Let \[\omega \] is an imaginary cube roots of unity then the value of\[2(\omega +1)({{\omega }^{2}}+1)+3(2\omega +1)(2{{\omega }^{2}}+1)+.....\]\[+(n+1)(n\omega +1)(n{{\omega }^{2}}+1)\]is [Orissa JEE 2002]

\[{{\left[ \frac{n(n+1)}{2} \right]}^{2}}\]

\[{{\left[ \frac{n(n+1)}{2} \right]}^{2}}+n\]

\[{{\left[ \frac{n(n+1)}{2} \right]}^{2}}\]

\[{{\left[ \frac{n(n+1)}{2} \right]}^{2}}-n\]

Let \[a,b,c\] be real numbers \[a\ne 0\]. If \[\alpha \]is a root \[{{a}^{2}}{{x}^{2}}+bx+c=0\], \[\beta \] is a root of \[{{a}^{2}}{{x}^{2}}-bx-c=0\] and \[0

\[\gamma =\frac{\alpha +\beta }{2}\]

\[\gamma =\alpha +\frac{\beta }{2}\]

The roots of the equation\[4{{x}^{4}}-24{{x}^{3}}+57{{x}^{2}}+18x-45=0\], If one of them is\[3+i\sqrt{6}\], are

\[3-i\sqrt{6},\pm \sqrt{\frac{3}{2}}\]

\[3-i\sqrt{6},\pm \frac{3}{\sqrt{2}}\]

\[3-i\sqrt{6},\pm \frac{\sqrt{3}}{2}\]

If\[a{{x}^{2}}+bx+c=0\], then x = [MP PET 1995]

\[\frac{b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]

\[\frac{-b\pm \sqrt{{{b}^{2}}-ac}}{2a}\]

\[\frac{2c}{-b\pm \sqrt{{{b}^{2}}-4ac}}\]

If a, b be the roots of \[{{x}^{2}}+px+q=0\] and \[\alpha +h,\,\beta +h\] are the roots of \[{{x}^{2}}+rx+s=0\], then [AMU 2001]

\[2h=\left[ \frac{p}{q}+\frac{r}{s} \right]\]

If \[{{(1+x)}^{n}}={{C}_{0}}+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}+.....+{{C}_{n}}{{x}^{n}},\] then the value of \[{{C}_{0}}-{{C}_{2}}+{{C}_{4}}-{{C}_{6}}+.....\]is

\[{{2}^{n}}\cos \frac{n\pi }{2}\]

\[{{2}^{n}}\sin \frac{n\pi }{2}\]

\[{{2}^{n/2}}\cos \frac{n\pi }{4}\]

In a triangle \[ABC\] the value of \[\angle A\] is given by \[5\cos A+3=0\], then the equationwhose roots are \[\sin A\] and \[\tan A\] will be [Roorkee 1972]

\[15{{x}^{2}}-8\sqrt{2}x+16=0\]

The values of \[x\] and \[y\] for which the numbers \[3+i{{x}^{2}}y\] and \[{{x}^{2}}+y+4i\] are conjugate complex can be

\[(-1,\text{ }2)\]or \[(-2,\text{ }1)\]

If\[|{{z}_{1}}|\,=\,|{{z}_{2}}|\] and \[amp\,{{z}_{1}}+amp\,\,{{z}_{2}}=0\], then [MP PET 1999]

\[{{\bar{z}}_{1}}={{\bar{z}}_{2}}\]

For any two complex numbers \[{{z}_{1}},{{z}_{2}}\]we have \[|{{z}_{1}}+{{z}_{2}}{{|}^{2}}=\] \[|{{z}_{1}}{{|}^{2}}+|{{z}_{2}}{{|}^{2}}\] then

\[\operatorname{Im}\left( \frac{{{z}_{1}}}{{{z}_{2}}} \right)=0\]

\[\operatorname{Re}\left( \frac{{{z}_{1}}}{{{z}_{2}}} \right)=0\]

\[\operatorname{Im}\left( \frac{{{z}_{1}}}{{{z}_{2}}} \right)=0\]

\[\operatorname{Re}({{z}_{1}}{{z}_{2}})=0\]

\[\operatorname{Im}({{z}_{1}}{{z}_{2}})=0\]

91 + Mcqs in Complex Numbers and Quadratic Equations in Mathematics Page 2 McqOptions

51.	\[\sinh ix\] is [EAMCET 2002]
A.	\[i\sin (ix)\]
B.	\[i\sin x\]
C.	\[-i\sin x\]
D.	\[\sin (ix)\]
Answer» C. \[-i\sin x\]

Discussion

52.	If \[1,\,\omega ,\,{{\omega }^{2}}\] are the roots of unity, then \[{{(1-2\omega +{{\omega }^{2}})}^{6}}\] is equal to [Pb. CET 2001]
A.	729
B.	246
C.	243
D.	81
Answer» B. 246

Discussion

53.	If \[\omega \] is a complex cube root of unity, then\[225+\]\[{{(3\omega +8{{\omega }^{2}})}^{2}}\]\[+{{(3{{\omega }^{2}}+8\omega )}^{2}}=\] [EAMCET 2003]
A.	72
B.	192
C.	200
D.	248
Answer» E.

Discussion

54.	If \[1,\omega ,{{\omega }^{2}}\] are three cube roots of unity, then \[{{(a+b\omega +c{{\omega }^{2}})}^{3}}\] + \[{{(a+b{{\omega }^{2}}+c\omega )}^{3}}\] is equal to, if \[a+b+c=0\] [West Bengal JEE 1992]
A.	\[27\,abc\]
B.	0
C.	\[3\,abc\]
D.	None of these
Answer» B. 0

Discussion

55.	If \[\omega (\ne 1)\] is a cube root of unity, then \[\left\| \begin{matrix} 1 & 1+i+{{\omega }^{2}} & {{\omega }^{2}}\\ 1-i & -1 & {{\omega }^{2}}-1\\ -i & -i+\omega -1 & -1\\ \end{matrix} \right\|\] is equal to [IIT 1995]
A.	0
B.	1
C.	\[\omega \]
D.	\[i\]
Answer» B. 1

Discussion

56.	The cube roots of unity when represented on the Argand plane form the vertices of an [IIT 1988; Pb. CET 2004]
A.	Equilateral triangle
B.	Isosceles triangle
C.	Right angled triangle
D.	None of these
Answer» B. Isosceles triangle

Discussion

57.	If 1, \[\omega ,\,{{\omega }^{2}}\] are the cube roots of unity then \[{{\omega }^{2}}{{(1+\omega )}^{3}}-(1+{{\omega }^{2}})\omega =\] [Orissa JEE 2005]
A.	1
B.	-1
C.	i
D.	0
Answer» E.

Discussion

58.	If \[(1+i)(1+2i)(1+3i).....(1+ni)=a+ib\], then2.5.10....\[(1+{{n}^{2}})\]is equal to [Karnataka CET 2002; Kerala (Engg.) 2002]
A.	\[{{a}^{2}}-{{b}^{2}}\]
B.	\[{{a}^{2}}+{{b}^{2}}\]
C.	\[\sqrt{{{a}^{2}}+{{b}^{2}}}\]
D.	\[\sqrt{{{a}^{2}}-{{b}^{2}}}\]
Answer» C. \[\sqrt{{{a}^{2}}+{{b}^{2}}}\]

Discussion

59.	Let \[\omega \] is an imaginary cube roots of unity then the value of\[2(\omega +1)({{\omega }^{2}}+1)+3(2\omega +1)(2{{\omega }^{2}}+1)+.....\]\[+(n+1)(n\omega +1)(n{{\omega }^{2}}+1)\]is [Orissa JEE 2002]
A.	\[{{\left[ \frac{n(n+1)}{2} \right]}^{2}}+n\]
B.	\[{{\left[ \frac{n(n+1)}{2} \right]}^{2}}\]
C.	\[{{\left[ \frac{n(n+1)}{2} \right]}^{2}}-n\]
D.	None of these
Answer» B. \[{{\left[ \frac{n(n+1)}{2} \right]}^{2}}\]

Discussion

60.	Let \[{{z}_{1}}\] and \[{{z}_{2}}\] be nth roots of unity which are ends of a line segment that subtend a right angle at the origin.Then n must be of the form[IIT Screening 2001; Karnataka 2002]
A.	4k + 1
B.	4k + 2
C.	4k + 3
D.	4k
Answer» E.

Discussion

61.	If \[a=\cos (2\pi /7)+i\,\sin (2\pi /7),\] then the quadratic equation whose roots are \[\alpha =a+{{a}^{2}}+{{a}^{4}}\] and \[\beta ={{a}^{3}}+{{a}^{5}}+{{a}^{6}}\] is [RPET 2000]
A.	\[{{x}^{2}}-x+2=0\]
B.	\[{{x}^{2}}+x-2=0\]
C.	\[{{x}^{2}}-x-2=0\]
D.	\[{{x}^{2}}+x+2=0\]
Answer» E.

Discussion

62.	Let \[a,b,c\] be real numbers \[a\ne 0\]. If \[\alpha \]is a root \[{{a}^{2}}{{x}^{2}}+bx+c=0\], \[\beta \] is a root of \[{{a}^{2}}{{x}^{2}}-bx-c=0\] and \[0<\alpha <\beta \], then the equation \[{{a}^{2}}{{x}^{2}}+2bx+2c=0\]has a root \[\gamma \]that always satisfies [IIT 1989]
A.	\[\gamma =\frac{\alpha +\beta }{2}\]
B.	\[\gamma =\alpha +\frac{\beta }{2}\]
C.	\[\gamma =\alpha \]
D.	\[\alpha <\gamma <\beta \]
Answer» E.

Discussion

63.	The values of a for which \[2{{x}^{2}}-2\,(2a+1)\,\,x+a(a+1)=0\] may have one root less than a and other root greater than a are given by [UPSEAT 2001]
A.	\[1>a>0\]
B.	\[-1<a<0\]
C.	\[a\ge 0\]
D.	\[a>0\,\,\text{or}a<-1\]
Answer» E.

Discussion

64.	The roots of the equation\[4{{x}^{4}}-24{{x}^{3}}+57{{x}^{2}}+18x-45=0\], If one of them is\[3+i\sqrt{6}\], are
A.	\[3-i\sqrt{6},\pm \sqrt{\frac{3}{2}}\]
B.	\[3-i\sqrt{6},\pm \frac{3}{\sqrt{2}}\]
C.	\[3-i\sqrt{6},\pm \frac{\sqrt{3}}{2}\]
D.	None of these
Answer» D. None of these

Discussion

65.	The value of 'a' for which one root of the quadratic equation \[({{a}^{2}}-5a+3){{x}^{2}}+(3a-1)x+2=0\] is twice as large as the other, is [AIEEE 2003]
A.	\[\frac{2}{3}\]
B.	\[-\frac{2}{3}\]
C.	\[\frac{1}{3}\]
D.	\[-\frac{1}{3}\]
Answer» B. \[-\frac{2}{3}\]

Discussion

66.	If \[{{x}^{2}}+px+q=0\] is the quadratic equation whose roots are a - 2 and b - 2 where a and b are the roots of \[{{x}^{2}}-3x+1=0\], then [Kerala (Engg.) 2002]
A.	\[p=1,\,q=5\]
B.	\[p=1,\,q=-5\]
C.	\[p=-1,\,\,q=1\]
D.	None of these
Answer» E.

Discussion

67.	If the equations \[2{{x}^{2}}+3x+5\lambda =0\] and \[{{x}^{2}}+2x+3\lambda =0\] have a common root, then \[\lambda =\] [RPET 1989]
A.	0
B.	-1
C.	\[0,-1\]
D.	2,-1
Answer» D. 2,-1

Discussion

68.	If\[a{{x}^{2}}+bx+c=0\], then x = [MP PET 1995]
A.	\[\frac{b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]
B.	\[\frac{-b\pm \sqrt{{{b}^{2}}-ac}}{2a}\]
C.	\[\frac{2c}{-b\pm \sqrt{{{b}^{2}}-4ac}}\]
D.	None of these
Answer» D. None of these

Discussion

69.	For the equation \[\|{{x}^{2}}\|+\|x\|-6=0\], the roots are [EAMCET 1988, 93]
A.	One and only one real number
B.	Real with sum one
C.	Real with sum zero
D.	Real with product zero
Answer» D. Real with product zero

Discussion

70.	If a, b be the roots of \[{{x}^{2}}+px+q=0\] and \[\alpha +h,\,\beta +h\] are the roots of \[{{x}^{2}}+rx+s=0\], then [AMU 2001]
A.	\[\frac{p}{r}=\frac{q}{s}\]
B.	\[2h=\left[ \frac{p}{q}+\frac{r}{s} \right]\]
C.	\[{{p}^{2}}-4q={{r}^{2}}-4s\]
D.	\[p{{r}^{2}}=q{{s}^{2}}\]
Answer» D. \[p{{r}^{2}}=q{{s}^{2}}\]

Discussion

71.	For the equation \[3{{x}^{2}}+px+3=0,\,p>0\] if one of the root is square of the other, then p is equal to[IIT Screening 2000]
A.	\[\frac{1}{3}\]
B.	1
C.	3
D.	\[\frac{2}{3}\]
Answer» D. \[\frac{2}{3}\]

Discussion

72.	The product of all real roots of the equation \[{{x}^{2}}-\|x\|-\,6=0\] is [Roorkee 2000]
A.	-9
B.	6
C.	9
D.	36
Answer» B. 6

Discussion

73.	If \[x=\cos \theta +i\sin \theta \] and \[y=\cos \varphi +i\sin \varphi \],then \[{{x}^{m}}{{y}^{n}}+{{x}^{-m}}{{y}^{-n}}\] is equal to
A.	\[\cos (m\theta +n\varphi )\]
B.	\[\cos (m\theta +n\varphi )\]
C.	\[2\cos (m\theta +n\varphi )\]
D.	\[2\cos (m\theta -n\varphi )\]
Answer» D. \[2\cos (m\theta -n\varphi )\]

Discussion

74.	If \[{{(1+x)}^{n}}={{C}_{0}}+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}+.....+{{C}_{n}}{{x}^{n}},\] then the value of \[{{C}_{0}}-{{C}_{2}}+{{C}_{4}}-{{C}_{6}}+.....\]is
A.	\[{{2}^{n}}\]
B.	\[{{2}^{n}}\cos \frac{n\pi }{2}\]
C.	\[{{2}^{n}}\sin \frac{n\pi }{2}\]
D.	\[{{2}^{n/2}}\cos \frac{n\pi }{4}\]
Answer» E.

Discussion

75.	Let \[z,w\]be complex numbers such that \[\overline{z}+i\overline{w}=0\]and \[arg\,\,zw=\pi \]. Then arg z equals [AIEEE 2004]
A.	\[5\pi /4\]
B.	\[\pi /2\]
C.	\[3\pi /4\]
D.	\[\pi /4\]
Answer» D. \[\pi /4\]

Discussion

76.	If one root of the equation \[a{{x}^{2}}+bx+c=0\]the square of the other, then\[a{{(c-b)}^{3}}=cX\], where X is
A.	\[{{a}^{3}}+{{b}^{3}}\]
B.	\[{{(a-b)}^{3}}\]
C.	\[{{a}^{3}}-{{b}^{3}}\]
D.	None of these
Answer» C. \[{{a}^{3}}-{{b}^{3}}\]

Discussion

77.	In a triangle \[ABC\] the value of \[\angle A\] is given by \[5\cos A+3=0\], then the equationwhose roots are \[\sin A\] and \[\tan A\] will be [Roorkee 1972]
A.	\[15{{x}^{2}}-8x+16=0\]
B.	\[15{{x}^{2}}+8x-16=0\]
C.	\[15{{x}^{2}}-8\sqrt{2}x+16=0\]
D.	\[15{{x}^{2}}-8x-16=0\]
Answer» C. \[15{{x}^{2}}-8\sqrt{2}x+16=0\]

Discussion

78.	If a root of the given equation \[a(b-c){{x}^{2}}+b(c-a)x+c(a-b)=0\]is 1, then the other will be [RPET 1986]
A.	\[\frac{a(b-c)}{b(c-a)}\]
B.	\[\frac{b(c-a)}{a(b-c)}\]
C.	\[\frac{c(a-b)}{a(b-c)}\]
D.	None of these
Answer» D. None of these

Discussion

79.	If one root of the equation \[a{{x}^{2}}+bx+c=0\]be \[n\] times the other root, then
A.	\[n{{a}^{2}}=bc{{(n+1)}^{2}}\]
B.	\[n{{b}^{2}}=ac{{(n+1)}^{2}}\]
C.	\[n{{c}^{2}}=ab{{(n+1)}^{2}}\]
D.	None of these
Answer» C. \[n{{c}^{2}}=ab{{(n+1)}^{2}}\]

Discussion

80.	The coefficient of \[x\] in the equation \[{{x}^{2}}+px+q=0\]was taken as 17 in place of 13, its roots were found to be -2 and -15, The roots of the original equation are[IIT 1977, 79]
A.	3, 10
B.	- 3, - 10
C.	- 5, - 18
D.	None of these
Answer» C. - 5, - 18

Discussion

81.	For positive integers \[{{n}_{1}},{{n}_{2}}\]the value of the expression\[{{(1+i)}^{{{n}_{1}}}}+{{(1+{{i}^{3}})}^{{{n}_{1}}}}+{{(1+{{i}^{5}})}^{{{n}_{2}}}}+{{(1+{{i}^{7}})}^{{{n}_{2}}}}\]where \[i=\sqrt{-1}\]is a real number if and only if [IIT 1996]
A.	\[{{n}_{1}}={{n}_{2}}+1\]
B.	\[{{n}_{1}}={{n}_{2}}-1\]
C.	\[{{n}_{1}}={{n}_{2}}\]
D.	\[{{n}_{1}}>0,{{n}_{2}}>0\]
Answer» E.

Discussion

82.	The number of real values of \[a\] satisfying the equation \[{{a}^{2}}-2a\sin x+1=0\] is
A.	Zero
B.	One
C.	Two
D.	Infinite
Answer» D. Infinite

Discussion

83.	If \[(3+i)z=(3-i)\bar{z},\] then complex number z is [AMU 2005]
A.	\[x\,(3-i),\,x\in R\]
B.	\[\frac{x}{3+i},\,x\in R\]
C.	\[x(3+i),\,x\in R\]
D.	\[x(-3+i),\,x\in R\]
Answer» B. \[\frac{x}{3+i},\,x\in R\]

Discussion

84.	The values of \[x\] and \[y\] for which the numbers \[3+i{{x}^{2}}y\] and \[{{x}^{2}}+y+4i\] are conjugate complex can be
A.	\[(-2,-1)\]or \[(2,-1)\]
B.	\[(-1,\text{ }2)\]or \[(-2,\text{ }1)\]
C.	\[(1,\,2)\]or \[(-1,-2)\]
D.	None of these
Answer» B. \[(-1,\text{ }2)\]or \[(-2,\text{ }1)\]

Discussion

85.	If\[\|{{z}_{1}}\|\,=\,\|{{z}_{2}}\|\] and \[amp\,{{z}_{1}}+amp\,\,{{z}_{2}}=0\], then [MP PET 1999]
A.	\[{{z}_{1}}={{z}_{2}}\]
B.	\[{{\bar{z}}_{1}}={{z}_{2}}\]
C.	\[{{z}_{1}}+{{z}_{2}}=0\]
D.	\[{{\bar{z}}_{1}}={{\bar{z}}_{2}}\]
Answer» C. \[{{z}_{1}}+{{z}_{2}}=0\]

Discussion

86.	For any two complex numbers \[{{z}_{1}},{{z}_{2}}\]we have \[\|{{z}_{1}}+{{z}_{2}}{{\|}^{2}}=\] \[\|{{z}_{1}}{{\|}^{2}}+\|{{z}_{2}}{{\|}^{2}}\] then
A.	\[\operatorname{Re}\left( \frac{{{z}_{1}}}{{{z}_{2}}} \right)=0\]
B.	\[\operatorname{Im}\left( \frac{{{z}_{1}}}{{{z}_{2}}} \right)=0\]
C.	\[\operatorname{Re}({{z}_{1}}{{z}_{2}})=0\]
D.	\[\operatorname{Im}({{z}_{1}}{{z}_{2}})=0\]
Answer» B. \[\operatorname{Im}\left( \frac{{{z}_{1}}}{{{z}_{2}}} \right)=0\]

Discussion

87.	Let \[z\] be a purely imaginary number such that \[\operatorname{Im}(z)<0\]. Then\[arg\,(z)\] is equal to
A.	\[\pi \]
B.	\[\frac{\pi }{2}\]
C.	0
D.	\[-\frac{\pi }{2}\]
Answer» E.

Discussion

88.	The moduli of two complex numbers are less than unity, then the modulus of the sum of these complex numbers
A.	Less than unity
B.	Greater than unity
C.	Equal to unity
D.	Any
Answer» E.

Discussion

89.	If \[{{x}^{2}}-hx-21=0,{{x}^{2}}-3hx+35=0\]\[(h>0)\]has a common root, then the value of \[h\] is equal to [EAMCET 1986]
A.	1
B.	2
C.	3
D.	4
Answer» E.

Discussion

90.	The maximum possible number of real roots of equation \[{{x}^{5}}-6{{x}^{2}}-4x+5=0\] is [EAMCET 2002]
A.	0
B.	3
C.	4
D.	5
Answer» C. 4

Discussion

91.	Let \[\alpha ,\beta \] be the roots of \[{{x}^{2}}+(3-\lambda )x-\lambda =0.\]The value of \[\lambda \] for which \[{{\alpha }^{2}}+{{\beta }^{2}}\] is minimum, is [AMU 2002]
A.	0
B.	1
C.	2
D.	3
Answer» D. 3

Discussion

Explore topic-wise MCQs in Mathematics.

\[\sinh ix\] is [EAMCET 2002]

If \[1,\,\omega ,\,{{\omega }^{2}}\] are the roots of unity, then \[{{(1-2\omega +{{\omega }^{2}})}^{6}}\] is equal to [Pb. CET 2001]

If \[\omega \] is a complex cube root of unity, then\[225+\]\[{{(3\omega +8{{\omega }^{2}})}^{2}}\]\[+{{(3{{\omega }^{2}}+8\omega )}^{2}}=\] [EAMCET 2003]

If \[1,\omega ,{{\omega }^{2}}\] are three cube roots of unity, then \[{{(a+b\omega +c{{\omega }^{2}})}^{3}}\] + \[{{(a+b{{\omega }^{2}}+c\omega )}^{3}}\] is equal to, if \[a+b+c=0\] [West Bengal JEE 1992]

If \[\omega (\ne 1)\] is a cube root of unity, then \[\left| \begin{matrix} 1 & 1+i+{{\omega }^{2}} & {{\omega }^{2}}\\ 1-i & -1 & {{\omega }^{2}}-1\\ -i & -i+\omega -1 & -1\\ \end{matrix} \right|\] is equal to [IIT 1995]

The cube roots of unity when represented on the Argand plane form the vertices of an [IIT 1988; Pb. CET 2004]

If 1, \[\omega ,\,{{\omega }^{2}}\] are the cube roots of unity then \[{{\omega }^{2}}{{(1+\omega )}^{3}}-(1+{{\omega }^{2}})\omega =\] [Orissa JEE 2005]

If \[(1+i)(1+2i)(1+3i).....(1+ni)=a+ib\], then2.5.10....\[(1+{{n}^{2}})\]is equal to [Karnataka CET 2002; Kerala (Engg.) 2002]

Let \[\omega \] is an imaginary cube roots of unity then the value of\[2(\omega +1)({{\omega }^{2}}+1)+3(2\omega +1)(2{{\omega }^{2}}+1)+.....\]\[+(n+1)(n\omega +1)(n{{\omega }^{2}}+1)\]is [Orissa JEE 2002]

Let \[{{z}_{1}}\] and \[{{z}_{2}}\] be nth roots of unity which are ends of a line segment that subtend a right angle at the origin.Then n must be of the form[IIT Screening 2001; Karnataka 2002]

If \[a=\cos (2\pi /7)+i\,\sin (2\pi /7),\] then the quadratic equation whose roots are \[\alpha =a+{{a}^{2}}+{{a}^{4}}\] and \[\beta ={{a}^{3}}+{{a}^{5}}+{{a}^{6}}\] is [RPET 2000]

Let \[a,b,c\] be real numbers \[a\ne 0\]. If \[\alpha \]is a root \[{{a}^{2}}{{x}^{2}}+bx+c=0\], \[\beta \] is a root of \[{{a}^{2}}{{x}^{2}}-bx-c=0\] and \[0<\alpha <\beta \], then the equation \[{{a}^{2}}{{x}^{2}}+2bx+2c=0\]has a root \[\gamma \]that always satisfies [IIT 1989]

The values of a for which \[2{{x}^{2}}-2\,(2a+1)\,\,x+a(a+1)=0\] may have one root less than a and other root greater than a are given by [UPSEAT 2001]

The roots of the equation\[4{{x}^{4}}-24{{x}^{3}}+57{{x}^{2}}+18x-45=0\], If one of them is\[3+i\sqrt{6}\], are

The value of 'a' for which one root of the quadratic equation \[({{a}^{2}}-5a+3){{x}^{2}}+(3a-1)x+2=0\] is twice as large as the other, is [AIEEE 2003]

If \[{{x}^{2}}+px+q=0\] is the quadratic equation whose roots are a - 2 and b - 2 where a and b are the roots of \[{{x}^{2}}-3x+1=0\], then [Kerala (Engg.) 2002]

If the equations \[2{{x}^{2}}+3x+5\lambda =0\] and \[{{x}^{2}}+2x+3\lambda =0\] have a common root, then \[\lambda =\] [RPET 1989]

If\[a{{x}^{2}}+bx+c=0\], then x = [MP PET 1995]

For the equation \[|{{x}^{2}}|+|x|-6=0\], the roots are [EAMCET 1988, 93]

If a, b be the roots of \[{{x}^{2}}+px+q=0\] and \[\alpha +h,\,\beta +h\] are the roots of \[{{x}^{2}}+rx+s=0\], then [AMU 2001]

For the equation \[3{{x}^{2}}+px+3=0,\,p>0\] if one of the root is square of the other, then p is equal to[IIT Screening 2000]

The product of all real roots of the equation \[{{x}^{2}}-|x|-\,6=0\] is [Roorkee 2000]

If \[x=\cos \theta +i\sin \theta \] and \[y=\cos \varphi +i\sin \varphi \],then \[{{x}^{m}}{{y}^{n}}+{{x}^{-m}}{{y}^{-n}}\] is equal to

If \[{{(1+x)}^{n}}={{C}_{0}}+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}+.....+{{C}_{n}}{{x}^{n}},\] then the value of \[{{C}_{0}}-{{C}_{2}}+{{C}_{4}}-{{C}_{6}}+.....\]is

Let \[z,w\]be complex numbers such that \[\overline{z}+i\overline{w}=0\]and \[arg\,\,zw=\pi \]. Then arg z equals [AIEEE 2004]

If one root of the equation \[a{{x}^{2}}+bx+c=0\]the square of the other, then\[a{{(c-b)}^{3}}=cX\], where X is

In a triangle \[ABC\] the value of \[\angle A\] is given by \[5\cos A+3=0\], then the equationwhose roots are \[\sin A\] and \[\tan A\] will be [Roorkee 1972]

If a root of the given equation \[a(b-c){{x}^{2}}+b(c-a)x+c(a-b)=0\]is 1, then the other will be [RPET 1986]

If one root of the equation \[a{{x}^{2}}+bx+c=0\]be \[n\] times the other root, then

The coefficient of \[x\] in the equation \[{{x}^{2}}+px+q=0\]was taken as 17 in place of 13, its roots were found to be -2 and -15, The roots of the original equation are[IIT 1977, 79]

For positive integers \[{{n}_{1}},{{n}_{2}}\]the value of the expression\[{{(1+i)}^{{{n}_{1}}}}+{{(1+{{i}^{3}})}^{{{n}_{1}}}}+{{(1+{{i}^{5}})}^{{{n}_{2}}}}+{{(1+{{i}^{7}})}^{{{n}_{2}}}}\]where \[i=\sqrt{-1}\]is a real number if and only if [IIT 1996]

The number of real values of \[a\] satisfying the equation \[{{a}^{2}}-2a\sin x+1=0\] is

If \[(3+i)z=(3-i)\bar{z},\] then complex number z is [AMU 2005]

The values of \[x\] and \[y\] for which the numbers \[3+i{{x}^{2}}y\] and \[{{x}^{2}}+y+4i\] are conjugate complex can be

If\[|{{z}_{1}}|\,=\,|{{z}_{2}}|\] and \[amp\,{{z}_{1}}+amp\,\,{{z}_{2}}=0\], then [MP PET 1999]

For any two complex numbers \[{{z}_{1}},{{z}_{2}}\]we have \[|{{z}_{1}}+{{z}_{2}}{{|}^{2}}=\] \[|{{z}_{1}}{{|}^{2}}+|{{z}_{2}}{{|}^{2}}\] then

Let \[z\] be a purely imaginary number such that \[\operatorname{Im}(z)<0\]. Then\[arg\,(z)\] is equal to

The moduli of two complex numbers are less than unity, then the modulus of the sum of these complex numbers

If \[{{x}^{2}}-hx-21=0,{{x}^{2}}-3hx+35=0\]\[(h>0)\]has a common root, then the value of \[h\] is equal to [EAMCET 1986]

The maximum possible number of real roots of equation \[{{x}^{5}}-6{{x}^{2}}-4x+5=0\] is [EAMCET 2002]

Let \[\alpha ,\beta \] be the roots of \[{{x}^{2}}+(3-\lambda )x-\lambda =0.\]The value of \[\lambda \] for which \[{{\alpha }^{2}}+{{\beta }^{2}}\] is minimum, is [AMU 2002]