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

Let Xn={z=x+iy:|z|21n} for all integers n1. Then, n=1Xn 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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Practice Class 11 Commerce Applied Mathematics Quiz Questions and Answers