Complex Numbers & Quadratic Equations · JEE Main previous year questions

2021 30 questions

1. Q102021 · 1 Sept 2021 · Shift 2

   The number of pairs (a, b) of real numbers, such that whenever $\alpha$ is a root of the equation $x^2+ax+b=0$, ${\alpha }^2-2$ is also a root of this equation, is :

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

2. Q232021 · 1 Sept 2021 · Shift 2· Numerical answer

   If for the complex numbers z satisfying $|z-2-2i|\le 1$, the maximum value of $|3iz+6|$ is attained at $a+ib$, then $a+b$ is equal to ___________.

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   Answer: 5

3. Q262021 · 1 Sept 2021 · Shift 2· Numerical answer

   All the arrangements, with or without meaning, of the word FARMER are written excluding any word that has two R appearing together. The arrangements are listed serially in the alphabetic order as in the English dictionary. Then the serial number of the word FARMER in this list is ___________.

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   Answer: 77

4. Q122021 · 26 Aug 2021 · Shift 2

   If $(\sqrt{3}+i{)}^{100}=2^{99}(p+iq)$, then $p$ and $q$ are roots of the equation :

   1. A.$x^2-(\sqrt{3}-1)x-\sqrt{3}=0$
   2. B.$x^2+(\sqrt{3}+1)x+\sqrt{3}=0$
   3. C.$x^2+(\sqrt{3}-1)x-\sqrt{3}=0$
   4. D.$x^2-(\sqrt{3}+1)x+\sqrt{3}=0$
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   Answer: A

5. Q252021 · 26 Aug 2021 · Shift 2· Numerical answer

   Let $\lambda \ne 0$ be in $\mathbb{R}$. If $\alpha$ and $\beta$ are the roots of the equation $x^2-x+2\lambda =0$, and $\alpha$ and $\gamma$ are the roots of equation $3x^2-10x+27\lambda =0$, then $\frac{\beta \gamma }{\lambda }$ is equal to ________.

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   Answer: 18

6. Q42021 · 27 Jul 2021 · Shift 2

   Let C be the set of all complex numbers. Let $S_1=\{z\in C:|z-2|\le 1\}$ and $S_2=\{z\in C:z(1+i)+\bar{z}(1-i)\ge 4\}$. Then, the maximum value of ${\left|z-\frac{5}{2}\right|}^2$ for $z\in S_1\cap S_2$ is equal to :

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

7. Q112021 · 27 Jul 2021 · Shift 2

   Let $\alpha =\underset{x\in R}{\max }\{8^{2\sin 3x}.4^{4\cos 3x}\}$ and $\beta =\underset{x\in R}{\min }\{8^{2\sin 3x}.4^{4\cos 3x}\}$. If $8x^2+bx+c=0$ is a quadratic equation whose roots are ${\alpha }^{1/5}$ and ${\beta }^{1/5}$, then the value of $c-b$ is equal to :

   1. A.42
   2. B.47
   3. C.43
   4. D.50
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   Answer: A

8. Q212021 · 27 Jul 2021 · Shift 2· Numerical answer

   If the real part of the complex number $z=\frac{3+2i\cos \theta }{1-3i\cos \theta },\theta \in \left(0,\frac{\pi }{2}\right)$ is zero, then the value of ${\sin }^{2}3\theta +{\cos }^2\theta$ is equal to ____.

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   Answer: 1

9. Q232021 · 27 Jul 2021 · Shift 2· Numerical answer

   The number of real roots of the equation $e^{4x}-e^{3x}-4e^{2x}-e^x+1=0$ is equal to ____.

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   Answer: 2

10. Q132021 · 25 Jul 2021 · Shift 1

    The number of real roots of the equation $e^{6x}-e^{4x}-2e^{3x}-12e^{2x}+e^x+1=0$ is :

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

11. Q262021 · 25 Jul 2021 · Shift 1· Numerical answer

    If $\alpha ,\beta$ are roots of the equation $x^2+5(\sqrt{2})x+10=0,\alpha >\beta$ and $P_n={\alpha }^n-{\beta }^n$ for each positive integer n, then the value of $\left(\frac{P_{17}{P}_{20}+5\sqrt{2}{P}_{17}{P}_{19}}{P_{18}{P}_{19}+5\sqrt{2}{P}_{18}^2}\right)$ is equal to ____.

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    Answer: 1

12. Q122021 · 25 Jul 2021 · Shift 2

    The number of real solutions of the equation, $x^2-|x|-12=0$ is :

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

13. Q62021 · 22 Jul 2021 · Shift 2

    The values of $\lambda$ and $\mu$ such that the system of equations $x+y+z=6$, $3x+5y+5z=26$, $x+2y+\lambda z=\mu$ has no solution, are :

    1. A.$\lambda =3,\mu =5$
    2. B.$\lambda =3,\mu \ne 10$
    3. C.$\lambda \ne 2,\mu =10$
    4. D.$\lambda =2,\mu \ne 10$
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    Answer: D

14. Q112021 · 22 Jul 2021 · Shift 2

    Let n denote the number of solutions of the equation $z^2+3\bar{z}=0$, where z is a complex number. Then the value of ${\sum }_{k=0}^{\infty }\frac{1}{n^k}$ is equal to :

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

15. Q42021 · 20 Jul 2021 · Shift 1

    If $\alpha$ and $\beta$ are the distinct roots of the equation $x^2+(3{)}^{1/4}x+3^{1/2}=0$, then the value of ${\alpha }^{96}({\alpha }^{12}-1)+{\beta }^{96}({\beta }^{12}-1)$ is equal to :

    1. A.$56\times 3^{25}$
    2. B.$56\times 3^{24}$
    3. C.$52\times 3^{24}$
    4. D.$28\times 3^{25}$
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    Answer: C

16. Q62021 · 20 Jul 2021 · Shift 1

    If z and $\omega$ are two complex numbers such that $|z\omega |=1$ and $\arg (z)-\arg (\omega )=\frac{3\pi }{2}$, then $\arg \left(\frac{1-2\bar{z}\omega }{1+3\bar{z}\omega }\right)$ is : (Here $\arg (z)$ denotes the principal argument of complex number z)

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

17. Q32021 · 18 Mar 2021 · Shift 1

    Let $\alpha ,\beta ,\gamma$ be the real roots of the equation, $x^3+a{x}^2+bx+c=0$, $(a,b,c\in \mathrm{R}\text{ and }a,b\ne 0)$. If the system of equations (in $u,v,w$) given by $\alpha u+\beta v+\gamma w=0$; $\beta u+\gamma v+\alpha w=0$; $\gamma u+\alpha v+\beta w=0$ has non-trivial solution, then the value of $\frac{a^2}{b}$ is

    1. A.5
    2. B.3
    3. C.1
    4. D.0
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    Answer: B

18. Q172021 · 18 Mar 2021 · Shift 1

    Let $(1+x+2x^2{)}^{20}=a_0+a_{1}x+a_2{x}^2+\cdots +a_{40}{x}^{40}$. Then $a_1+a_3+a_5+\cdots +a_{37}$ is equal to

    1. A.$2^{20}(2^{20}-21)$
    2. B.$2^{19}(2^{20}-21)$
    3. C.$2^{19}(2^{20}+21)$
    4. D.$2^{20}(2^{20}+21)$
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    Answer: B

19. Q242021 · 18 Mar 2021 · Shift 1· Numerical answer

    Let $z_1,z_2$ be the roots of the equation $z^2+az+12=0$ and $z_1,z_2$ form an equilateral triangle with origin. Then, the value of $|a|$ is :

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    Answer: 6

20. Q22021 · 18 Mar 2021 · Shift 2

    Let the system of linear equations $4x+\lambda y+2z=0$ $2x-y+z=0$ $\mu x+2y+3z=0,\lambda ,\mu \in \mathrm{R}$. has a non-trivial solution. Then which of the following is true?

    1. A.$\mu =6,\lambda \in \mathrm{R}$
    2. B.$\lambda =3,\mu \in \mathrm{R}$
    3. C.$\mu =-6,\lambda \in \mathrm{R}$
    4. D.$\lambda =2,\mu \in \mathrm{R}$
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    Answer: A

21. Q62021 · 18 Mar 2021 · Shift 2

    Let a complex number be $w=1-\sqrt{3}i$. Let another complex number z be such that $|zw|=1$ and $\arg (z)-\arg (w)=\frac{\pi }{2}$. Then the area of the triangle with vertices origin, z and w is equal to :

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

22. Q222021 · 18 Mar 2021 · Shift 2· Numerical answer

    If f(x) and g(x) are two polynomials such that the polynomial $\mathrm{P}(x)=f(x^3)+xg(x^3)$ is divisible by $x^2+x+1$, then P(1) is equal to ____.

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    Answer: 0

23. Q52021 · 16 Mar 2021 · Shift 1

    Let a complex number z, $|z|\ne 1$, satisfy ${\log }_{\frac{1}{\sqrt{2}}}\left(\frac{|z|+11}{(|z|-1{)}^2}\right)\le 2$. Then, the largest value of $|z|$ is equal to _________.

    1. A.5
    2. B.8
    3. C.6
    4. D.7
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    Answer: D

24. Q262021 · 16 Mar 2021 · Shift 1· Numerical answer

    Let z and $\omega$ be two complex numbers such that $\omega =z\bar{z}-2z+2$, $\left|\frac{z+i}{z-3i}\right|=1$ and $\mathrm{Re}(\omega )$ has minimum value. Then, the minimum value of $n\in \mathbb{N}$ for which ${\omega }^n$ is real, is equal to _________.

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    Answer: 4

25. Q232021 · 26 Feb 2021 · Shift 1· Numerical answer

    The sum of $162^{\text{th}}$ power of the roots of the equation $x^3-2x^2+2x-1=0$ is _________.

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    Answer: 3