Complex Numbers & Quadratic Equations · JEE Main previous year questions

2023 35 questions

1. Q192023 · 30 Jan 2023 · Shift 1

   The number of points on the curve $y=54x^5-135x^4-70x^3+180x^2+210x$ at which the normal lines are parallel to $x+90y+2=0$ is :

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

2. Q232023 · 30 Jan 2023 · Shift 1· Numerical answer

   Let $z=1+i$ and $z_1=\frac{1+i\bar{z}}{\bar{z}(1-z)+\frac{1}{z}}$. Then $\frac{12}{\pi }\arg (z_1)$ is equal to ____.

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   Answer: 9

3. Q12023 · 29 Jan 2023 · Shift 1

   Let $\lambda \ne 0$ be a real number. Let $\alpha ,\beta$ be the roots of the equation $14x^2-31x+3\lambda =0$ and $\alpha ,\gamma$ be the roots of the equation $35x^2-53x+4\lambda =0$. Then $\frac{3\alpha }{\beta }$ and $\frac{4\alpha }{\gamma }$ are the roots of the equation

   1. A.$7x^2-245x+250=0$
   2. B.$49x^2-245x+250=0$
   3. C.$49x^2+245x+250=0$
   4. D.$7x^2+245x-250=0$
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   Answer: B

4. Q102023 · 29 Jan 2023 · Shift 1

   For two non-zero complex numbers $z_1$ and $z_2$, if $\mathrm{Re}(z_1{z}_2)=0$ and $\mathrm{Re}(z_1+z_2)=0$, then which of the following are possible? A. $\mathrm{Im}(z_1)>0$ and $\mathrm{Im}(z_2)>0$ B. $\mathrm{Im}(z_1)<0$ and $\mathrm{Im}(z_2)>0$ C. $\mathrm{Im}(z_1)>0$ and $\mathrm{Im}(z_2)<0$ D. $\mathrm{Im}(z_1)<0$ and $\mathrm{Im}(z_2)<0$ Choose the correct answer from the options given below :

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

5. Q172023 · 29 Jan 2023 · Shift 1

   Consider the following system of equations $\alpha x+2y+z=1$ $2\alpha x+3y+z=1$ $3x+\alpha y+2z=\beta$ for some $\alpha ,\beta \in \mathbb{R}$. Then which of the following is NOT correct.

   1. A.It has a solution for all $\alpha \ne -1$ and $\beta =2$
   2. B.It has no solution if $\alpha =-1$ and $\beta \ne 2$
   3. C.It has no solution for $\alpha =-1$ and for all $\beta \in \mathbb{R}$
   4. D.It has no solution for $\alpha =3$ and for all $\beta \ne 2$
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   Answer: C

6. Q242023 · 29 Jan 2023 · Shift 2· Numerical answer

   Let ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_7$ be the roots of the equation $x^7+3x^5-13x^3-15x=0$ and $|{\alpha }_1|\ge |{\alpha }_2|\ge \dots \ge |{\alpha }_7|$. Then ${\alpha }_1{\alpha }_2-{\alpha }_3{\alpha }_4+{\alpha }_5{\alpha }_6$ is equal to ____.

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   Answer: 9

7. Q132023 · 25 Jan 2023 · Shift 1

   Let ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$ be respectively the sets of all $a\in \mathbb{R}-\{0\}$ for which the system of linear equations $ax+2ay-3az=1$ $(2a+1)x+(2a+3)y+(a+1)z=2$ $(3a+5)x+(a+5)y+(a+2)z=3$ has unique solution and infinitely many solutions. Then

   1. A.$\mathrm{n}({\mathrm{S}}_1)=2$ and ${\mathrm{S}}_2$ is an infinite set
   2. B.${\mathrm{S}}_1=\Phi$ and ${\mathrm{S}}_2=\mathbb{R}-\{0\}$
   3. C.${\mathrm{S}}_1=\mathbb{R}-\{0\}$ and ${\mathrm{S}}_2=\Phi$
   4. D.${\mathrm{S}}_1$ is an infinite set and $\mathrm{n}({\mathrm{S}}_2)=2$
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   Answer: C

8. Q252023 · 25 Jan 2023 · Shift 1· Numerical answer

   Let $S=\left\{\alpha :{\log }_2(9^{2\alpha -4}+13)-{\log }_2\left(\frac{5}{2}\cdot 3^{2\alpha -4}+1\right)=2\right\}$. Then the maximum value of $\beta$ for which the equation $x^2-2{\left(\sum _{\alpha \in S}\alpha \right)}^{2}x+\sum _{\alpha \in S}(\alpha +1{)}^2\beta =0$ has real roots, is ____.

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   Answer: 25

9. Q282023 · 25 Jan 2023 · Shift 2· Numerical answer

   Let $a\in \mathbb{R}$ and let $\alpha ,\beta$ be the roots of the equation $x^2+60^{\frac{1}{4}}x+a=0$. If ${\alpha }^4+{\beta }^4=-30$, then the product of all possible values of $a$ is ____.

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   Answer: 45

10. Q12023 · 24 Jan 2023 · Shift 1

    Let $p,q\in \mathbb{R}$ and $(1-\sqrt{3}i{)}^{200}=2^{199}(p+iq)$, $i=\sqrt{-1}$ then $p+q+q^2$ and $p-q+q^2$ are roots of the equation.

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