Number Theory Test 39

Result

Number Theory Test 39

Score
-
out of -
Rank
-
out of -

TIME Taken - -

SHARING IS CARING

If our Website helped you a little, then kindly spread our voice using Social Networks. Spread our word to your readers, friends, teachers, students & all those close ones who deserve to know what you know now.

Weekly Quiz Competition

Question 1
1 / -0

Choose the composite numbers from the following numbers $$87, 67, 45, 34, 23, 27, 33$$.

A
$$45, 87, 34, 27, 33$$

B
$$45, 87, 67, 33$$

C
$$33, 27, 23, 34$$

D
All the above

E
None of these

Solution

$$87, 45, 34, 33, 27$$ are composite numbers
[A composite number define as except own and one which are divisible by any numbers such that (4,6,8,9,....)]
Question 2
1 / -0

If $$z$$ is a complex number, then $$z^{2}+\bar{z}^{2}=2$$ represents-

A
a circle

B
a straight line

C
a hyperbola

D
an ellipse

Solution

Let $$z=x+iy$$
$$\bar {z} =x-iy$$
$$\therefore z^{2}+{\bar {z}}^{2}={ \left( x+iy \right) }^{ 2 }+{ \left( x-iy \right) }^{ 2 }$$
$$=x^2+2xyi-y^2+x^2-2xyi-y^2=2$$
$$\therefore 2\left( { x }^{ 2 }-{ y }^{ 2 } \right) =2$$
$$\therefore \boxed {x^2-y^2=1}$$
Equation of hyperbola.
Question 3
1 / -0

The modulus of the complex number $$z=\frac { \left( 1-i\sqrt { 3 } \right) \left( \cos { \theta } +isin\theta \right) }{ 2\left( 1-i \right) \left( \cos { \theta } -isin\theta \right) } $$ is-

A
$$\frac { 1 }{ 2\sqrt { 2 } } $$

B
$$\frac { 1 }{ \sqrt { 3 } } $$

C
$$\frac { 1 }{ \sqrt { 2 } } $$

D
$$\frac { 1 }{ 2\sqrt { 3 } } $$

Solution
Question 4
1 / -0

If $$x^{2}+y^{2}=1$$ and $$x \neq -1$$ then $$\dfrac {1+y+ix}{1+y-ix}$$

A
$$1$$

B
$$y+ix$$

C
$$2$$

D
$$x+ix$$

Solution
Question 5
1 / -0

If $$\left|z\right|=1$$ and $$\varpi=\dfrac{z-1}{z+1}$$, where $$z\neq-1$$, then $$Re\left(\varpi\right)$$ is

A
$$0$$

B
$$-\dfrac{1}{|z+1|^{2}}$$

C
$$\dfrac{1}{\sqrt{2}}{\left|z+1\right|^{2}}$$

D
$$\dfrac{\sqrt{2}}{|z+1|^{2}}$$

Solution
Question 6
1 / -0

What is the modulus of following complex number:$$-2+2\sqrt { 3i } $$

A
4

B
5

C
2

D
3

Solution
Question 7
1 / -0

If $$z=(3+7i)(p+iq)$$ where $$p,q\in I-\left\{ 0 \right\} $$, is purely imaginary then minimum value of $${ \left| z \right| }^{ 2 }$$ is

A
$$0$$

B
$$58$$

C
$$\dfrac{3364}{3}$$

D
$$3364$$

Solution
Question 8
1 / -0

If $$z=(3+7i)(p+iq)$$, where $$p,q\in I-\left\{ 0 \right\} $$, is a purely imaginary, then minimum value of $${ \left| z \right| }^{ 2 }$$ is

A
$$0$$

B
$$58$$

C
$$\cfrac{3364}{3}$$

D
$$3364$$

Solution

$$z=(3+7i)(p+iq) \quad \dots (1)$$

$$z=(3p-7q)+i(3q+7p)$$

Now, $$z$$ is purely imaginary

$$3p-7q=0$$

or $$\cfrac{p}{q}=\cfrac{7}{3}$$

$$\cfrac{p}{q}+i=\cfrac{7}{3}+i$$

Therefore, $$\cfrac{(p+qi)}{q}=\cfrac{7+3i}{3}$$

$$p+iq=7+3i \quad \dots (2)$$

Substitute $$(2)$$ in $$(1)$$

$$z=21+9i+49i-21$$

Thus, $$z=58i$$

$$\left| { z }^{ 2 } \right| =3364$$
Question 9
1 / -0

$$z=a+ib$$, $$a,b,\in R$$, $$b\ne 0$$ and $$\left| z \right| =1$$, then $$z=\cfrac{c+i}{c-i}$$, where $$c$$ is equal to

A
$$\cfrac{a}{b}$$

B
$$\cfrac{a-1}{b}$$

C
$$\cfrac{a+1}{b}$$

D
$$\cfrac{a+1}{b+1}$$

Solution
Question 10
1 / -0

If $$z=\dfrac{\sqrt{3}}{2}+\dfrac{1}{2}i$$ then $$z\bar{z}$$ is

A
$$-1$$

B
$$0$$

C
$$1$$

D
$$2$$

Solution

Conjugate of $$\dfrac{\sqrt{3}+i}{2}=\dfrac{\sqrt{3}-i}{2}$$
$$\therefore\,\bar{z}=\dfrac{\sqrt{3}-i}{2}$$
Now,$$z\bar{z}=\left(\dfrac{\sqrt{3}+i}{2}\right)\left(\dfrac{\sqrt{3}-i}{2}\right)=\dfrac{3-{i}^{2}}{4}=\dfrac{3+1}{4}=\dfrac{4}{4}=1$$