Complex Numbers & Quadratic Equations · JEE Main previous year questions

2025 19 questions

1. Q62025 · 8 Apr 2025 · Shift 2

   Let $\alpha$ be a solution of $x^2+x+1=0$, and for some $a$ and $b$ in $\mathbb{R}$, $\left[\begin{array}{ccc}4& a& b\end{array}\right]\left[\begin{array}{ccc}1& 16& 13\\ -1& -1& 2\\ -2& -14& -8\end{array}\right]=\left[\begin{array}{ccc}0& 0& 0\end{array}\right]$. If $\frac{4}{{\alpha }^4}+\frac{m}{{\alpha }^a}+\frac{n}{{\alpha }^b}=3$, then $m+n$ is equal to $\underline{}$

   1. A.$11$
   2. B.$3$
   3. C.$8$
   4. D.$7$
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   Answer: A

2. Q252025 · 4 Apr 2025 · Shift 1· Numerical answer

   Let $A=\{z\in \mathrm{C}:|z-2-i|=3\},B=\{z\in \mathrm{C}:\mathrm{Re}(z-iz)=2\}$ and $S=A\cap B$. Then ${\sum }_{z\in S}|z{|}^2$ is equal to ____.

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   Answer: 22

3. Q92025 · 3 Apr 2025 · Shift 1

   Let $\alpha$ and $\beta$ be the roots of $x^2+\sqrt{3}x-16=0$, and $\gamma$ and $\delta$ be the roots of $x^2+3x-1=0$. If $P_n={\alpha }^n+{\beta }^n$ and $Q_n={\gamma }^n+{\delta }^n$, then $\frac{P_{25}+\sqrt{3}{P}_{24}}{2P_{23}}+\frac{Q_{25}-Q_{23}}{Q_{24}}$ is equal to

   1. A.4
   2. B.3
   3. C.5
   4. D.7
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   Answer: C

4. Q92025 · 2 Apr 2025 · Shift 1

   Let $z$ be a complex number such that $|z|=1$. If $\frac{2+k^{2}z}{k+\bar{z}}=kz$, $k\in \mathbf{R}$, then the maximum distance of $k+i{k}^2$ from the circle $|z-(1+2i)|=1$ is :

   1. A.$\sqrt{5}+1$
   2. B.3
   3. C.$\sqrt{3}+1$
   4. D.2
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   Answer: A

5. Q122025 · 2 Apr 2025 · Shift 1

   Let $P_n={\alpha }^n+{\beta }^n,n\in \mathbf{N}$. If $P_{10}=123,P_9=76,P_8=47$ and $P_1=1$, then the quadratic equation having roots $\frac{1}{\alpha }$ and $\frac{1}{\beta }$ is :

   1. A.$x^2+x-1=0$
   2. B.$x^2-x+1=0$
   3. C.$x^2+x+1=0$
   4. D.$x^2-x-1=0$
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   Answer: A

6. Q142025 · 2 Apr 2025 · Shift 1

   If the system of linear equations $$3x+y+\beta z=3$$ $$2x+\alpha y-z=-3$$ $$x+2y+z=4$$ has infinitely many solutions, then the value of $22\beta -9\alpha$ is :

   1. A.31
   2. B.37
   3. C.43
   4. D.49
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   Answer: A

7. Q52025 · 2 Apr 2025 · Shift 2

   If the system of equations $2x+\lambda y+3z=5$ $3x+2y-z=7$ $4x+5y+\mu z=9$ has infinitely many solutions, then $({\lambda }^2+{\mu }^2)$ is equal to :

   1. A.30
   2. B.26
   3. C.22
   4. D.18
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   Answer: B

8. Q62025 · 29 Jan 2025 · Shift 2

   Let $\alpha ,\beta (\alpha \ne \beta )$ be the values of $m$, for which the equations $x+y+z=1$, $x+2y+4z=m$ and $x+4y+10z=m^2$ have infinitely many solutions. Then the value of ${\sum }_{n=1}^{10}(n^{\alpha }+n^{\beta })$ is equal to :

   1. A.$3410$
   2. B.$560$
   3. C.$3080$
   4. D.$440$
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   Answer: D

9. Q32025 · 28 Jan 2025 · Shift 1

   The sum, of the squares of all the roots of the equation $x^2+|2x-3|-4=0$, is

   1. A.$6(2-\sqrt{2})$
   2. B.$3(3-\sqrt{2})$
   3. C.$3(2-\sqrt{2})$
   4. D.$6(3-\sqrt{2})$
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   Answer: A

10. Q192025 · 28 Jan 2025 · Shift 1

    Let $O$ be the origin, the point $A$ be $z_1=\sqrt{3}+2\sqrt{2}i$, the point $B\left(z_2\right)$ be such that $\sqrt{3}\left|z_2\right|=\left|z_1\right|$ and $\arg \left(z_2\right)=\arg \left(z_1\right)+\frac{\pi }{6}$. Then

    1. A.area of triangle ABO is $\frac{11}{4}$
    2. B.area of triangle ABO is $\frac{11}{\sqrt{3}}$
    3. C.ABO is a scalene triangle
    4. D.ABO is an obtuse angled isosceles triangle
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    Answer: D

11. Q202025 · 28 Jan 2025 · Shift 1

    The area (in sq. units) of the region $\left\{(x,y):0\le y\le 2|x|+1,0\le y\le x^2+1,|x|\le 3\right\}$ is

    1. A.$\frac{32}{3}$
    2. B.$\frac{64}{3}$
    3. C.$\frac{17}{3}$
    4. D.$\frac{80}{3}$
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    Answer: B

12. Q32025 · 24 Jan 2025 · Shift 1

    The product of all the rational roots of the equation ${\left(x^2-9x+11\right)}^2-(x-4)(x-5)=3$, is equal to

    1. A.$7$
    2. B.$21$
    3. C.$28$
    4. D.$14$
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    Answer: D

13. Q72025 · 24 Jan 2025 · Shift 1

    If $\alpha$ and $\beta$ are the roots of the equation $2z^2-3z-2i=0$, where $i=\sqrt{-1}$, then $16\cdot \mathrm{Re}\left(\frac{{\alpha }^{19}+{\beta }^{19}+{\alpha }^{11}+{\beta }^{11}}{{\alpha }^{15}+{\beta }^{15}}\right)\cdot \mathrm{Im}\left(\frac{{\alpha }^{19}+{\beta }^{19}+{\alpha }^{11}+{\beta }^{11}}{{\alpha }^{15}+{\beta }^{15}}\right)$ is equal to

    1. A.$441$
    2. B.$312$
    3. C.$409$
    4. D.$398$
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    Answer: A

14. Q152025 · 24 Jan 2025 · Shift 1

    The area of the region $\left\{(x,y):x^2+4x+2\le y\le |x+2|\right\}$ is equal to

    1. A.$7$
    2. B.$\frac{24}{5}$
    3. C.$\frac{20}{3}$
    4. D.$5$
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    Answer: C

15. Q192025 · 24 Jan 2025 · Shift 1

    If the system of equations $2x-y+z=4$ $5x+\lambda y+3z=12$ $100x-47y+\mu z=212$ has infinitely many solutions, then $\mu -2\lambda$ is equal to

    1. A.$56$
    2. B.$59$
    3. C.$57$
    4. D.$55$
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    Answer: C

16. Q22025 · 23 Jan 2025 · Shift 2

    The number of complex numbers $z$, satisfying $|z|=1$ and $\left|\frac{z}{\bar{z}}+\frac{\bar{z}}{z}\right|=1$, is :

    1. A.8
    2. B.10
    3. C.4
    4. D.6
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    Answer: A

17. Q102025 · 23 Jan 2025 · Shift 2

    The system of equations $x+y+z=6$, $x+2y+5z=9$, $x+5y+\lambda z=\mu$, has no solution if

    1. A.$\lambda =17,\mu =18$
    2. B.$\lambda =17,\mu \ne 18$
    3. C.$\lambda =15,\mu \ne 17$
    4. D.$\lambda \ne 17,\mu \ne 18$
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    Answer: B

18. Q212025 · 23 Jan 2025 · Shift 2· Numerical answer

    Let $\alpha ,\beta$ be the roots of the equation $x^2-\mathrm{a}x-\mathrm{b}=0$ with $\mathrm{Im}(\alpha )<\mathrm{Im}(\beta )$. Let ${\mathrm{P}}_{\mathrm{n}}={\alpha }^{\mathrm{n}}-{\beta }^{\mathrm{n}}$. If ${\mathrm{P}}_3=-5\sqrt{7}i,{\mathrm{P}}_4=-3\sqrt{7}i,{\mathrm{P}}_5=11\sqrt{7}i$ and ${\mathrm{P}}_6=45\sqrt{7}i$, then $\left|{\alpha }^4+{\beta }^4\right|$ is equal to _________.

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    Answer: 31

19. Q32025 · 22 Jan 2025 · Shift 1

    Let $z_1,z_2$ and $z_3$ be three complex numbers on the circle $|z|=1$ with $\arg (z_1)=\frac{-\pi }{4}$, $\arg (z_2)=0$ and $\arg (z_3)=\frac{\pi }{4}$. If $|z_1{\bar{z}}_2+z_2{\bar{z}}_3+z_3{\bar{z}}_1{|}^2=\alpha +\beta \sqrt{2}$, $\alpha ,\beta \in Z$, then the value of ${\alpha }^2+{\beta }^2$ is :

    1. A.41
    2. B.29
    3. C.24
    4. D.31
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    Answer: B