8666 + Mcqs in Mathematics Page 125 McqOptions

6201.	If \[^{n}{{C}_{r}}=84,{{\ }^{n}}{{C}_{r-1}}=36\] and \[^{n}{{C}_{r+1}}=126\], then \[n\] equals [RPET 1997; MP PET 2001]
A.	8
B.	9
C.	10
D.	5
Answer» C. 10

Discussion

6202.	If \[^{10}{{C}_{r}}{{=}^{10}}{{C}_{r+2}}\], then \[^{5}{{C}_{r}}\] equals [RPET 1996]
A.	120
B.	10
C.	360
D.	5
Answer» E.

Discussion

6203.	There are 12 volleyball players in all in a college, out of which a team of 9 players is to be formed. If the captain always remains the same, then in how many ways can the team be formed
A.	36
B.	108
C.	99
D.	165
Answer» E.

Discussion

6204.	In an examination there are three multiple choice questions and each question has 4 choices. Number of ways in which a student can fail to get all answers correct, is [Pb. CET 1990; UPSEAT 2001]
A.	11
B.	12
C.	27
D.	63
Answer» E.

Discussion

6205.	In a city no two persons have identical set of teeth and there is no person without a tooth. Also no person has more than 32 teeth. If we disregard the shape and size of tooth and consider only the positioning of the teeth, then the maximum population of the city is
A.	\[{{2}^{32}}\]
B.	\[{{(32)}^{2}}-1\]
C.	\[{{2}^{32}}-1\]
D.	\[{{2}^{32-1}}\]
Answer» D. \[{{2}^{32-1}}\]

Discussion

6206.	On the occasion of Deepawali festival each student of a class sends greeting cards to the others. If there are 20 students in the class, then the total number of greeting cards exchanged by the students is
A.	\[^{20}{{C}_{2}}\]
B.	\[2\ .{{\ }^{20}}{{C}_{2}}\]
C.	\[2\ .{{\ }^{20}}{{P}_{2}}\]
D.	None of these
Answer» C. \[2\ .{{\ }^{20}}{{P}_{2}}\]

Discussion

6207.	If \[\alpha {{=}^{m}}{{C}_{2}}\], then \[^{\alpha }{{C}_{2}}\]is equal to
A.	\[^{m+1}{{C}_{4}}\]
B.	\[^{m-1}{{C}_{4}}\]
C.	\[3\,.{{\ }^{m+2}}{{C}_{4}}\]
D.	\[3\ .{{\ }^{m+1}}{{C}_{4}}\]
Answer» E.

Discussion

6208.	In a football championship, there were played 153 matches. Every team played one match with each other. The number of teams participating in the championship is [WB JEE 1992; Kurukshetra CEE 1998]
A.	17
B.	18
C.	9
D.	13
Answer» C. 9

Discussion

6209.	\[\sum\limits_{r=0}^{m}{^{n+r}{{C}_{n}}=}\] [Pb. CET 2003]
A.	\[^{n+m+1}{{C}_{n+1}}\]
B.	\[^{n+m+2}{{C}_{n}}\]
C.	\[^{n+m+3}{{C}_{n-1}}\]
D.	None of these
Answer» B. \[^{n+m+2}{{C}_{n}}\]

Discussion

6210.	A man has 7 friends. In how many ways he can invite one or more of them for a tea party
A.	128
B.	256
C.	127
D.	130
Answer» D. 130

Discussion

6211.	The solution set of \[^{10}{{C}_{x-1}}>2\ .{{\ }^{10}}{{C}_{x}}\] is
A.	{1, 2, 3}
B.	{4, 5, 6}
C.	{8,9, 10}
D.	{9, 10, 11}
Answer» D. {9, 10, 11}

Discussion

6212.	If \[^{20}{{C}_{n+2}}{{=}^{n}}{{C}_{16}}\], then the value of \[n\] is [MP PET 1984]
A.	7
B.	10
C.	13
D.	No value
Answer» E.

Discussion

6213.	The value of \[^{15}{{C}_{3}}{{+}^{15}}{{C}_{13}}\] is [MP PET 1983]
A.	\[^{16}{{C}_{3}}\]
B.	\[^{30}{{C}_{16}}\]
C.	\[^{15}{{C}_{10}}\]
D.	\[^{15}{{C}_{15}}\]
Answer» B. \[^{30}{{C}_{16}}\]

Discussion

6214.	\[^{n}{{C}_{r}}{{+}^{n}}{{C}_{r-1}}\] is equal to [MP PET 1984; Kerala (Engg.) 2002]
A.	\[^{n+1}{{C}_{r}}\]
B.	\[^{n}{{C}_{r+1}}\]
C.	\[^{n+1}{{C}_{r+1}}\]
D.	\[^{n-1}{{C}_{r-1}}\]
Answer» B. \[^{n}{{C}_{r+1}}\]

Discussion

6215.	If \[^{n}{{C}_{r-1}}=36,{{\ }^{n}}{{C}_{r}}=84\] and \[^{n}{{C}_{r+1}}=126\], then the value of \[r\] is [IIT 1979; Pb. CET 1993, 2003; DCE 1999, 2000; MP PET 2001]
A.	1
B.	2
C.	3
D.	None of these
Answer» D. None of these

Discussion

6216.	\[^{n}{{C}_{r}}+{{2}^{n}}{{C}_{r-1}}{{+}^{n}}{{C}_{r-2}}=\]
A.	\[^{n+1}{{C}_{r}}\]
B.	\[^{n+1}{{C}_{r+1}}\]
C.	\[^{n+2}{{C}_{r}}\]
D.	\[^{n+2}{{C}_{r+1}}\]
Answer» D. \[^{n+2}{{C}_{r+1}}\]

Discussion

6217.	If \[^{{{n}^{2}}-n}{{C}_{2}}{{=}^{{{n}^{2}}-n}}{{C}_{10}}\], then \[n=\]
A.	12
B.	4 only
C.	\[-3\]only
D.	4 or \[-3\]
Answer» E.

Discussion

6218.	If \[n\] is even and the value of \[^{n}{{C}_{r}}\] is maximum, then \[r=\]
A.	\[\frac{n}{2}\]
B.	\[\frac{n+1}{2}\]
C.	\[\frac{n-1}{2}\]
D.	None of these
Answer» B. \[\frac{n+1}{2}\]

Discussion

6219.	\[\cos (x+iy)\]is equal to [RPET 2001]
A.	\[\sin \,x\,\,\cosh \,y+i\,\cos \,x\,\,\sinh \,y\]
B.	\[\cos \,x\,\,\cosh \,y+i\,\sin \,x\,\,\sinh \,y\]
C.	\[\cos \,x\,\,\cosh \,y-i\,\sin \,x\,\,\sinh \,y\]
D.	None of these
Answer» D. None of these

Discussion

6220.	The imaginary part of \[\cosh (\alpha +i\beta )\]is [RPET 2000]
A.	\[\cosh \,\alpha \,\,\cos \,\beta \]
B.	\[\sinh \,\alpha \,\,\sin \,\beta \]
C.	\[\cos \alpha \cosh \beta \]
D.	\[\cos \alpha \cos \beta \]
Answer» C. \[\cos \alpha \cosh \beta \]

Discussion

6221.	\[\cosh (\alpha +i\beta )-\cosh (\alpha -i\beta )\] is equal to [RPET 2000]
A.	\[2\,\,\sinh \,\alpha \,\,\sinh \,\beta \]
B.	\[2\,\,\cosh \,\alpha \,\,\cosh \,\beta \]
C.	\[2i\,\,\sinh \,\alpha \,\,\sin \,\beta \]
D.	\[2\,\,\cosh \,\alpha \,\,\cos \,\beta \]
Answer» D. \[2\,\,\cosh \,\alpha \,\,\cos \,\beta \]

Discussion

6222.	If \[\cos (u+iv)=\alpha +i\beta ,\] then \[{{\alpha }^{2}}+{{\beta }^{2}}+1\] equals [RPET 1999]
A.	\[{{\cos }^{2}}u+{{\sinh }^{2}}v\]
B.	\[{{\sin }^{2}}u+{{\cosh }^{2}}v\]
C.	\[{{\cos }^{2}}u+{{\cosh }^{2}}v\]
D.	\[{{\sin }^{2}}u+{{\sinh }^{2}}v\]
Answer» D. \[{{\sin }^{2}}u+{{\sinh }^{2}}v\]

Discussion

6223.	The real part of \[{{\sin }^{-1}}({{e}^{i\theta }})\] is [RPET 1997]
A.	\[{{\cos }^{-1}}(\sqrt{\sin \theta })\]
B.	\[{{\sinh }^{-1}}(\sqrt{\sin \theta })\]
C.	\[{{\sin }^{-1}}(\sqrt{\sin \theta })\]
D.	\[{{\sin }^{-1}}(\sqrt{\cos \theta })\]
Answer» B. \[{{\sinh }^{-1}}(\sqrt{\sin \theta })\]

Discussion

6224.	If \[\omega \] is a complex cube root of unity, then the value of \[{{\omega }^{99}}+{{\omega }^{100}}+{{\omega }^{101}}\] is [Pb. CET 2004]
A.	1
B.	-1
C.	3
D.	0
Answer» E.

Discussion

6225.	If \[\omega =\frac{-1+\sqrt{3}i}{2}\]then \[{{(3+\omega +3{{\omega }^{2}})}^{4}}\]= [Karnataka CET 2004; Pb. CET 2000]
A.	16
B.	-16
C.	16 \[\omega \]
D.	16\[{{\omega }^{2}}\]
Answer» D. 16\[{{\omega }^{2}}\]

Discussion

6226.	If \[(\cos \theta +i\sin \theta )(\cos 2\theta +i\sin 2\theta )........\] \[(\cos n\theta +i\sin n\theta )=1\], then the value of \[\theta \] is[Karnataka CET 1992; Kurukshetra CEE 2002]
A.	\[4m\pi \]
B.	\[\frac{2m\pi }{n(n+1)}\]
C.	\[\frac{4m\pi }{n(n+1)}\]
D.	\[\frac{m\pi }{n(n+1)}\]
Answer» D. \[\frac{m\pi }{n(n+1)}\]

Discussion

6227.	If \[1,\omega ,{{\omega }^{2}}\] are the cube roots of unity, then\[\Delta =\left\| \,\begin{matrix} 1\,\,\,\, & {{\omega }^{n}} & {{\omega }^{2n}} \\ {{\omega }^{n}}\,\, & \,\,\,{{\omega }^{2n}}\,\, & 1 \\ {{\omega }^{2n}}\, & 1\,\, & {{\omega }^{n}} \\ \end{matrix} \right\|\]= [AIEEE 2003]
A.	0
B.	1
C.	\[\omega \]
D.	\[{{\omega }^{2}}\]
Answer» B. 1

Discussion

6228.	Find the value of \[{{(1+2\omega +{{\omega }^{2}})}^{3n}}-{{(1+\omega +2{{\omega }^{2}})}^{3n}}=\] [UPSEAT 2002]
A.	0
B.	1
C.	\[\omega \]
D.	\[{{\omega }^{2}}\]
Answer» B. 1

Discussion

6229.	The value of (8)1/3 is [RPET 2003]
A.	\[-1+i\sqrt{3}\]
B.	\[-1-i\sqrt{3}\]
C.	2
D.	All of these
Answer» E.

Discussion

6230.	. Which of the following is a fourth root of \[\frac{1}{2}+\frac{i\sqrt{3}}{2}\] [Karnataka CET 2003]
A.	\[cis\left( \frac{\pi }{2} \right)\]
B.	\[cis\left( \frac{\pi }{12} \right)\]
C.	\[cis\left( \frac{\pi }{6} \right)\]
D.	\[cis\left( \frac{\pi }{3} \right)\]
Answer» C. \[cis\left( \frac{\pi }{6} \right)\]

Discussion

6231.	If \[\frac{1+\sqrt{3}\,i}{2}\] is a root of equation \[{{x}^{4}}-{{x}^{3}}+x-1=0\] then its real roots are [EAMCET 2002]
A.	1, 1
B.	- 1, - 1
C.	1, - 1
D.	1, 2
Answer» D. 1, 2

Discussion

6232.	If \[z+{{z}^{-1}}=1,\,\text{then }\,{{z}^{100}}+{{z}^{-100}}\] is equal to [UPSEAT 2001]
A.	i
B.
C.	1
D.	-1
Answer» E.

Discussion

6233.	If \[z=\frac{\sqrt{3}+i}{-2}\], then \[{{z}^{69}}\] is equal to [RPET 2001]
A.	1
B.	-1
C.	i
D.
Answer» D.

Discussion

6234.	If \[a=\sqrt{2i}\] then which of the following is correct [Roorkee 1989]
A.	\[a=1+i\]
B.	\[a=1-i\]
C.	\[a=-(\sqrt{2})i\]
D.	None of these
Answer» B. \[a=1-i\]

Discussion

6235.	If \[1,\omega ,{{\omega }^{2}}\] are the cube roots of unity, then their product is [Karnataka CET 1999, 2001]
A.	0
B.	\[\omega \]
C.	-1
D.	1
Answer» E.

Discussion

6236.	If cube root of 1 is \[\omega \], then the value of \[{{(3+\omega +3{{\omega }^{2}})}^{4}}\] is [MP PET 2001]
A.	0
B.	16
C.	\[16\,\omega \]
D.	\[16\,{{\omega }^{2}}\]
Answer» D. \[16\,{{\omega }^{2}}\]

Discussion

6237.	If \[\pi /3\] is a complex root of the equation \[{{z}^{3}}=1\], then \[\omega +{{\omega }^{\left( \frac{1}{2}\,+\,\frac{3}{8}\,+\,\frac{9}{32}\,+\,\frac{27}{128}\,+... \right)}}\] is equal to [Roorkee 2000; AMU 2005]
A.	-1
B.	0
C.	9
D.	i
Answer» B. 0

Discussion

6238.	\[\frac{{{(-1+i\sqrt{3})}^{15}}}{{{(1-i)}^{20}}}+\frac{{{(-1-i\sqrt{3})}^{15}}}{{{(1+i)}^{20}}}\] is equal to [AMU 2000]
A.	-64
B.	-32
C.	-16
D.	\[\frac{1}{16}\]
Answer» B. -32

Discussion

6239.	If \[\omega \] is an imaginary cube root of unity, \[{{(1+\omega -{{\omega }^{2}})}^{7}}\]equals [IIT 1998; MP PET 2000]
A.	\[128\omega \]
B.	\[-128\omega \]
C.	\[128{{\omega }^{2}}\]
D.	\[-128{{\omega }^{2}}\]
Answer» E.

Discussion

6240.	If \[\omega \] is an imaginary cube root of unity, then the value of \[\sin \,\left[ ({{\omega }^{10}}+{{\omega }^{23}})\,\pi -\frac{\pi }{4} \right]\] is [IIT Screening 1994]
A.	\[-\sqrt{3}/2\]
B.	\[-1/\sqrt{2}\]
C.	\[1/\sqrt{2}\]
D.	\[\sqrt{3}/2\]
Answer» D. \[\sqrt{3}/2\]

Discussion

6241.	The following in the form of \[A+iB\] \[{{(\cos 2\theta +i\sin 2\theta )}^{-5}}\] \[{{(\cos 3\theta -i\sin 3\theta )}^{6}}\]\[{{(\sin \theta -i\cos \theta )}^{3}}\] in the form of \[A+iB\] is [MNR 1991]
A.	\[(\cos 25\theta +i\sin 25\theta )\]
B.	\[i(\cos 25\theta +i\sin 25\theta )\]
C.	\[i\,(\cos 25\theta -i\sin 25\theta )\]
D.	\[(\cos 25\theta -i\sin 25\theta )\]
Answer» E.

Discussion

6242.	If \[\omega \] is the cube root of unity, then \[{{(3+5\omega +3{{\omega }^{2}})}^{2}}\] + \[{{(3+3\omega +5{{\omega }^{2}})}^{2}}\] = [MP PET 1999]
A.	4
B.	0
C.	-4
D.	None of these
Answer» D. None of these

Discussion

6243.	If \[\alpha \] and \[\beta \] are imaginary cube roots of unity, then the value of \[{{\alpha }^{4}}+{{\beta }^{28}}+\frac{1}{\alpha \beta }\],is [MP PET 1998]
A.	1
B.	\[-1\]
C.	0
D.	None of these
Answer» D. None of these

Discussion

6244.	\[{{\left( \frac{-1+i\sqrt{3}}{2} \right)}^{20}}+{{\left( \frac{-1-i\sqrt{3}}{2} \right)}^{20}}=\]
A.	\[20\sqrt{3}i\]
B.	1
C.	\[\frac{1}{{{2}^{19}}}\]
D.	\[-1\]
Answer» E.

Discussion

6245.	If \[\alpha \] is an imaginary cube root of unity, then for \[n\in N\], the value of \[{{\alpha }^{3n+1}}+{{\alpha }^{3n+3}}+{{\alpha }^{3n+5}}\] is [MP PET 1996; Pb. CET 2000]
A.	\[-1\]
B.	0
C.	1
D.	3
Answer» C. 1

Discussion

6246.	If \[{{z}_{1}},{{z}_{2}}{{z}_{3}},{{z}_{4}}\]are the roots of the equation \[{{z}^{4}}=1\], then the value of \[\sum\limits_{i=1}^{4}{z_{i}^{3}}\]is [Kurukshetra CEE 1996]
A.	0
B.	1
C.	\[i\]
D.	\[1+i\]
Answer» B. 1

Discussion

6247.	The common roots of the equations \[{{x}^{12}}-1=0\], \[{{x}^{4}}+{{x}^{2}}+1=0\] are [EAMCET 1989]
A.	\[\pm \omega \]
B.	\[\pm {{\omega }^{2}}\]
C.	\[\pm \omega ,\,\pm {{\omega }^{2}}\]
D.	None of these
Answer» D. None of these

Discussion

6248.	If \[{{z}_{1}},{{z}_{2}},{{z}_{3}}......{{n}_{n}}\] are nth, roots of unity, then for \[k=1,\,2,.....,n\]
A.	\[\|{{z}_{k}}\|=k\|{{z}_{k+1}}\|\]
B.	\[\|{{z}_{k+1}}\|=k\|{{z}_{k}}\|\]
C.	\[\|{{z}_{k+1}}\|\,=\,\|{{z}_{k}}\|+\|{{z}_{k+1}}\|\]
D.	\[\|{{z}_{k}}\|=\|{{z}_{k+1}}\|\]
Answer» E.

Discussion

6249.	If \[\omega \] is an nth root of unity, other than unity, then the value of \[1+\omega +{{\omega }^{2}}+...+{{\omega }^{n-1}}\] is [Karnataka CET 1999]
A.	0
B.	1
C.	\[-1\]
D.	None of these
Answer» B. 1

Discussion

6250.	If \[n\] is a positive integer greater than unity and \[z\] is a complex number satisfying the equation \[{{z}^{n}}={{(z+1)}^{n}}\], then
A.	\[\operatorname{Re}(z)<0\]
B.	\[\operatorname{Re}(z)>0\]
C.	\[\operatorname{Re}(z)=0\]
D.	None of these
Answer» B. \[\operatorname{Re}(z)>0\]

Discussion

Explore topic-wise MCQs in Joint Entrance Exam - Main (JEE Main).

If \[^{n}{{C}_{r}}=84,{{\ }^{n}}{{C}_{r-1}}=36\] and \[^{n}{{C}_{r+1}}=126\], then \[n\] equals [RPET 1997; MP PET 2001]

If \[^{10}{{C}_{r}}{{=}^{10}}{{C}_{r+2}}\], then \[^{5}{{C}_{r}}\] equals [RPET 1996]

There are 12 volleyball players in all in a college, out of which a team of 9 players is to be formed. If the captain always remains the same, then in how many ways can the team be formed

In an examination there are three multiple choice questions and each question has 4 choices. Number of ways in which a student can fail to get all answers correct, is [Pb. CET 1990; UPSEAT 2001]

In a city no two persons have identical set of teeth and there is no person without a tooth. Also no person has more than 32 teeth. If we disregard the shape and size of tooth and consider only the positioning of the teeth, then the maximum population of the city is

On the occasion of Deepawali festival each student of a class sends greeting cards to the others. If there are 20 students in the class, then the total number of greeting cards exchanged by the students is

If \[\alpha {{=}^{m}}{{C}_{2}}\], then \[^{\alpha }{{C}_{2}}\]is equal to

In a football championship, there were played 153 matches. Every team played one match with each other. The number of teams participating in the championship is [WB JEE 1992; Kurukshetra CEE 1998]

\[\sum\limits_{r=0}^{m}{^{n+r}{{C}_{n}}=}\] [Pb. CET 2003]

A man has 7 friends. In how many ways he can invite one or more of them for a tea party

The solution set of \[^{10}{{C}_{x-1}}>2\ .{{\ }^{10}}{{C}_{x}}\] is

If \[^{20}{{C}_{n+2}}{{=}^{n}}{{C}_{16}}\], then the value of \[n\] is [MP PET 1984]

The value of \[^{15}{{C}_{3}}{{+}^{15}}{{C}_{13}}\] is [MP PET 1983]

\[^{n}{{C}_{r}}{{+}^{n}}{{C}_{r-1}}\] is equal to [MP PET 1984; Kerala (Engg.) 2002]

If \[^{n}{{C}_{r-1}}=36,{{\ }^{n}}{{C}_{r}}=84\] and \[^{n}{{C}_{r+1}}=126\], then the value of \[r\] is [IIT 1979; Pb. CET 1993, 2003; DCE 1999, 2000; MP PET 2001]

\[^{n}{{C}_{r}}+{{2}^{n}}{{C}_{r-1}}{{+}^{n}}{{C}_{r-2}}=\]

If \[^{{{n}^{2}}-n}{{C}_{2}}{{=}^{{{n}^{2}}-n}}{{C}_{10}}\], then \[n=\]

If \[n\] is even and the value of \[^{n}{{C}_{r}}\] is maximum, then \[r=\]

\[\cos (x+iy)\]is equal to [RPET 2001]

The imaginary part of \[\cosh (\alpha +i\beta )\]is [RPET 2000]

\[\cosh (\alpha +i\beta )-\cosh (\alpha -i\beta )\] is equal to [RPET 2000]

If \[\cos (u+iv)=\alpha +i\beta ,\] then \[{{\alpha }^{2}}+{{\beta }^{2}}+1\] equals [RPET 1999]

The real part of \[{{\sin }^{-1}}({{e}^{i\theta }})\] is [RPET 1997]

If \[\omega \] is a complex cube root of unity, then the value of \[{{\omega }^{99}}+{{\omega }^{100}}+{{\omega }^{101}}\] is [Pb. CET 2004]

If \[\omega =\frac{-1+\sqrt{3}i}{2}\]then \[{{(3+\omega +3{{\omega }^{2}})}^{4}}\]= [Karnataka CET 2004; Pb. CET 2000]

If \[(\cos \theta +i\sin \theta )(\cos 2\theta +i\sin 2\theta )........\] \[(\cos n\theta +i\sin n\theta )=1\], then the value of \[\theta \] is[Karnataka CET 1992; Kurukshetra CEE 2002]

If \[1,\omega ,{{\omega }^{2}}\] are the cube roots of unity, then\[\Delta =\left| \,\begin{matrix} 1\,\,\,\, & {{\omega }^{n}} & {{\omega }^{2n}} \\ {{\omega }^{n}}\,\, & \,\,\,{{\omega }^{2n}}\,\, & 1 \\ {{\omega }^{2n}}\, & 1\,\, & {{\omega }^{n}} \\ \end{matrix} \right|\]= [AIEEE 2003]

Find the value of \[{{(1+2\omega +{{\omega }^{2}})}^{3n}}-{{(1+\omega +2{{\omega }^{2}})}^{3n}}=\] [UPSEAT 2002]

The value of (8)1/3 is [RPET 2003]

. Which of the following is a fourth root of \[\frac{1}{2}+\frac{i\sqrt{3}}{2}\] [Karnataka CET 2003]

If \[\frac{1+\sqrt{3}\,i}{2}\] is a root of equation \[{{x}^{4}}-{{x}^{3}}+x-1=0\] then its real roots are [EAMCET 2002]

If \[z+{{z}^{-1}}=1,\,\text{then }\,{{z}^{100}}+{{z}^{-100}}\] is equal to [UPSEAT 2001]

If \[z=\frac{\sqrt{3}+i}{-2}\], then \[{{z}^{69}}\] is equal to [RPET 2001]

If \[a=\sqrt{2i}\] then which of the following is correct [Roorkee 1989]

If \[1,\omega ,{{\omega }^{2}}\] are the cube roots of unity, then their product is [Karnataka CET 1999, 2001]

If cube root of 1 is \[\omega \], then the value of \[{{(3+\omega +3{{\omega }^{2}})}^{4}}\] is [MP PET 2001]

If \[\pi /3\] is a complex root of the equation \[{{z}^{3}}=1\], then \[\omega +{{\omega }^{\left( \frac{1}{2}\,+\,\frac{3}{8}\,+\,\frac{9}{32}\,+\,\frac{27}{128}\,+... \right)}}\] is equal to [Roorkee 2000; AMU 2005]

\[\frac{{{(-1+i\sqrt{3})}^{15}}}{{{(1-i)}^{20}}}+\frac{{{(-1-i\sqrt{3})}^{15}}}{{{(1+i)}^{20}}}\] is equal to [AMU 2000]

If \[\omega \] is an imaginary cube root of unity, \[{{(1+\omega -{{\omega }^{2}})}^{7}}\]equals [IIT 1998; MP PET 2000]

If \[\omega \] is an imaginary cube root of unity, then the value of \[\sin \,\left[ ({{\omega }^{10}}+{{\omega }^{23}})\,\pi -\frac{\pi }{4} \right]\] is [IIT Screening 1994]

The following in the form of \[A+iB\] \[{{(\cos 2\theta +i\sin 2\theta )}^{-5}}\] \[{{(\cos 3\theta -i\sin 3\theta )}^{6}}\]\[{{(\sin \theta -i\cos \theta )}^{3}}\] in the form of \[A+iB\] is [MNR 1991]

If \[\omega \] is the cube root of unity, then \[{{(3+5\omega +3{{\omega }^{2}})}^{2}}\] + \[{{(3+3\omega +5{{\omega }^{2}})}^{2}}\] = [MP PET 1999]

If \[\alpha \] and \[\beta \] are imaginary cube roots of unity, then the value of \[{{\alpha }^{4}}+{{\beta }^{28}}+\frac{1}{\alpha \beta }\],is [MP PET 1998]

\[{{\left( \frac{-1+i\sqrt{3}}{2} \right)}^{20}}+{{\left( \frac{-1-i\sqrt{3}}{2} \right)}^{20}}=\]

If \[\alpha \] is an imaginary cube root of unity, then for \[n\in N\], the value of \[{{\alpha }^{3n+1}}+{{\alpha }^{3n+3}}+{{\alpha }^{3n+5}}\] is [MP PET 1996; Pb. CET 2000]

If \[{{z}_{1}},{{z}_{2}}{{z}_{3}},{{z}_{4}}\]are the roots of the equation \[{{z}^{4}}=1\], then the value of \[\sum\limits_{i=1}^{4}{z_{i}^{3}}\]is [Kurukshetra CEE 1996]

The common roots of the equations \[{{x}^{12}}-1=0\], \[{{x}^{4}}+{{x}^{2}}+1=0\] are [EAMCET 1989]

If \[{{z}_{1}},{{z}_{2}},{{z}_{3}}......{{n}_{n}}\] are nth, roots of unity, then for \[k=1,\,2,.....,n\]

If \[\omega \] is an nth root of unity, other than unity, then the value of \[1+\omega +{{\omega }^{2}}+...+{{\omega }^{n-1}}\] is [Karnataka CET 1999]

If \[n\] is a positive integer greater than unity and \[z\] is a complex number satisfying the equation \[{{z}^{n}}={{(z+1)}^{n}}\], then