CBSE Questions for Class 11 Commerce Applied Mathematics Number Theory Quiz 5 - MCQExams.com

If $$\displaystyle\ \alpha,\beta$$ be two complex numbers then $$\displaystyle\ |\alpha|^{2}+|\beta|^{2}$$ is equal to
  • $$\displaystyle\ \frac{1}{2}(|\alpha+\beta|^{2}-|\alpha-\beta|^{2})$$
  • $$\displaystyle\ \frac{1}{2}(|\alpha+\beta|^{2}+|\alpha-\beta|^{2})$$
  • $$\displaystyle\ |\alpha+\beta|^{2}+|\alpha-\beta|^{2}$$
  • None of these
For $${ { Z }_{ 1 }=\sqrt [ 6 ]{ \dfrac { 1-i }{ 1+i\sqrt { 3 }  }  }  };\quad { { Z }_{ 2 }=\sqrt [ 6 ]{ \dfrac { 1-i }{ \sqrt { 3 } +i }  } ;\quad { { Z }_{ 3 }=\sqrt [ 6 ]{ \dfrac { 1-i }{ \sqrt { 3 } -i }  }  } }$$ which of the following holds good?
  • $$\sum { { \left| { Z }_{ 1 } \right| }^{ 2 } } =\dfrac { 3 }{ 2 } $$
  • $${ \left| { Z }_{ 1 } \right| }^{ 4 }+{ \left| { Z }_{ 2 } \right| }^{ 4 }={ \left| { Z }_{ 3 } \right| }^{ -8 }$$
  • $$\sum { { \left| { Z }_{ 1 } \right| }^{ 3 }+ } { \left| { Z }_{ 2 } \right| }^{ 3 }={ \left| { Z }_{ 3 } \right| }^{ -6 }$$
  • $${ \left| { Z }_{ 1 } \right| }^{ 4 }+{ \left| { Z }_{ 2 } \right| }^{ 4 }={ \left| { Z }_{ 3 } \right| }^{ 8 }$$
if $$\displaystyle\ z=1+i\ \tan \alpha $$, where $$\displaystyle\ \pi < \alpha < \frac{3\pi }{2}$$ is $$|z|$$ is equal to 
  • $$\displaystyle\ \sec \alpha$$
  • $$\displaystyle\ -\sec \alpha $$
  • $$\displaystyle\ cosec\alpha $$
  • none of these
If $$\displaystyle u_{i}=1-\frac{1}{i}$$ then $$\displaystyle u_{2}\cdot u_{3}\cdot ... \cdot u_{n}$$ is equal to 
  • $$\displaystyle \frac{1}{n}$$
  • $$\displaystyle \frac{1}{n!}$$
  • $$1$$
  • none of these
If $$\displaystyle\ z=x-iy\displaystyle\ $$ such that $$\displaystyle\ |z+1|=|z-1|$$  and amp $$\displaystyle\ \frac{z-1}{z+1}= \frac{\pi}{4}$$ then   
  • $$\displaystyle\ x=\sqrt{2}+1, y=0$$
  • $$\displaystyle\ x=0, y=\sqrt{2}+1$$
  • $$\displaystyle\ x=0, y=\sqrt{2}-1 $$
  • $$\displaystyle\ x=\sqrt{2}-1, y=0$$
The complex number which satisfies the equation $$z+\sqrt { 2 } \left| z+1 \right| +i=0$$ is
  • $$2-i$$
  • $$-2-i$$
  • $$2+i$$
  • $$-2+i$$
If $$z$$ be a complex number, then the minimum value of $$\left | z-7 \right |+\left | z \right |$$ is
  • $$\displaystyle \frac{\sqrt{3}-1}{2}$$
  • $$-\sqrt{7}$$
  • $$\displaystyle \frac{7+\sqrt{2}}{2}$$
  • $$7$$
If $$\displaystyle z= 1+i\sqrt{3}$$,then $$\displaystyle z^{6}$$ equals
  • 32
  • -32
  • 64
  • None of these
Write all the composite numbers between $$10$$ and $$35.$$
  • $$7, 12, 14, 15, 16, 17, 20, 21, 22, 24, 25, 26, 27, 28, 30,33$$ and $$34$$
  • $$7, 12, 14, 15, 16, 18, 20, 21, 21, 24, 25, 26, 27, 28, 32, 33$$ and $$34$$
  • $$7, 12, 14, 15, 16, 19, 20, 21, 22, 24, 25, 26, 27, 28, 32, 33$$ and $$34$$
  • None of these
The locus of $$\displaystyle z= x+iy$$ which satisfying the inequality $$\displaystyle \log_{1/2}\left | z-1 \right |> \log_{1/2}\left | z-i \right |$$ is given by
  • $$\displaystyle x+y< 0$$
  • $$\displaystyle x-y> 0$$
  • $$\displaystyle x-y< 0$$
  • $$\displaystyle x+y> 0$$
The complex number $$z$$ satisfying the equations $$\left| z \right| -4=\left| z-i \right| -\left| z+5i \right| =0$$, is
  • $$\sqrt { 3 } -i$$
  • $$2\sqrt { 3 } -2i$$
  • $$-2\sqrt { 3 } -2i$$
  • $$0$$
Solve : $$(2-\sqrt{-100})(1+\sqrt{-36})$$
  • $$62+2i$$
  • $$52+2i$$
  • $$-52+2i$$
  • $$-88+2i$$
If $$\displaystyle z=1+i\cot\alpha,-\frac{\pi}{2}<\alpha<0,$$ then $$|z|$$ is equal to 
  • $$cosec\alpha$$
  • $$-cosec\alpha$$
  • $$cosec\alpha$$ or $$-cosec\alpha$$
  • none of these
If $$z=4+i\sqrt7$$, then value of $$z^3-4z^2-9z+91$$ equals
  • $$0$$
  • $$1$$
  • $$-1$$
  • $$2$$
$$\displaystyle \left [ \left ( \cos \theta +i \sin \theta \right )\left ( \cos \theta -i\sin \theta  \right ) \right ]^{-1}$$
  • $$\displaystyle i$$
  • $$\displaystyle 1$$
  • $$\displaystyle -i$$
  • $$\displaystyle -1$$
If $$\left| z \right| \ge 5$$ then the least value $$\left| {z + \frac{2}{z}} \right|$$ is 
  • $$\frac{{23}}{5}$$
  • $$\frac{{24}}{5}$$
  • $$5$$
  • none of these
For a complex number $$z$$, the minimum value of $$\left | z \right |+\left | z-\cos\alpha-i\sin\alpha \right |$$ is
  • 0
  • 1
  • 2
  • None of these
Find the modulus and amplitude of $$-2 + 2 \sqrt 3i$$
  • $$|z|=2\sqrt [ 2 ]{ 2 } ; amp(z)=\dfrac {\pi}{3}$$
  • $$|z|=2\sqrt [ 2 ]{ 2 } ; amp(z)=\dfrac {2\pi}{3}$$
  • $$|z|=4; amp(z)=\dfrac {2\pi}{3}$$
  • $$|z|=4; amp(z)=\dfrac {\pi}{3}$$
The inequality $$|z-4| < |z-2|$$ represents the region given by 
  • $$Re(z) > 1$$
  • $$Re(z) < 2$$
  • $$Re(z) > 0$$
  • None of these
Solve:
$$\displaystyle \left ( x+iy \right )\left ( 2-3i \right )= 4+i$$
  • $$\displaystyle x= \left ( 8/13 \right ), y= -\left ( 14/13 \right ).$$
  • $$\displaystyle x= \left ( 5/13 \right ), y= \left ( 14/13 \right ).$$
  • $$\displaystyle x= -\left ( 14/13 \right ), y= \left ( 5/13 \right ).$$
  • $$\displaystyle x= \left ( 14/13 \right ), y=- \left ( 8/13 \right ).$$
Find the value of $$x^3 + 7x^2 -x + 16$$, where $$x = 1 + 2i$$
  • $$-17 + 24i$$
  • $$24 + 17i$$
  • $$17 - 24i$$
  • $$24 - 17i$$
Consider the following statements:
A. The sum of two prime numbers is a prime number. 
B. The product of two prime numbers is a prime number. 
Which of these statements is/are correct ?
  • Neither A nor B
  • A alone
  • B alone
  • Both A and B
Which one of the following is the largest prime number of three digits?
  • $$997$$
  • $$999$$
  • $$991$$
  • $$993$$
If $$z = re^{i\theta}$$, then the value of $$|e^{iz}|$$ is equal to
  • $$e^{rcos\theta}$$
  • $$e^{-rcos\theta}$$
  • $$e^{rsin\theta}$$
  • $$e^{-rsin\theta}$$
Find the modulus and amplitude of $$-2i$$
  • $$|z|=2; amp(z)=-\dfrac {3\pi}{2}$$
  • $$|z|=2i; amp(z)=\dfrac {\pi}{2}$$
  • $$|z|=2; amp(z)=\dfrac {\pi}{2}$$
  • $$|z|=2; amp(z)=-\dfrac {\pi}{2}$$
A student was asked to find the sum of all the prime numbers between $$10$$ and $$40.$$He found the sum as $$180.$$Which of the following statements is true?
  • He missed one prime number between $$10$$ and $$20$$
  • He missed one prime number between $$20$$ and $$30$$
  • He added one extra prime number between $$10$$ and $$20$$
  • None of these
Which of the following statement is true?
  • 1 is the smallest prime number
  • Every prime number is an odd number
  • The sum of two prime numbers is always a prime number
  • None of these
The number $$10$$ has four factors: $$1,2, 5$$ and $$10$$. The table below lists the number of factors for some numbers
NumbersNumber of factors
$$21$$$$4$$
$$23$$$$2$$
$$25$$$$3$$
$$27$$$$4$$
$$29$$$$2$$
From this, we can say that the number of prime numbers between $$20$$ and $$30$$ is:
  • $$0$$
  • $$2$$
  • $$3$$
  • $$4$$
Find the modulus and amplitude of $$-\sqrt 3-i$$
  • $$|z|=2; amp(z)=-\dfrac {5\pi}{6}$$
  • $$|z|=4; amp(z)=\dfrac {5\pi}{6}$$
  • $$|z|=4; amp(z)=-\dfrac {\pi}{6}$$
  • none of these
(i) Every prime number is odd.
(ii) The product of any two prime numbers is odd.
Which of the above statements(s) is/are correct? 
  • Statement I alone.
  • Statement II alone.
  • Both statements I and II.
  • Neither statement I nor statement II.
If $$z_1, z_2, \varepsilon C$$ are such that $$|z_1 + z_2|^2 = |z_1|^2 + |z_2|^2$$ then $$\displaystyle \frac{z_1}{z_2}$$ is
  • zero
  • purely real
  • purely imaginary
  • complex
Let $$\displaystyle \left| Z_{r}-r\right| \leq r,\forall r=1,2,3,.......n.$$ Then $$\displaystyle \left| \sum_{r=1}^{n} Z_{r} \right| $$ is less than
  • $$n$$
  • $$2n$$
  • $$n(n+1)$$
  • $$\displaystyle \frac{n(n+1)}{2}$$
$$(1 + i)^8 + (1 -i)^8 =$$
  • $$16$$
  • $$-16$$
  • $$32$$
  • $$-32$$
If z = x + iy and |z 1 + 2i | = | z + 1 2i |,then the locus of z is
  • circle
  • parabola
  • straight line
  • None of these
Modulus of $$\displaystyle \frac{cos \theta- isin\theta }{sin\theta - icos\theta} is$$
  • 0
  • 2$$\theta$$
  • $$\pi - 2\theta$$
  • none of these
If $$z_1$$ and $$z_2$$ are any two complex numbers, then $$\displaystyle \frac{z_2 + z_1}{||z_2| - |z_1||}$$ is
  • $$\leq 1$$ 
  • $$\geq 1$$
  • $$\geq -1$$
  • none of these
Find the value of: $$i^2 + i^4 + i^6$$ +..... upto $$(2n +1)$$ terms.
  • $$i$$
  • $$-i$$
  • $$1$$
  • $$-1$$
$$i^n + i^{n + 1} + i^{n + 2}+ i^{n + 3} (n   \in   N) $$ is equal to
  • $$4$$
  • $$1$$
  • $$0$$
  • $$2$$
Solve:
 $$\displaystyle \left|(1 + i)\frac{(2+i)}{(3 + i)}\right| $$
  • $$\dfrac{1}{2}$$
  • $$\dfrac{-1}{2}$$
  • $$1$$
  • $$-1$$
If x and y are real then which one of the following is true
  • |x - y| = |x| - |y|
  • |x + y| $$\leq$$ |x| - |y|
  • |x - y| $$\geq$$ |x| - |y|
  • |x + y| = |x| + |y|
The modulus of (1 + i) (1 + 2i) (1 + 3i) is equal to
  • $$\sqrt10$$
  • $$\sqrt 5$$
  • 5
  • 10
If $$|z -2 + i| = |z-3-i|$$, then the locus of z is
  • $$2x -4y -5 = 0$$
  • $$2x + 4y -5 = 0$$
  • $$x -2y + 5 = 0$$
  • none of these
The value of $$(x - 1) \displaystyle \left ( x + \frac{1}{2} - \frac{\sqrt{3}}{2} i \right )\left ( x + \frac{1}{2} + \frac{\sqrt 3}{2} i \right )$$ is
  • $$x^3 + x^2 + x 1$$
  • $$x^3 -1$$
  • $$x^3 + 1$$
  • $$x^3 - x^2 + x + 1$$
Number of complex numbers $$z$$ satisfying $$\left| 2z \right| =\left| 2z-1 \right| =\left| 2z+1 \right| $$ is equal to-
  • $$0$$
  • $$1$$
  • $$2$$
  • $$3$$
The locus represented by $$|z -1|=|z + i|$$ is
  • a circle of radius $$1$$ unit
  • an ellipse with foci at $$(1, 0)$$ and $$(0, 1)$$
  • a straight line through the origin
  • a circle on the line joining $$(1, 0)$$ and $$(0, 1)$$ as diameter
Fill in the blanks.
Every composite number can be expressed as a product of ____, and this factorization is unique except for the order in which the prime factors occur.
  • Co-primes
  • Twin primes
  • Primes
  • All of these
If $$\displaystyle z= (i)^{(i)^{(i)}}$$ where $$ i = \sqrt{-1}$$, then $$z$$ is equal to
  • $$-i$$
  • $$-1$$
  • $$1$$
  • $$i$$
If $${z}_{1},{z}_{2}$$ are two complex numbers and $$c>0$$ such that $${ \left| { z }_{ 1 }+{ z }_{ 2 } \right|  }^{ 2 }\le \left( 1+c \right) { \left| { z }_{ 1 } \right|  }^{ 2 }+k{ \left| { z }_{ 2 } \right|  }^{ 2 },$$ then $$k=$$
  • $$1-c$$
  • $$c-1$$
  • $$1+{c}^{-1}$$
  • $$1-{c}^{-1}$$
Which of the following number is composite?
  • $$1$$
  • $$4$$
  • $$2$$
  • $$3$$
Find the product and write the answer in standard form.
$$\left( 2-4i \right) \left( 3+7i \right) $$
  • $$34+2i$$
  • $$-5-3i$$
  • $$5+3i$$
  • $$5-3i$$
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