Complex Number and Quadratic Equations Quiz-25
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 a,b,c∈Z and ax2+bx+c=0 has an
irrational root, then |f(λ)|≥1/q2, where
λ∈(λ=p/q;p,q∈Z) and f(x)=ax2+bx+c
Statement 2: If a,b,c∈Q and b2-4ac is positive but not a perfect square, then roots of equation ax2+bx+c=0 are irrational and always occur in conjugate pair like 2+√3 and 2-√3
Statement 2: If a,b,c∈Q and b2-4ac is positive but not a perfect square, then roots of equation ax2+bx+c=0 are irrational and always occur in conjugate pair like 2+√3 and 2-√3
Solution
Q2. Statement 1: If equations ax2+bx+c=0 and
x2-3x+4=0 have exactly one root common, then at least
one of a,b,c is imaginary
Statement 2: If a,b,c are not all real, then equation ax2+bx+c=0 can have one root real and one root imaginary
Statement 2: If a,b,c are not all real, then equation ax2+bx+c=0 can have one root real and one root imaginary
Solution
Q3. Statement 1: Locus of z, satisfying the equation |z-1|+|z-8|=5 is
an ellipse
Statement 2: Sum of focal distances of any point on ellipse is constant
Statement 2: Sum of focal distances of any point on ellipse is constant
Solution
Q4. Statement 1: If cos2Ï€/8 is a root of the equation x2+ax+b=0 where a,b∈Q, then ordered pair (a,b) is [-1,(1/8)]
Statement 2: If a+mb=0 and m is irrational, then a,b=0
Solution
Q5. Consider the function
f(x)=loge(ax3+(a+b)x2+(b+c)x+c)
Statement 1: Domain of the functions is (-1,∞)~{-(b/2a)}, where a>0,b2-4ac=0
Statement 2: ax2+bx+c=0 has equal roots when b2-4ac=0
Statement 1: Domain of the functions is (-1,∞)~{-(b/2a)}, where a>0,b2-4ac=0
Statement 2: ax2+bx+c=0 has equal roots when b2-4ac=0
Solution
Q6. If z1≠-z2 and |z1+z2
|=|(1/z1)+(1/z2)| then
Statement 1: z1z2 is unimodular
Statement 2: z1 and z2 both are unimodular
Statement 1: z1z2 is unimodular
Statement 2: z1 and z2 both are unimodular
Solution
Q7. Statement 1: If z1+z2=a and
z1z2=b, where a=¯a and b=¯b, then
arg(z1z2)=0
Statement 2: The sum and product of two complex numbers are real if and only if they are conjugate of each other
Statement 2: The sum and product of two complex numbers are real if and only if they are conjugate of each other
Solution
Q8. Statement 1: If all real values of x obtained from the equation
4x-(a-3) 2x+(a-4)=0 are non-positive, then
a∈(4,5]
Statement 2: If ax2+bx+c is non-positive for all real values of x, then b2-4ac must be negative or zero and ‘a’ must be negative
Statement 2: If ax2+bx+c is non-positive for all real values of x, then b2-4ac must be negative or zero and ‘a’ must be negative
Solution
Q9.
Statement 1: If px2+qx+r=0 is a quadratic equation
(p,q,r∈R) such that its roots are α,β and p+q+r<0,p-q+r<0
and r>0, then [α]+[β]=-1, where [∙] denotes greatest integer
function
Statement 2: If for any two real numbers a and b, function f(x) is such that f(a)f(b)<0⇒f(x) has at least one real root lying in (a,b)
Statement 2: If for any two real numbers a and b, function f(x) is such that f(a)f(b)<0⇒f(x) has at least one real root lying in (a,b)
Solution
Q10. Statement 1: If a>0 and b2-ac<0, then domain of the
function f(x)=√(ax2+2bx+c) is R
Statement 2: If b2-ac<0, then ax2+2bx+c=0 has imaginary roots
Statement 2: If b2-ac<0, then ax2+2bx+c=0 has imaginary roots
Solution
