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

An example for twin primes is _____.
  • $$5, 11$$
  • $$3, 5$$
  • $$11, 17$$
  • $$3, 7$$
If $$z = x+iy$$ and $$w = \dfrac{(1-iz)}{(z-i)}$$, then $$|w| = 1$$ implies that, in the complex plane
  • $$z$$ lies on the imaginary axis
  • $$z$$ lies on the real axis
  • $$z$$ lies on the unit circle
  • None of these
$$\displaystyle { \left( \frac { \sqrt { 3 } +i }{ 2 }  \right)  }^{ 6 }+{ \left( \frac { i-\sqrt { 3 }  }{ 2 }  \right)  }^{ 6 }=$$
  • $$-2$$
  • $$2$$
  • $$-1$$
  • $$1$$
If $$z_1$$ and $$z_2$$ are two complex numbers such that $$| z_1 | = | z_2 | + | z_1 z_2 |$$, then $$arg(z_1) - arg(z_2)$$ is
  • $$0$$
  • $$\pi$$/2
  • $$-\pi$$/2
  • none of these
Dividing $$ f(z) $$ by $$ z - i$$ we obtain the remainder i and dividing it by $$ z + i$$ we get the remainder $$1 + i$$ then remainder upon the division of $$ f(z)$$ by $$ z^{2} + 1$$ is
  • $$\displaystyle \frac{1}{2}(z+1)+i$$
  • $$\displaystyle \frac{1}{2}(iz+1)+i$$
  • $$\displaystyle \frac{1}{2}(z-1)+i $$
  • $$\displaystyle \frac{1}{2}(z+1)+1$$
If $$(\sqrt 3-i)^n=2^n, n\in N$$, then $$n$$ is a multiple of
  • $$6$$
  • $$10$$
  • $$9$$
  • $$12$$
Evaluate in standard form: $$\dfrac {(2-3i)}{(2-2i)}$$, where $${i}^{2}=-1$$.
  • $$\dfrac {5}{4}-\dfrac {i}{4}$$
  • $$\dfrac {5}{4}+\dfrac {i}{4}$$
  • $$-\dfrac {5}{4}-\dfrac {i}{4}$$
  • $$-\dfrac {5}{4}+\dfrac {i}{4}$$
If $$z_1, z_2, ..., z_n$$ lie on $$|z|=r$$ and $$Re\left(\displaystyle\sum_{j=1}^n\displaystyle\sum_{k=1}^n{\displaystyle\frac{z_j}{z_k}}\right) = 0$$, then
  • $$\displaystyle\sum_{j=1}^n{z_j}=0$$
  • $$\left|\displaystyle\sum_{j=1}^n{z_j}\right|=0$$
  • $$\displaystyle\sum_{j=1}^n{\displaystyle\frac{1}{z_j}}=0$$
  • None of these
Given that $$i = \sqrt {-1}$$, find the multiplicative inverse of $$5 - i$$.
  • $$5 + i$$
  • $$\dfrac {5 + i}{26}$$
  • $$\dfrac {1}{5 + i}$$
  • $$\dfrac {5 + i}{24}$$
  • $$\dfrac {5 - i}{24}$$
Write the complete number $$- 2 - 2i$$ in polar form.
  • $$2(cos\dfrac{\pi}{4}+i\,sin\dfrac{\pi}{4})$$
  • $$-2(cos\dfrac{\pi}{4}-i\,sin\dfrac{\pi}{4})$$
  • $$2\sqrt{2}(cos\dfrac{3\pi}{4}+i\,sin\dfrac{3\pi}{4})$$
  • $$2\sqrt{2}(cos\dfrac{7\pi}{4}+i\,sin\dfrac{7\pi}{4})$$
  • $$2\sqrt{2}(cos\dfrac{5\pi}{4}+i\,sin\dfrac{5\pi}{4})$$
What is the product of the complex numbers $$\left( -3i+4 \right) $$ and $$\left( 3i+4 \right) $$?
  • $$1$$
  • $$7$$
  • $$25$$
  • $$-7+24i$$
  • $$7+24i$$
Add and express in the form of a complex number $$a+bi$$
$$(2+3i)+(-4+5i)-\dfrac {(9-3i)}{3}$$
  • $$-4+9i$$
  • $$-5+9i$$
  • $$2+9i$$
  • $$-5+8i$$
What is the next-highest prime number after 67?
  • 68
  • 69
  • 71
  • 73
  • 76
If $$z$$ is a complex number such that $$|z|\geq 2$$ then the minimum value of $$\left |z + \dfrac {1}{2}\right |$$ is
  • Is strictly greater than $$\dfrac {5}{2}$$
  • Is strictly greater than $$\dfrac 32$$ but less than $$\dfrac {5}{2}$$
  • Is equal to $$\dfrac {5}{2}$$
  • Lies in the interval $$(1, 2)$$
If $$\alpha $$ and $$\beta$$ are two different complex numbers with $$|\beta|=1$$, then $$\left | \dfrac{\beta -\alpha}{1-\bar{\alpha }\beta } \right |$$ is equal to.
  • $$0$$
  • $$1$$
  • $$\dfrac{1}{2}$$
  • $$-1$$
If $$z_ 1 = 2 \sqrt 2 (1 + i)$$ and $$z = 1 + i \sqrt 3$$, then $$z_1^2 z_2^3$$ is equal to
  • $$128 i$$
  • $$64 i$$
  • $$-64 i$$
  • $$- 128 i$$
  • $$256$$
Express in the form of a complex number $$a+bi$$
$$-(7-i)(-4-2i)(2-i)$$
  • $$60-10i$$
  • $$70-10i$$
  • $$28-14i$$
  • $$70-15i$$
The numbers which have more than two factors are called ______.
  • Prime
  • Composite
  • Both $$(A)$$ and $$(B)$$
  • None of these
If z be any complex number such that $$|3z-2|+|3z+2|=4$$, then locus of z is
  • An ellipse
  • A circle
  • A line-segment
  • None of these
In binary system the highest value of a 8-bit number is
  • 255
  • 256
  • 253
  • none of these
The $$3$$ bit operation code for ADD operation is $$001$$ and indirect memory address is $$23$$ then $$16$$-bit instruction code can be written as-
  • $$00001000000010111$$
  • $$1001000000010111$$
  • $$100010000000010111$$
  • None of the above
Let $$z_1$$ = 18 + 83i, $$z_2$$ = 18 + 39i, ana $$z_3 $$= 78 + 99i. where i = $$\sqrt-1$$. Let z be a unique comlpex number with the properties that $$\dfrac{z_3 - z_1}{z_2 - z_1}$$ $$\cdot$$ $$\dfrac{z - z_2}{z - z_3}$$ is a real number and the imaginary part of the size z is the greatest possible.
  • $$Re (z) = 56$$
  • $$Re (z) = 61$$
  • $$Re (z) = 54$$
  • $$Re (z) = 59$$
What is $${ i }^{ 1000 }+{ i }^{ 1001 }+{ i }^{ 1002 }+{ i }^{ 1003 }$$ equal to (where $$i=\sqrt { -1 } $$)?
  • $$0$$
  • $$i$$
  • $$-i$$
  • $$1$$
If $$z_1+ z_2 + z_3 = 0$$ and $$|z_1|=|z_2|=|z_3|= 1$$, then area of triangle whose vertices are $$z_1, z_2$$ and $$z_3$$ is:
  • $$\dfrac{3 \sqrt{3}}{4}$$
  • $$\dfrac{\sqrt{3}}{4}$$
  • $$1$$
  • $$2$$
The least positive integer $$n$$ for which $$\left(\dfrac {1+i}{1-i}\right)^{n}=\dfrac {2}{\pi} \sin^{-1}\left(\dfrac {1+x^{2}}{2x}\right)$$, where $$x>0$$, is 
  • $$2$$
  • $$4$$
  • $$8$$
  • $$12$$
If $${z}_{1},{z}_{2},..{z}_{n}$$ lie on the circle $$|z|=2$$ then the value of $$|{z}_{1},{z}_{2},..{z}_{n}|-4|\dfrac {1}{{z}_{1}}+\dfrac {1}{{z}_{2}}++\dfrac {1}{{z}_{n}}|=$$
  • $$0$$
  • $$n$$
  • $$-n$$
  • $$1$$
If $${z}_{1},{z}_{2}$$ and $${z}_{3}$$ be three complex numbers such that $$\left| { z }_{ 1 }+1 \right| \le 1,\left| { z }_{ 2 }+2 \right| \le 2$$ and $$\left| { z }_{ 3 }+4 \right| \le 4$$, then the maximum value of $$\left| { z }_{ 1 } \right| +\left| { z }_{ 2 } \right| +\left| { z }_{ 3 } \right|$$ is ?
  • $$7$$
  • $$10$$
  • $$12$$
  • $$14$$
If $$Z_{1},Z_{2}$$ are two complex numbers satisfying $$|\dfrac{Z_{1}-3Z_{2}}{3-Z_{1}Z_{2}}|=1|z_{1}|\neq 3$$ then $$|z_{2}|=$$
  • $$1$$
  • $$2$$
  • $$3$$
  • $$4$$
A complex number z is said to be unimodular if $$|z| =$$. Suppose $$z_1$$ and $$z_2$$ are complex numbers such that $$\frac{z_1-2z_2}{2-z_1\overline {z}_2}$$ is unimolecolar and $$z_2$$ is not unimodular. Then the point $$z_1$$ lies on a:
  • straight line parallel to y-axis
  • circle of radius 2.
  • Circle of radius $$\sqrt2$$.
  • Staright line parallel to x-axis.
If $$\sqrt{3}+i(a+ib)(c+id)$$, then $$\tan^{-1}\left(\dfrac{b}{a}\right)+\tan^{-1}\left(\dfrac{d}{c}\right)$$ has the value
  • $$\dfrac{\pi}{3}+2n\pi, n \notin 1$$
  • $$n\pi+\dfrac{\pi}{6}, n \in 1$$
  • $$n\pi-\dfrac{\pi}{3}, n \in 1$$
  • $$2n\pi-\dfrac{\pi}{3}, n \in 1$$
If $$z$$ is a complex number such that $$|z|\ge\ 2$$, then the minimum value of $$\left| z+\dfrac { 1 }{ 2 }  \right|$$ 
  • is equal to $$\dfrac{5}{2}$$
  • is strictly greater than $$\dfrac{5}{2}$$
  • lies in the interval $$(1,2)$$
  • is strictly greater than $$\dfrac{3}{2}$$ but less than $$\dfrac{5}{2}$$
The value of $$(z+3) (\overline{z} +3)$$ is eqquivalent to
  • $$|z +3 | ^2$$
  • $$| z- 3|$$
  • $$z^2+3$$
  • None of these
The imaginary part of $$(z - 1)(\cos \, \alpha - i \, \sin \, \alpha) + (z - 1)^{-1} \times (\cos \, \alpha + i \, \sin \, \alpha ) $$ is zero, if 
  • $$|z - 1 | = 2$$
  • $$\text{arg} \, (z - 1) = 2 \alpha$$
  • $$\text{arg} \, (z - 1) = \alpha$$
  • $$|z | = 1$$
If  $$   | z _ { 1 } | < 1 \text { and } | \frac { z _ { 1 } - z _ { 2 } } { 1 - \overline { z } _ { 1 } z _ { 2 } } | < 1 , \text { then  } | z _ { 2 } | > 1$$
  • True
  • False
If $$\left| z \right| =1$$ and $$\left| \omega -1 \right| =1$$ where $$z,\omega \in C$$ then the largest set of values of $${ \left| 2z-1 \right|  }^{ 2 }+{ \left| 2\omega -1 \right|  }^{ 2 }$$ equals 
  • $$\left[1,9\right]$$
  • $$\left[2,6\right]$$
  • $$\left[2,12\right]$$
  • $$\left[2,18\right]$$
If $$\dfrac {3+2i \sin x}{1-2i \sin x}$$ is purely imaginary then $$x=$$ ?
  • $$n \pi \pm \dfrac {\pi}{6}$$
  • $$n \pi \pm \dfrac {\pi}{3}$$
  • $$2n \pi \pm \dfrac {\pi}{3}$$
  • $$2n \pi \pm \dfrac {\pi}{6}$$
If $$\alpha$$ and $$\beta$$ are different complex number with $$|\beta|=1$$, then $$\left |\dfrac {\beta-\alpha}{1-\overline {\alpha }\beta}\right|$$ is equal to
  • $$0$$
  • $$1/2$$
  • $$1$$
  • $$2$$
Which of the following pairs are twin primes?
  • $$(19,21)$$
  • $$(29,31)$$
  • $$(39,41)$$
  • $$(49,51)$$
If $$z^4+1=\sqrt{3}$$i then?
  • $$z^3$$ is purely real
  • z represents vertices of a square of side $$2^{\dfrac{1}{4}}$$
  • $$z^9$$ is purely imaginary
  • z represents vertices of a square of side $$2^{\dfrac{3}{4}}$$
If $$a,b,c$$ are non-zero numbers and the equation $$ax^{2}+bx+c+i=0$$ has purely imaginary roots, then
  • $$a=bc$$
  • $$a=b^{2}c$$
  • $$a=\sqrt {bc}\ if\ b > 0$$
  • $$none\ of\ these$$
If the roots $$a^2x^2 +2bx+c^2 = 0$$ are imaginary then the roots of $$b(x^2+1)+2acx = 0$$ are 
  • complex number
  • real and unequal
  • real and equal
  • none
If $$z$$ is a complex number of unit modulus and argument $$\theta $$, then the real part of 
$$\dfrac { z\left( 1-\bar { z }  \right)  }{ \bar { z } \left( 1+2 \right)  } $$ is:
  • $$1+cos\dfrac { \theta }{ 2 } $$
  • $$1-sin{ \dfrac { \theta }{ 2 } }$$
  • $$-2\sin { ^{ 2 } } \dfrac { \theta }{ 2 } $$
  • $$2cos^{ 2 }\dfrac { \theta }{ 2 } $$
If $$a > b > c$$, $$a\neq 0$$ and the system of equations
$$ax+by+cz=0$$, $$bx+cy+az=0$$, $$cx+ay+bz=0$$ has non-trivial solutions, then the roots of the quadratic equation $$at^2+bt+c=0$$.
  • Are imaginary
  • Are real and equal
  • Are real and distinct
  • May be real of imaginary
The value of $$(sin\frac{\pi }{8}+i\cos \frac{\pi }{8})^{8}{(sin\frac{\pi }{8}-i cos \frac{\pi }{8})^{8}}$$ is 
  • -1
  • 0
  • 1
  • 2i
The equation $$x(x+2)(x^{2}-x)=-1$$, has 
  • All roots imaginary
  • All roots negative
  • Two roots real and two roots imaginary
  • All roots real
If $$\overline { \Delta  } =\begin{vmatrix} -1 & 2-3i & 5+4i \\ 2+3i & 8 & 1-i \\ 5-4i & 1+i & 3 \end{vmatrix}$$ then $$\Delta =$$
  • Purely real
  • Purely imaginary
  • Complex
  • $$0$$
If $$\theta$$ real then the modulus of $$\dfrac{1}{1+\cos\theta+i\sin\theta}$$ is
  • $$\dfrac{1}{2}\sec\dfrac{\theta}{2}$$
  • $$\dfrac{1}{2}\cos\dfrac{\theta}{2}$$
  • $$\sec\dfrac{\theta}{2}$$
  • $$\cos\dfrac{\theta}{2}$$
The function of imaginary roots of the equation $$(x-1)(x-2)(3x+1)=32$$ is 
  • $$0$$
  • $$1$$
  • $$2$$
  • $$4$$
The number of imaginar roots of the equation $$(x-1)(x-2)(3x-2)(3x+1)=32$$ is
  • $$Zero$$
  • $$1$$
  • $$2$$
  • $$4$$
If $$\mathrm{{z} _ { 1 }} = 10 + 6\mathrm{i} ,  \mathrm{{ z } _ { 2 }}= 4 + 6 \mathrm { i }$$ and $$\mathrm{ z}$$ is a complex number such that $$\operatorname { amp } \left( \dfrac { \mathrm { z } - \mathrm { z } _ { 1 } } { \mathrm { z } - \mathrm { z } _ { 2 } } \right) = \dfrac { \pi } { 4 }$$ , then the value of $$\left| \mathrm{z} - 7 - 9 \mathrm { i } \right|$$ is equal to
  • $$\sqrt { 2 }$$
  • $$2\sqrt { 2 }$$
  • $$3\sqrt { 2 }$$
  • $$2\sqrt { 3 }$$
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