CBSE Questions for Class 11 Commerce Applied Mathematics Number Theory Quiz 9 - MCQExams.com

What is the modulus of following complex number:$$-2+2\sqrt { 3i } $$
  • 4
  • 5
  • 2
  • 3
If $$ a+ib=\dfrac { c+1 }{ c-1 } $$, where $$c$$ is real number, then $$ { a }^{ 2 }+{ b }^{ 2 }=1$$ and $$ \dfrac { b }{ a } =\dfrac { 2c }{ { c }^{ 2 }-1 } $$
  • True
  • False
Which of the following is a pair of twin-prime number ? 
  • $$ 19 , 21 $$
  • $$ 43 , 47 $$
  • $$ 59 , 61 $$
  • $$ 73 , 79 $$
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
  • $$0$$
  • $$58$$
  • $$\cfrac{3364}{3}$$
  • $$3364$$
If $$z=\dfrac{\sqrt{3}}{2}+\dfrac{1}{2}i$$ then $$z\bar{z}$$ is
  • $$-1$$
  • $$0$$
  • $$1$$
  • $$2$$
Let $$A = \left\{ {z \in c:\left| z \right|} \right. = 2\left. 5 \right\}$$ and $$B = \left\{ {z \in c:\left| {z + 5 + 12i} \right|} \right. = 4$$. Then the minimum value of $$\left| {z - w} \right|$$ for $$Z \in A$$ and $$\omega  \in B$$ is :
  • 7
  • 8
  • 9
  • 6
Solve $$i^{57}+\dfrac{1}{i^{125}}$$
  • $$0$$
  • $$2i$$
  • $$-2i$$
  • $$2$$
The value of the sum $$\displaystyle\sum^{13}_{n=1}\left(i^n+i^{n+1}\right)$$, where $$i=\sqrt{-1}$$, is?
  • i
  • $$i-1$$
  • $$-i$$
  • $$0$$
If $${z_1}$$ and $${z_2}$$ be complex numbers such that $${z_1} \ne {z_2}$$ and $$\left| {{z_1}} \right| = \left| {{z_2}} \right|$$. If $${z_1}$$ has positive real part and $${z_2}$$ has negative imarinary part, then $$\frac{{\left( {{z_1} + {z_2}} \right)}}{{\left( {{z_1} - {z_2}} \right)}}$$ may be
  • Purely imaginary
  • Real and positive
  • Real and negative
  • zero
If $$z_{1} and z_{2} are on straight line$$ $$\left| \frac { 1 } { 2 } \left( z _ { 1 } + z _ { 2 } \right) + \sqrt { z _ { 1 } z _ { 2 } } \right| + \left| \frac { 1 } { 2 } \left( z _ { 1 } + z _ { 2 } \right) - \sqrt { z _ { 1 } z _ { 2 } } \right| =$$
  • $$\left| z _ { 1 } + z _ { 2 } \right|$$
  • $$\left| z _ { 1 } - z _ { 2 } \right|$$
  • $$\left| z _ { 1 } \right| + \left| z _ { 2 } \right|$$
  • $$\left| z _ { 1 } \right| - \left| z _ { 2 } \right|$$
If $$|z_{1}+z_{2}|=|z_{1}-z_{2}|$$, then the different in the amplitudes of $$z_{1}$$ and $$z_{2}$$ is
  • $$\dfrac {\pi}{4}$$
  • $$\dfrac {\pi}{3}$$
  • $$\dfrac {\pi}{2}$$
  • $$0$$
For any two complex numbers $$z_{1}$$ and $$z_{2}$$, then $$Re(z_{1}z_{2})=Rez_{1} Rez_{2}-\ln z_{1} \ln z_{2}$$
  • True
  • False
The value of $$2x^{4}+5x^{3}+7x^{2}-x+41$$, when $$x=-2-\sqrt{3i}$$ is:
  • -4
  • 4
  • -6
  • 6
If the six solutions of $$x^6 = -64$$ are written in the form $$a + bi$$, where $$a$$ and $$b$$ are real, then the product those solution with $$a < 0$$, is
  • $$4$$
  • $$8$$
  • $$16$$
  • $$64$$
If $$z_{1}$$ and $$z_{2}$$ two non-zero complex number such that $$|z_{1}+z_{2}|=|z_{1}|+|z_{2}|$$, then $$arg z_{1}-arg z_{2}$$ is equal to
  • $$-p$$
  • $$p/2$$
  • $$-p/2$$
  • $$0$$
If$${ z }_{ 1 }$$ and $${ z }_{ 2 }$$ are two complex number such that Im$$({ z }_{ 1 }={ z }_{ 2 })$$= 0= Im $$({ z }_{ 1 }{ z }_{ 2 })$$, then 
  • $${ z }_{ 1 }={ z }_{ 2 }$$
  • $${ z }_{ 1 }={ \overline { z } }_{ 2 }$$
  • $${ z }_{ 1 }={ -z }_{ 2 }$$
  • $${ z }_{ 1 }=-{ \overline { z } }_{ 2 }$$
$$z$$ is a complex number. If $$a = | x | + | y |$$ and
$$b = \sqrt { 2 } | x + i y |$$ then which of the following is
true
  • $$a \leq b$$
  • $$a > b$$
  • none of these
  • $$a - b + 2$$
If $$z=(3+4i)^6+(3-4i)^6,$$ where $$i=\sqrt { -1 }, $$ then $$Im(z)$$ equals to 
  • $$-6$$
  • $$0$$
  • $$6$$
  • None of these
Number of complex numbers $$z$$ such that $$|z|=1$$ and $$\left|\dfrac {z}{z}+\dfrac {\bar {z}}{z}\right|=1$$ is
  • $$4$$
  • $$1$$
  • $$8$$
  • $$more\ then\ 8$$
If a complex number z and $$z+\dfrac { 1 }{ z } $$ have same argument then- 
  • z must be purely real
  • z must be purely imaginary
  • z cannot be imaginary
  • z must be raal
Let P$$\left( x \right) ={ x }^{ 3 }-6{ x }^{ 2 }+Bx+C$$ has 1+5i as a zero and B,C real number, then value of (B+C) is
  • -70
  • 70
  • 24
  • 138
A value of $$\theta $$ for which$$\dfrac { 2+3isin\theta  }{ 1-2isin\theta  } $$ is purely imaginary, is:
  • $${ sin }^{ -1 }\left( \dfrac { 1 }{ \sqrt { 3 } } \right) $$
  • $$\dfrac { \pi }{ 3 } $$
  • $$cos^{-1}\sqrt-1$$
  • $$None of these$$
Let  $$\left| z _ { i } \right| = i , i = 1,2,3,4$$  and  $$\left| 16 z _ { 1 } z _ { 2 } z _ { 3 } + 9 z _ { 1 } z _ { 2 } z _ { 4 } + 4 z _ { 1 } z _ { 3 } z _ { 4 } + z _ { 2 } z _ { 3 } z _ { 4 } \right| = 48 ,$$  then the value of    $$\left| \dfrac { 1 }{ \overline { z } _{ { 1 } } } +\dfrac { 4 }{ \overline { z } _{ { 2 } } } +\dfrac { 9 }{ \overline { z } _{ { 3 } } } +\dfrac { 16 }{ \overline { z } _{ { 4 } } }  \right| .$$
  • $$1$$
  • $$2$$
  • $$4$$
  • $$8$$
If $$z$$ satisfies $$|z - 2+ 2i| \le 1$$
  • $$|z|_{least} = 2\sqrt{2} - 1$$
  • $$|z|_{least} = 2\sqrt{2} + 1$$
  • $$|z - 1| \le 1$$
  • None of these
If $$\dfrac { z+1 }{ z+i }$$ is purely imaginary, then z lies on a 
  • straight lone
  • circle
  • Circle with radius 1
  • circle passing through (1, 1).
Purely imaginary then find the sum of statement i $$a,b$$ 
  • $$\dfrac {5\pi}{6}$$
  • $$\pi$$
  • $$\dfrac {3\pi}{4}$$
  • $$\dfrac {2\pi}{3}$$
If $$\alpha$$ and $$\beta$$ are the roots of $${ 4x }^{ 2 }-16x+c=0,$$ c>0 such that $$1<\alpha <2<\beta <3$$, then the no.of integer values of c is 
  • $$17$$
  • $$14$$
  • $$18$$
  • $$15$$
$$\sqrt { \left( \log _ { 3 } \tan x \right) }$$  is real for:
  • $$n \pi + \pi / 4 \leq x < n \pi + \pi / 2$$
  • $$n \pi < x < n \pi + \pi / 2$$
  • $$n \pi \pm \pi / 4 \leq x < n \pi \pm \pi / 2$$
  • None of these.
All even numbers are prime numbers.
  • True
  • False
Which of the following is not a composite

number?
  • $$2 \times 3 \times 5\times 13\times 17 + 13$$
  • $$7\times 6\times 5\times 4\times 3\times 2\times 1 + 5$$
  • $$17\times 41\times 43\times 61 + 43$$
  • $$2 \times 3\times 43 + 13$$
If $$\left| {z - 3 + 2i} \right|\, \le 4$$ then the difference between the greatest value and the least value of $$\left| z \right|$$ is :
  • $$2\sqrt {13} $$
  • $$8$$
  • $$4 + \sqrt {13} $$
  • $$\sqrt {13} $$
Find the value of $${x}^{3}+7{x}^{2}-x+16$$, when $$x=1+2i$$
  • $$-11+24i$$
  • $$-17+24i$$
  • $$-17-24i$$
  • $$-1+24i$$
Let 'z' be a complex number satisfying $$|z-2-i|\le 5,$$ Then |z-14-6i| lies in 
  • {8,18}
  • {2,8}
  • {0,2}
  • {3,7}
If  $$w = \dfrac { z } { z - \dfrac { 1 } { 3 } i }$$  and  $$| w | = 1$$  then  $$z$$  lies on
  • a circle
  • an ellipse
  • a parabola
  • a straight line
The real part of  $$\left[ 1 + \cos \left( \dfrac { \pi } { 5 } \right) + i \sin \left( \dfrac { \pi } { 5 } \right) \right] ^ { - 1 }$$  is
  • $$1$$
  • $$\dfrac { 1 } { 2 }$$
  • $$\dfrac { 1 } { 2 } \cos \left( \dfrac { \pi } { 10 } \right)$$
  • $$\dfrac { 1 } { 2 } \cos \left( \dfrac { \pi } { 5 } \right)$$
$$\dfrac{1-2i}{2+i}+\dfrac{4-i}{3+2i}=$$
  • $$\dfrac{24}{13}+\dfrac{11}{13}i$$
  • $$\dfrac{24}{13}-\dfrac{11}{13}i$$
  • $$\dfrac{10}{13}+\dfrac{24}{13}i$$
  • $$\dfrac{10}{13}-\dfrac{24}{13}i$$
The principle amplitude of $$(\sin 40^{o}+i \cos 40^{o})^{5}$$ is
  • $$70^{o}$$
  • $$-1100^{o}$$
  • $$70^{110}$$
  • $$70^{-70}$$
If $$|z-3i|<\sqrt{5}$$, then $$|i(z+1)+1|<2\sqrt{5}$$.
  • True
  • False
IF $$z_1=1+i,z_2=1-i$$ find $$z_1z_2$$
  • $$z_1+z_2$$
  • $$z_1-z_2$$
  • $$z_1/z_2$$
  • None.
The value of the sum $$\sum _{ n=1 }^{ 13 }{ ({ i }^{ n }+{ i }^{ n+1 }) } $$ , where $$i=\sqrt { -1 } $$ , equals
  • $$i$$
  • $$i-1$$
  • $$-i$$
  • $$0$$
$$z_1$$ and $$z_2$$ are two non-zero complex numbers such that $$z_1=2+4i\\z_2=5-6i$$, then $$z_2-z_1$$ equals
  • $$3-10i$$
  • $$3+10i$$
  • $$7-2i$$
  • $$10-24i$$
The imaginary part of $$t ; t \in R$$ is 
  • $$0$$
  • $$1$$
  • $$2$$
  • $$-1$$
How many prime numbers are there in the following series.
$$1,2,7,9,13,15,21,23,27,29$$
  • 7
  • 6
  • 5
  • 4
The real value of $$'\theta '$$, for which the expression $$\frac { 1+i\cos { \theta  }  }{ 1-2i\cos { \theta  }  } $$ is a real number is
  • $$2n\pi +\frac { 3\pi }{ 2 } ,n\in I$$
  • $$2n\pi -\frac { 3\pi }{ 2 } ,n\in I$$
  • $$2n\pi \pm \frac { \pi }{ 2 } ,n\in I$$
  • $$2n\pi +\frac { \pi }{ 4 } ,n\in I$$
The greatest and least value of $$\left | z \right |$$ if $$z$$ satisfies $$\left | z - 5 + 5i \right |$$ $$\leq 5$$ are 
  • $$10$$ , $$5\sqrt{2}$$
  • $$5\sqrt{2}$$ , $$5$$
  • $$10$$ , $$0$$
  • $$5 + 5\sqrt{2}$$ , $$5\sqrt{2} - 5$$
let $$z=\left| \begin{matrix} 1 & 1+2i & -5i \\ 1-2i & -3 & 5+3i \\ 5i & 5-3i & 7 \end{matrix} \right|$$ then
  • z is purely real
  • z is purely imaginary
  • $$\left( z-\bar { z } \right) $$ i is real and imaginary both
  • $$\left( z+\bar { z } \right) =0$$
Which of the following is a prime number 
  • 391
  • 899
  • 621
  • 199
Given $$ z _ { 1 } + 3 z _ { 2 } - 4 z _ { 3 } = 0 $$ then $$ z _ { 1 } , z _ { 2 } , z _ { 3 } $$ are
  • collinear
  • can form sides of equilateral $$ \Delta $$
  • lie on circle
  • none of these
The imaginary roots of the equation $${ ({ x }^{ 2 }+2) }^{ 2 }+8{ x }^{ 2 }=6x({ x }^{ 2 }+2)$$ are ____________.
  • $$1+i$$
  • $$2\pm i$$
  • $$-1\pm i$$
  • $$none of these$$
If z be a complex number satisfying $$z^{4}+z^{3}+2z^{2}+z+1=0$$ then $$\left|z\right|=$$
  • $$\dfrac{1}{2}$$
  • $$\dfrac{3}{4}$$
  • $$1$$
  • none of these
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