Complex Numbers and Quadratic Equations-Quiz-30

Complex Number & Quadratic Equation
Linked Comprehension Type
Quiz-30

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. Let z be a complex number satisfying z²+2zλ+1=0, where λ is a parameter which can take any real value

The roots of this equation lie on a certain circle if

•  -1<λ<1
•  b) λ>1
•  b) λ<1
•  None of these

Solution

Q2.Consider the equation az²+z+1=0 having purely imaginary root where a=cos⁡θ+i sin⁡θ,i=√(-1) and function f(x)=x³-3x²+3(1+cos⁡θ )x+5, then answer the following questions

Which of the following is true about f(x)?

•  f(x) decreases for x∈[2nπ,(2n+1)π],n∈Z
•  f(x) decreases for x∈[ (2n-1) π/2, (2n+1) π/2 ],n∈Z
•  f(x) is non-monotonic function
•  f(x) increases for x∈R

Solution

Q3.  Complex numbers z satisfy the equation |z-(4/z)|=2

The difference between the least and the greatest moduli of complex numbers is

•  2
•  4
•  1
•  3

Solution

Q4. Consider ∆ABC in Argand plane. Let A(0),B(1) and C(1+i) be its vertices and M be the mid-point of CA. Let z be a variable complex number on the line BM. Let u be another variable complex number defined as u=z²+1

Locus of u is?

•  Parabola
•  Ellipse
•  Hyperbola
•  None of these

Solution

Q5.In an Argand plane z₁,z₂ and z₃ are respectively, the vertices of an isosceles triangle ABC with Ac=BC and ∠CAB=θ. If z₄ is the centre of triangle, then

The value of AB×AC/(IA)² is

•  ((z₃-z₁ )(z₃-z₁))/(z₄-z₁ )²
•  ((z₂-z₁ )(z₁-z₃))/(z₄-z₁ )²
•  ((z₄-z₁ ))/((z₂-z₁)(z₃-z₁))
•  None of these

Solution
 

Q6. A(z₁ ),B(z₂ ),C(z₃) are the vertices of a triangle ABC inscribed in the circle |z|=2. Internal angle bisector of the angle A meets the circumcircle again at D(z₄)

Complex number representing point D is

•  z₄=1/z₂ +1/z₃
•  √( ( z₂+z₃ )/z₁ )
• √( ( z₂ z₃ )/z₁ )
•  z₄=√(z₂ z₃ )

Solution

Q7. Consider an unknown polynomial which when divided by (x-3) and by (x-4) leaves remainders as 2 and 1, respectively. Let R(x) be the remainder when this polynomial is divided by (x-3)(x-4)

If equation R(x)=x²+ax+1 has two distinct real root then exhaustive values of a are

•  (-2,2)
•  (-∞,-2)∪(2,∞)
•  (-2,∞)
•  All real numbers

Solution

Q8. Consider the quadratic equation ax²-bx+c=0,a,b,c∈N, which has two distinct real root belonging to the interval (1, 2)

The least value of a is

•  4
•  6
•  7
•  5

Solution

Q9. Consider the equation x⁴+2ax³+x²+2ax+1=0, where a∈R. Also range of function f(x)=x+1/x is (-∞,-2]∪[2,∞)

If equation has at least two distinct positive real roots then all possible values of a are

•  (-∞,-1/4)
•  (5/4,∞)
•  (-∞,-3/4)
•  None of these

Solution

Q10. Let f(x)=x²+b₁ x+c₁,g(x)=x²+b₂ x+c₂. Let the real roots of f(x)=0 be α,β and real roots of g(x)=0 be α+h,β+h. The least value of f(x) is -1/4. The least value of g(x) occurs at x=7/2

The least value of g(x) is

•  -1/4
•  -1
•  -1/3
• -1/2

Solution

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