CBSE Questions for Class 11 Commerce Applied Mathematics Number Theory Quiz 4 - MCQExams.com

$$\left| \dfrac { (3+i)(2-i) }{ 1+i }  \right|=$$
  • $$\sqrt{5}$$
  • $$5\sqrt{2}$$
  • $$\sqrt{10}$$
  • $$5$$
If $$z_1$$ and $$z_2$$ are two complex numbers, then $$Re(z_1z_2)$$ is:
  • $$Re(z_1)Re(z_2)$$
  • $$Re(z_1).Re(z_2)-Im(z_1).Im(z_2)$$
  • $$Im(z_1).Re(z_2)$$
  • $$Re(z_1).Im(z_2)$$
The real part of $$\left ( \dfrac{1+i}{3-i} \right )^2=$$
  • $$1$$
  • $$16$$
  • $$16\omega ^2$$
  • $$\displaystyle \frac{-3}{25}$$
If $$z=-1+3i$$ then $$z^2+2z+10=$$
  • $$0$$
  • $$1$$
  • $$–1$$
  • $$2$$
The value of $$1+(1+i)+(1+i)^2+(1+i)^3=$$
  • $$0$$
  • $$5i$$
  • $$4i$$
  • $$3i$$
If $$\left ( 5+3i \right )(x+iy)=3-4i$$ then $$34x = $$
  • $$1$$
  • $$2$$
  • $$3$$
  • $$4$$
The simplified value of $$\displaystyle \frac{1-i}{1+i}$$ is:
  • $$i$$
  • $$-i$$
  • $$1$$
  • $$-2i$$

The minimum value of $$|\mathrm{z}|+|\mathrm{z}-1|+|\mathrm{z}-2|$$ is
  • $$0$$
  • $$1$$
  • $$2$$
  • $$4$$

lf $$z_{1},\ z_{2}$$ are roots of equation $$z^{2}-az+a^{2}=0$$, then $$|\displaystyle \frac{z_{1}}{z_{2}}|=$$
  • $$a^{2}$$
  • a
  • 2a
  • 1

 lf $$\log_{\frac{1}{2}}|\mathrm{z}-2|>\log_{\frac{1}{2}}|z|$$ then
  • $$x>1$$
  • $$x<1$$
  • $$x<2$$
  • $$x>2$$
The modulus of $$(1 + i) (3 + 4i) =$$
  • $$\sqrt{50}$$
  • $$\sqrt{25}$$
  • $$10$$
  • $$10 \sqrt{2}$$

The region represented by z such that $$\left | \dfrac{\mathrm{z}-a}{z+a} \right |=1({\rm Im} (a) = 0)$$ is
  • $$y=0$$
  • $$x=0$$
  • $$x+y=0$$
  • $$x-y=0$$

$$ \left | \displaystyle \frac{1}{(1-i)^{2}}-\frac{1}{(1+i)^{2}}\right |=$$
  • 4
  • 3
  • 2
  • 1

lf $$z_{1},\ z_{2}$$ are any two complex numbers then $$\left |z_{1^{+}}\sqrt{\mathrm{z}_{1}^{2}-\mathrm{z}_{2}^{2}} \right |+\left|\mathrm{z}_{1}-\sqrt{\mathrm{z}_{1}^{2}-\mathrm{z}_{2}^{2}} \right|$$ is equal to
  • $$|\mathrm{z}_{1}|$$
  • $$|\mathrm{z}_{2}|$$
  • $$|z_{1}+\mathrm{z}_{2}|+|\mathrm{z}_{1}-\mathrm{z}_{2}|$$
  • $$|\mathrm{z}_{1}+\mathrm{z}_{2}|-|\mathrm{z}_{1}-\mathrm{z}_{2}|$$
If $$z_1$$, $$z_2$$, $$z_3$$ are complex numbers such that $$\left| { z }_{ 1 } \right| =\left| { z }_{ 2 } \right| =\left| { z }_{ 3 } \right| =\left| \dfrac { 1 }{ { z }_{ 1 } } +\dfrac { 1 }{ { z }_{ 2 } } +\dfrac { 1 }{ { z }_{ 3 } }  \right| =1$$, then $$|z_1+z_2+z_3|$$ is:
  • Equal to 1
  • Less than 1
  • Greater than 3
  • Equal to 3
I. $$\left| \frac { { z }_{ 1 }+{ z }_{ 2 } }{ { z }_{ 1 }-{ z }_{ 2 } }  \right| =1$$ if $$\frac{z_1}{z_2}$$ is purely imaginary
II. If z is purely real then $$z=\overline{z}$$
  • Only II is true
  • Only I is true
  • Both I and II are true
  • I is false but II is true
lf $$(x+iy)(2+cos\theta+isin\theta)=3$$ then $$x^{2}+y^{2}-4x+3$$ is
  • $$0$$
  • $$1$$
  • $$3$$
  • $$4$$
If $$z=2-i\sqrt{3}$$ then $$z^{4}-4z^{2}+8z+35$$ is :
  • $$6$$
  • $$0$$
  • $$1$$
  • $$2$$

lf $$Z_{1},Z_{2}$$ are two unimodular Complex numbers then $$ \left |\displaystyle \frac{1}{Z_{1}}+\frac{1}{Z_{2}} \right|=$$
  • $$1$$
  • $$|Z_{1}+Z_{2}|$$
  • $$|Z_{1}|+|Z_{2}|$$
  • $$2$$
The real value of $$\theta$$ for which the expression, $$\displaystyle \frac{1+i\cos\theta}{1-2i\cos\theta}$$ is real number is
  • $$n\displaystyle \pi\pm\frac{\pi}{2}$$
  • $$n\displaystyle \pi-\frac{\pi}{2}$$
  • $$n\displaystyle \pi+\frac{\pi}{2}$$
  • $$2n\displaystyle \pi\pm \cfrac {\pi}{2}$$

If $$\alpha,\ \beta,\ \gamma$$ are modulus of the complex number $$3+4i, -5+12i,\ 1-i$$, then the increasing order for $$\alpha, \beta $$ and $$\gamma$$ is
  • $$\alpha,\ \gamma,\ \beta$$
  • $$\alpha,\ \beta,\ \gamma$$
  • $$\gamma,\ \alpha,\ \beta$$
  • can't be determined

$$\mathrm{l}\mathrm{n}$$ a G. $$\mathrm{P}$$ first term is $$\sqrt{3}+i$$ and common ratio is $$\sqrt{3}-i$$ then the modulus of the $$n^{th}$$ term of the G.$$\mathrm{P}$$. is
  • 4
  • $$2^{n-1}$$
  • $$2^{n}$$
  • $$2^{n+1}$$

lf $$\displaystyle \log_{(\frac{1}{3})}|z+1|>\log_{(\frac{1}{3})}|z-1|$$, then
  • $${\rm Re}(z)\geq 0$$
  • $${\rm Re}(z)<0$$
  • $${\rm Im}(z)>0$$
  • $${\rm Im}(z)\leq 0$$
If $$z_{1},z_{2}$$ are complex numbers and a.b are real numbers then $$|az_{1}-bz_{2}|^{2}+|bz_{1}+az_{2}|^{2}=$$
  • $$(a^{2}+b^{2})|z_{1}^{2}+z_{2}^{2}|$$
  • $$(a^{2}+b^{2})(|z_{1}|^{2}+|z_{2}|^{2})$$
  • $$\displaystyle \frac{1}{a^{2}+b^{2}}(|z_{1}|^{2}+|z_{2}|^{2})$$
  • $$(a^{2}+b^{2})|z_{1}|+|z_{2}|$$
If $$\dfrac{x+3i}{2+iy}=1-i$$, then the value of $$\left ( 5x-7y \right )^2$$ is
  • $$1$$
  • $$0$$
  • $$2$$
  • $$4$$
Number of solutions of the equation $$|z|^{2}+7{z}=0$$ is
  • $$1$$
  • $$2$$
  • $$4$$
  • $$6$$
The given figure represents a multiplication operation, where each alphabet represents a different number, then what is the value of A.


99751_363f64b2356043bab765ef568d463828.png
  • 0
  • 3
  • 2
  • 4
Solve the equation $$\left|z\right| = z + 1 + 2i$$.
  • $$ 3 - 2i$$
  • $$ 2 - \displaystyle \frac{3}{2}i$$
  • $$ 2 +\displaystyle \frac{3}{2}i$$
  • $$ \displaystyle \frac{3}{2} - 2i$$
If $$z = x + iy$$ and $$w = \displaystyle \frac{1 - zi}{z - i}, |w| = 1$$, then find the locus of z.
  • z lies on the imaginary axis.
  • z lies only on positive real axis.
  • z lies only on negative real axis.
  • z lies on the real axis.
If $$z = x + iy$$ and $$x^2 + y^2 = 16$$, then the range of $$\left|\left|x\right| - \left|y\right|\right|$$ is 
  • [0, 4]
  • [0,2]
  • [2, 4]
  • none of these
If $$\left|z\right |^2 - 3 = 3\left|z\right|$$, then the value of $$\left|z\right|$$ is
  • $$1$$
  • $$\displaystyle \frac{3 + \sqrt{21}}{2}$$
  • $$\displaystyle \frac{\sqrt{21}- 3}{2}$$
  • none of these
Find the complex numbers z which simultaneously satisfy the equation $$\displaystyle \left | \frac{z - 12}{z - 8 i} \right | = \frac{5}{3}$$ and $$\displaystyle \left | \frac{z - 4}{z - 8} \right | = 1$$.
  • 6 + 8 i or 6 + 17 i
  • 6 + 8 i or 6 - 17 i
  • 6 - 8 i or 6 + 17 i
  • 6 - 8 i or 6 - 17 i
  • Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
  • Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
  • Assertion is correct but Reason is incorrect
  • Assertion is incorrect and Reason is correct
Locate the complex number $$z = x + iy$$ for which $$log_{1/3} \{ log_{1/2} (|z|^2 + 4 |z| + 3) \} < 0$$
  • Empty set
  • circle of radius 6 and center at origin
  • circle of radius 3 and center at origin
  • circle of radius 2 and center at origin
If $$i{ z }^{ 3 }+{ z }^{ 2 }-z+i=0$$, then 
  • $$\left| z \right| <1$$
  • $$\left| z \right| >1$$
  • $$\left| z \right| =1$$
  • $$\left| z \right| =0$$
Number of roots of the equation $$z^{10} - z^5 - 992 = 0$$ where real parts are negative is
  • $$3$$
  • $$4$$
  • $$5$$
  • $$6$$
Find the greatest and the least value of $$\left|z_1 + z_2\right|$$ if $$z_1 = 24 + 7i$$ and $$\left|z_2\right| = 6.$$
  • least value is 25, greatest value is 31
  • least value is 19, greatest value is 31
  • least value is 19, greatest value is 25
  • least value is 13, greatest value is 25
If$$\displaystyle\ z_{1}\neq-z_{2}$$ and $$\displaystyle\ |z_{1}+z_{2}|=\left | \frac{1}{z_{1}}+\frac{1}{z_{2}} \right |$$ then
  • at least one of $$\displaystyle\ z_{1}, z_{2}$$ is unimodular
  • both$$\displaystyle\ z_{1}, z_{2}$$ are unimodular
  • $$\displaystyle\ z_{1}. z_{2} = 1$$
  • None of these
$$\left( \frac{1+cos\dfrac{\pi}{8}-i  sin \dfrac{\pi}{8}}{1+cos \dfrac{\pi}{8}+i  sin \dfrac{\pi}{8}}\right)^8 =$$
  • $$1$$
  • $$-1$$
  • $$2$$
  • $$\frac{1}{2}$$
If $$(w - \overline{w}z)/(1-z)$$ is purely real where $$w = \alpha + i\beta, \beta \neq 0$$ and $$z \neq 1$$, then set of the values of  $$z$$ is 
  • $${z : \left|z\right| = 1}$$
  • $${z : z = \overline{z}}$$
  • $${z : z \neq 1}$$
  • $${z : \left|z\right| = 1, z \neq 1}$$
If $$(\sqrt{8} + i)^{50} = 3^{49}(a + ib)$$, then find the value of $$a^2 + b^2$$. 
  • $$(a^2 + b^2) = 9$$
  • $$(a^2 + b^2) = 27$$
  • $$(a^2 + b^2) = 3$$
  • $$(a^2 + b^2) = 1$$
Find the minimum value of $$|z-1|$$ if $$\left|\left|z - 3\right| - \left|z + 1\right|\right| = 2$$.
  • $$\left|z - 1\right| \ge 0$$
  • $$\left|z - 1\right| \ge 1$$
  • $$\left|z - 1\right| \ge 2$$
  • $$\left|z - 1\right| \ge 3$$
If z is a complex number, then find the minimum value of $$\left|z\right| + \left|z - 1\right| + \left|2z - 3\right|.$$
  • $$E = 1$$
  • $$E = 2$$
  • $$E = 3$$
  • $$E = 4$$
Let z be a complex number such that the imaginary part of z is nonzero and a = $$z^2 + z + 1$$ is real. Then a cannot take the value
  • $$-1$$
  • $$\dfrac{1}{3}$$
  • $$\dfrac{1}{2}$$
  • $$\dfrac{3}{4}$$
If $$\left|z_1 - 1\right| \le 1, \left|z_2 - 2\right| \le 2, \left|z_3 - 3\right| \le 3$$, then find the greatest value of $$\left|z_1 +  z_2 + z_3\right|$$.
  • the greatest value is 6.
  • the greatest value is 7.
  • the greatest value is 9.
  • the greatest value is 12.
For all complex numbers $$z_1, z_2$$ satisfying $$\left|z_1\right| = 12$$ and $$\left|z_2 - 3 - 4i\right| = 5$$, then minimum value of $$\left|z_1 - z_2\right|$$ is  
  • $$0$$
  • $$2$$
  • $$7$$
  • $$17$$
The value of the sum $$\displaystyle\ \sum _{n=1}^{13}\left ( i^{n}+i^{n+1} \right ) $$, where $$\displaystyle\ i=\sqrt{-1} $$
  • $$ i $$
  • $$ i-1$$
  • $$ -i$$
  • $$ 0$$
If $$z_{1}$$ and $$z_{2}$$ are two nonzero complex number such that $$\displaystyle |z_1|+|z_{2}|=|z_{1}+z_{2}|$$ then $$arg\: z_{1}-arg\: z_{2}$$ is equal to 
  • $$-\pi$$
  • $$\displaystyle \frac{\pi}{2}$$
  • $$\displaystyle -\frac{\pi}{2}$$
  • $$0$$
If $$\displaystyle \displaystyle\ |z_{1}-1|<1, |z_{2}-2|<2, |z_{3}-3|<3$$ then $$\displaystyle\ |z_{1}+z_{2}+z_{3}|$$
  • $$\displaystyle\ $$ is less than 6
  • $$\displaystyle\ $$ is more than 3
  • $$\displaystyle\ $$ is less than 12
  • $$\displaystyle\ $$ lies between 6 and 12
If $$\displaystyle\ z_{1}, z_{2}$$ are two nonzero complex numbers such that $$\displaystyle\ |z_{1}+z_{2}|=|z_{1}|+\left| z_{ 2 } \right| $$ then amp $$\displaystyle\ \frac{z_{1}}{z_{2}}$$ is equal to
  • $$\displaystyle\ \pi$$
  • $$\displaystyle\ -\pi$$
  • $$0$$
  • $$\displaystyle\ \frac{\pi}{2}$$
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