Geometrical Representation of Argument and Modulus

Q.

If ${\sin }^{-1}\left(x\right)=\theta +\beta$ and ${\sin }^{-1}\left(y\right)=\theta -\beta$, then $1+xy=$

1. ${\sin }^2\theta +{\sin }^2\beta$

2. ${\sin }^2\theta +{\cos }^2\beta$

3. ${\cos }^2\theta +{\cos }^2\beta$

4. ${\cos }^2\theta +{\sin }^2\beta$

Q.

If $\tan \theta =\frac{1}{\sqrt{3}}$, then find the value of $sec\theta .$

Q.

The period of the function $f\left(\theta \right)=4+4\sin 3\theta -3\sin \theta$ is:

1. $\frac{2\mathrm{\pi }}{3}$

2. $\frac{\mathrm{\pi }}{3}$

3. $\frac{\mathrm{\pi }}{2}$

4. $\mathrm{\pi }$

Q.

Let the function $f:(0,\pi )\to R$ be defined by $f\left(\theta \right)={(\sin \theta +\cos \theta )}^2+{(\sin \theta –\cos \theta )}^4$. Suppose, the function $f$ has a local minimum at $\theta$ precisely when $\theta \in \{{\lambda }_1\pi ,...,{\lambda }_r\pi \}$, where $0<{\lambda }_1<\cdots <{\lambda }_r<1$. Then the value of ${\lambda }_1+\cdots +{\lambda }_r$ is _____.

Q. Let $z$ be a complex number such that $|2z+\frac{1}{z}|=1$ and $\arg \left(z\right)=\theta$, then minimum value of $8{\sin }^2\theta$ is

1. $0$
2. $5$
3. $7$
4. $8$

Q. The imaginary part of $(z-1)(\cos \alpha -i\sin \alpha )+(z-1{)}^{-1}\times (\cos \alpha +i\sin \alpha )$ is zero, if

1. $|z-1|=1$
2. $\arg (z-1)=2\alpha$
3. $\arg (z-1)=\alpha$
4. $|z-1|=2$

Q. If ${}^{\prime }{z}^{\prime }$ lies on the circle $|z-2i|=2\sqrt{2}$, then the value of $\arg \left(\frac{z-2}{z+2}\right)$ is equal to

1. $\frac{\pi }{3}$
2. $\frac{\pi }{6}$
3. $\frac{\pi }{4}$
4. $\frac{\pi }{2}$

Q. If $z$ and $\omega$ are two complex numbers such that $|z\omega |=1$ and $\arg \left(z\right)-\arg \left(\omega \right)=\frac{3\pi }{2},$ then $\arg \left(\frac{1-2\overline{z}\omega }{1+3\overline{z}\omega }\right)$ is
(Here $\arg \left(z\right)$ denotes the principal argument of complex number $z$ )

Q. Polar form of $z=\frac{(1+7i)}{(2-i{)}^2}$ is

1. $\sqrt{2}(\cos \frac{3\pi }{4}+i\sin \frac{3\pi }{4})$
2. $2(\cos \frac{3\pi }{4}+i\sin \frac{3\pi }{4})$
3. $\sqrt{2}(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4})$
4. $2(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4})$

Q. $\sin 2(\alpha -\beta )+\sin 2(\beta -\gamma )+\sin 2(\gamma -\alpha )$ $=-4\sin (\alpha -\beta )\sin (\beta -\gamma )\sin (\gamma -\alpha )$.

Q. The continued product of the four values of ${[\cos \left(\frac{\pi }{3}\right)+i\sin \left(\frac{\pi }{3}\right)]}^{3/4}$ is

1. $-1$
2. $1$
3. $2$
4. $-2$

Q.

Prove the identity

$\sqrt{\frac{1+\sin \theta }{1-\sin \theta }}-\sqrt{\frac{1-\sin \theta }{1+\sin \theta }}=\frac{2}{cot\theta }$

Q. If $\left|\frac{z_1}{z_2}\right|=1$ and $\arg \left(z_1{z}_2\right)=0$, then

1. $z_1=z_2$
2. $|z_2{|}^2=z_1{z}_2$
3. $z_1{z}_2=1$
4. $z_1=2z_2$