Complex Number and Quadratic Equations Quiz-26
Assertion - Reasoning Type
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.
Statement 1: If |z1 |=1,|z2 |=2,|z3
|=3 and |z1+2z2+3z3 |=6, then the
value of |z2z3+8z3
z1+27z1z2 | is 36
Statement 2: | z1+z2+z3 |≤|z1 |+|z2 |+|z3|
Statement 2: | z1+z2+z3 |≤|z1 |+|z2 |+|z3|
Solution
Q2. Statement 1: If f(x) is a quadratic polynomial satisfying
f(2)+f(4)=0. If unity is a root of f(x)=0, then the other root is
3.5
Statement 2: If g(x)=px2+qx+r=0 has roots α,β, then α+β=-q/p and αβ=(r/p)
Statement 2: If g(x)=px2+qx+r=0 has roots α,β, then α+β=-q/p and αβ=(r/p)
Solution
Q3. Statement 1: If both roots of the equation
4x2-2x+a=0,a∈R lie in the interval (-1,1),
then-2<a≤1/4.
Statement 2: If f(x)=4x2-2x+a,then D≥0,f(-1)>0 and f(1)>0⇒-2<a≤1/4.
Statement 2: If f(x)=4x2-2x+a,then D≥0,f(-1)>0 and f(1)>0⇒-2<a≤1/4.
Solution
Q4. Statement 1: If a2+b2+c2<0, then if roots of the equation ax2+bx+c=0 are imaginary, then they are not complex conjugates
Statement 2: equation ax2+bx+c=0 has complex conjugate roots when a,b,c are real
Solution
Q5. Statement 1: The equation x2+(2m+1)x+(2n+1)=0, where m
and n are integer cannot have any rational roots
Statement 2: The quantity (2m+1)2-4(2n+1), where m,n∈I can never be a perfect square
Statement 2: The quantity (2m+1)2-4(2n+1), where m,n∈I can never be a perfect square
Solution
Q6. Statement 1: If n is an odd integer greater than 3 but not a
multiple of 3, then (x+1)n-xn-1 is divisible
by x3+x2+x
Statement 2: If n is an odd integer greater than 3 but not a multiple of 3, we have 1+ωn+ω2n=3
Statement 2: If n is an odd integer greater than 3 but not a multiple of 3, we have 1+ωn+ω2n=3
Solution
Q7. Statement 1: If x+(1/x)=1 and
p=x4000+(1/x4000) and q be the digit at unit
place in the number 2(2n)+1,n∈N and n>1,
then the value of p+q=8
Statement 2: If ω,ω2 are the roots of x+1/x=-1, then x2+1/x2=-1,x3+(1/x3)=2
Statement 2: If ω,ω2 are the roots of x+1/x=-1, then x2+1/x2=-1,x3+(1/x3)=2
Solution
Q8. Statement 1: If 0<α<(π/4), then the equation (x-sinα
)×(x-cosα )-2=0 has both roots in (sinα,cosα)
Statement 2: If f(a) and f(b) possess opposite signs, then there exists at least one solution of the equation f(x)=0 in open interval (a,b)
Statement 2: If f(a) and f(b) possess opposite signs, then there exists at least one solution of the equation f(x)=0 in open interval (a,b)
Solution
Q9.
Statement 1: Let z1 and z2 are two complex
numbers such that |z1-z2
|=|z1+z2 | then the orthocenter of ∆AOB is
[(z1+z2)/2] (where O is origin)
Statement 2: In case of right-angled triangle, orthocenter is that point at which the triangle is right angled
Statement 2: In case of right-angled triangle, orthocenter is that point at which the triangle is right angled
Solution
Q10. Statement 1: If
|(zz1-z2)/(zz1+z2
)|=k,(z1,z2≠0), then the locus of z is
circle
Statement 2: As |(z-z1)/(z-z2 )|=λ represents a circle, if λ∉{0,1}
Statement 2: As |(z-z1)/(z-z2 )|=λ represents a circle, if λ∉{0,1}
Solution
