CBSE Questions for Class 11 Engineering Maths Complex Numbers And Quadratic Equations Quiz 3 - MCQExams.com

If $$z=2-i\sqrt{3}$$ then $$z^{4}-4z^{2}+8z+35$$ is :
  • $$6$$
  • $$0$$
  • $$1$$
  • $$2$$

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
  • $$x>1$$
  • $$x<1$$
  • $$x<2$$
  • $$x>2$$
For n an integer, the argument of 

$$Z=\displaystyle \frac{(\sqrt{3}+i)^{4n+1}}{(1-i\sqrt{3})^{4n}}$$ is
  • $$\displaystyle \frac{\pi}{6}$$
  • $$\displaystyle \frac{\pi}{3}$$
  • $$\displaystyle \frac{\pi}{2}$$
  • $$\displaystyle \frac{2\pi}{3}$$
The amplitude of $$1 + \cos x - i \sin x$$ is
  • $$\displaystyle \frac{x}{2}$$
  • $$\displaystyle x$$
  • $$-\displaystyle \frac{x}{2}$$
  • $$\displaystyle 2 x$$

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

The principal argument of $$\displaystyle \frac{i-3}{i-1}$$ is
  • $$\tan^{-1}\dfrac{1}{2}$$
  • $$\tan^{-1}\dfrac{3}{2}$$
  • $$\tan^{-1}\dfrac{5}{2}$$
  • $$\tan^{-1}\dfrac{7}{2}$$
The principal argument of

$$\displaystyle \sqrt{2}[\cos\frac{5\pi}{3}+i\sin\frac{5\pi}{3}]$$ is
  • $$\displaystyle \frac{5\pi}{3}$$
  • $$\displaystyle \frac{\pi}{3}$$
  • $$-\displaystyle \frac{\pi}{3}$$
  • $$-\displaystyle \frac{\pi}{2}$$

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

The principal amplitude of $$(2-i)(1-2i)^{2}$$ is in the interval
  • $$\left (0,\displaystyle \frac{\pi}{2} \right)$$
  • $$\left (-\displaystyle \frac{\pi}{2},0 \right)$$
  • $$\left (-\displaystyle \pi, -\frac{\pi}{2} \right)$$
  • $$\left (-\displaystyle \frac{\pi}{2},\frac{\pi}{2} \right)$$

The principal argument of $$1+\sqrt{2}+i$$ is
  • $$\displaystyle \frac{\pi}{3}$$
  • $$\displaystyle \frac{\pi}{6}$$
  • $$\displaystyle \frac{\pi}{8}$$
  • $$\displaystyle \frac{\pi}{4}$$

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
  • $$2^{n-1}$$
  • $$2^{n}$$
  • $$2^{n+1}$$

If $$z=\sqrt{\dfrac{1-i}{1+i}}$$ , then $$\arg \mathrm{z}=$$
  • $$\displaystyle \dfrac{\pi}{4},\dfrac{\pi}{2}$$
  • $$-\displaystyle \dfrac{\pi}{4},\dfrac{\pi}{2}$$
  • $$\displaystyle \dfrac{3\pi}{4},\pi$$
  • $$-\displaystyle \dfrac{\pi}{4},\dfrac{3\pi}{4}$$

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 $$\dfrac{x+3i}{2+iy}=1-i$$, then the value of $$\left ( 5x-7y \right )^2$$ is
  • $$1$$
  • $$0$$
  • $$2$$
  • $$4$$

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
  • $$|\mathrm{z}_{1}|$$
  • $$|\mathrm{z}_{2}|$$
  • $$|z_{1}+\mathrm{z}_{2}|+|\mathrm{z}_{1}-\mathrm{z}_{2}|$$
  • $$|\mathrm{z}_{1}+\mathrm{z}_{2}|-|\mathrm{z}_{1}-\mathrm{z}_{2}|$$
lf $$z\neq 0$$, then $$\displaystyle \int_{0}^{50}\arg(-|z|)dx$$ equals
  • 50
  • not defined
  • 0
  • $$ 50\pi$$
Number of solutions of the equation $$|z|^{2}+7{z}=0$$ is
  • $$1$$
  • $$2$$
  • $$4$$
  • $$6$$
Which of the following equations have no real roots ?
  • $$x^2-2\sqrt{3}+5=0$$
  • $$2x^2+6\sqrt{2}x+9=0$$
  • $$x^2-2\sqrt{3}-5=0$$
  • $$2x^2-6\sqrt{2}x-9=0$$
The values of $$k$$ for which the equation $$2x^2 + kx + x + 8 = 0$$ will have real and equal roots are
  • $$10$$ and $$-6$$
  • $$7$$ and $$-9$$
  • $$6$$ and $$-10$$
  • $$-7$$ and $$9$$
The equation $$2x^2 + 2(p + 1) x +p= 0$$, where $$p$$ is real, always has roots that are
  • Equal
  • Equal in magnitude but opposite in sign
  • Irrational
  • Real
If the equation $$x^2- (2 + m) x + 1 (m^2 -4m + 4) = 0 $$ has coincident roots, then:
  • $$m=0$$
  • $$m=6$$
  • $$m=2$$
  • $$m=\cfrac{2}{3}$$
Let us consider a quadratic equation $$x^2+\lambda x+\lambda +1.25=0$$,  where $$\lambda$$ is a constant. 
The value of $$\lambda$$ such that the above quadratic equation has two coincident roots
  • $$\lambda =5$$ or $$\lambda=-1$$
  • $$\lambda =1$$ or $$\lambda=5$$
  • $$\lambda =-5$$ or $$\lambda=1$$
  • None of these
If the equation $$(m^2+n^2)x^2-2(mp+nq)x+p^2+q^2=0$$ has equal roots , then
  • $$mp = nq$$
  • $$mq =np$$
  • $$mn=pq$$
  • $$mq=\sqrt{np}$$
If the roots of the equation $$px^2+2qx+r=0$$ and $$qx^2-2\sqrt{pr}x+q=0$$ be real, then
  • $$p = q$$
  • $$q^2=pr$$
  • $$p^2=qr$$
  • $$r^2=pq$$
Which of the following equations have no real roots ?
  • $$x^2 -2\sqrt{3}x+5=0$$
  • $$-2x^2 +6\sqrt{2}+11=0$$
  • $$x^2 -2\sqrt{3}x-5=0$$
  • $$-2x^2 -6\sqrt{2}x-9=0$$
If the equation $$2x^2-6x+p=0$$ has real and different roots, then the values of $$p$$ are given by
  • $$p<\cfrac{9}{2}$$
  • $$p\leq \cfrac{9}{2}$$
  • $$p>\cfrac{9}{2}$$
  • $$p\geq \cfrac{9}{2}$$
Let us consider a quadratic equation   $$x^2+3ax+2a^2=0$$ 
If this equation has roots $$\alpha ,\beta $$ and it is given that $$\alpha^2 +\beta ^2=5$$, then value of discriminant, $$D$$, for the above quadratic equation is
  • $$D>0$$
  • $$D<0$$
  • $$D=0$$
  • none of these
Solve the equation $$\left|z\right| = z + 1 + 2i$$.
  • $$ 3 - 2i$$
  • $$ 2 - \displaystyle \frac{3}{2}i$$
  • $$ 2 +\displaystyle \frac{3}{2}i$$
  • $$ \displaystyle \frac{3}{2} - 2i$$
Both the roots of the given equation $$(x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0$$ are always
  • Positive
  • Negative
  • Real
  • Imaginary
If $$\sqrt {5-12i}+\sqrt {-5-12i}=z$$, then principal value of arg z can be
  • $$-\dfrac {\pi}{4}$$
  • $$\dfrac {\pi}{4}$$
  • $$\dfrac {3\pi}{4}$$
  • $$-\dfrac {3\pi}{4}$$
The nature of roots of $$x^2 + x + 1$$ is
  • real and equal
  • real and unequal
  • imaginary and distinct
  • imaginary and equal
The greatest positive argument of complex number satisfying $$|z-4|=Re(z)$$ is
  • $$\displaystyle \frac {\pi}{3}$$
  • $$\displaystyle \frac {2\pi}{3}$$
  • $$\displaystyle \frac {\pi}{2}$$
  • $$\displaystyle \frac {\pi}{4}$$
If $$k+\mid k+z^2\mid =\mid z\mid^2 (k\epsilon R^-)$$, then possible argument of $$z$$ is
  • $$0$$
  • $$\pi$$
  • $$\displaystyle \dfrac {\pi}{2}$$
  • None of these
If $$b^2-4ac\geq 0$$, then the root of quadratic equation $$ax^2+bx+c=0$$ is
  • $$\frac {b}{2a}\pm \frac {\sqrt {b^2-4a}}{2a}$$
  • $$-\frac {b}{2a}\pm \frac {\sqrt {b^2+4ac}}{2a}$$
  • $$\frac {b}{2a}\pm \frac {\sqrt {b^2+4ac}}{2a}$$
  • $$-\frac {b}{2a}\pm \frac {\sqrt {b^2-4ac}}{2a}$$
The principal value of arg z where $$z = 1 + \cos \displaystyle \frac{6 \pi}{5} +  i \sin\frac{6 \pi}{5} $$ is given by
  • $$ \displaystyle \frac{3 \pi}{5}$$
  • $$- \displaystyle \frac{ \pi}{5}$$
  • $$- \displaystyle \frac{3 \pi}{5}$$
  • $$ \displaystyle \frac{ \pi}{5}$$
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
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
  • $$3$$
  • $$4$$
  • $$5$$
  • $$6$$
For any integer $$n$$, the argument of $$\displaystyle z=\frac { { \left( \sqrt { 3 } +i \right)  }^{ 4n+1 } }{ { \left( 1-i\sqrt { 3 }  \right)  }^{ 4n } } $$ is 
  • $$\displaystyle \frac { \pi  }{ 6 } $$
  • $$\displaystyle \frac { \pi  }{ 3 } $$
  • $$\displaystyle \frac { \pi  }{ 2 } $$
  • $$\displaystyle \frac { 2\pi  }{ 3 } $$
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
For $$\left|z - 1\right| = 1$$, find tan $$[arg\dfrac{((z -1)}{(2 - 2 \frac{i}{z}}))].$$
  • $$i$$
  • $$1$$
  • $$-i$$
  • $$-1$$
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 $$(w - \overline{w}z)/(1-z)$$ is purely real where $$w = \alpha + i\beta, \beta \neq 0$$ and $$z \neq 1$$, then set of the values of  $$z$$ is 
  • $${z : \left|z\right| = 1}$$
  • $${z : z = \overline{z}}$$
  • $${z : z \neq 1}$$
  • $${z : \left|z\right| = 1, z \neq 1}$$
Complex number $$z$$ satisfy the equation $$\left|z - (4/z)\right| = 2$$. Locus of z if $$\left|z - z_1\right| = \left|z - z_2\right|$$, where $$z_1$$ and $$z_2$$ are complex numbers with the greatest and the least moduli, is  
  • line parallel to the real axis
  • line parallel to the imaginary axis
  • line having a positive slope
  • line having a negative slope
  • 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
If $$(\sqrt{8} + i)^{50} = 3^{49}(a + ib)$$, then find the value of $$a^2 + b^2$$. 
  • $$(a^2 + b^2) = 9$$
  • $$(a^2 + b^2) = 27$$
  • $$(a^2 + b^2) = 3$$
  • $$(a^2 + b^2) = 1$$
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