CBSE Questions for Class 11 Commerce Applied Mathematics Number Theory Quiz 6 - MCQExams.com

Let $${ X }_{ n }=\left\{ z=x+iy:{ \left| z \right|  }^{ 2 } \le \dfrac { 1 }{ n }  \right\} $$ for all integers $$n\ge 1$$. Then, $$\displaystyle\bigcap _{ n=1 }^{ \infty  }{ { X }_{ n } } $$ is
  • A singleton set
  • Not a finite set
  • An empty set
  • A finite set with more than one element
If $$z_1, z_2$$ and $$z_3$$ are complex numbers such that $$|z_1| = |z_2| = |z_3| = \left | \displaystyle \frac{1}{z_1} + \frac{1}{z_2} + \frac{1}{z_3} \right | = 1,$$ then $$|z_1 + z_2 + z_3|$$ is
  • equal to 1
  • less than 1
  • greater than 3
  • equal to 3
The modulus of $$\dfrac { 1-i }{ 3+i } +\dfrac { 4i }{ 5 } $$ is
  • $$\sqrt { 5 } $$ unit
  • $$\dfrac { \sqrt { 11 } }{ 5 } $$ unit
  • $$\dfrac { \sqrt { 5 } }{ 5 } $$ unit
  • $$\dfrac { \sqrt { 12 } }{ 5 } $$ unit
Find the product. Write the answer in standard form.
$$i\left( 6-2i \right) \left( 7-5i \right) $$
  • $$52+16i$$
  • $$10{i}^{3}+44{i}^{2}+42i$$
  • $$-44-32i$$
  • $$44+32i$$
The number of composite number between $$101$$ and $$120$$ are 
  • $$11$$
  • $$12$$
  • $$13$$
  • $$None$$
The product of $$(3-2i)$$ and $$\left(\dfrac { 5 }{ 2 } -4i\right)$$, if $$i=\sqrt { -1 } $$ , is:
  • $$-\dfrac { 1 }{ 2 } -17i$$
  • $$14+\dfrac{9}{2}i$$
  • $$2-8i-14{i}^{2}$$
  • $$i\left(8+\dfrac{9}{2}\right)$$
The resultant complex number when $$(4+6i)$$ is divided by $$(10-5i)$$ is
  • $$\dfrac {2}{25} + \dfrac {16}{25}i$$
  • $$\dfrac {2}{25} - \dfrac {16}{25}i$$
  • $$\dfrac {2}{5} + \dfrac {6}{5}i$$
  • $$\dfrac {2}{5} - \dfrac {6}{5}i$$
If $$A = (3 - 4i)$$ and $$B = (9 + ki)$$, where $$k$$ is a constant. 
If $$AB - 15 = 60$$, then the value of $$k$$ is
  • $$6$$
  • $$24$$
  • $$12$$
  • $$3$$
The simplest form of $$\sqrt {-18} \times \sqrt {-50}$$ is
  • $$-30$$
  • $$-30i$$
  • $$30$$
  • $$30i$$
Express  $$\dfrac {(-5-i)(-7+8i)}{(2-4i)}$$  in the form of a complex number $$a+bi$$.
  • $$\dfrac{109}{10}-\dfrac{53}{10}i$$
  • $$-\dfrac{109}{10}-\dfrac{53}{10}i$$
  • $$\dfrac{109}{10}+\dfrac{53}{10}i$$
  • $$-\dfrac{109}{10}+\dfrac{53}{10}i$$
Simplify $$(2+8i)(1-4i)-(3-2i)(6+4i)$$
 (Note$$:i=\sqrt{-1}$$)
  • $$8$$
  • $$26$$
  • $$34$$
  • $$50$$
$$\displaystyle \left|\dfrac{\sqrt{3}+i}{(1+i)(1+\sqrt{3}i)}\right|=$$
  • $$1$$
  • $$\sqrt{2}$$
  • $$\dfrac{1}{2}$$
  • $$\dfrac{1}{\sqrt{2}}$$
What is the approximate magnitude of $$8 + 4i$$?
  • 4.15
  • 8.94
  • 12.00
  • 18.64
  • 32.00
The imaginary number $$i$$ is defined such that $$i^2=-1$$. What is the value of $$(1 - i \sqrt {5}) ( 1 + i\sqrt {5})$$?
  • $$\sqrt5$$
  • $$5$$
  • $$6$$
  • $$\sqrt6$$
How many of the prime factors of $$30$$ are greater than $$2$$? 
  • One
  • Two
  • Three
  • Four
  • Five
If $$i = \sqrt {-1}$$, find the values of $$n$$ such that $$i^{n} + (i)^{n}$$ have a positive value.
  • $$23$$
  • $$24$$
  • $$25$$
  • $$26$$
  • $$27$$
The value of $$(a+2i)(b-i)$$ is
  • $$a+b-i$$
  • $$ab+2$$
  • $$ab+(2b-a)i+2$$
  • $$ab-2$$
  • $$ab+(2b-a)i-2$$
Find $$(5 + 2i)(5 - 2i)$$
  • $$25 - 4i$$
  • $$25 - 20i$$
  • $$21$$
  • $$29$$
  • $$0$$
Find the number of prime numbers between 301 and 320?
  • 6
  • 5
  • 4
  • 3
If $$(2 - i)\times(a - bi) = 2 + 9i$$, where i is the imaginary unit and a and b are real numbers, then a equals
  • $$3$$
  • $$2$$
  • $$1$$
  • $$0$$
  • $$-1$$
Every composite number has _____________.
  • no prime divisor
  • one and only one prime divisor
  • atleast one prime divisor
  • atleast two prime divisor
The modulus of the complex quantity $$(2-3i)(-1+7i)$$.
  • $$5\sqrt{13}$$
  • $$5\sqrt{26}$$
  • $$13\sqrt{5}$$
  • $$26\sqrt{5}$$
The real part of $${ \left( 1-\cos { \theta  } +i\sin { \theta  }  \right)  }^{ -1 }$$ is
  • $$\cfrac{1}{2}$$
  • $$\cfrac { 1 }{ 1+\cos { \theta } } $$
  • $$\tan { \cfrac { \theta }{ 2 } } $$
  • $$\cot { \cfrac { \theta }{ 2 } } $$
If $$z = \dfrac {(\sqrt {3} + i)^{3} (3i + 4)^{2}}{(8 + 6i)^{2}}$$, then $$|z|$$ is equal to
  • $$0$$
  • $$1$$
  • $$2$$
  • $$3$$
Given : $$u = 1+i \sqrt{3}$$ and $$v = \sqrt{3} + i$$

Calculate $$\dfrac{u^3 }{ v^4}$$
  • $$(1/4) - i \sqrt{1/4}$$
  • $$(3/4) - i \sqrt{3}/4$$
  • $$(1/4) - i \sqrt{3}/4$$
  • none of these
The expression $$\dfrac{3-4i}{5+3i}$$ is equivalent to
  • $$\dfrac{27-29i}{34}$$
  • $$\dfrac{27-29i}{16}$$
  • $$\dfrac{3-29i}{34}$$
  • $$\dfrac{1}{8}$$
  • $$15-8i$$
The fraction $$\dfrac{1}{1+i}$$ is equivalent to
  • $$1-i$$
  • $$\dfrac{1+i}{2}$$
  • $$\dfrac{1-i}{2}$$
  • $$i$$
  • $$-i$$
$$p + iq = (2 - 3i) (4 + 2i)$$ then $$q$$ is
  • $$14$$
  • $$-14$$
  • $$-8$$
  • $$8$$
Perform the indicated operations:
$$(5+3i)(3-2i)$$
  • $$21-2i$$
  • $$19-3i$$
  • $$11-2i$$
  • $$21-i$$
If $${ x }^{ 2 }+{ y }^{ 2 }=1$$ then value of $$\dfrac { 1+x+iy }{ 1+x-iy } $$ is
  • $$x-iy$$
  • $$2x$$
  • $$-2iy$$
  • $$x+iy$$
If $$f\left( z \right) =\dfrac { 1-{ z }^{ 3 } }{ 1-z } $$, where $$z=x+iy$$ with $$z\neq 1$$, then $$Re\overline { \left\{ f\left( z \right)  \right\}  } =0$$ reduces to
  • $${ x }^{ 2 }+{ y }^{ 2 }+x+1=0$$
  • $${ x }^{ 2 }-{ y }^{ 2 }+x-1=0$$
  • $${ x }^{ 2 }-{ y }^{ 2 }-x+1=0$$
  • $${ x }^{ 2 }-{ y }^{ 2 }+x+1=0$$
  • $${ x }^{ 2 }-{ y }^{ 2 }+x+2=0$$
If $$x + i  y = \dfrac{3}{2 + cos  \theta + i  sin  \theta}$$, then $$x^2 + y^2$$ is equal to
  • $$3x - 4$$
  • $$4x - 3$$
  • $$4x +3$$
  • None of these
The modulus of $$\dfrac { \left( 3+2i \right) ^{ 2 } }{ \left( 4-3i \right)  } $$ is:
  • $$\frac { 13 }{ 5 } $$
  • $$\frac { 11 }{ 5 } $$
  • $$\frac { 9 }{ 5 } $$
  • $$\frac { 7 }{ 5 } $$
Let $$z = x + iy$$, where $$x$$ and $$y$$ are real. The points $$(x, y)$$ in the $$X-Y$$ plane for which $$\dfrac {z + i}{z - i}$$ is purely imaginary lie on
  • A straight line
  • An ellipse
  • A hyperbola
  • A circle
The complex number $$z$$ satisfying the equation $$|z - i| = |z + 1| = 1$$ is
  • $$0$$
  • $$1 + i$$
  • $$-1 + i$$
  • $$1 - i$$
The expression $$\dfrac {(1 + i)^{n}}{(1 - i)^{n - 2}}$$ equals.
  • $$-i^{n + 1}$$
  • $$i^{n + 1}$$
  • $$-2i^{n + 1}$$
  • $$1$$
If $$\left( \dfrac{1 + i}{1 - i} \right)^m = 1$$, then the least positive integral value of m is
  • 1
  • 4
  • 2
  • 3
Let Z and w be complex numbers. If $$Re(z)=|z-2|, Re(w) = |w-z|$$ and $$arg(z-w)=\dfrac{\pi}{3}$$, then the value of $$Im(z+w)$$, is
  • $$\dfrac{1}{\sqrt{3}}$$
  • $$\dfrac{2}{\sqrt{3}}$$
  • $$\sqrt{3}$$
  • $$\dfrac{4}{\sqrt{3}}$$
If $$z_1$$ and $$z_2$$ are complex numbers with $$|z_1|=|z_2|$$, then which of the following is/are correct?
1. $$z_1=z_2$$
2. Real part of $$z_1 =$$ Real part of $$z_2$$
3. Imaginary part of $$z_1 =$$ Imaginary part of $$z_2$$
Select the correct answer using the statements given below :
  • 1 only
  • 2 only
  • 3 only
  • None
If $$iz^{3} + z^{2} - z + i = 0$$, then $$|z|$$ is equal to
  • $$0$$
  • $$1$$
  • $$2$$
  • None of these
If '$$\omega$$' is a complex cube root of unity,then $$\omega ^{ \begin{pmatrix} \frac { 1 }{ 3 }  & +\frac { 2 }{ 9 } +\frac { 4 }{ 27 } ...\infty  \end{pmatrix} }+\omega^{ \begin{pmatrix} \frac { 1 }{ 2 }  & +\frac { 3 }{ 8 } +\frac { 9 }{ 32 } ...\infty  \end{pmatrix} }=$$
  • $$1$$
  • $$-1$$
  • $$\omega$$
  • $$i$$
The principal argument of the complex number $$z=\cfrac { 1+\sin { \cfrac { \pi  }{ 3 }  } +i\cos { \cfrac { \pi  }{ 3 }  }  }{ 1+\sin { \cfrac { \pi  }{ 3 }  } -i\cos { \cfrac { \pi  }{ 3 }  }  } $$ is?
  • $$\cfrac { \pi }{ 3 } $$
  • $$\cfrac { \pi }{ 6 } $$
  • $$\cfrac { 2\pi }{ 3 } $$
  • $$\cfrac { \pi }{ 2 } $$
  • $$\cfrac { \pi }{ 4 } $$
The value of $$ \sum _{ k=0 }^{ n }{ (i^k + i^{k+1} ) } , $$ where $$ i^2 = -1 ,$$ is equal to :
  • $$ i - i^n $$
  • $$ - i + i^{n+1} $$
  • $$ i - i^{n+1} $$
  • $$ i - i^{n+2} $$
  • $$ - i - i^{n} $$
If $$z_{1}$$ and $$z_{2}$$ be complex numbers such that $$z_{1} + i(\overline {z_{2}}) = 0$$ and $$arg (\overline {z_{1}}z_{2}) = \dfrac {\pi}{3}$$. Then, $$arg (\overline {z_{1}})$$ is equal to
  • $$\dfrac {\pi}{3}$$
  • $$\pi$$
  • $$\dfrac {\pi}{2}$$
  • $$\dfrac {5\pi}{12}$$
  • $$\dfrac {5\pi}{6}$$
The inequality $$\left| z-4 \right| <\left| z-2 \right| $$ represents the region given by:
  • $$Re(z)\ge 0$$
  • $$Re(z)<0$$
  • $$Re(z)>0$$
  • None of these
Let$$ z$$ = $$\cos\theta + i \sin\theta$$. Then the value of $$\sum\limits_{m=1}^15Im( z^{2m-1})$$ at $$\theta = 2^0$$ is 
  • $$\dfrac{1}{sin2^0}$$
  • $$\dfrac{1}{3sin2^0}$$
  • $$\dfrac{1}{2sin2^0}$$
  • $$\dfrac{1}{4sin2^0}$$
If the complex numbers $$z_1, z_2$$ and $$z_3$$ denote the vertices of an isosceles triangle, right angled at $$z_1$$, then $$(z_1 - z_2)^2 + (z_1 - z_3)^2$$ is equal to
  • $$0$$
  • $$(z_2 + z_3)^2$$
  • $$2$$
  • $$3$$
  • $$(z_2 - z_3)^2$$
If $$ z = \dfrac {-1}{2} + i \dfrac {\sqrt3}{2} $$, then $$ 8 + 10z + 7z^2 $$ is equal to :
  • $$ - \dfrac {1} {2} - i \dfrac {\sqrt3}{2} $$
  • $$ \dfrac {1} {2} + i \dfrac {\sqrt3}{2} $$
  • $$ - \dfrac {1} {2} + i \dfrac {3\sqrt3}{2} $$
  • $$ \dfrac {\sqrt3}{2} i $$
  • $$ - \dfrac {\sqrt3}{2} i $$
If $$z$$ is a complex number such that $$z + |z| = 8 + 12i$$, then the value of $$|z^{2}|$$ is
  • $$228$$
  • $$144$$
  • $$121$$
  • $$169$$
  • $$189$$
Let $$P(e^{i\theta_1})$$,  $$Q(e^{i\theta_2})$$  and  $$R(e^{i\theta_3})$$ be the vertices of a triangle PQR in the Argand Plane. The orthocenter of the triangle PQR is 
  • $$2e^(\theta_1+\theta_2+\theta_3)$$
  • $$\frac{2}{3}e^({\theta_1+\theta_2+\theta_3})$$
  • $$e^{\theta_1}$$+$$e^{\theta_2}$$+$$e^{\theta_3}$$
  • None of these
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