CBSE Questions for Class 11 Engineering Maths Complex Numbers And Quadratic Equations Quiz 1 - MCQExams.com

Express $$\dfrac{1}{(1 -  cos  \theta  +  2  i  sin  \theta)}$$ in the form $$x + iy$$
  • $$\left(\displaystyle \frac{1}{5 + 3 cos \theta}\right) + \left(\displaystyle \frac{2 cot \theta/2}{5 + 3 cos \theta}\right)i$$
  • $$\left(\displaystyle \frac{1}{5 - 3 cos \theta}\right) + \left(\displaystyle \frac{-2 cot \theta/2}{5 - 3 cos \theta}\right)i$$
  • $$\left(\displaystyle \frac{1}{5 + 3 cos \theta}\right) + \left(\displaystyle \frac{-2 cot \theta/2}{5 + 3 cos \theta}\right)i$$
  • $$\left(\displaystyle \frac{1}{5 - 3 cos \theta}\right) + \left(\displaystyle \frac{2 cot \theta/2}{5 - 3 cos \theta}\right)i$$
If $$z = x + iy$$ and $$\omega = \dfrac{(1 -iz)}{(z-i)}$$, then $$\left|\omega\right| = 1$$ 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 $$a, b$$ and $$c$$ are real numbers then the roots of the equation $$(x - a)(x - b) + (x - b)(x - c) + (x - c)(x - a) = 0$$ are always
  • Real
  • Imaginary
  • Positive
  • Negative
If $$(x+iy)(2-3i)=4+i$$ then (x, y) =
  • $$\left ( 1,\dfrac{1}{13} \right )$$
  • $$\left ( -\dfrac{5}{13},\dfrac{14}{13} \right )$$
  • $$\left ( \dfrac{5}{13},\dfrac{14}{13} \right )$$
  • $$\left ( -\dfrac{5}{13},-\dfrac{14}{13} \right )$$
If $$z =3+5i$$, then $$z^3+z+198=$$
  • $$3 - 15i$$
  • $$-3 - 15i$$
  • $$-3 + 15i$$
  • $$3 + 15i$$
If $$z=2-3i$$ then $$z^2-4z+13=$$
  • $$0$$
  • $$1$$
  • $$2$$
  • $$3$$
The complex number $$\displaystyle \frac{1+2i}{1-i}$$ lies in the quadrant :
  • I
  • II
  • III
  • IV
$$\sqrt{-3}\sqrt{-75}=$$
  • $$15$$
  • $$15i$$
  • $$-15$$
  • $$-15i$$
The sum of two complex numbers $$a + ib$$ and $$c +id$$ is a real number if
  • $$a + c = 0$$
  • $$b + d = 0$$
  • $$a + b= 0$$
  • $$b + c = 0$$
The locus of complex number z such that z is purely real and real part is equal to - 2 is
  • Negative y-axis
  • Negative x-axis
  • The point (-2, 0)
  • The point (2, 0)
$$\dfrac{1}{i-1}+\dfrac{1}{i+1}$$ is
  • positive rational number
  • purely imaginary
  • positive Integer
  • negative integer
The argument of every complex number is
  • Double valued
  • Single valued
  • Many valued
  • Triple valued
The sum of two complex numbers $$a + ib$$ and $$c+ id$$ is purely imaginary if
  • $$a + c = 0$$
  • $$a + d = 0$$
  • $$b + d = 0$$
  • $$b + c = 0$$
For $$a < 0$$, arg $$(ia) = $$ 
  • $$\dfrac{\pi }{2}$$
  • $$-\dfrac{\pi }{2}$$
  • $$\pi $$
  • $$-\pi $$
The principal value of the argument of $$-\sqrt{3}+i$$ is :
  • $$\dfrac{\pi }{6}$$
  • $$\dfrac{3\pi }{6}$$
  • $$\dfrac{5\pi }{6}$$
  • $$\dfrac{7\pi }{6}$$
Amplitude of $$\dfrac{1+i}{1-i}$$ is :
  • $$0$$
  • $$\pi $$
  • $$\dfrac{\pi }{2}$$
  • $$-\pi $$
Which of the following equations has two distinct real roots ?
  • $$2x^2-3\sqrt 2 x+\dfrac 94=0$$
  • $$x^{2}+x-5=0$$
  • $$x^{2}+3x+2\sqrt{2}=0$$
  • $$5x^{2}-3x+1=0$$
A quadratic equation $$ax^2 + bx+c=0$$ has two distinct real roots, if 
  • $$a=0$$
  • $$b^2-4ac = 0$$
  • $$b^2-4ac < 0$$
  • $$b^2-4ac > 0$$
For $$a > 0$$, arg $$(ia) =$$
  • $$\dfrac{\pi }{2}$$
  • $$-\dfrac{\pi }{2}$$
  • $$\pi $$
  • $$-\pi $$
The modulus of $$\sqrt{2}i-\sqrt{-2}i$$ is:
  • 2
  • $$\sqrt{2}$$
  • 0
  • $$2\sqrt{2}$$
The roots of the equation $$3x^{2} - 4x + 3 = 0$$ are :
  • real and unequal
  • real and equal
  • imaginary
  • none of these
For $$a<0$$,  arg $$a=$$
  • $$\dfrac{\pi }{2}$$
  • $$\dfrac{-\pi }{2}$$
  • $$\pi $$
  • $$-\pi $$
If the square of $$(a + ib)$$ is real, then $$ ab=$$
  • $$0$$
  • $$1$$
  • $$-1$$
  • $$2$$
Find the argument of $$-1 - i\sqrt{3}$$
  • $$\theta= -2\pi/3$$
  • $$\theta= 2\pi/3$$
  • $$\theta= -4\pi/3$$
  • $$\theta= 4\pi/3$$
The roots of $$x^{2}-x+1=0$$ are:
  • Real and equal
  • Real and not equal
  • Imaginary
  • Reciprocals
Nature of the roots of the quadratic equation $$2x^{2}-2\sqrt{6}x+3=0$$ is:
  • Real, irrational, unequal
  • Real, rational, equal
  • Real, rational, unequal
  • Complex
Determine the nature of roots of the equation $$x^2 + 2x\sqrt{3}+3=0$$.
  • Real and distinct
  • Non-real and distinct
  • Real and equal
  • Non-real and equal
Find the value of $$x$$ of the equation $${ \left( 1-i \right)  }^{ x }={ 2 }^{ x }$$ 
  • $$1$$
  • $$2$$
  • $$0$$
  • none of these
If the discriminant of a quadratic equation is negative, then its roots are:
  • unequal
  • equal
  • inverse
  • imaginary
Solve $$\displaystyle \left ( 1-i \right )x+\left ( 1+i \right )y= 1-3i,$$
  • $$\displaystyle x= -1, y= 2.$$
  • $$\displaystyle x= 2, y= -1.$$
  • $$\displaystyle x= 2, y= 1.$$
  • $$\displaystyle x= 1, y= 2.$$
The roots of $$4x^{2}-2x+8=0$$ are:
  • Real and equal
  • Rational and not equal
  • Irrational
  • Not real
Evaluate :
 $$\sqrt{-25} + 3 \sqrt{-4} +2 \sqrt{-9}$$
  • $$-17i$$
  • $$5i$$
  • $$17i$$
  • $$6i$$
If $$x^{2}-2px+8p-15=0$$ has equal roots, then $$p=$$
  • $$3$$ or $$-5$$
  • $$3$$ or $$5$$
  • $$-3$$ or $$5$$
  • $$-3$$ or $$-5$$
Determine the values of $$p$$ for which the quadratic equation $$2x^2 + px + 8 = 0$$ has equal roots.
  • $$p=\pm 64$$
  • $$p=\pm 8$$
  • $$p=\pm 4$$
  • $$p=\pm 16$$
Find the values of $$k$$ for the following quadratic equation, so that they have two real and equal roots:
$$2x^2 + k x + 3 = 0$$
  • $$k = \pm 2\sqrt 3$$
  • $$k = \pm 2\sqrt 6$$
  • $$k = \pm \sqrt 6$$
  • $$k = \pm \sqrt 3$$
$$\displaystyle \frac{\displaystyle i^{4n + 3} + (-i)^{8n - 3}}{\displaystyle(i)^{12 n- 1} - i^{2 - 16 n}}, n    \varepsilon N$$ is equal to
  • 1 + i
  • 2i
  • -2i
  • -1 - i
1+$$i^2 + i^4 + i^6 + ........+ i^{2n}$$ is
  • Positive
  • Negative
  • Zero
  • Cannot be determined
Find the modulus and the principal value of the argument of the number $$1-i$$
  • $$\displaystyle \sqrt{2},\pi/4$$
  • $$\displaystyle \sqrt{2},-\pi/4$$
  • $$\displaystyle \sqrt{2},-\pi/3$$
  • $$\displaystyle \sqrt{2},3\pi/4$$
If $$i^2 = - 1$$, then the value of $$\displaystyle \sum^{200}_{n = 1} i^n $$ is
  • 50
  • -50
  • 0
  • 100
If i = $$\sqrt {-1}, then  1 + i^2 + i^3 -i^6 + i^8 $$ is equal to -
  • 2- i
  • 1
  • -3
  • -1
Check whether $$2x^2 - 3x + 5 = 0$$ has real roots or no.
  • The equation has real roots.
  • The equation has no real roots.
  • Data insufficient
  • None of these
($$i^{10}+1) (i^9 + 1)(i^8 +1).......(i+1)$$  equal to 
  • -1
  • 1
  • 0
  • i
If $$i^2$$ $$= -1$$, then find the odd one out of the following expressions.
  • $$-i^2$$
  • $$(-i)^2$$
  • $$i^4$$
  • $$(-i)^4$$
  • $$-i^6$$
When $$(3-2i)$$ is subtracted from $$(4 + 7i)$$, then the result is
  • $$1 + 5i$$
  • $$1 + 9i$$
  • $$7 + 5i$$
  • $$7 + 9i$$
The value of k for which polynomial $$x^{2} - kx + 4$$ has equal zeroes is
  • $$4$$
  • $$2$$
  • $$-4$$
  • $$-2$$
If the discriminant of a quadratic equation is negative, then its roots are
  • Unequal
  • Equal
  • Inverses
  • Imaginary
If the equation $$(1 + m^{2}) x^{2} + 2mcx + (c^{2} - a^{2}) = 0$$ has equal roots, then $$c^{2} =$$ 
  • $$ a^{2} (1 + m^{2})$$
  • $$ a (1 + m^{2})$$
  • $$ a^{4} (1 - m^{2})$$
  • $$ a^{2} (1 - m^{2})$$
Amplitude of $$\displaystyle \frac{1 +\sqrt 3i}{ \sqrt3 + i} $$is
  • $$\displaystyle \frac {\pi}{3}$$
  • $$\displaystyle \frac {\pi}{2}$$
  • 0
  • $$\displaystyle \frac {\pi}{6}$$
For $$i=\sqrt{-1}$$, what is the sum $$\left(7+3i\right) + \left(-8+9i\right)$$?
  • $$-1+12i$$
  • $$-1-6i$$
  • $$15+12i$$
  • $$15-6i$$
If the equation $$\displaystyle x^{2}-bx+1=0$$ does not possess real roots then
  • $$-3 < b < 3$$
  • $$-2 < b < 2$$
  • $$b > 2$$
  • $$b < -2$$
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