Cube Roots and nth Roots of Unity

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JEE Main 2011

LEVELBoard

If $ω (\neq = 1)$ is a cube root of unity, and $(1 + ω)^{7} = A + B ω$ . Then $(A, B)$ equals

If $α$ and $β$ are the roots of the equation $x^{2} - x + 1 = 0$ , then $α^{2009} + β^{2009} =$

(A)

- 1

(B)

(C)

(D)

- 2

JEE Main 2006

LEVELJEE Main

The value of $\sum_{k = 1}^{10} (sin \frac{2 π k}{11} + i cos \frac{2 π k}{11})$ is

If $z^{2} + z + 1 = 0$ , where $z$ is complex number, then the value of $(z + 1/ z)^{2} + (z^{2} + 1/ z^{2})^{2} + (z^{3} + 1/ z^{3})^{2} + \dots + (z^{6} + 1/ z^{6})^{2}$ is

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

(A)

-1, -1 + 2\omega, -1 - 2\omega^2

(B)

-1, -1, -1

(C)

-1, 1 - 2\omega, 1 - 2\omega^2

(D)

-1, 1 + 2\omega, 1 + 2\omega^2

JEE Main 2003

LEVELJEE Main

If $1, ω, ω^{2}$ are the cube roots of unity, then $Δ = 1 ω^{n} ω^{2 n} ω^{n} ω^{2 n} 1 ω^{2 n} 1 ω^{n}$ is equal to

(A)

ω^{2}

(B)

0

(C)

1

(D)

ω

JEE Advanced 2010

LEVELJEE Main

Let $ω$ be a complex cube root of unity with $ω \neq = 1$ . A fair die is thrown three times. If $r_{1}, r_{2}$ and $r_{3}$ are the numbers obtained on the die, then the probability that $ω^{r_{1}} + ω^{r_{2}} + ω^{r_{3}} = 0$ is

$a, b, c$ are integers, not all simultaneously equal and $ω$ is cube root of unity ( $ω \neq = 1$ ), then minimum value of $∣ a + bω + c ω^{2} ∣$ is

(A)

(B)

(C)

3 /2

If $ω (\neq = 1)$ be a cube root of unity and $(1 + ω^{2})^{n} = (1 + ω^{4})^{n}$ , then the least positive value of $n$ is

Let $ω = - 1/2 + i 3 /2$ , then the value of the det. $111 1 - 1 - ω^{2} ω^{2} 1 ω^{2} ω^{4}$ is

(A)

3 ω

(B)

3 ω (ω - 1)

(C)

3 ω^{2}

(D)

3 ω (1 - ω)

JEE Advanced 2002

LEVELJEE Main

Let $ω = - \frac{1}{2} + i \frac{3}{2}$ . Then the value of the determinant $111 1 - 1 - ω^{2} ω^{2} 1 ω^{2} ω^{4}$ is

(A)

(a)

3 ω

(B)

(b)

3 ω (ω - 1)

(C)

(c)

3 ω^{2}

(D)

(d)

3 ω (1 - ω)

JEE Advanced 2001S

LEVELJEE Main

Let $z_{1}$ and $z_{2}$ be $n^{t h}$ roots of unity which subtend a right angle at the origin. Then $n$ must be of the form

(A)

4 k + 1

(B)

4 k + 2

(C)

4 k + 3

(D)

4 k

JEE Advanced 1999

LEVELJEE Main

If $i = - 1$ , then $4 + 5 (- \frac{1}{2} + \frac{i 3}{2})^{334} + 3 (- \frac{1}{2} + \frac{i 3}{2})^{365}$ is equal to

(A)

1 - i 3

(B)

- 1 + i 3

(C)

i 3

(D)

- i 3

JEE Advanced 1998

LEVELJEE Main

If $ω$ is an imaginary cube root of unity, then $(1 + ω - ω^{2})^{7}$ equals

(A)

128 ω

(B)

- 128 ω

(C)

128 ω^{2}

(D)

- 128 ω^{2}

JEE Advanced 1995S

LEVELBoard

If $ω (\neq = 1)$ is a cube root of unity and $(1 + ω)^{7} = A + B ω$ then $A$ and $B$ are respectively

If $ω$ is an imaginary cube root of unity then the value of $sin {(ω^{10} + ω^{23}) π - \frac{π}{4}}$ is

(A)

3 /2

(B)

- 1/ 2

(C)

1/ 2

(D)

3 /2

JEE Advanced 1987

LEVELJEE Main

The value of $\sum_{k = 1}^{6} (sin \frac{2 π k}{7} - i cos \frac{2 π k}{7})$ is

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

(A)

- 1, 1 + 2 ω, 1 + 2 ω^{2}

(B)

- 1, 1 - 2 ω, 1 - 2 ω^{2}

(C)

- 1, - 1, - 1

The value of the expression $1 \cdot (2 - ω) (2 - ω^{2}) + 2 \cdot (3 - ω) (3 - ω^{2}) + \dots + (n - 1) \cdot (n - ω) (n - ω^{2})$ , where $ω$ is an imaginary cube root of unity, is $\dots$

JEE Advanced 2002

LEVELJEE Main

Let a complex number $α, α \neq = 1$ , be a root of the equation $z^{p + q} - z^{p} - z^{q} + 1 = 0$ , where $p, q$ are distinct primes. Show that either $1 + α + α^{2} + \dots + α^{p - 1} = 0$ or $1 + α + α^{2} + \dots + α^{q - 1} = 0$ , but not both together.

JEE Advanced 1997

LEVELJEE Advanced

Prove that $\sum_{k = 1}^{n - 1} (n - k) cos \frac{2 k π}{n} = - n /2$ , where $n \geq 3$ is an integer.

JEE Advanced 1984

LEVELJEE Main

If $1, a_{1}, a_{2}, \dots, a_{n - 1}$ are the $n$ roots of unity, then show that $(1 - a_{1}) (1 - a_{2}) (1 - a_{3}) \dots (1 - a_{n - 1}) = n$ .

JEE Advanced 1978

LEVELBoard

If $x = a + b, y = aγ + b β$ and $z = a β + bγ$ where $γ$ and $β$ are the complex cube roots of unity, show that $x y z = a^{3} + b^{3}$ .

JEE Advanced 2014

LEVELJEE Advanced

Let $z_{k} = cos (\frac{2 k π}{10}) + i sin (\frac{2 k π}{10}); k = 1, 2, \dots, 9$ . Match the statements in List-I with those in List-II.

List-I

(P)

For each

z_{k}

there exists a

z_{j}

such that

z_{k} \cdot z_{j} = 1

(Q)

There exists a

k \in {1, 2, \dots, 9}

such that

z_{1} \cdot z = z_{k}

has no solution

z

in the set of complex numbers

(R)

\frac{∣1 - z _{1} ∣∣1 - z _{2} ∣ \dots ∣1 - z _{9} ∣}{10}

equals

(S)

1 - \sum_{k = 1}^{9} cos (\frac{2 k π}{10})

equals

List-II

Let $ω$ be a complex number such that $2 ω + 1 = z$ where $z = - 3$ . If $111 1 - ω^{2} - 1 ω^{2} 1 ω^{2} ω^{7} = 3 k$ , then k is equal to:

(A)

- z

(B)

z

(C)

- 1

(D)

1

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If ω(=1) is a cube root of unity, and (1+ω)7=A+Bω. Then (A,B) equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If α and β are the roots of the equation x2−x+1=0, then α2009+β2009=

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of ∑k=110​(sin112πk​+icos112πk​) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If z2+z+1=0, where z is complex number, then the value of (z+1/z)2+(z2+1/z2)2+(z3+1/z3)2+⋯+(z6+1/z6)2 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the cube roots of unity are 1,ω,ω2 then the roots of the equation (x−1)3+8=0 are

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If 1,ω,ω2 are the cube roots of unity, then Δ=​1ωnω2n​ωnω2n1​ω2n1ωn​​ is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a,b,c are integers, not all simultaneously equal and ω is cube root of unity (ω=1), then minimum value of ∣a+bω+cω2∣ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If ω(=1) be a cube root of unity and (1+ω2)n=(1+ω4)n, then the least positive value of n is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let ω=−1/2+i3​/2, then the value of the det. ​111​1−1−ω2ω2​1ω2ω4​​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let ω=−21​+i23​​. Then the value of the determinant ​111​1−1−ω2ω2​1ω2ω4​​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let z1​ and z2​ be nth roots of unity which subtend a right angle at the origin. Then n must be of the form

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If i=−1​, then 4+5(−21​+2i3​​)334+3(−21​+2i3​​)365 is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If ω is an imaginary cube root of unity, then (1+ω−ω2)7 equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If ω(=1) is a cube root of unity and (1+ω)7=A+Bω then A and B are respectively

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If ω is an imaginary cube root of unity then the value of sin{(ω10+ω23)π−4π​} is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of ∑k=16​(sin72πk​−icos72πk​) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the cube roots of unity are 1,ω,ω2, then the roots of the equation (x−1)3+8=0 are

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of the expression 1⋅(2−ω)(2−ω2)+2⋅(3−ω)(3−ω2)+⋯+(n−1)⋅(n−ω)(n−ω2), where ω is an imaginary cube root of unity, is …

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Prove that ∑k=1n−1​(n−k)cosn2kπ​=−n/2, where n≥3 is an integer.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If 1,a1​,a2​,…,an−1​ are the n roots of unity, then show that (1−a1​)(1−a2​)(1−a3​)…(1−an−1​)=n.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If x=a+b,y=aγ+bβ and z=aβ+bγ where γ and β are the complex cube roots of unity, show that xyz=a3+b3.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let zk​=cos(102kπ​)+isin(102kπ​);k=1,2,…,9. Match the statements in List-I with those in List-II.

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let ω be a complex number such that 2ω+1=z where z=−3​. If ​111​1−ω2−1ω2​1ω2ω7​​=3k, then k is equal to:

If $ω (\neq = 1)$ is a cube root of unity, and $(1 + ω)^{7} = A + B ω$ . Then $(A, B)$ equals

If $α$ and $β$ are the roots of the equation $x^{2} - x + 1 = 0$ , then $α^{2009} + β^{2009} =$

The value of $\sum_{k = 1}^{10} (sin \frac{2 π k}{11} + i cos \frac{2 π k}{11})$ is

If $z^{2} + z + 1 = 0$ , where $z$ is complex number, then the value of $(z + 1/ z)^{2} + (z^{2} + 1/ z^{2})^{2} + (z^{3} + 1/ z^{3})^{2} + \dots + (z^{6} + 1/ z^{6})^{2}$ is

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

If $1, ω, ω^{2}$ are the cube roots of unity, then $Δ = 1 ω^{n} ω^{2 n} ω^{n} ω^{2 n} 1 ω^{2 n} 1 ω^{n}$ is equal to

$a, b, c$ are integers, not all simultaneously equal and $ω$ is cube root of unity ( $ω \neq = 1$ ), then minimum value of $∣ a + bω + c ω^{2} ∣$ is

If $ω (\neq = 1)$ be a cube root of unity and $(1 + ω^{2})^{n} = (1 + ω^{4})^{n}$ , then the least positive value of $n$ is

Let $ω = - 1/2 + i 3 /2$ , then the value of the det. $111 1 - 1 - ω^{2} ω^{2} 1 ω^{2} ω^{4}$ is

Let $ω = - \frac{1}{2} + i \frac{3}{2}$ . Then the value of the determinant $111 1 - 1 - ω^{2} ω^{2} 1 ω^{2} ω^{4}$ is

Let $z_{1}$ and $z_{2}$ be $n^{t h}$ roots of unity which subtend a right angle at the origin. Then $n$ must be of the form

If $i = - 1$ , then $4 + 5 (- \frac{1}{2} + \frac{i 3}{2})^{334} + 3 (- \frac{1}{2} + \frac{i 3}{2})^{365}$ is equal to

If $ω$ is an imaginary cube root of unity, then $(1 + ω - ω^{2})^{7}$ equals

If $ω (\neq = 1)$ is a cube root of unity and $(1 + ω)^{7} = A + B ω$ then $A$ and $B$ are respectively

If $ω$ is an imaginary cube root of unity then the value of $sin {(ω^{10} + ω^{23}) π - \frac{π}{4}}$ is

The value of $\sum_{k = 1}^{6} (sin \frac{2 π k}{7} - i cos \frac{2 π k}{7})$ is

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

The value of the expression $1 \cdot (2 - ω) (2 - ω^{2}) + 2 \cdot (3 - ω) (3 - ω^{2}) + \dots + (n - 1) \cdot (n - ω) (n - ω^{2})$ , where $ω$ is an imaginary cube root of unity, is $\dots$

Prove that $\sum_{k = 1}^{n - 1} (n - k) cos \frac{2 k π}{n} = - n /2$ , where $n \geq 3$ is an integer.

If $1, a_{1}, a_{2}, \dots, a_{n - 1}$ are the $n$ roots of unity, then show that $(1 - a_{1}) (1 - a_{2}) (1 - a_{3}) \dots (1 - a_{n - 1}) = n$ .

If $x = a + b, y = aγ + b β$ and $z = a β + bγ$ where $γ$ and $β$ are the complex cube roots of unity, show that $x y z = a^{3} + b^{3}$ .

Let $z_{k} = cos (\frac{2 k π}{10}) + i sin (\frac{2 k π}{10}); k = 1, 2, \dots, 9$ . Match the statements in List-I with those in List-II.

Let $ω$ be a complex number such that $2 ω + 1 = z$ where $z = - 3$ . If $111 1 - ω^{2} - 1 ω^{2} 1 ω^{2} ω^{7} = 3 k$ , then k is equal to: