Complex Number and Quadratic Equations Quiz-25 Assertion - Reasoning Type

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(Î»)|≥1/q2, where Î»∈(Î»=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 1 is True, Statement 2 is True; Statement 2 is correct explanation for Statement 1
•  Statement 1 is True, Statement 2 is True; Statement 2 is not correct explanation for Statement 1
•  Statement 1 is True, Statement 2 is False
•  Statement 1 is False, Statement 2 is True
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 1 is True, Statement 2 is True; Statement 2 is correct explanation for Statement 1
•  Statement 1 is True, Statement 2 is True; Statement 2 is not correct explanation for Statement 1
•  Statement 1 is True, Statement 2 is False
•  Statement 1 is False, Statement 2 is True
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 1 is True, Statement 2 is True; Statement 2 is correct explanation for Statement 1
•  Statement 1 is True, Statement 2 is True; Statement 2 is not correct explanation for Statement 1
•  Statement 1 is True, Statement 2 is False
•  Statement 1 is False, Statement 2 is True
Solution

Q4. Statement 1: If cos2⁡Ï€/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
•  Statement 1 is True, Statement 2 is True; Statement 2 is correct explanation for Statement 1
•  Statement 1 is True, Statement 2 is True; Statement 2 is not correct explanation for Statement 1
•  Statement 1 is True, Statement 2 is False
•  Statement 1 is False, Statement 2 is True
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 is True, Statement 2 is True; Statement 2 is correct explanation for Statement 1
•  Statement 1 is True, Statement 2 is True; Statement 2 is not correct explanation for Statement 1
•  Statement 1 is True, Statement 2 is False
•  Statement 1 is False, Statement 2 is True
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 is True, Statement 2 is True; Statement 2 is correct explanation for Statement 1
•  Statement 1 is True, Statement 2 is True; Statement 2 is not correct explanation for Statement 1
•  Statement 1 is True, Statement 2 is False
•  Statement 1 is False, Statement 2 is True
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 1 is True, Statement 2 is True; Statement 2 is correct explanation for Statement 1
•  Statement 1 is True, Statement 2 is True; Statement 2 is not correct explanation for Statement 1
•  Statement 1 is True, Statement 2 is False
•  Statement 1 is False, Statement 2 is True
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 1 is True, Statement 2 is True; Statement 2 is correct explanation for Statement 1
•  Statement 1 is True, Statement 2 is True; Statement 2 is not correct explanation for Statement 1
•  Statement 1 is True, Statement 2 is False
•  Statement 1 is False, Statement 2 is True
Solution

Q9.  Statement 1: If px2+qx+r=0 is a quadratic equation (p,q,r∈R) such that its roots are Î±,Î² and p+q+r<0,p-q+r<0 and r>0, then [Î±]+[Î²]=-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 1 is True, Statement 2 is True; Statement 2 is correct explanation for Statement 1
•  Statement 1 is True, Statement 2 is True; Statement 2 is not correct explanation for Statement 1
•  Statement 1 is True, Statement 2 is False
•  Statement 1 is False, Statement 2 is True
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 1 is True, Statement 2 is True; Statement 2 is correct explanation for Statement 1
•  Statement 1 is True, Statement 2 is True; Statement 2 is not correct explanation for Statement 1
•  Statement 1 is True, Statement 2 is False
•  Statement 1 is False, Statement 2 is True
Solution

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