Logarithm of a Complex Number

Q.

If $3p^2=5p+2$ and $3q^2=5q+2$, where $p\ne q$, then the equation, whose roots are $3p-2q$ and $3q-2p$, is

1. $3x^2–5x–100=0$

2. $5x^2+3x+100=0$

3. $3x^2–5x+100=0$

4. $5x^2–3x–100=0$

Q. The value of ${\log }_2(-3)$ is

1. $\frac{\log 2-i\pi }{\log 3}$
2. $\frac{\log 3+2i\pi }{\log 2}$
3. $\frac{\log 3+i\pi }{\log 2}$
   
4. $\frac{\log 2+i\pi }{\log 3}$

Q.

Let $z$ be a complex number with modulus $2$ and argument $\frac{2\pi }{3}$ then $z$ is equal to:

1. $-1+i\sqrt{3}$

2. $\frac{-1+i\sqrt{3}}{2}$

3. $-1-i\sqrt{3}$

4. $\frac{-1-i\sqrt{3}}{2}$

Q.

If $a=i-k,b=xi+j+(1-x)k$ and $c=yi+xj+(1+x-y)k$. Then, $[abc]$ depends on?

1. neither $x$ nor $y$

2. both $x$ and $y$

3. only $x$

4. only $y$

Q.

Write the absolute value equation representing all numbers $\mathrm{x}$ whose distance from $4$ is $8$ units?

Q. If the sum of the roots of an equation is –2 and the sum of their cubes is 82, then the equation is

यदि एक समीकरण के मूलों का योगफल –2 है तथा इसके घनों का योगफल 82 है, तब समीकरण है

1. x² + 2x + 9 = 0
2. x² + 2x + 15 = 0
3. x² + 2x + 19 = 0
4. x² + 2x + 10 = 0

Q.

Find the general value of ${log}_e$(i).

1. $\frac{i\pi }{2}(2n+1)$ n $ϵ$ I

2. $\frac{i\pi }{2}(2n-1)$ n $ϵ$ I

3. $\frac{i\pi }{2}(4n+1)$ n $ϵ$ I

4. $\frac{i\pi }{2}(n+1)$ n $ϵ$ I

Q.

If z = $i^i$ . Express ${log}_e$z in A+iB form.Find the value of A and B.

1. A = $\frac{1}{e^{\frac{\pi }{2}}}$, B = $\frac{1}{e^{\frac{\pi }{2}}}$

2. A = 0 , B = $\frac{1}{e^{\frac{\pi }{2}}}$

3. A = $\frac{1}{e^{\frac{\pi }{2}}}$ , B = 0

4. A = $e^{\frac{\pi }{2}}$ , B = $e^{\frac{\pi }{2}}$

Q.

If $a,b,h,k$ are constants, while $U$ and $V$ are $U=\frac{X-a}{h},V=\frac{Y-b}{k}$, then

1. $Cov(X,Y)=Cov(U,V)$

2. $Cov(X,Y)=hkCov(U,V)$

3. $Cov(X,Y)=abCov(U,V)$

4. $Cov(U,V)=hkCov(X,Y)$

Q. If z₁ = 1 + 3i and z₂ = 2 + i, then $\mathrm{Re}\left(\frac{{\overline{z}}_2{z}_1}{z_2}\right)$ is

यदि z₁ = 1 + 3i तथा z₂ = 2 + i है, तब $\mathrm{Re}\left(\frac{{\overline{z}}_2{z}_1}{z_2}\right)$ का मान है

1. 3
2. –3
3. 2
4. –2

Q. Paragraph for below question
नीचे दिए गए प्रश्न के लिए अनुच्छेद

Let z = a + ib (a, b ∈ R) be a complex number. Modulus of z defined as $|z|=\sqrt{a^2+b^2}$ and argument of z is $\arg \left(z\right)={\tan }^{-1}\frac{b}{a}$.

माना z = a + ib (a, b ∈ R) एक सम्मिश्र संख्या है। z का मापांक $|z|=\sqrt{a^2+b^2}$ के रूप में परिभाषित है तथा z का कोणांक $\arg \left(z\right)={\tan }^{-1}\frac{b}{a}$ है।

Q. If z = –1 + 2i, then the principal argument of z is

प्रश्न - यदि z = –1 + 2i, तब z का मुख्य कोणांक है

1. –tan^–12
2. tan^–12
3. π – tan^–12
4. –π + tan^–12

Q. Paragraph for below question
नीचे दिए गए प्रश्न के लिए अनुच्छेद

Roots of x² + 6x + 12 = 0 are α and β, where α and β are complex numbers, then

समीकरण x² + 6x + 12 = 0 के मूल α व β हैं, जहाँ α व β सम्मिश्र संख्याएं हैं, तब

Q. α³ + β³ is

प्रश्न - α³ + β³ का मान है

1. 6
2. –12
3. 1
4. Zero
   शून्य

Q. Paragraph for below question
नीचे दिए गए प्रश्न के लिए अनुच्छेद

Given f(x) = ax² + bx + c, a, b, c ∈ R and a ≠ 0. α and β are roots of f(x) = 0.

दिया है f(x) = ax² + bx + c, a, b, c ∈ R तथा a ≠ 0 है। α तथा β, f(x) = 0 के मूल हैं।

Q. The value of f(α + β) is

प्रश्न - f(α + β) का मान है

1. a
2. b
3. c
4. $\frac{b^2}{a}$

Q.

Find the value of i.$i^i$ is _______.

1. i.$e^{\frac{\pi }{2}}$

2. i.$e^{-\frac{\pi }{2}}$

3. i.$e^{\frac{\pi }{4}}$

4. i.$e^{-\frac{\pi }{4}}$

Q.

What is the value of sin(${log}_e{i}^i)$ ?

1. 0

2. 1

3. -1

4. 2

Q. ${\log }_{\frac{\pi }{4}}(-1+\sqrt{3}i)$ can be expressed in cartesian form as

1. $\frac{\ln 2+\left(\frac{2\pi }{3}\right)i}{\ln \frac{\pi }{4}}$
2. $\frac{\ln 2-\left(\frac{2\pi }{3}\right)i}{\ln \frac{\pi }{4}}$
3. $\frac{\ln 2+\left(\frac{\pi }{3}\right)i}{\ln \frac{\pi }{2}}$
4. $\frac{\ln 2-\left(\frac{2\pi }{3}\right)i}{\ln \frac{\pi }{2}}$

Q.

Find the value of i.$i^i$ is _______.

1. i.$e^{\frac{\pi }{2}}$

2. i.$e^{-\frac{\pi }{2}}$

3. i.$e^{\frac{\pi }{4}}$

4. i.$e^{-\frac{\pi }{4}}$