CBSE Questions for Class 11 Engineering Maths Complex Numbers And Quadratic Equations Quiz 5 - MCQExams.com

Find the modulus and amplitude of $$-\sqrt 3-i$$
  • $$|z|=2; amp(z)=-\dfrac {5\pi}{6}$$
  • $$|z|=4; amp(z)=\dfrac {5\pi}{6}$$
  • $$|z|=4; amp(z)=-\dfrac {\pi}{6}$$
  • none of these
Find the value of $$x^3 + 7x^2 -x + 16$$, where $$x = 1 + 2i$$
  • $$-17 + 24i$$
  • $$24 + 17i$$
  • $$17 - 24i$$
  • $$24 - 17i$$

The amplitude of

$$\displaystyle sin \frac{\pi}{5} + i \left ( 1 - cos \frac{\pi}{5} \right )$$
  • $$\displaystyle \frac{\pi}{5}$$
  • $$\displaystyle \frac{2\pi}{5}$$
  • $$\displaystyle \frac{\pi}{10}$$
  • $$\displaystyle \frac{\pi}{15}$$
If $$z_1, z_2, \varepsilon C$$ are such that $$|z_1 + z_2|^2 = |z_1|^2 + |z_2|^2$$ then $$\displaystyle \frac{z_1}{z_2}$$ is
  • zero
  • purely real
  • purely imaginary
  • complex
$$(1 + i)^8 + (1 -i)^8 =$$
  • $$16$$
  • $$-16$$
  • $$32$$
  • $$-32$$
Modulus of $$\displaystyle \frac{cos \theta- isin\theta }{sin\theta - icos\theta} is$$
  • 0
  • 2$$\theta$$
  • $$\pi - 2\theta$$
  • none of these
The number of real solutions of the equation $$\displaystyle{\left(\frac{5}{7}\right)^2}=-x^2+2x-3$$ is equal to 
  • 2
  • zero
  • 1
  • none of these
Number of integer values of k for which the quadratic equation $$\displaystyle 2x^{2}+kx-4=0$$ will have two rational solutions is
  • $$1$$
  • $$2$$
  • $$4$$
  • $$5$$
Find the value of: $$i^2 + i^4 + i^6$$ +..... upto $$(2n +1)$$ terms.
  • $$i$$
  • $$-i$$
  • $$1$$
  • $$-1$$
$$i^n + i^{n + 1} + i^{n + 2}+ i^{n + 3} (n   \in   N) $$ is equal to
  • $$4$$
  • $$1$$
  • $$0$$
  • $$2$$
The argument of the complex number $$z = sin  \alpha + i (1 - cos  \alpha)$$ is
  • $$2 sin \displaystyle \frac{\alpha}{2}$$
  • $$\displaystyle \frac{\alpha}{2}$$
  • $$\alpha$$
  • none of these
Solve:
 $$\displaystyle \left|(1 + i)\frac{(2+i)}{(3 + i)}\right| $$
  • $$\dfrac{1}{2}$$
  • $$\dfrac{-1}{2}$$
  • $$1$$
  • $$-1$$
In the argand diagram, if $$O,P$$ and $$Q$$ represents respectively the origin and the complex numbers $$z$$ and $$z+iz$$ then the $$\angle OPQ$$ is
  • $$\displaystyle\frac {\pi}{4}$$
  • $$\displaystyle\frac {\pi}{3}$$
  • $$\displaystyle\frac {\pi}{2}$$
  • $$\displaystyle\frac {2\pi}{3}$$
The modulus of (1 + i) (1 + 2i) (1 + 3i) is equal to
  • $$\sqrt10$$
  • $$\sqrt 5$$
  • 5
  • 10
The number of real roots of the equation $$\displaystyle \frac{A^{2}}{x} +\frac{B^{2}}{x-1}=1$$ where A and B are real numbers not equal to zero simultaneously, is :
  • $$1 \;or\; 2 $$
  • $$1$$
  • $$2$$
  • None
Find the value of $$k$$ for which the quadratic equation $$\displaystyle \left ( k-2 \right )x^{2}+2\left ( 2k-3 \right )x+5k-6=0$$ has equal roots
  • 1
  • 3
  • A and B both
  • none of these
Number of complex numbers $$z$$ satisfying $$\left| 2z \right| =\left| 2z-1 \right| =\left| 2z+1 \right| $$ is equal to-
  • $$0$$
  • $$1$$
  • $$2$$
  • $$3$$
If both a and b belong to the set $$\displaystyle \left\{ 1,2,3,4 \right\}$$ then the number of equations of the form $$ ax^{2}+bx+1=0$$ having real roots is :
  • $$10$$
  • $$7$$
  • $$6$$
  • $$12$$
If 0 $$\leq$$ argz $$\leq \displaystyle \frac{\pi}{4}$$, then the least value of $$\sqrt 2 |2z - 4|$$ is
  • 6
  • 1
  • 4
  • 2
If $$\displaystyle z= (i)^{(i)^{(i)}}$$ where $$ i = \sqrt{-1}$$, then $$z$$ is equal to
  • $$-i$$
  • $$-1$$
  • $$1$$
  • $$i$$
The roots of the equations $$\displaystyle x^{2}-x-3=0$$ are
  • Imaginary
  • Rational
  • Irrational
  • None of these
If the roots of the equation $$\displaystyle \left ( a^{2}+b^{2} \right )x^{2}-2b\left ( a+c \right )x+\left ( b^{2}+c^{2} \right )=0 $$ are equal then
  • $$2b = ac$$
  • $$\displaystyle b^{2}=ac $$
  • $$\displaystyle b=\frac{2ac}{a+c} $$
  • b = ac
The value (s) of $$k$$ for which the quadratic equation $$\displaystyle kx^{2}-kx+1=0$$ has equal roots is
  • $$k = 0$$ only
  • $$k = 4$$ only
  • $$k = 0, 4$$
  • $$k = -4$$
Let $$z = x + iy$$ & amp $$(e^{z^2})$$ = amp $$(e^{(z+i)})$$. If $$y = (x)$$ is a function, then $$y(3)$$ is equal to 
  • $$\displaystyle \frac{1}{2}$$
  • $$\displaystyle \frac{1}{3}$$
  • $$\displaystyle \frac{1}{4}$$
  • $$\displaystyle \frac{1}{5}$$
The quadratic equations $$\displaystyle x^{2}-5x+3=0$$ has 
  • no real roots
  • two distinct real roots
  • two equal real roots
  • more than two real roots
Find the value of 
$$arg\left ( \left ( 1+i \right )^{i} \right )$$
  • $$\displaystyle \dfrac{1}{4}ln\left ( 2 \right )$$
  • $$\displaystyle \dfrac{1}{2}ln\left ( 2 \right )$$
  • $$\displaystyle \dfrac{1}{2}ln\left (\dfrac{1}{ 2} \right )$$
  • $$\displaystyle ln\left ( 2 \right )$$
Find the value of $$k$$ for which the equation $$(k+1){x}^{2}-2(k-1)x+1=0$$ has real and equal roots
  • $$1,-2$$
  • $$0,4$$
  • $$-1,3$$
  • $$0,3$$
If the roots of the equation $$\displaystyle x^{2}+px-6=0$$ are $$6$$ and $$-1$$ then the value of $$p$$ is
  • $$2$$
  • $$3$$
  • $$-5$$
  • $$5$$
The roots of the equation, where $$\displaystyle a\: \: \epsilon \: \: R$$  are 
$$\displaystyle x^{2}+ax-4=0$$
  • Real and distinct
  • Equal
  • Imaginary
  • Real
If the roots of the equation $$\displaystyle (a^{2}+b^{2})x^{2}-2b(a+c)x+(b^{2}+c^{2})=0$$ are equal then =
  • $$2b = ac$$
  • $$\displaystyle b^{2}=ac$$
  • $$\displaystyle b=\frac{2ac}{a+c}$$
  • $$b = ac$$
If the equation $$\displaystyle x^{2}-2kx-2x+k^{2}=0 $$ has equal roots the value of k must be
  • zero
  • either zero or $$\displaystyle -\frac{1}{2} $$
  • $$\displaystyle -\frac{1}{2} $$
  • either $$\displaystyle \frac{1}{2} $$ or $$\displaystyle -\frac{1}{2} $$
For what value of k will $$\displaystyle x^{2}-\left ( 3k-1 \right )x+2k^{2}+2k=11$$ have equal roots?
  • $$9, -5$$
  • $$-9, 5$$
  • $$9, 5$$
  • $$-9, -5$$
If the equation $$\displaystyle 16x^{2}+6kx+4=0$$ has equal roots, then the value of $$k$$ is 
  • $$\displaystyle \pm 8$$
  • $$\displaystyle \pm \frac{8}{3}$$
  • $$\displaystyle \pm \frac{3}{8}$$
  • $$0$$
Which of the following equation has two equal real toots ?
  • $$\displaystyle 3x^{2}+14x-5=0$$
  • $$\displaystyle 4x^{2}+2x-1=0$$
  • $$\displaystyle 9x^{2}-6x+1=0$$
  • $$\displaystyle x^{2}-5x+4=0$$
For what values of $$k$$ will the quadratic equation : $$\displaystyle { 2x }^{ 2 }-kx+1=0$$ have real and equal roots?
  • $$\displaystyle \pm 2\sqrt { 2 } $$
  • $$\displaystyle \pm \sqrt { 2 } $$
  • $$\displaystyle \pm 3\sqrt { 2 } $$
  • $$\displaystyle \pm \sqrt { 3 } $$
The roots of $$2{x}^{2}-6x+3=0$$ are
  • rational
  • equal
  • irrational
  • imaginary
Given that $$kx(x-2)+6=0$$ has real and equal roots, the root is
  • $$2$$
  • $$-1$$
  • $$1$$
  • $$\cfrac{1}{2}$$
Find the value of 'm' so that the equation $$\displaystyle { 9x }^{ 2 }-8mx-9=0$$ has one root as the negative of the other. 
  • 0
  • 1
  • 2
  • None of these
The value of $$k$$ for which $$4{x}^{2}-4\sqrt {3}x+k=0$$ is satisfied by only one real value of $$x$$ is
  • $$\sqrt 6$$
  • $$-3$$
  • $$3$$
  • $$\cfrac{1}{\sqrt {3}}$$
$$kx^2-2\sqrt 5x +4 = 0$$
For what value of $$k$$ will the quadratic equation have real and equal roots ?
  • $$\displaystyle \frac { 2 }{ 3 } $$
  • $$\displaystyle \frac { 4 }{ 5 } $$
  • $$\displaystyle \frac { 5 }{ 4 } $$
  • $$\displaystyle \frac { 3 }{ 2 } $$
For what value of 'k' the equation $$\displaystyle \left( k+3 \right) { x }^{ 2 }-\left( 5-k \right) x+1=0$$ has coincident roots ?
  • $$1, 13$$
  • $$1, 12$$
  • $$3, 13$$
  • None of these
For what value of $$k$$ will the quadratic equation $$\displaystyle { 2x }^{ 2 }+3x+k=0$$ have real equal roots ?
  • $$\displaystyle \frac { 3 }{ 8 } $$
  • $$\displaystyle \frac { 8 }{ 3 } $$
  • $$\displaystyle \frac { 9 }{ 8 } $$
  • $$\displaystyle \frac { 8 }{ 9 } $$
Find the value of $$k$$ for which the equation
$${x}^{2}+kx+81=0$$ and $${x}^{2}-6\sqrt {2}x+k=0$$ has both real roots, $$k> 0$$
  • $$k=9$$
  • $$k=18$$
  • $$k=3\sqrt {2}$$
  • No such value
Find the value of $$k$$ for which the equation $${x}^{2}-6x+k=0$$ has distinct roots.
  • $$k> 9$$
  • $$k=6,7$$ only
  • $$k<9$$
  • $$k=9$$
The values of $$b$$ for which the equation $$\displaystyle { 2bx }^{ 2 }-40x+25=0$$ has equal root is :
  • $$5$$
  • $$6$$
  • $$7$$
  • $$8$$
For what value of $$k$$ will the quadratic equation:  $$\displaystyle { kx }^{ 2 }+4x+1=0$$ have real and equal roots ?
  • 2
  • 3
  • 4
  • 1
If $$z_1, z_2$$ and $$z_3$$ are complex numbers such that $$|z_1| = |z_2| = |z_3| = \left | \displaystyle \frac{1}{z_1} + \frac{1}{z_2} + \frac{1}{z_3} \right | = 1,$$ then $$|z_1 + z_2 + z_3|$$ is
  • equal to 1
  • less than 1
  • greater than 3
  • equal to 3
If k be the ratio of the roots of the equation $$ \displaystyle x^{2}-px+q=0   $$   , the value of $$ \displaystyle \frac{k}{1+k^{2}}  $$  is
  • $$ \displaystyle \frac{q^{2}-2p}{p} $$
  • $$ \displaystyle \frac{q}{p^{2}-2q} $$
  • $$ \displaystyle \frac{p}{q^{2}-2p} $$
  • $$ \displaystyle \frac{p}{p^{2}-2q} $$
For what values of k, will quadratic equation $$\displaystyle { 9x }^{ 2 }+3kx+4=0$$ have real and equal roots? 
  • $$\displaystyle \pm 4$$
  • $$\displaystyle \pm 3$$
  • $$\displaystyle \pm 2$$
  • $$\displaystyle \pm 1$$
For what values of k will the quadratic equation :  $$\displaystyle { 3x }^{ 2 }+kx+3=0$$ have real equal roots? 
  • $$\displaystyle \pm \sqrt { 3 } $$
  • $$\displaystyle \pm \sqrt { 6 } $$
  • $$\displaystyle \pm 6$$
  • $$\displaystyle \pm 3$$
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