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  • Question 1
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    \(\text { Find } \theta \in\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right] \text { satisfying } 2 \cos ^{2} \theta+\sin \theta \leq 2\)

  • Question 2
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    The locus of the midpoints of the chords of the circle \(x^{2} y^{2}=16\) which are tangents to the hyperbola \(9 x^{2}-16 y^{2}=144\) is

  • Question 3
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    In an acute - angled triangle ABC, points D, E and F are the feet of the perpendiculars from A, B and C onto BC, AC and AB, respectively. H is the orthocentre of ∆ABC . If sin A = 3/5 and BC = 39, then the length of AH is:

  • Question 4
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    lf Z=x+iy is a complex number then |x|+|y|≤ ?

  • Question 5
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    Let \(\omega\) be a complex number such that \(2 \omega+1=z\) where \(z=\sqrt{-3}\), if \(\left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & -\omega^{2}-1 & \omega^{2} \\ 1 & \omega^{2} & \omega^{7}\end{array}\right|=3 k,\) then \(k\) ie equal to.

  • Question 6
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    Let ω be a cube root of unity not equal to 1. Then the maximum possible value of |a+bω+cω2| where a, b, c ϵ {+1,-1} is:

  • Question 7
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    Number of solutions of the equation |z|2+7z=0 is:

  • Question 8
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    If ω is a cube root of unity but not equal to 1, then minimum value of ∣a+b+cω2∣ (where a,b,ca,b,c are integers but not all equal) is:

  • Question 9
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    If \(\left|z_{1}\right|=1,\left|z_{2}\right|=2,\left|z_{3}\right|=3\) and \(\left|9 z_{1} z_{2}+4 z_{1} z_{3}+z_{2} z_{3}\right|=12,\) then the value of \(\left|z_{1}+z_{2}+z_{3}\right|\)
    is

  • Question 10
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    Let \(z=x+i y\) be a complex number where \(x\) and \(y\) are integers. Then the area of the
    rectangle whose vertices are the roots of the equation \(\bar{z} z^{3}+z \bar{z}^{3}=350\) is:

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