Complex Numbers & Quadratic Equations · JEE Main previous year questions

2026 32 questions

1. Q22026 · 8 Apr 2026 · Shift 2

   The number of values of $z\in \mathbb{C}$, satisfying the equations $|z-(4+8i)|=\sqrt{10}$ and $|z-(3+5i)|+|z-(5+11i)|=4\sqrt{5}$, is:

   1. A.$0$
   2. B.$2$
   3. C.$1$
   4. D.$4$
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   Answer: B

2. Q32026 · 8 Apr 2026 · Shift 2

   If the system of linear equations : $$x+y+z=6,$$ $$x+2y+5z=10,$$ $$2x+3y+\lambda z=\mu$$ has infinitely many solutions, then the value of $\lambda +\mu$ equals :

   1. A.$12$
   2. B.$16$
   3. C.$22$
   4. D.$28$
   Show answer

   Answer: C

3. Q212026 · 8 Apr 2026 · Shift 2· Numerical answer

   The sum of squares of all the real solutions of the equation ${\log }_{(x+1)}(2x^2+5x+3)=4-{\log }_{(2x+3)}(x^2+2x+1)$ is equal to ____ .

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   Answer: 2

4. Q12026 · 6 Apr 2026 · Shift 2

   Let $f:\mathbf{R}\to \mathbf{R}$ be defined as $f(x)=\frac{2x^2-3x+2}{3x^2+x+3}$. Then $f$ is :

   1. A.both one-one and onto
   2. B.one-one but not onto
   3. C.onto but not one-one
   4. D.neither one-one nor onto
   Show answer

   Answer: D

5. Q22026 · 6 Apr 2026 · Shift 2

   Consider the quadratic equation $(n^2-2n+2)x^2-3x+(n^2-2n+2{)}^2=0$, $n\in \mathbf{R}$. Let $\alpha$ be the minimum value of the product of its roots and $\beta$ be the maximum value of the sum of its roots. Then the sum of the first six terms of the G.P., whose first term is $\alpha$ and the common ratio is $\frac{\alpha }{\beta }$, is :

   1. A.$\frac{61}{37}$
   2. B.$\frac{121}{81}$
   3. C.$\frac{364}{243}$
   4. D.$\frac{1093}{729}$
   Show answer

   Answer: C

6. Q192026 · 6 Apr 2026 · Shift 2

   The area of the region $\{(x,y):x^2-8x\le y\le -x\}$ is :

   1. A.$\frac{343}{6}$
   2. B.$\frac{637}{6}$
   3. C.$\frac{437}{6}$
   4. D.$\frac{523}{6}$
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   Answer: A

7. Q12026 · 5 Apr 2026 · Shift 1

   Let $a,b\in \mathbb{C}$. Let $\alpha ,\beta$ be the roots of the equation $x^2+ax+b=0$. If $\beta -\alpha =\sqrt{11}$ and ${\beta }^2-{\alpha }^2=3i\sqrt{11}$ , then $({\beta }^3-{\alpha }^3{)}^2$ is equal to:

   1. A.160
   2. B.176
   3. C.194
   4. D.187
   Show answer

   Answer: B

8. Q12026 · 5 Apr 2026 · Shift 2

   Let $\alpha$, $\beta$ be the roots of the equation $x^2-x+p=0$ and $\gamma$, $\delta$ be the roots the equation $x^2-4x+q=0$; $p,q\in \mathbb{Z}$. If $\alpha ,\beta ,\gamma ,\delta$ are in G.P., then $|p+q|$ equals :

   1. A.$16$
   2. B.$32$
   3. C.$34$
   4. D.$38$
   Show answer

   Answer: C

9. Q22026 · 5 Apr 2026 · Shift 2

   Let $z_1,z_2\in \mathbb{C}$ be the distinct solutions of the equation $z^2+4z-(1+12i)=0$. Then $|z_1{|}^2+|z_2{|}^2$ is equal to :

   1. A.$18$
   2. B.$22$
   3. C.$29$
   4. D.$34$
   Show answer

   Answer: D

10. Q32026 · 4 Apr 2026 · Shift 1

    Let $z$ be a complex number such that $|z+2|=|z-2|$ and $\arg \left(\frac{z+3}{z-i}\right)=\frac{\pi }{4}$. Then $|z{|}^2$ is equal to:

    1. A.9
    2. B.4
    3. C.5
    4. D.1
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    Answer: A

11. Q22026 · 4 Apr 2026 · Shift 2

    Let $S=\{z\in \mathbb{C}:z^2+4z+16=0\}$. Then ${\sum }_{z\in S}|z+\sqrt{3}\mathrm{i}{|}^2$ is equal to:

    1. A.42
    2. B.23
    3. C.27
    4. D.38
    Show answer

    Answer: D

12. Q32026 · 4 Apr 2026 · Shift 2

    If the system of equations: $$x+y+z=5$$ $$x+2y+3z=9$$ $$x+3y+\lambda z=\mu$$ has infinitely many solutions, then the value of $\lambda +\mu$ is:

    1. A.16
    2. B.18
    3. C.19
    4. D.21
    Show answer

    Answer: B

13. Q42026 · 4 Apr 2026 · Shift 2

    If $\alpha =1$ and $\beta =1+\mathrm{i}\sqrt{2}$, where $\mathrm{i}=\sqrt{-1}$ are two roots of the equation $x^3+a{x}^2+bx+c=0$, $a,b,c\in \mathbb{R}$, then ${\int }_{-1}^1(x^3+a{x}^2+bx+c)\mathrm{d}x$ is equal to:

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

14. Q52026 · 4 Apr 2026 · Shift 2

    If the quadratic equation $(\lambda +2)x^2-3\lambda x+4\lambda =0$, $\lambda \ne -2$, has two positive roots, then the number of possible integral values of $\lambda$ is:

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

15. Q12026 · 2 Apr 2026 · Shift 1

    Let $\alpha ,\alpha +2,\alpha \in \mathbb{Z}$, be the roots of the quadratic equation $x(x+2)+(x+1)(x+3)+(x+2)(x+4)+\cdots +(x+n-1)(x+n+1)=4n\text{for}\text{some}n\in \mathbb{N}$. Then $n+\alpha$ is equal to :

    1. A.0
    2. B.1
    3. C.2
    4. D.3
    Show answer

    Answer: C

16. Q32026 · 2 Apr 2026 · Shift 1

    Let $\alpha ,\beta \in \mathbb{R}$ be such that the system of linear equations $x+2y+z=5$ $2x+y+\alpha z=5$ $8x+4y+\beta z=18$ has no solution. Then $\frac{\beta }{\alpha }$ is equal to :

    1. A.$-4$
    2. B.4
    3. C.8
    4. D.$-8$
    Show answer

    Answer: B

17. Q12026 · 2 Apr 2026 · Shift 2

    Let $\alpha ,\beta$ be the roots of the equation $x^2-3x+r=0$, and $\frac{\alpha }{2},2\beta$ be the roots of the equation $x^2+3x+r=0$. If the roots of the equation $x^2+6x=m$ are $2\alpha +\beta +2r$ and $\alpha -2\beta -\frac{r}{2}$, then $m$ is equal to :

    1. A.$-135$
    2. B.$-567$
    3. C.$135$
    4. D.$567$
    Show answer

    Answer: D

18. Q242026 · 2 Apr 2026 · Shift 2· Numerical answer

    If the area of the region bounded by $16x^2-9y^2=144$ and $8x-3y=24$ is A, then $3(\mathrm{A}+6{\log }_{\mathrm{e}}(3))$ is equal to ____.

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    Answer: 24

19. Q22026 · 28 Jan 2026 · Shift 1

    If $g(x)=3x^2+2x-3$, $f(0)=-3$ and $4g(f(x))=3x^2-32x+72$, then $f(g(2))$ is equal to:

    1. A.$\frac{7}{2}$
    2. B.$-\frac{25}{6}$
    3. C.$\frac{25}{6}$
    4. D.$-\frac{7}{2}$
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    Answer: A

20. Q52026 · 28 Jan 2026 · Shift 1

    If $\alpha ,\beta$, where $\alpha <\beta$, are the roots of the equation $\lambda x^2-(\lambda +3)x+3=0$ such that $\frac{1}{\alpha }-\frac{1}{\beta }=\frac{1}{3}$, then the sum of all possible values of $\lambda$ is

    1. A.2
    2. B.6
    3. C.8
    4. D.4
    Show answer

    Answer: B

21. Q122026 · 28 Jan 2026 · Shift 1

    Let $z$ be a complex number such that $|z-6|=5$ and $|z+2-6i|=5$. Then the value of $z^3+3z^2-15z+141$ is equal to :

    1. A.61
    2. B.37
    3. C.42
    4. D.50
    Show answer

    Answer: D

22. Q32026 · 28 Jan 2026 · Shift 2

    Let $\text{Q}(a,b,c)$ be the image of the point $\text{P}(3,2,1)$ in the line $\frac{x-1}{1}=\frac{y}{2}=\frac{z-1}{1}$. Then the distance of Q from the line $\frac{x-9}{3}=\frac{y-9}{2}=\frac{z-5}{-2}$ is

    1. A.8
    2. B.7
    3. C.6
    4. D.5
    Show answer

    Answer: B

23. Q32026 · 24 Jan 2026 · Shift 1

    Let ${\text{A}}_1$ be the bounded area enclosed by the curves $y=x^2+2,x+y=8$ and $y$-axis that lies in the first quadrant. Let ${\text{A}}_2$ be the bounded area enclosed by the curves $y=x^2+2,y^2=x,x=2$, and $y$-axis that lies in the first quadrant. Then ${\text{A}}_1-{\text{A}}_2$ is equal to

    1. A.$\frac{2}{3}(2\sqrt{2}+1)$
    2. B.$\frac{2}{3}(3\sqrt{2}+1)$
    3. C.$\frac{2}{3}(\sqrt{2}+1)$
    4. D.$\frac{2}{3}(4\sqrt{2}+1)$
    Show answer

    Answer: A

24. Q42026 · 23 Jan 2026 · Shift 1

    If $\alpha$ and $\beta (\alpha <\beta )$ are the roots of the equation $(-2+\sqrt{3})(|\sqrt{x}-3|)+(x-6\sqrt{x})+(9-2\sqrt{3})=0,x⩾0$, then $\sqrt{\frac{\beta }{\alpha }}+\sqrt{\alpha \beta }$ is equal to :

    1. A.8
    2. B.10
    3. C.9
    4. D.11
    Show answer

    Answer: B

25. Q52026 · 23 Jan 2026 · Shift 1

    Let $\mathrm{S}=\{z:3⩽|2z-3(1+\mathrm{i})|⩽7\}$ be a set of complex numbers. Then ${\mathrm{Min}}_{z\in S}\left|\left(z+\frac{1}{2}(5+3i)\right)\right|$ is equal to :

    1. A.$\frac{1}{2}$
    2. B.$\frac{5}{2}$
    3. C.2
    4. D.$\frac{3}{2}$
    Show answer

    Answer: D