CBSE Questions for Class 11 Engineering Maths Complex Numbers And Quadratic Equations Quiz 10 - MCQExams.com

$$\left ( x^{2} + 1 \right )^{2} - x^{2} =0$$ has
  • four real roots
  • two real roots
  • no real roots
  • one real root
The argument of the complex number $$\sin \frac{6\pi }{5}+i\left ( 1+\cos \frac{6\pi }{5} \right )$$ is
  • $$\displaystyle \frac{6\pi }{5}$$
  • $$\displaystyle\frac{5\pi }{6}$$
  • $$\displaystyle\frac{9\pi }{10}$$
  • $$\displaystyle\frac{2\pi }{5}$$
The equation $$\sqrt {3x^2+x+5}=x-3$$, where $$x$$ is real, has 
  • no solution.
  • exactly one solution.
  • exactly two solutions.
  • exactly four solutions.
If $$ \displaystyle z_{0}=\frac{1-i}{2}$$,  then $$ \displaystyle \left (1+z_{0}  \right )\left (1+z_{0}^{{2}^{1}}  \right )\left (1+z_{0}^{{2}^{2}}  \right ).......... \left (1+z_{0}^{2^n}  \right )$$  must be
  • $$(1-i)(1+\dfrac{1}{2^{2^n}})$$ for $$n>1$$
  • $$(1-i)(1-\dfrac{1}{2^{2^n}})$$ for $$n>1$$
  • $$\dfrac{1+i}{2}$$ for $$n>1$$
  • $$(1-i)(1-\dfrac{1}{2^{2^{n+1}}})$$ for $$n>1$$
The number of solutions of $$log _{\frac{1}{5}}log_{\frac{1}{2}}(\left | z \right |^{2}+4\left | z \right |+3)< 0$$ is/are?
  • $$0$$
  • $$2$$
  • $$4$$
  • infinite
Let $$\alpha,\ \beta$$ be real and $$\mathrm{z}$$ be a complex number. If $$\mathrm{z}^{2}+\alpha \mathrm{z}+\beta=0$$ has two distinct roots on the line $$Re(z) =1$$, then it is necessary that: 
  • $$\beta\in(0,1)$$
  • $$\beta\in(-1,0)$$
  • $$|\beta|=1$$
  • $$\beta\in(1, \infty)$$
Let $$z$$ be a complex number and $$c$$ be a real number $$\geq $$ 1 such that z + $$c\left | z+1 \right |+i=0 ,$$ then $$c$$ belongs to 
  • $$[2, 3]$$
  • $$(3, 4)$$
  • $$[1,\sqrt{2}]$$
  • None of these
lf $$\displaystyle \log_{\tan 30^{\circ}}\left(\frac{2|Z|^{2}+2|Z|-3}{|z|+1}\right) <-2$$ then
  • $$|\displaystyle \mathrm{z}|<\frac{3}{2}$$
  • $$|z|>\displaystyle \frac{3}{2}$$
  • $$|z|>2$$
  • $$|z|<2$$
If $$x = 2 + 5i($$where $$1 i = \sqrt{-1})$$ and $$2(\displaystyle \frac{1}{1! 9!}  + \frac{1}{3! 7!}) + \frac{1}{5! 5!} = \frac{2^{a}}{b!}$$ then $$ x^{3}-5x^{2}+33x-10 = $$
  • $$a + b$$
  • $$b - a$$
  • $$a-b$$
  • $$-a-b$$
  • $$(a - b)(a + b)$$
If $$x=9^{\frac {1}{3}} 9^{\frac {1}{9}} 9^{\frac {1}{27}} .....\infty, y=4^{\frac {1}{3}} 4^{\frac {-1}{9}} 4^{\frac {1}{27}}....\infty,$$ and $$z=\sum_{r=1}^{\infty} (1+i)^{-r}$$, then $$arg (x+yz)$$ is equal to
  • 0
  • $$\tan^{-1}(\frac {\sqrt 2}{3})$$
  • $$-\tan^{-1}(\frac {\sqrt 2}{3})$$
  • $$- \tan^{-1}(\frac {2}{\sqrt 3})$$
If $$\left | \log_{\sqrt{3}} \frac{\left | z \right |^{2}-\left | z \right |+1}{2+{}\left | z \right |}\right |< 2$$, then
  • $$|\displaystyle \mathrm{z}|<\frac{1}{3}$$
  • $$|\mathrm{z}|=1$$
  • $$|\mathrm{z}|=5$$
  • $$1<|\mathrm{z}|<5$$
Given $$z$$ is a complex number with modulus $$1$$. Then the equation in $$a$$, $$\left(\dfrac{1+ia}{1-ia}\right )^4=z$$ has
  • all roots real and distinct.
  • two real and two imaginary.
  • three roots real and one imaginary.
  • one root real and three imaginary.
Find the value of $$x$$ such that $$\displaystyle \frac{(x + \alpha)^2 - (x + \beta)^2}{ \alpha + \beta} = \frac{sin  2 \theta}{sin^2  \theta}$$. when $$\alpha$$ and $$\beta $$ are the roots of $$t^2 - 2t + 2 = 0$$
  • $$x = icot \, \, \, \theta - 1$$
  • $$x = -(icot \, \, \, \theta + 1$$)
  • $$x = icot \, \, \, \theta $$
  • $$x = itan \, \, \, \theta - 1$$
The complex numbers $$\sin  x + i  \cos  2x$$  and  $$\cos  x - i  \sin  2x$$ are conjugate to each other, for
  • $$x = n\pi$$
  • $$x \displaystyle = \left ( n + \frac{1}{2} \right ) \pi$$
  • $$x = 0$$
  • No value of $$x$$
If $$z_1, z_2$$ be two non zero complex numbers satisfying the equation $$\displaystyle \left | \frac{z_1 + z_2}{z_1 - z_2} \right | = 1$$ then $$\displaystyle \frac{z_1}{z_2} + \left ( \frac{z_1}{z_2} \right )$$ is
  • zero
  • 1
  • purely imaginary
  • 2
Interpret the following equations geometrically on the Argand plane.
$$1 < |z - 2 - 3 i| < 4$$
  • Annular
  • Straight line
  • A point
  • Ring
If n is a natural number $$\geq 2$$, such that $$z^n=(z+1)^n$$, then
  • Roots of equation lie on a straight line parallel to the $$y-axis$$
  • Roots of equation lie on a straight line parallel to the $$x-axis$$
  • Sum of the real parts of the roots is $$-[(n-1)/2]$$
  • 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
  • $$\displaystyle \frac {1}{2}(z+1)+i$$
  • $$\displaystyle \frac {1}{2}(iz+1)+i$$
  • $$\displaystyle \frac {1}{2}(iz-1)+i$$
  • $$\displaystyle \frac {1}{2}(z+i)+1$$
If $$b_1b_2=2(c_1+c_2)$$, then at least one of the equations $$x^2+b_1x+c_1=0$$ and $$x^2+b_2x+c_2=0$$ has
  • imaginary roots
  • real roots
  • purely imaginary roots
  • none of these
Find the regions of the z-plane for which $$\displaystyle \left | \frac{z - a}{z + \overline a} \right | < 1, = 1$$ or $$> 1$$. when the real part of a is positive.
  • The required regions are the right half of the z-pane, the imaginary axis and the left half of the z-plane respectively.
  • The required regions are the left half of the z-pane, the imaginary axis and the right half of the z-plane respectively.
  • The required regions are the right half of the z-pane the real axis and the left half of the z-plane respectively.
  • The required regions are the left half of the z-pane the real axis and the right half of the z-plane respectively.
If $$ \alpha, \beta  $$ are the roots of equation $$(k + 1) x^{2} - (20k + 14)x + 91k + 40 = 0 ; (\alpha < \beta)$$, $$k > 0$$, then 
The nature of the roots of this equation is
  • imaginary
  • real and distinct
  • equal real roots
  • None of these
If z be a complex number satisfying$$\displaystyle\ z^{4}+z^{3}+2z^{2}+z+1=0$$ then $$\displaystyle\ |z|$$ is 
  • $$\displaystyle\ \frac{1}{2}$$
  • $$\displaystyle\ \frac{3}{4}$$
  • $$\displaystyle\ 1$$
  • None of these
Find the range of real number $$\alpha$$ for which the equation $$z + \alpha |z - 1| + 2i = 0;  z= x + iy$$ has a solution. Find the solution.
  • $$\displaystyle x = 5/2, y = - 2$$
  • $$\displaystyle x = -2, y = 5/2$$
  • $$\displaystyle x = -5/2, y = 2$$
  • $$\displaystyle x = 2, y = -5/2$$
$$\displaystyle { \left( \frac { \sqrt { 3 } +i }{ 2 }  \right)  }^{ 6 }+{ \left( \frac { i-\sqrt { 3 }  }{ 2 }  \right)  }^{ 6 }=$$
  • $$-2$$
  • $$2$$
  • $$-1$$
  • $$1$$
Let $$\displaystyle\ z_{1}= a+ib, z_{2}= p+iq$$ be two unimodular complex numbers such that $$\displaystyle\ Im(z_{1}z_{2})=1$$. If$$\displaystyle\ \omega_{1}= a+ip, \omega_{2}=b+iq$$ then
  • $$\displaystyle\ Re(\omega_{1}\omega_{2})=1$$
  • $$\displaystyle\ Im(\omega_{1}\omega_{2})=1$$
  • $$\displaystyle\ Rm(\omega_{1}\omega_{2})=0$$
  • $$\displaystyle\ Im(\omega_{1}\bar{\omega_{2}})=0$$
Find all complex numbers satisfying the equation $$2|z|^2 + z^2 - 5 + i \sqrt{3} = 0$$
  • $$\displaystyle \pm \left ( \frac{\sqrt{6}}{2} + \frac{1}{\sqrt{2}} i \right ); \pm \left ( \frac{1}{\sqrt{6}} + \frac{3}{2} i \right )$$
  • $$\displaystyle \pm \left ( \frac{\sqrt{6}}{2} - \frac{1}{\sqrt{2}} i \right ); \pm \left ( \frac{2}{\sqrt{6}} - \frac{3}{2} i \right )$$
  • $$\displaystyle \pm \left ( \frac{\sqrt{6}}{2} - \frac{1}{\sqrt{3}} i \right ); \pm \left ( \frac{1}{\sqrt{6}} - \frac{3}{2} i \right )$$
  • $$\displaystyle \pm \left ( \frac{\sqrt{6}}{2} - \frac{1}{\sqrt{2}} i \right ); \pm \left ( \frac{1}{\sqrt{6}} - \frac{3}{\sqrt{2}} i \right )$$
Let $$\displaystyle z=1+i\:b=(a,b)$$  be any complex number, $$\displaystyle a,b,\epsilon R$$ and $$\displaystyle \sqrt{-1}=i.$$ Let $$\displaystyle z\neq 0+0i,arg z=\tan^{-1}\left (\frac{Im\:z}{Re\:z}\right)$$ where $$\displaystyle -\pi<arg z\leq \pi$$ 

$$\displaystyle arg(\bar{z})+arg(-z)=\left\{\begin{matrix}\pi, \; if\: arg (z)<0 & \\ -\pi, \; if\: arg (z)>0 & \end{matrix}\right.$$

Let $$z$$ & $$w$$ be non-zero complex numbers such that they have equal modulus values and $$\displaystyle arg z- arg  \bar{w} =\pi,$$ then z equals
  • $$-w$$
  • $$w$$
  • $$\displaystyle -\bar{w}$$
  • $$\displaystyle \bar{w}$$
The root of $$(x + a) (x + b) - 8k = (k - 2)^2$$ are real and equal, when $$a,b,c$$ $$\epsilon$$ R, then
  • $$a + b = 0$$
  • $$a =b$$
  • $$k = -3$$
  • $$k = 0$$
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
If $$a, b, c$$ are real distinct numbers satisfying the condition $$a + b + c = 0$$, then the roots of the quadratic equation $$3ax^2 + 5bx + 7c = 0$$ are
  • positive
  • negative
  • real and distinct
  • imaginary
Find the modulus, argument and the principal argument of the complex numbers.
$$z=1+cos\frac {10\pi}{9}+i sin \left (\frac {10\pi}{9}\right )$$
  • Principal Arg $$z=-\frac {4\pi}{9}; |z|=2 cos \frac {4\pi}{9}; Arg z=2 k\pi -\frac {4\pi}{9} k\epsilon l$$
  • Principal Arg $$z=-\frac {10\pi}{9}; |z|=2 cos \frac {10\pi}{9}; Arg z=2 k\pi -\frac {10\pi}{9} k\epsilon l$$
  • Principal Arg $$z=-\frac {-10\pi}{9}; |z|=2 cos \frac {-10\pi}{9}; Arg z=2 k\pi -\frac {4\pi}{9} k\epsilon l$$
  • Principal Arg $$z=-\frac {-4\pi}{9}; |z|=2 cos \frac {-4\pi}{9}; Arg z=2 k\pi -\frac {4\pi}{9} k\epsilon l$$
Find the modulus, argument and the principal argument of the complex numbers.
$$(tan 1-i)^2$$
  • $$Modulus =\sec^21, Arg (z) = 2 n\pi +(2-  \pi), Principal\ Arg (z) = (2 -\pi)$$
  • $$Modulus =\text{cosec}^21, Arg (z) = 2 n\pi -(2-  \pi), Principal\ Arg (z) = (-2-  \pi)$$
  • $$Modulus =\sec^21, Arg (z) = 2 n\pi -(2-  \pi), Principal\ Arg (z) = (-2-  \pi)$$
  • $$Modulus =\text{cosec}^21, Arg (z) = 2 n\pi +(2-  \pi), Principal\ Arg (z) = (2-  \pi)$$
The set of all real values of $$p$$ for which the equation $$x + 1 = \displaystyle \sqrt{px}$$ has exactly one root is
  • {0}
  • {4}
  • {0, 4}
  • {0,2}
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.
  • $$2(cos\dfrac{\pi}{4}+i\,sin\dfrac{\pi}{4})$$
  • $$-2(cos\dfrac{\pi}{4}-i\,sin\dfrac{\pi}{4})$$
  • $$2\sqrt{2}(cos\dfrac{3\pi}{4}+i\,sin\dfrac{3\pi}{4})$$
  • $$2\sqrt{2}(cos\dfrac{7\pi}{4}+i\,sin\dfrac{7\pi}{4})$$
  • $$2\sqrt{2}(cos\dfrac{5\pi}{4}+i\,sin\dfrac{5\pi}{4})$$
If $$a> 0$$ and discriminant of $$a{ x }^{ 2 }+2bx+c$$ is -ve then
$$\begin{vmatrix} a & b & ax+b \\ b & c & bx+c \\ ax+b & bx+c & 0 \end{vmatrix}$$ is equal to
  • +ve
  • $$\left( ac-{ b }^{ 2 } \right) \left( a{ x }^{ 2 }+2bx+c \right) $$
  • -ve
  • $$0$$
If $$(cos\theta+i\, sin \theta)\,\,( cos\,2\theta+i\,sin\,\theta)$$ ....
$$(cos\, n\theta +i\,sin\,n\theta) = 1$$, then the value of $$\theta$$ is
  • $$\dfrac{2m\pi}{n\,(n+1)}$$
  • $$4\,m\,\pi$$
  • $$\dfrac{4\,m\pi}{n\,(n+1)}$$
  • $$\dfrac{m\pi}{n\,(n+1)}$$
Add and express in the form of a complex number $$a+bi$$
$$(2+3i)+(-4+5i)-\dfrac {(9-3i)}{3}$$
  • $$-4+9i$$
  • $$-5+9i$$
  • $$2+9i$$
  • $$-5+8i$$
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 the equation $$ \displaystyle 4x^{2}+x\left ( p+1 \right )+1=0 $$  has exactly two equal roots , then one of the value of $$p$$ is
  • $$5$$
  • $$-3$$
  • $$0$$
  • $$3$$
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 the circle $$|z|=2$$ then the value of $$|{z}_{1},{z}_{2},..{z}_{n}|-4|\dfrac {1}{{z}_{1}}+\dfrac {1}{{z}_{2}}++\dfrac {1}{{z}_{n}}|=$$
  • $$0$$
  • $$n$$
  • $$-n$$
  • $$1$$
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.
  • $$Re (z) = 56$$
  • $$Re (z) = 61$$
  • $$Re (z) = 54$$
  • $$Re (z) = 59$$
If $$z=\sqrt{20i-21}+\sqrt{21+20i}$$, then the principal value of arg 'z' can be 
  • $$\dfrac{\pi}{4}$$
  • $$\dfrac{3\pi}{4}$$
  • $$-\dfrac{3\pi}{4}$$
  • $$-\dfrac{\pi}{4}$$
What is $${ i }^{ 1000 }+{ i }^{ 1001 }+{ i }^{ 1002 }+{ i }^{ 1003 }$$ equal to (where $$i=\sqrt { -1 } $$)?
  • $$0$$
  • $$i$$
  • $$-i$$
  • $$1$$
If $$Z_{1},Z_{2}$$ 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 $$z_1+ z_2 + z_3 = 0$$ and $$|z_1|=|z_2|=|z_3|= 1$$, then area of triangle whose vertices are $$z_1, z_2$$ and $$z_3$$ is:
  • $$\dfrac{3 \sqrt{3}}{4}$$
  • $$\dfrac{\sqrt{3}}{4}$$
  • $$1$$
  • $$2$$
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