Euler's Representation

Q.

If $co{t}^{-1}\left(\alpha \right)+co{t}^{-1}\left(\beta \right)=co{t}^{-1}\left(x\right)$, then $x$

1. $\alpha +\beta$

2. $\alpha –\beta$

3. $\frac{(1+\alpha \beta )}{(\alpha +\beta )}$

4. $\frac{(\alpha \beta –1)}{(\alpha +\beta )}$

Q.

Prove that cos 4x = 1 - 8 sin${}^{2}x{\text{cos}}^{2}x$.

Q. The value of $e^{2+i}$ is

1. $e[\cos 1+i\sin 1]$
2. $e^2[\cos 1+i\sin 1]$
3. $\cos 1+i\sin 1$
4. $e^{-2}[\cos 1+i\sin 1]$

Q.

The expression $tan(ilog(\frac{a-ib}{a+ib}))$ reduces to:

1. 

2. 

3. 

4. 

Q.

If sin $\theta =\frac{-4}{5}$ and $\pi <0<\frac{3\pi }{2}$, find the values of all the other five trigonometric functions.

Q. $cos\left(ilog\frac{a-ib}{a+ib}\right)$ is equal to

1. ab
2. 
3. 
4. 

Q.

$\theta$ ∈ [0, 2$\pi$] and $z_1,z_2,z_3$ are three complex numbers such that they are collinear and

$(1+|sin\theta |)z_1$ + $(|cos\theta |-1)z_2$ - $\sqrt{2}$$z_3$ =0. If at least one of the complex numbers $z_1,z_2,z_3$ is

non-zero then number of possible values of $\theta$ is

1. Infinite

2. 4

3. 2

4. 8

Q.

If a tan$\theta$= b then a cos 2$\theta$ + b sin 2$\theta$ is equal to

1. b

2. a

3. -a

4. -b

Q.

If ${\left(\frac{1+cos\theta +isin\theta }{1+cos\theta -isin\theta }\right)}^4$ = cosn$\theta$+isinn$\theta$ , then n is

equal to

1. 1

2. 3

3. 4

4. 2

Q.

The polar form of $\frac{-\sqrt{3}}{2}-\frac{i}{2}$ (where i = $\sqrt{-1}$) is

1. 

2. 

3. 

4. 

Q.

The Side of $△$ ABC are $AB=\sqrt{13}cm,BC=4\sqrt{3}cm,CA=7cm$, Then $sin\theta$,
Where $\theta$ is the smallest angle of the triangle, is equal to

1. 

2. 

3. None of these

4. 

Q.

If $\cos \theta =\frac{-5}{13}$ and $\frac{\pi }{2}<\theta <\pi$ then find the values of

(i) $\sin 3\theta +\sin 5\theta$

(ii) $\tan 3\theta$

Q.

Express $(1+i{)}^i$ in the A+iB form, Where i = $\sqrt{-1}$

1. 

2. 

3. 

4. 

Q.

if $\omega$ is complex number such that $|$ $\omega$ $|$ $\ne 1$ then the complex number z = $\omega$ + $\frac{1}{\omega }$ describes

1. parabola

2. straight line

3. ellipse

4. circle

Q.

If z1=a+ib and z1=c+id are complex numbers such that |$z_1$|=|$z_2$|=1 and R($z_1\overline{z_2}$) =0, then the pair of complex numbers w1=a+ic and w2=b+id satisfies

1. 

2. 

3. 

4. All the above

Q.

if z=r$e^{i\theta }$, then |$e^{iz}$|=

1. 

2. 

3. 

4. 

Q.

For $z=x+iy$, where $x,y\in R$ and $i=\sqrt{-1}$, what is the polar form of representation?

1. $\sqrt{x^2+y^2}[cos\theta -isin\theta ]$

2. $\sqrt{x^2-y^2}[cos\theta -isin\theta ]$

3. $\sqrt{x^2+y^2}[cos\theta +isin\theta ]$

4. $\sqrt{x^2–y^2}[cos\theta +isin\theta ]$

Q.

If $|u|<1,|v|<1$, and $z=\frac{u-v}{1+\overline{u}v}$, then least value of $|z|$ is

1. $\frac{|u|-|v|}{1+|u||v|}$

2. None of these

3. $\frac{|u|+|v|}{1+|u||v|}$

4. $\frac{||u|-|v||}{1-|u||v|}$

Q.

Evaluate $\underset{x\to \frac{\pi }{4}}{lim}\frac{(sinx-cosx)}{(x-\frac{\pi }{4})}$

Q. If $\tan (\alpha +i\beta )=e^{i\theta };$ where $\alpha ,\beta \in R$, $\theta \ne (2n+1)\frac{\pi }{2},n\in Z$ and $i=\sqrt{-1}$, then

1. $\alpha$ is an odd multiple of $\frac{\pi }{2}$
2. $\alpha$ is an even multiple of $\frac{\pi }{2}$
3. $\alpha$ is an odd multiple of $\frac{\pi }{4}$
4. $\alpha$ is an even multiple of $\frac{\pi }{4}$

Q. If $\cos A+\cos B=\cos C,\sin A+\sin B=\sin C$
then the value of expression $\frac{\sin (A+B)}{\sin 2C}$ is

Q.

If $\theta$ is in radian, area of the sector = $r^2\theta$

1. False

2. True

Q.

If a tan $\theta$ = b, then a cos 2$\theta$ + b sin 2$\theta$ =

[EAMCET 1981, 82; MP PET 1996; J & K 2005]

1. b

2. a

3. -a

4. -b

Q.

Express $(cos2\theta +isin2\theta {)}^{-5}$ $(cos3\theta -isin3\theta {)}^6$ $(sin\theta -icos\theta {)}^3$ in the form of A+iB is

1. 

2. 

3. 

4. 

Q.

The number of values of $\theta$ in the interval (- $\frac{\pi }{2}$, $\frac{\pi }{2}$) satisfying the eqautions

(1 - tan$\theta$)(1 + tan$\theta$)$se{c}^2$$\theta$ + $2^{ta{n}^2\theta }$ = 0

1. 4

2. 3

3. 5

4. 1

Q. The value of ${\sum }_{k=1}^6(sin\frac{2k\pi }{7}-icos\frac{2k\pi }{7})$ is

1. -i
2. i
3. 0
4. -1

Q. If $|z-1|=2,\arg (z-1)=\frac{\pi }{6}$, then $z$ is

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

Q.

If $\theta$ = $\frac{\pi }{3}$ + $\frac{\pi }{3^2}$ + $\frac{\pi }{3^3}$ + $\frac{\pi }{3^4}+\dots \dots -\infty$ , Find the vvalue of $e^{i\theta }$ .

___

Q. If $si{n}^2(\theta -\alpha )cos\alpha =co{s}^2(\theta -\alpha )sin\alpha =msin\alpha cos\alpha$ then

1. 
2. 
3. 
4. 

Q. If $z=i\log (2-\sqrt{3})$, then $\cos z=$

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