Complex Numbers & Quadratic Equations · JEE Main previous year questions

2005 7 questions

1. Q12005 · 28 Apr 2005 · Shift 1

   If the cube roots of unity are 1, $\omega$, ${\omega }^2$ then the roots of the equation $(x-1{)}^3+8=0$, are :

   1. A.$-1,-1+2\omega ,-1-2{\omega }^2$
   2. B.$-1,-1,-1$
   3. C.$-1,1-2\omega ,1-2{\omega }^2$
   4. D.$-1,1+2\omega ,1+2{\omega }^2$
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   Answer: C

2. Q22005 · 28 Apr 2005 · Shift 1

   If $z_1$ and $z_2$ are two non-zero complex numbers such that $|z_1+z_2|=|z_1|+|z_2|$, then arg $z_1$ - arg $z_2$ is equal to :

   1. A.$\frac{\pi }{2}$
   2. B.$-\pi$
   3. C.0
   4. D.$\frac{-\pi }{2}$
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   Answer: C

3. Q52005 · 28 Apr 2005 · Shift 1

   If both the roots of the quadratic equation $x^2-2kx+k^2+k-5=0$ are less than 5, then $k$ lies in the interval

   1. A.$(5,6]$
   2. B.$(6,\infty )$
   3. C.$(-\infty ,4)$
   4. D.$[4,5]$
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   Answer: C

4. Q182005 · 28 Apr 2005 · Shift 1

   The value of $a$ for which the sum of the squares of the roots of the equation $x^2-(a-2)x-a-1=0$ assumes the least value is

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

5. Q192005 · 28 Apr 2005 · Shift 1

   If the roots of the equation $x^2-bx+c=0$ be two consecutive integers, then $b^2-4c$ equals

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

6. Q212005 · 28 Apr 2005 · Shift 1

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

   1. A.$2\sin 2\alpha$
   2. B.$4$
   3. C.$4{\sin }^2\alpha$
   4. D.$-4{\sin }^2\alpha$
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   Answer: C

7. Q252005 · 28 Apr 2005 · Shift 1

   The system of equations $\alpha x+y+z=\alpha -1$ $x+\alpha y+z=\alpha -1$ $x+y+\alpha z=\alpha -1$ has no solutions, if $\alpha$ is :

   1. A.$-2$
   2. B.either $-2$ or $1$
   3. C.not $-2$
   4. D.$1$
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   Answer: A