Complex Numbers and Quadratic Equations-Quiz-10

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Complex Number and Quadratic Equations Quiz-10

Dear Readers,

Complex numbers and quadratic equations is a segment of maths that deals with crucial theorems and concepts along with various formulae. It comprises of linear and quadratic equations along with roots related to the complex number's set (known as complex roots)..

Q1.  The number of integral values of a for which the quadratic equation (x+a)(x+1991)+1=0 has integral roots are

•  3
•  0
•  1
•  2

Solution
(d)
(x+a)(x+1991)+1=0
⇒(x+a)(x+1991)=-1
⇒(x+a)=1 and x+1991=-1
⇒a=1993
or x+a=-1 and x+1991=1 ⇒a=1989

Q2. Let z and ω be two complex numbers such that |z| ≤ 1, |ω| ≤ 1 and |z - iω|= |z - i¯ω|=2 then z equals

•  1 or i
•  i or –i
•  1 or -1
•  i or -1

Solution
(c)
We have, 2 = |z-iω| ≤ |z| + |ω| (∵|z₁+z₂ |≤|z₁ |+|z₂ |)
∴|z| + |ω| ≥ 2 (i)
But given that |z|≤1 and |ω|≤1. Hence
⇒|z| + |ω|≤2 (ii)
From (i) and (ii),
|z|=|ω|=1
Also, |z+iω|=|z-i¯ω|
⇒|z-(-iω)|=|z-i¯ω|
Hence, z lies on perpendicular bisector of the line segment joining (-iω) and (i¯ω), which is a real axis, as (-iω) and (i¯ω) are conjugate to each other. For z,Im(z)=0. If z= x, then
|z| ≤ 1 ⇒ x²≤1
⇒ -1 ≤ x ≤ 1

Q3. z₁ and z₂ lie on a circle with centre at the origin. The point of intersection z₃ of the tangents at z₁ and z₂ is given by

•  1/2 (¯z₁+¯z₂)
•  (2z₁ z₂ )/( z₁+z₂ )
•  1/2 ( 1/z₁ + 1/z₂ )
•  (z₁+z₂)/(¯z₁ ¯z₂ )

Solution
(b)

Q4. If α and β be the roots of the equation x²+px-1/(2p²)=0 where p∈R. Then the minimum value of α⁴+β⁴ is

•  2√2
•  2 - √2
•  2
•  2 + √2

Solution
(d)

Q5. If α,β are the roots of ax² + c = bx, then the equation (a+ cy)² = b² y in y has the roots

•  αβ^(-1) ,α^(-1) β
•  α^(-2) ,β^(-2)
•  α^(-1) ,β^(-1)
•  α² ,β²

Solution
  (b)

Q6. If (cos⁡θ+i sin⁡θ )(cos2θ+i sin⁡2θ )⋯(cos⁡nθ+i sin⁡nθ ) = 1, then the value of θ is, m∈N

•  4mπ
•  2mπ/( n(n+1) )
• 4mπ/(n(n+1))
•  mπ/( n(n + 1))

Solution
(c)

Q7. The roots of the cubic equation(z + ab)³=a³, such that a ≠ 0, represent the vertices of a triangle of sides of length

•  1/√3 |ab|
•  √3|a|
•  √3|b|
•  |a|

Solution
(b)

Q8. A quadratic equation whose product of roots x₁ and x₂ is equal to 4 and satisfying the relation x₁/(x₁-1) + x₂/( x₂-1)=2 is

•  x²-2x+4=0
•  x²+2x+4=0
•  x²+4x+4=0
•  x²-4x+4=0

Solution
(a)

Q9.  If the equation cot⁴⁡x -2cosec²x + a² = 0 has at least one solution then, sum of all possible integral values of a is equal to

•  4
•  3
•  2
•  0

Solution
(d)

Q10. The number of irrational roots of the equation 4x/(x²+x+3)+5x/(x²-5x+3)=-3/2 is

•  4
•  0
•  1
• 2

Solution
(d)
Here, x=0 is not a root. Divide both the numerator and denominator by x and put x+3/x=y to obtain
4/(y+1)+5/(y-5)=-3/2⇒y=-5,3
x+3/x=-5 has two irrational roots and x+3/x=3 has imaginary roots

Written by: AUTHORNAME

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