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question_answer1)
The expression \[{{\left[ \frac{1+\sin \frac{\pi }{8}+i\cos \frac{\pi }{8}}{1+\sin \frac{\pi }{8}-i\cos \frac{\pi }{8}} \right]}^{8}}=\]
A)
1 done
clear
B)
-1 done
clear
C)
i done
clear
D)
-i done
clear
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question_answer2)
If p and q are the roots of the equation \[{{x}^{2}}+px+q=0\], then
A)
p =1, q=-2 done
clear
B)
p =0, q=2 done
clear
C)
p =-2, q=0 done
clear
D)
p =-2, q=1 done
clear
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question_answer3)
If for complex numbers, \[{{z}_{1}}\]and \[{{z}_{2}}\]arg (\[{{z}_{1}}\]) - arg(\[{{z}_{2}}\])=0 then \[\left| {{z}_{1}}-{{z}_{2}} \right|\]is equal to
A)
\[\left| {{z}_{1}} \right|+\left| {{z}_{2}} \right|\] done
clear
B)
\[\left| {{z}_{1}} \right|-\left| {{z}_{2}} \right|\] done
clear
C)
\[\left\| {{z}_{1}}-{{z}_{2}} \right\|\] done
clear
D)
0 done
clear
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question_answer4)
The curve y=(\[\lambda \]+1)\[{{x}^{2}}\]+2 intersects the curve y=\[\lambda x+3\]in exactly one point, if \[\lambda \]equals
A)
\[\left\{ -2,2 \right\}\] done
clear
B)
\[\left\{ 1 \right\}\] done
clear
C)
\[\left\{ -2 \right\}\] done
clear
D)
\[\left\{ 2 \right\}\] done
clear
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question_answer5)
If \[k+\left| k+{{z}^{2}} \right|={{\left| z \right|}^{2}}(k\in {{R}^{-}})\], then possible argument of z is
A)
0 done
clear
B)
\[\pi \] done
clear
C)
\[\pi /2\] done
clear
D)
none of these done
clear
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question_answer6)
Let \[a\ne 0\]and p(x) be a polynomial of degree greater than 2. If p(x) leaves remainders a and -a when divided respectively, by x+a and x-a, the remainder when p(x) is divided by \[{{x}^{2}}-{{a}^{2}}\]is
A)
2x done
clear
B)
-2X done
clear
C)
x done
clear
D)
-x done
clear
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question_answer7)
If \[z(1+a)=b+ic\]and \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}=1\],then \[[(1+iz)/(1-iz)]=\]
A)
\[\frac{a+ib}{1+c}\] done
clear
B)
\[\frac{b-ic}{1+a}\] done
clear
C)
\[\frac{a+ic}{1+b}\] done
clear
D)
none of these done
clear
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question_answer8)
If \[{{b}_{1}}{{b}_{2}}\]=2(\[{{c}_{1}}+{{c}_{2}}\]), then at least one of the equations \[{{x}^{2}}+{{b}_{1}}x+{{c}_{1}}=0\]and \[{{x}^{2}}+{{b}_{2}}x+{{C}_{2}}=0\]has
A)
imaginary done
clear
B)
real roots done
clear
C)
purely imaginary roots done
clear
D)
none of these done
clear
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question_answer9)
If \[\alpha ,\beta \]are the roots of the equation \[{{u}^{2}}-2u+2=0\]and if \[\cot \theta =x+1\], then \[[{{(x+\alpha )}^{n}}-{{(x+\beta )}^{n}}]/[\alpha -\beta ]\]is equal to
A)
\[\frac{\sin n\theta }{{{\sin }^{n}}\theta }\] done
clear
B)
\[\frac{\cos n\theta }{{{\cos }^{n}}\theta }\] done
clear
C)
\[\frac{\sin n\theta }{{{\cos }^{n}}\theta }\] done
clear
D)
\[\frac{\cos n\theta }{{{\sin }^{n}}\theta }\] done
clear
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question_answer10)
If \[a{{(p+q)}^{2}}+2bpq+c=0\,\,and\,\,a{{(p+r)}^{2}}+2bpr+c=0\,\,(a\ne 0),\] then
A)
\[qr={{p}^{2}}\] done
clear
B)
\[qr={{p}^{2}}+\frac{c}{a}\] done
clear
C)
\[qr=-{{p}^{2}}\] done
clear
D)
none of these done
clear
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question_answer11)
Sum of common roots the equations \[{{z}^{3}}+2{{z}^{2}}+2z+1=0\]and \[{{z}^{1985}}+{{z}^{100}}+1=0\]
A)
-1 done
clear
B)
1 done
clear
C)
0 done
clear
D)
1 done
clear
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question_answer12)
If the roots of the equation \[{{x}^{2}}+2ax+b=0\]are real and distinct and they differ by at most 2m then b lies in the interval
A)
\[({{a}^{2}},{{a}^{2}}+{{m}^{2}})\] done
clear
B)
\[({{a}^{2}}-{{m}^{2}},{{a}^{2}})\] done
clear
C)
[\[{{a}^{2}}-{{m}^{2}},{{a}^{2}}\]) done
clear
D)
none of these done
clear
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question_answer13)
If \[{{z}_{1}}\],\[{{z}_{2}}\],\[{{z}_{3}}\]are the vertices of an equilateral triangle ABC such that \[\left| {{z}_{1}}-i \right|\]=\[\left| {{z}_{2}}-i \right|\]=\[\left| {{z}_{3}}-i \right|\],then \[\left| {{z}_{1}}+{{z}_{2}}+{{z}_{3}} \right|\]equals to
A)
\[3\sqrt{3}\] done
clear
B)
\[\sqrt{3}\] done
clear
C)
3 done
clear
D)
\[\frac{1}{3\sqrt{3}}\] done
clear
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question_answer14)
If a, b, c, d\[\in \]R, then the equation \[({{x}^{2}}+ax-3b)\]\[({{x}^{2}}-cx+b)\]\[({{x}^{2}}-dx+2b)\]=0 has
A)
6 real roots done
clear
B)
at least 2 real roots done
clear
C)
4 real toots done
clear
D)
3 real roots done
clear
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question_answer15)
If \[\left| z \right|\]=1, then the point representing the complex number -1+3z will lie on
A)
a circle done
clear
B)
a straight line done
clear
C)
a parabola done
clear
D)
a hyperbola done
clear
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question_answer16)
If a, b\[\in \]R, \[a\ne 0\]and the quadratic equation \[a{{x}^{2}}-bx+1=0\] has imaginary roots then \[(a+b+1)\]is
A)
positive done
clear
B)
negative done
clear
C)
zero done
clear
D)
dependent on the sign of b done
clear
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question_answer17)
Let \[z=1-t+i \sqrt{{{t}^{2}}+t+2}\], where t is a real parameter. The locus of z in the argand plane is
A)
a hyperbola done
clear
B)
an ellipse done
clear
C)
a straight line done
clear
D)
none of these done
clear
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question_answer18)
Let p(x) =0 be a polynomial equation of the least possible degree, with rational coefficients, having \[\sqrt[3]{7}+\sqrt[3]{49}\]as one of its roots. Then the product of all the roots of p(x)=0 is
A)
56 done
clear
B)
63 done
clear
C)
7 done
clear
D)
49 done
clear
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question_answer19)
If z and \[\omega \]are two non-zero complex numbers such that \[\left| z \right|=\left| \omega \right|\]and arg z + arg \[\omega \]=\[\pi \], then z equals
A)
\[\bar{\omega }\] done
clear
B)
-\[\bar{\omega }\] done
clear
C)
\[\omega \] done
clear
D)
-\[\omega \] done
clear
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question_answer20)
The difference between the corresponding roots of \[{{x}^{2}}+ax+b=0\]and \[{{x}^{2}}+bx+a=0\]is same and \[a\ne b\], then
A)
a+b+4=0 done
clear
B)
a+b-4=0 done
clear
C)
a-b-4=0 done
clear
D)
a-b+4=0 done
clear
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question_answer21)
If \[\left| {{z}_{2}}+i{{z}_{1}} \right|=\left| {{z}_{1}} \right|+\left| {{z}_{2}} \right|\]and \[\left| {{z}_{1}} \right|\]=3 and \[\left| {{z}_{2}} \right|\]=4 then the area of \[\Delta \]ABC. If vertices of A, B and C are \[{{z}_{1}},{{z}_{2}}\] and \[[({{z}_{2}}-i{{z}_{1}})/(1-i)]\] respectively, is _________.
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question_answer22)
If arg\[\left( \frac{{{z}_{1}}-\frac{z}{\left| z \right|}}{\frac{z}{\left| z \right|}} \right)=\frac{\pi }{2}\]and \[\left| \frac{z}{\left| z \right|}-{{z}_{1}} \right|\]=3, then \[\left| {{z}_{1}} \right|\] equals to _______.
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question_answer23)
If \[\alpha .\beta ,\gamma \]are the roots of \[{{x}^{3}}-{{x}^{2}}-1=0\] then the value of \[(1+\alpha )/(1-a)+(1+\beta )/(1-\beta )+(1+\gamma )/(1-\gamma )\] is equal to _________.
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question_answer24)
If \[\alpha \] and \[\beta \] be the roots of the equation\[{{x}^{2}}+px-1/(2{{P}^{2}})=0\], where p\[\in \]R. Then the minimum value of \[{{a}^{4}}+{{\beta }^{4}}\] is _________.
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question_answer25)
The number of real solutions to the equation\[{{x}^{2}}-3\left| x \right|+2=0\] is ________.
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