numbers and $L$ be the set of solutions $z=x+iy$ ($x,y\in\mathbb{R}$…

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21. Q21 2017 Paper 1

    Let $a, b, x$ and $y$ be real numbers such that $a-b=1$ and $y\neq0$. If the complex number $z=x+iy$…

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22. Q22 2017 Paper 2

    PARAGRAPH 2…

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23. Q23 2017 Paper 2

    PARAGRAPH 2…

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24. Q24 2016 Paper 2

    Let $a,b\in\mathbb{R}$ and $a^2+b^2\neq0$. Suppose…

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25. Q25 2015 Paper 2

    For any integer $k$, let $\alpha_k = \cos\left(\frac{k\pi}{7}\right) + i\sin\left(\frac{k\pi}{7}\right)$…

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26. Q26 2014 Paper 2

    Let $z_k = \cos\left(\frac{2k\pi}{10}\right) + i\sin\left(\frac{2k\pi}{10}\right)$; $k = 1,2,\dots,9$.

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27. Q27 2013 Paper 1

    Let complex numbers $\alpha$ and $\dfrac{1}{\bar{\alpha}}$ lie on circles $(x - x_0)^2 + (y - y_0)^2 = r^2$…

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28. Q28 2013 Paper 2

    Let $w = \frac{\sqrt{3}+i}{2}$ and $P = \{w^n : n = 1, 2, 3, \ldots\}$. Further…

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29. Q29 2013 Paper 2

    Paragraph for Questions 49 and 50…

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30. Q30 2013 Paper 2

    Paragraph for Questions 49 and 50…

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31. Q31 2012 Paper 1

    Let $z$ be a complex number such that the imaginary part of $z$ is nonzero and $a=z^2+z+1$ is real. Then $a$…

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32. Q32 2011 Paper 1

    Let $\alpha$ and $\beta$ be the roots of $x^2-6x-2=0$, with $\alpha>\beta$. If $a_n=\alpha^n-\beta^n$ for…

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33. Q33 2011 Paper 1

    If $z$ is any complex number satisfying $|z - 3 - 2i| \le 2$, then the minimum value of $|2z - 6 + 5i|$ is

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34. Q34 2011 Paper 2

    Let $\omega = e^{i\pi/3}$, and $a, b, c, x, y, z$ be non-zero complex numbers such that…

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35. Q35 2011 Paper 2

    The number of distinct real roots of $x^4-4x^3+12x^2+x-1=0$ is

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36. Q36 2010 Paper 1

    Let $z_1$ and $z_2$ be two distinct complex numbers and let $z = (1-t)z_1 + tz_2$ for some real number $t$…

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37. Q37 2010 Paper 2

    Match the statements in Column-I with those in Column-II.…

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38. Q38 2009 Paper 1

    Let $z=x+iy$ be a complex number where $x$ and $y$ are integers. Then the area of the rectangle whose…

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39. Q39 2009 Paper 1

    Let $z=\cos\theta+i\sin\theta$. Then the value of…

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40. Q40 2008 Paper 1

    Paragraph for Question Nos. 21 to 23…

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41. Q41 2008 Paper 1

    Paragraph for Question Nos. 21 to 23…

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42. Q42 2008 Paper 1

    Paragraph for Question Nos. 21 to 23…

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43. Q43 2008 Paper 2

    A particle $P$ starts from the point $z_0 = 1+2i$, where $i=\sqrt{-1}$. It moves first horizontally away from…

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44. Q44 2007 Paper 1

    Let $\alpha$, $\beta$ be the roots of the equation $x^2 - px + r = 0$ and $\dfrac{\alpha}{2}$, $2\beta$ be…

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45. Q45 2007 Paper 1

    A man walks a distance of 3 units from the origin towards the north-east ($N\,45^\circ E$) direction. From…

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46. Q46 2007 Paper 2

    If $|z| = 1$ and $z \neq \pm 1$, then all the values of $\dfrac{z}{1-z^2}$ lie on

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