CBSE Questions for Class 11 Engineering Maths Complex Numbers And Quadratic Equations Quiz 4 - MCQExams.com

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
  • $$-1$$
  • $$\dfrac{1}{3}$$
  • $$\dfrac{1}{2}$$
  • $$\dfrac{3}{4}$$
Complex number $$z$$ satisfy the equation $$\left|z - (4/z)\right| = 2$$. Then the value of $$arg(z_1/z_2)$$, where $$z_1$$ and $$z_2$$ are complex numbers with the greatest and the least moduli, can be
  • 2$$\pi$$
  • $$\pi$$
  • $$\dfrac {\pi}{2}$$
  • none of these
Find the argument of  $$sin  \alpha + i(1 - cos  \alpha), 0 < \alpha < \pi$$
  • $$\displaystyle \frac{\alpha}{2}$$
  • $$\displaystyle \frac{\alpha}{4}$$
  • $$2\alpha$$
  • $$\alpha$$
Find the modulus and the principal argument of the complex number $$(\tan 1 - i)^2$$
  • $$|z|=(\tan1)^2+1$$,z lies in 4rd quadrant, $$arg(z)=2-\pi/2$$
  • $$|z|=(\tan1)^2+1$$,z lies in 4rd quadrant, $$arg(z)=2-\pi$$
  • $$|z|=(\tan1)^2+1$$,z lies in 3rd quadrant, $$arg(z)=2-\pi/2$$
  • $$|z|=(\tan1)^2+1$$,z lies in 3rd quadrant, $$arg(z)=2-\pi$$
Find the argument of $$\displaystyle \frac{1 + \sqrt{3}i}{\sqrt{3} +i}$$
  • $$\displaystyle \frac{\pi}{3}$$
  • $$\displaystyle \frac{\pi}{6}$$
  • $$\displaystyle \frac{\pi}{2}$$
  • $$\pi$$
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 $$\left|z\right| + \left|z - 1\right| + \left|2z - 3\right|.$$
  • $$E = 1$$
  • $$E = 2$$
  • $$E = 3$$
  • $$E = 4$$
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.
Find the modulus and the principal argument of the complex number $$\displaystyle \frac{i - 1}{i\left(1 - cos \frac{2\pi}{5}\right) + sin \frac{2\pi}{5}}$$
  • $$\displaystyle \frac{cosec(\frac{\pi}{5})}{\sqrt2}$$,$$ \frac{9\pi}{20}$$
  • $$\displaystyle \frac{sin(\frac{\pi}{5})}{\sqrt2}$$,$$\displaystyle \frac{11\pi}{20}$$
  • $$\displaystyle \frac{cosec(\frac{\pi}{5})}{\sqrt2}$$,$$\displaystyle \frac{11\pi}{20}$$
  • $$\displaystyle \frac{sin(\frac{\pi}{5})}{\sqrt2}$$,$$ \displaystyle \frac{9\pi}{20}$$
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
Let $$\displaystyle\ z= \frac{\cos \theta +i\sin \theta }{\cos \theta -i\sin \theta }$$, $$\displaystyle\ \frac{\pi}{4}< 0< \frac{\pi}{2}$$. Then arg z is 
  • $$\displaystyle\ 2\theta $$
  • $$\displaystyle\ 2\theta-\pi$$
  • $$\displaystyle\ \pi +2\theta$$
  • None of these
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 $$\displaystyle \displaystyle\ |z_{1}-1|<1, |z_{2}-2|<2, |z_{3}-3|<3$$ 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
The equation $$\displaystyle x^{2}-6x+8+\lambda (x^{2}-4x+3)=0,\lambda \in R$$, has 
  • real and unequal roots for all $$\lambda $$
  • real roots for $$\lambda < 0$$ only
  • real roots for $$\lambda > 0$$ only
  • real and unequl roots for $$\lambda =0$$ only
If  $$ { a }^{ 3 }+{ b }^{ 3 }+{ c }^{ 3 }-3abc=0$$ then the roots of the equation
$$ \left( { a }^{ 2 }-bc \right) { x }^{ 2 }+2\left( { b }^{ 2 }-ac \right) x+{ c }^{ 2 }-ab=0$$ are 
  • imaginary
  • real and unequal
  • real and equal
  • Cannot say
If $$ \displaystyle\ z=\frac{\sqrt{3}+i}{\sqrt{3}-i}$$ then the fundamental amplitude of z is 
  • $$\displaystyle\ -\frac{\pi}{3}$$
  • $$\displaystyle\ \frac{\pi}{3}$$
  • $$\displaystyle\ \frac{\pi}{6}$$
  • None of these
If $$l,m$$ are real $$l\neq m$$ then the roots of the equation $$\displaystyle (l-m)x^{2}-5(l+m)x-2(l-m)=0$$ are
  • real and equal
  • non real complex
  • real and unequal
  • none of these
In the Argand's plane, the locus of $$z (\neq 1)$$ such that $$arg\displaystyle \left\{\frac{3}{2}\left(\frac{2z^2-5z+3}{3z^2 -z-2}\right)\right\} = \frac{2\pi}{3}$$ is
  •  a hyperbola with the directrices at $$z = -3/2$$ and $$z= -2/3$$.
  • an ellipse with the directrices at $$z = 3/2$$ and $$z= 2/3$$.
  • a segment of a circle subtending angle $$\dfrac {2\pi}{3}$$ on arc between points $$z = -3/2$$ and $$z= 2/3$$ lying below real axis.
  • a segment of a circle subtending angle $$\dfrac {2\pi}{3}$$ on arc between points $$z = 3/2$$ and $$z= -2/3$$ lying above real axis.
Find the principal argument of the complex number $$sin \displaystyle \frac{6\pi}{5} + i\left(1 + cos \displaystyle \frac{6\pi}{5}\right)$$.  
  • $$arg(z)=\displaystyle \frac{9\pi}{10}, \left|z\right| = -2 cos \displaystyle \frac{3\pi}{5}$$
  • $$arg(z)=\displaystyle \frac{\pi}{10}, \left|z\right| = -2 cos \displaystyle \frac{3\pi}{5}$$
  • $$arg(z)=\displaystyle \frac{9\pi}{10}, \left|z\right| = 2 cos \displaystyle \frac{3\pi}{5}$$
  • $$arg(z)=\displaystyle \frac{9\pi}{10}, \left|z\right| = -2 cos \displaystyle \frac{2\pi}{5}$$
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
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 $$\displaystyle z= 1+i\sqrt{3}$$,then $$\displaystyle z^{6}$$ equals
  • 32
  • -32
  • 64
  • None of these
Find the value of $$k$$ for which given equation has real and equal roots.
$$(k\,-\,12)\,x^2\,+\,2\,(k\,-\,12)\,x\,+\,2\,=\,0$$
  • $$k\,=\,12$$
  • $$k\,=\,13$$
  • $$k\,=\,14$$
  • $$k\,=\,15$$
Determine the nature of the roots of the following equations from their discriminants.

$$y^2\,-\,4y\,-\,1\,=\,0$$
  • real and equal
  • real and unequal
  • complex
  • Cannot be determined
$$\text{arg(bi)}, \left( b>0 \right) $$ is
  • $$\pi$$
  • $$\displaystyle \frac { \pi  }{ 2 } $$
  • $$\displaystyle -\frac { \pi  }{ 2 } $$
  • $$0$$
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
  • $$\displaystyle x+y< 0$$
  • $$\displaystyle x-y> 0$$
  • $$\displaystyle x-y< 0$$
  • $$\displaystyle x+y> 0$$
$$\left| { z }_{ 1 }+{ z }_{ 2 } \right| =\left| { z }_{ 1 } \right| +\left| { z }_{ 2 } \right| $$ is possible if 
  • $${ z }_{ 2 }=\overline { { z }_{ 1 } } $$
  • $$\displaystyle { z }_{ 2 }=\frac { 1 }{ { z }_{ 1 } } $$
  • $$arg{ z }_{ 1 }=arg{ z }_{ 2 }$$
  • $$\left| { z }_{ 1 } \right| =\left| { z }_{ 2 } \right| $$
If arg $$\displaystyle z < 0$$ then arg $$\displaystyle \left (-z  \right )-$$ arg z is equal to
  • $$\displaystyle \pi$$
  • $$\displaystyle -\pi$$
  • $$\displaystyle -\frac{\pi }{2}$$
  • $$\displaystyle \frac{\pi }{2}$$
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
A quadratic equation with rational coefficients can have
  • both roots equal and irrational
  • one root rational and other irrational
  • one root real and other imaginary
  • none of these
$$\displaystyle \left [ \left ( \cos \theta +i \sin \theta \right )\left ( \cos \theta -i\sin \theta  \right ) \right ]^{-1}$$
  • $$\displaystyle i$$
  • $$\displaystyle 1$$
  • $$\displaystyle -i$$
  • $$\displaystyle -1$$
The inequality $$|z-4| < |z-2|$$ represents the region given by 
  • $$Re(z) > 1$$
  • $$Re(z) < 2$$
  • $$Re(z) > 0$$
  • None of these
Solve:
$$\displaystyle \left ( x+iy \right )\left ( 2-3i \right )= 4+i$$
  • $$\displaystyle x= \left ( 8/13 \right ), y= -\left ( 14/13 \right ).$$
  • $$\displaystyle x= \left ( 5/13 \right ), y= \left ( 14/13 \right ).$$
  • $$\displaystyle x= -\left ( 14/13 \right ), y= \left ( 5/13 \right ).$$
  • $$\displaystyle x= \left ( 14/13 \right ), y=- \left ( 8/13 \right ).$$
Solve : $$(2-\sqrt{-100})(1+\sqrt{-36})$$
  • $$62+2i$$
  • $$52+2i$$
  • $$-52+2i$$
  • $$-88+2i$$
Find arg($$\displaystyle 1+\sqrt{2}+i$$)
  • $$\displaystyle \pi /16.$$
  • $$\displaystyle \pi /8.$$
  • $$\displaystyle \pi /12.$$
  • $$\displaystyle \pi /10.$$
Determine the nature of roots of the following equation from the discriminant:
$$y^{2}\, -\, 5y\, +\, 11\, =\, 0$$
  • Real and equal
  • Real and unequal
  • nonreal.
  • None of these
Find the nature of the roots of the following quadratic equation. If the real roots exist, find them.
$$\displaystyle 2x^2 - 3x + 5 =0$$
  • $$x=0$$ and $$ x=-2$$
  • $$x=3, x=-6$$
  • No real root.
  • None of these
Find the value of $$p$$ for which the quadratic equation $$x^2 + p(4x + p - 1) + 2 = 0$$ has equal roots ?
  • $$\displaystyle -1, \frac{2}{3}$$
  • $$3 , 5$$
  • $$\displaystyle -1, \frac{4}{3}$$
  • $$\displaystyle \frac{3}{4}, 2$$
Find the values of $$k$$ for each of the following quadratic equation, so that they have two real equal roots.
$$\displaystyle 2x^2 + kx + 3 =0$$
  • $$k=\pm 2\sqrt{3}$$
  • $$k=\pm \sqrt{3}$$
  • $$k=\pm 2\sqrt{6}$$
  • $$k=\pm \sqrt{6}$$
The nature of the roots of the equation $$x^2 - 5x + 7 = 0$$ is
  • No real roots
  • 1 real root and 1 imaginary
  • Real and unequal
  • Real and equal
The equation $$x^2 - px + q = 0\  p, q \in R$$ has no real roots if 
  • $$p^2 > 4q$$
  • $$p^2 < 4q$$
  • $$p^2 = 4q$$
  • None of these
Find the roots of the following quadratic equation by using the quadratic formula
$$4x^2 + 3x + 5 =0$$
  • $$\displaystyle \frac { 3\pm \sqrt { 71 } i }{ 8 } $$
  • $$\displaystyle \frac { 3\pm \sqrt { 89 } i }{ 8 } $$
  • $$\displaystyle \frac { -3\pm \sqrt { 71 } i }{ 8 } $$
  • $$\displaystyle \frac { -3\pm \sqrt { 89 } i }{ 8 } $$
$$ax^2 + bx + c = 0,$$ where $$a, b, c$$ are real, has real roots if 
  • $$a, b, c$$ are integers
  • $$b^2> 3ac$$
  • $$ac > 0$$ 
  • $$c = 0$$
If $$p, q, r$$ are real and $$\displaystyle p\neq q,$$ then roots of the equation $$\displaystyle \left ( p-q \right )x^{2}+5\left ( p+q \right )x-2\left ( p-q \right )=0$$ are
  • Real and equal
  • Complex
  • Real and unequal
  • None of these
The roots of $$a^2x^2 + abx = b^2, a \neq 0, b \neq 0$$ are:
  • Equal
  • Non- real
  • Unequal
  • None of these
If $$-2$$ is a root of the quadratic equation $$x^2 + px + 2 = 0$$ and the quadratic equation $$2x^2 + px+ k = 0$$ has equal roots, find the value of $$k$$.
  • $$\displaystyle k = \frac{8}{9}$$
  • $$\displaystyle k = -\frac{8}{9}$$
  • $$\displaystyle k = -\frac{9}{8}$$
  • $$\displaystyle k = \frac{9}{8}$$
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 discriminant of the equation and the nature of roots. Also find the roots.
$$6x^2 + x - 2 = 0$$
  • $$D=49$$, Real and distinct roots: $$\displaystyle \frac{1}{5}, \frac{-2}{3}$$
  • $$D=39$$, Real and distinct roots: $$\displaystyle \frac{1}{2}, \frac{-2}{3}$$
  • $$D=49$$, Real and distinct roots: $$\displaystyle \frac{1}{3}, \frac{-7}{3}$$
  • $$D=49$$, Real and distinct roots: $$\displaystyle \frac{1}{2}, \frac{-2}{3}$$
Find the modulus and amplitude of $$-2i$$
  • $$|z|=2; amp(z)=-\dfrac {3\pi}{2}$$
  • $$|z|=2i; amp(z)=\dfrac {\pi}{2}$$
  • $$|z|=2; amp(z)=\dfrac {\pi}{2}$$
  • $$|z|=2; amp(z)=-\dfrac {\pi}{2}$$
Find the modulus and amplitude of $$-2 + 2 \sqrt 3i$$
  • $$|z|=2\sqrt [ 2 ]{ 2 } ; amp(z)=\dfrac {\pi}{3}$$
  • $$|z|=2\sqrt [ 2 ]{ 2 } ; amp(z)=\dfrac {2\pi}{3}$$
  • $$|z|=4; amp(z)=\dfrac {2\pi}{3}$$
  • $$|z|=4; amp(z)=\dfrac {\pi}{3}$$
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