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

(x2+1)2x2=0 has
  • four real roots
  • two real roots
  • no real roots
  • one real root
The argument of the complex number sin6π5+i(1+cos6π5) is
  • 6π5
  • 5π6
  • 9π10
  • 2π5
The equation 3x2+x+5=x3, 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
0:0:2


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