Complex Numbers & Quadratic Equations · JEE Main previous year questions

2019 25 questions

1. Q52019 · 12 Apr 2019 · Shift 1

   The equation $|z-i|=|z-1|$, $i=\sqrt{-1}$, represents :

   1. A.a circle of radius 1
   2. B.the line through the origin with slope $-1$
   3. C.a circle of radius $\frac{1}{2}$
   4. D.the line through the origin with slope 1
   Show answer

   Answer: D

2. Q32019 · 10 Apr 2019 · Shift 1

   If $\alpha$ and $\beta$ are the roots of the quadratic equation, $x^2+x\sin \theta -2\sin \theta =0$, $\theta \in \left(0,\frac{\pi }{2}\right)$, then $\frac{{\alpha }^{12}+{\beta }^{12}}{({\alpha }^{-12}+{\beta }^{-12})\cdot (\alpha -\beta {)}^{24}}$ is equal to :

   1. A.$\frac{2^{12}}{(\sin \theta -8{)}^6}$
   2. B.$\frac{2^6}{(\sin \theta +4{)}^{12}}$
   3. C.$\frac{2^{12}}{(\sin \theta +8{)}^{12}}$
   4. D.$\frac{2^{12}}{(\sin \theta -4{)}^{12}}$
   Show answer

   Answer: C

3. Q42019 · 10 Apr 2019 · Shift 1

   If $a>0$ and $z=\frac{(1+i{)}^2}{a-i}$, has magnitude $\sqrt{\frac{2}{5}}$, then $\bar{z}$ is equal to :

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

   Answer: B

4. Q172019 · 10 Apr 2019 · Shift 1

   If the system of linear equations $x+y+z=5$ $x+2y+2z=6$ $x+3y+\lambda z=\mu$, $(\lambda ,\mu \in \mathrm{R})$, has infinitely many solutions, then the value of $\lambda +\mu$ is :

   1. A.10
   2. B.9
   3. C.7
   4. D.12
   Show answer

   Answer: A

5. Q22019 · 10 Apr 2019 · Shift 2

   The number of real roots of the equation $5+|2^x-1|=2^x(2^x-2)$ is

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

   Answer: B

6. Q62019 · 10 Apr 2019 · Shift 2

   Let $\lambda$ be a real number for which the system of linear equations $x+y+z=6$, $4x+\lambda y-\lambda z=\lambda -2$, $3x+2y-4z=-5$ has infinitely many solutions. Then $\lambda$ is a root of the quadratic equation:

   1. A.${\lambda }^2+\lambda -6=0$
   2. B.${\lambda }^2-\lambda -6=0$
   3. C.${\lambda }^2-3\lambda -4=0$
   4. D.${\lambda }^2+3\lambda -4=0$
   Show answer

   Answer: B

7. Q72019 · 10 Apr 2019 · Shift 2

   If z and w are two complex numbers such that |zw| = 1 and $\arg (z)-\arg (w)=\frac{\pi }{2}$ , then :

   1. A.$\bar{z}w=\frac{1-i}{\sqrt{2}}$
   2. B.$\bar{z}w=i$
   3. C.$\bar{z}w=\frac{-1+i}{\sqrt{2}}$
   4. D.$\bar{z}w=-i$
   Show answer

   Answer: D

8. Q92019 · 10 Apr 2019 · Shift 2

   The sum of the real roots of the equation $\left|\begin{array}{ccc}x& -6& -1\\ 2& -3x& x-3\\ -3& 2x& x+2\end{array}\right|=0$, is equal to :

   1. A.- 4
   2. B.0
   3. C.1
   4. D.6
   Show answer

   Answer: B

9. Q212019 · 10 Apr 2019 · Shift 2

   If ${\cos }^{-1}x-{\cos }^{-1}\frac{y}{2}=\alpha$, where $-1\le x\le 1$, $-2\le y\le 2$, $x\le \frac{y}{2}$ , then for all x, y, $4x^2-4xy\cos \alpha +y^2$ is equal to :

   1. A.$4{\sin }^2\alpha$
   2. B.$2{\sin }^2\alpha$
   3. C.$4{\sin }^2\alpha -2x^2{y}^2$
   4. D.$4{\cos }^2\alpha +2x^2{y}^2$
   Show answer

   Answer: A

10. Q72019 · 9 Apr 2019 · Shift 1

    Let $p,q\in R$. If $2-\sqrt{3}$ is a root of the quadratic equation, $x^2+px+q=0$, then :

    1. A.$p^2-4q-12=0$
    2. B.$q^2-4p-16=0$
    3. C.$q^2+4p+14=0$
    4. D.$p^2-4q+12=0$
    Show answer

    Answer: A

11. Q162019 · 9 Apr 2019 · Shift 1

    Let $\alpha$ and $\beta$ be the roots of the equation $x^2+x+1=0$. Then for $y\ne 0$ in R, $$\left|\begin{array}{ccc}y+1& \alpha & \beta \\ \alpha & y+\beta & 1\\ \beta & 1& y+\alpha \end{array}\right|$$ is equal to

    1. A.$y(y^2-1)$
    2. B.$y(y^2-3)$
    3. C.$y^3$
    4. D.$y^3-1$
    Show answer

    Answer: C

12. Q22019 · 9 Apr 2019 · Shift 2

    If the system of equations 2x + 3y − z = 0, x + ky − 2z = 0 and 2x − y + z = 0 has a non-trival solution (x, y, z), then $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}+k$ is equal to :-

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

    Answer: C

13. Q62019 · 9 Apr 2019 · Shift 2

    Let $z\in C$ be such that |z| < 1. If $\omega =\frac{5+3z}{5(1-z)}$, then :

    1. A.$4\mathrm{Im}(\omega )>5$
    2. B.$5\mathrm{Im}(\omega )<1$
    3. C.$5\mathrm{Re}(\omega )>4$
    4. D.$5\mathrm{Re}(\omega )>1$
    Show answer

    Answer: D

14. Q152019 · 9 Apr 2019 · Shift 2

    If m is chosen in the quadratic equation $(m^2+1)x^2-3x+(m^2+1{)}^2=0$ such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is :-

    1. A.$4\sqrt{3}$
    2. B.$8\sqrt{3}$
    3. C.$8\sqrt{5}$
    4. D.$10\sqrt{5}$
    Show answer

    Answer: C

15. Q42019 · 8 Apr 2019 · Shift 2

    If the system of linear equations x − 2y + kz = 1 2x + y + z = 2 3x − y − kz = 3 has a solution (x,y,z), $z\ne 0$, then (x,y) lies on the straight line whose equation is :

    1. A.4x − 3y − 4 = 0
    2. B.3x − 4y − 1 = 0
    3. C.4x − 3y − 1 = 0
    4. D.3x − 4y − 4 = 0
    Show answer

    Answer: A

16. Q182019 · 12 Jan 2019 · Shift 1

    An ordered pair $(\alpha ,\beta )$ for which the system of linear equations $$\begin{array}{rl}(1+\alpha )x+\beta y+z& =2\\ \alpha x+(1+\beta )y+z& =3\\ \alpha x+\beta y+2z& =2\end{array}$$ has a unique solution, is :

    1. A.$(-3,1)$
    2. B.$(1,-3)$
    3. C.$(-4,2)$
    4. D.$(2,4)$
    Show answer

    Answer: D

17. Q212019 · 12 Jan 2019 · Shift 1

    If $\lambda$ be the ratio of the roots of the quadratic equation in x, $3m^2{x}^2+m(m-4)x+2=0$, then the least value of m for which $\lambda +\frac{1}{\lambda }=1$, is

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

    Answer: B

18. Q142019 · 12 Jan 2019 · Shift 2

    Let $z_1$ and $z_2$ be two complex numbers satisfying $|z_1|=9$ and $|z_2-3-4i|=4$. Then the minimum value of $|z_1-z_2|$ is :

    1. A.0
    2. B.1
    3. C.2
    4. D.$\sqrt{2}$
    Show answer

    Answer: A

19. Q202019 · 12 Jan 2019 · Shift 2

    The number of integral values of m for which the quadratic expression, $(1+2m)x^2-2(1+3m)x+4(1+m)$, $x\in \mathbb{R}$, is always positive, is :

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

    Answer: A

20. Q302019 · 12 Jan 2019 · Shift 2

    The tangent to the curve $y=x^2-5x+5$, parallel to the line $2y=4x+1$, also passes through the point :

    1. A.$(\frac{1}{4},\frac{7}{2})$
    2. B.$(-\frac{1}{8},7)$
    3. C.$(\frac{7}{2},\frac{1}{4})$
    4. D.$(\frac{1}{8},-7)$
    Show answer

    Answer: D

21. Q52019 · 11 Jan 2019 · Shift 2

    Let z be a complex number such that $|z|+z=3+i$ (where $i=\sqrt{-1}$). Then $|z|$ is equal to :

    1. A.$\frac{\sqrt{34}}{3}$
    2. B.$\frac{5}{3}$
    3. C.$\frac{5}{4}$
    4. D.$\frac{\sqrt{41}}{4}$
    Show answer

    Answer: B

22. Q62019 · 11 Jan 2019 · Shift 2

    Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^2\sin \theta -x(\sin \theta \cos \theta +1)+\cos \theta =0$ $(0<\theta <45^{\circ })$, and $\alpha <\beta$. Then ${\sum }_{n=0}^{\infty }\left({\alpha }^n+\frac{(-1{)}^n}{{\beta }^n}\right)$ is equal to :

    1. A.$\frac{1}{1+\cos \theta }+\frac{1}{1-\sin \theta }$
    2. B.$\frac{1}{1-\cos \theta }+\frac{1}{1+\sin \theta }$
    3. C.$\frac{1}{1-\cos \theta }-\frac{1}{1+\sin \theta }$
    4. D.$\frac{1}{1+\cos \theta }-\frac{1}{1-\sin \theta }$
    Show answer

    Answer: B

23. Q232019 · 11 Jan 2019 · Shift 2

    Let $(x+10{)}^{50}+(x-10{)}^{50}=a_0+a_{1}x+a_2{x}^2+\dots +a_{50}{x}^{50}$, for all $x\in \mathbb{R}$; then $\frac{a_2}{a_0}$ is equal to

    1. A.12.25
    2. B.12.75
    3. C.12.00
    4. D.12.50
    Show answer

    Answer: A

24. Q22019 · 9 Jan 2019 · Shift 1

    Let $\alpha$ and $\beta$ be two roots of the equation $x^2+2x+2=0$, then ${\alpha }^{15}+{\beta }^{15}$ is equal to :

    1. A.-256
    2. B.512
    3. C.-512
    4. D.256
    Show answer

    Answer: A

25. Q172019 · 9 Jan 2019 · Shift 1

    The system of linear equations $$\begin{array}{rl}x+y+z& =2\\ 2x+3y+2z& =5\\ 2x+3y+(a^2-1)z& =a+1\end{array}$$ then

    1. A.has infinitely many solutions for $a=4$
    2. B.has a unique solution for $|a|=\sqrt{3}$
    3. C.is inconsistent when $|a|=\sqrt{3}$
    4. D.is inconsistent when $a=4$
    Show answer

    Answer: C