Complex Numbers Questions

MathematicsComplex Numbers105 Questions Found

MathematicsCube Roots and nth Roots of UnityJEE Advanced 1984Moderate

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

MathematicsGeometrical Applications of Complex NumbersJEE Advanced 1986Moderate

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Show that the area of the triangle on the Argand diagram formed by the complex numbers $z, i z$ and $z + i z$ is $\frac{1}{2} ∣ z ∣^{2}$ .

MathematicsGeometrical Applications of Complex NumbersJEE Advanced 1990Difficult

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Let $z_{1} = 10 + 6 i$ and $z_{2} = 4 + 6 i$ . If $z$ is any complex number such that the argument of $\frac{z - z _{1}}{z - z _{2}}$ is $\frac{π}{4}$ , then prove that $∣ z - 7 - 9 i ∣ = 32$ .

MathematicsGeometrical Applications of Complex NumbersJEE Advanced 1995Moderate

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If $∣ Z ∣ \leq 1, ∣ W ∣ \leq 1$ , show that $∣ Z - W ∣^{2} \leq (∣ Z ∣ - ∣ W ∣)^{2} + (A r g Z - A r g W)^{2}$ .

MathematicsAlgebraic Operations on Complex NumbersJEE Advanced 1995Easy

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If $i z^{3} + z^{2} - z + i = 0$ , then show that $∣ z ∣ = 1$ .

MathematicsGeometrical Applications of Complex NumbersJEE Advanced 1997Moderate

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Let $z_{1}$ and $z_{2}$ be roots of the equation $z^{2} + p z + q = 0$ , where the coefficients $p$ and $q$ may be complex numbers. Let $A$ and $B$ represent $z_{1}$ and $z_{2}$ in the complex plane. If $∠ A O B = α \neq = 0$ and $O A = O B$ , where $O$ is the origin, prove that $p^{2} = 4 q cos^{2} (α /2)$ .

MathematicsCube Roots and nth Roots of UnityJEE Advanced 1997Moderate

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

MathematicsAlgebraic Operations on Complex NumbersJEE Advanced 1999Moderate

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For complex numbers $z$ and $w$ , prove that $∣ z ∣^{2} w - ∣ w ∣^{2} z = z - w$ if and only if $z = w$ or $z \overset{w}{ˉ} = 1$ .

MathematicsCube Roots and nth Roots of UnityJEE Advanced 2002Moderate

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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.

MathematicsGeometrical Applications of Complex NumbersJEE Advanced 2003Moderate

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Prove that there exists no complex number $z$ such that $∣ z ∣ < 1/3$ and $\sum_{r = 1}^{n} a_{r} z^{r} = 1$ where $∣ a_{r} ∣ < 2$ .

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.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; } Show that the area of the triangle on the Argand diagram formed by the complex numbers z,iz and z+iz is 21​∣z∣2.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let z1​=10+6i and z2​=4+6i. If z is any complex number such that the argument of z−z2​z−z1​​ is 4π​, then prove that ∣z−7−9i∣=32​.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If ∣Z∣≤1,∣W∣≤1, show that ∣Z−W∣2≤(∣Z∣−∣W∣)2+(ArgZ−ArgW)2.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If iz3+z2−z+i=0, then show that ∣z∣=1.

.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; } For complex numbers z and w, prove that ∣z∣2w−∣w∣2z=z−w if and only if z=w or zwˉ=1.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Prove that there exists no complex number z such that ∣z∣<1/3 and ∑r=1n​ar​zr=1 where ∣ar​∣<2.

.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; } Show that the area of the triangle on the Argand diagram formed by the complex numbers z,iz and z+iz is 21​∣z∣2.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let z1​=10+6i and z2​=4+6i. If z is any complex number such that the argument of z−z2​z−z1​​ is 4π​, then prove that ∣z−7−9i∣=32​.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If ∣Z∣≤1,∣W∣≤1, show that ∣Z−W∣2≤(∣Z∣−∣W∣)2+(ArgZ−ArgW)2.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If iz3+z2−z+i=0, then show that ∣z∣=1.

.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; } For complex numbers z and w, prove that ∣z∣2w−∣w∣2z=z−w if and only if z=w or zwˉ=1.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Prove that there exists no complex number z such that ∣z∣<1/3 and ∑r=1n​ar​zr=1 where ∣ar​∣<2.

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

Show that the area of the triangle on the Argand diagram formed by the complex numbers $z, i z$ and $z + i z$ is $\frac{1}{2} ∣ z ∣^{2}$ .

Let $z_{1} = 10 + 6 i$ and $z_{2} = 4 + 6 i$ . If $z$ is any complex number such that the argument of $\frac{z - z _{1}}{z - z _{2}}$ is $\frac{π}{4}$ , then prove that $∣ z - 7 - 9 i ∣ = 32$ .

If $∣ Z ∣ \leq 1, ∣ W ∣ \leq 1$ , show that $∣ Z - W ∣^{2} \leq (∣ Z ∣ - ∣ W ∣)^{2} + (A r g Z - A r g W)^{2}$ .

If $i z^{3} + z^{2} - z + i = 0$ , then show that $∣ z ∣ = 1$ .

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

For complex numbers $z$ and $w$ , prove that $∣ z ∣^{2} w - ∣ w ∣^{2} z = z - w$ if and only if $z = w$ or $z \overset{w}{ˉ} = 1$ .

Prove that there exists no complex number $z$ such that $∣ z ∣ < 1/3$ and $\sum_{r = 1}^{n} a_{r} z^{r} = 1$ where $∣ a_{r} ∣ < 2$ .

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

Show that the area of the triangle on the Argand diagram formed by the complex numbers $z, i z$ and $z + i z$ is $\frac{1}{2} ∣ z ∣^{2}$ .

Let $z_{1} = 10 + 6 i$ and $z_{2} = 4 + 6 i$ . If $z$ is any complex number such that the argument of $\frac{z - z _{1}}{z - z _{2}}$ is $\frac{π}{4}$ , then prove that $∣ z - 7 - 9 i ∣ = 32$ .

If $∣ Z ∣ \leq 1, ∣ W ∣ \leq 1$ , show that $∣ Z - W ∣^{2} \leq (∣ Z ∣ - ∣ W ∣)^{2} + (A r g Z - A r g W)^{2}$ .

If $i z^{3} + z^{2} - z + i = 0$ , then show that $∣ z ∣ = 1$ .

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

For complex numbers $z$ and $w$ , prove that $∣ z ∣^{2} w - ∣ w ∣^{2} z = z - w$ if and only if $z = w$ or $z \overset{w}{ˉ} = 1$ .

Prove that there exists no complex number $z$ such that $∣ z ∣ < 1/3$ and $\sum_{r = 1}^{n} a_{r} z^{r} = 1$ where $∣ a_{r} ∣ < 2$ .