Purely Real

Q.

If x, y, z are real and distinct, then $x^2+4y^2+9z^2-6yz-3zx-2xy$ =

1. Non-negative

2. Zero

3. positive only

4. Non-positive

Q. If $z_1$ and $z_2$ are two complex numbers satisfying the equation $\left|\frac{z_1+z_2}{z_1-z_2}\right|=1,$ then $\frac{z_1}{z_2}$ is a number which is

1. Positive real

2. Negative real

3. Zero or purely imaginary
4. None of these

Q.

Convert the given complex number in polar form: $\sqrt{3}+i$

Q.

If a, b, c are distinct positive real numbers such that b(a+c) = 2ac then the roots a$x^2$ + 2bx + c = 0 are :

1. Imaginary

2. None of these

3. Real and equal

4. Real and distinct

Q. Let $z$ be a unimodular complex number having the argument $\theta$, $0<\theta <\frac{\pi }{2}$ and satisfying the relation $|z-3i|=3,$ then $\arg (\cot \theta -\frac{6}{z})$ is

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

Q.

Convert each of the complex numbers in he polar form:

$1-i$

Q. If z is a complex number such that $z^2=\left(\overline{z}{)}^2\right),$ then

1. z is purely real

2. z is purely imaginary

3. Either z is purely real or purely imaginary

4. None of these

Q.

If $\theta$ lies in the second quadrant, then the value of $\sqrt{\frac{1-sin\theta }{1+sin\theta }}$ + $\sqrt{\frac{1+sin\theta }{1-sin\theta }}$.

1. -2 sec

2. 2 sec

3. 2 cosec

4. -2 cesec

Q.

Let z be a complex number such that the imaginary part of z is non-zero and $a=z^2+z+1$ is real. Then, $a$ cannot take the value

1. $-1$

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

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

4. $\frac{3}{4}$

Q. If $z_1$ and $z_2$ are two complex numbers satisfying the equation $\left|\frac{z_1+z_2}{z_1-z_2}\right|=1,$ then $\frac{z_1}{z_2}$ is a number which is

1. Positive real

2. Negative real

3. None of these
4. Zero or purely imaginary

Q.

If $cos\theta +2cos\varphi +3cos\Psi =sin\theta +2sin\varphi +3\Psi =0,$ then $sin3\theta +8sin3\varphi +27sin3\psi =$

1. 

2. 

3. 

4. 

Q.

The expression $nsi{n}^2\theta +2ncos(\theta +\alpha )sin\alpha sin\theta +cos2(\alpha +\theta )$ is independent of $\theta$, the value of n is

1. 2

2. 3

3. 4

4. 5

Q. $\frac{3+2isin\theta }{1-2isin\theta }$ will be real, if $\theta$ =
[IIT 1976; EAMCET 2002]

1. $2n\pi$
2. $n\pi +\frac{\pi }{2}$
3. $n\pi$
4. None of these

Q. Let z be a complex number such that the imaginary part of z is non - zero and $a=z^2+z+1$ is real. Then, $a$ cannot take the value

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

Q.

If one root of bi-quadratic equation $x^4+2x^3-16x^2-22x+7=0is2+\sqrt{3}$. Find the other three roots.

1. 

2. 

3. 

4. 

Q.

If one root of $5x^2$ + 13x + k = 0 is reciprocal of the other, then k is equal to :

1. 

2. 

3. 

4. 

Q.

a + ib > c + id can be explained only when

1. b = 0, c = 0

2. b = 0, d = 0

3. a = 0, c = 0

4. a = 0, d = 0

Q. If $a$ and $b$ are positive integers such that $N=(a+ib{)}^3-107i$ is a positive integer, then the value of $N$ is

Q. Let z be a complex number such that the imaginary part of z is non - zero and $a=z^2+z+1$ is real. Then, $a$ cannot take the value

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

Q. If $\arg \left(z\right)=\frac{\pi }{3}$ and $\arg (z-1)=\frac{2\pi }{3},$ then $z$ is

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

Q. If a + ib = , prove that a 2 + b 2 =

Q.

For the equation ${(x+\frac{1}{x})}^2-5(x+\frac{1}{x})+6=0,$

1. has all roots irrational

2. two roots are rational, two roots are irrational

3. has all roots integers

4. two roots are real, two rootsare imaginary

Q.

If z is a complex number such that $z^2$ = (${\overline{z}}^2)$, then

1. z is purely real

2. z is purely imaginary

3. Either z is purely real or purely imaginary

4. None of these

Q. From the prices of shares $X$ and $Y$ below find out which is more stable in value :

X	35	54	52	53	56	58	52	50	51	49
Y	108	107	105	105	106	107	104	103	104	101

Q.

The positive integer value of n>3 satisfying the equation

$\frac{1}{sin\left(\frac{\pi }{n}\right)}=\frac{1}{sin\left(\frac{2\pi }{n}\right)}+\frac{1}{sin\left(\frac{3\pi }{n}\right)}$ is ___