Complex Numbers & Quadratic Equations · JEE Main previous year questions

2026 32 questions

1. Q112026 · 23 Jan 2026 · Shift 2

   The sum of all the real solutions of the equation ${\log }_{(x+3)}\left(6x^2+28x+30\right)=5-2{\log }_{(6x+10)}\left(x^2+6x+9\right)$ is equal to :

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

2. Q202026 · 23 Jan 2026 · Shift 2

   The system of linear equations $$x+y+z=6$$ $$2x+5y+az=36$$ $$x+2y+3z=b$$ has :

   1. A.unique solution for $a=8$ and $b=16$
   2. B.infinitely many solutions for $a=8$ and $b=14$
   3. C.infinitely many solutions for $a=8$ and $b=16$
   4. D.unique solution for $a=8$ and $b=14$
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   Answer: B

3. Q132026 · 21 Jan 2026 · Shift 1

   The sum of all the roots of the equation $(x-1{)}^2-5|x-1|+6=0$, is :

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

4. Q212026 · 21 Jan 2026 · Shift 1· Numerical answer

   Let $f:\mathbf{R}\to \mathbf{R}$ be a twice differentiable function such that the quadratic equation $f(x)m^2-2f^{\prime }(x)m+f^{\prime \prime }(x)=0$ in m , has two equal roots for every $x\in \mathbf{R}$. If $f(0)=1$, $f^{\prime }(0)=2$, and $(\alpha ,\beta )$ is the largest interval in which the function $f({\log }_{e}x-x)$ is increasing, then $\alpha +\beta$ is equal to ____ .

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   Answer: 1

5. Q42026 · 21 Jan 2026 · Shift 2

   If the system of equations $3x+y+4z=3$ $2x+\alpha y-z=-3$ $x+2y+z=4$ has no solution, then the value of $\alpha$ is equal to:

   1. A.13
   2. B.4
   3. C.19
   4. D.23
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   Answer: C

6. Q52026 · 21 Jan 2026 · Shift 2

   Let $z$ be the complex number satisfying $|z-5|\le 3$ and having maximum positive principal argument. Then $34{\left|\frac{5z-12}{5iz+16}\right|}^2$ is equal to:

   1. A.20
   2. B.26
   3. C.12
   4. D.16
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   Answer: A

7. Q92026 · 21 Jan 2026 · Shift 2

   Let $A=\{x:|x^2-10|\le 6\}$ and $B=\{x:|x-2|>1\}$. Then

   1. A.$A\cup B=(-\infty ,1]\cup (2,\infty )$
   2. B.$B-A=(-\infty ,-4)\cup (-2,1)\cup (4,\infty )$
   3. C.$A-B=[2,3)$
   4. D.$A\cap B=[-4,-2]\cup [3,4]$
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   Answer: B