IIT-JEE 2010 Paper 1 Offline | Quadratic Equation and Inequalities Question 23 | Mathematics

IIT-JEE 2010 Paper 1 Offline

MCQ (Single Correct Answer)

-0.75

Let $$p$$ and $$q$$ be real numbers such that $$p \ne 0,\,{p^3} \ne q$$ and $${p^3} \ne - q.$$ If $${p^3} \ne - q.$$ and $$\,\beta $$ are nonzero complex numbers satisfying $$\alpha \, + \beta = - p\,$$ and $${\alpha ^3} + {\beta ^3} = q,$$ then a quadratic equation having $${\alpha \over \beta }$$ and $${\beta \over \alpha }$$ as its roots is

$$\left( {{p^3} + q} \right){x^2} - \left( {{p^3} + 2q} \right)x + \left( {{p^3} + q} \right) = 0$$

$$\left( {{p^3} + q} \right){x^2} - \left( {{p^3} - 2q} \right)x + \left( {{p^3} + q} \right) = 0$$

$$\left( {{p^3} - q} \right){x^2} - \left( {5{p^3} - 2q} \right)x + \left( {{p^3} - q} \right) = 0$$

$$\left( {{p^3} - q} \right){x^2} - \left( {5{p^3} + 2q} \right)x + \left( {{p^3} - q} \right) = 0$$

IIT-JEE 2008 Paper 2 Offline

MCQ (Single Correct Answer)

-1

Let $$a,\,b,c$$, $$p,q$$ be real numbers. Suppose $$\alpha ,\,\beta $$ are the roots of the equation $${x^2} + 2px + q = 0$$ and $$\alpha ,{1 \over \beta }$$ are the roots of the equation $$a{x^2} + 2bx + c = 0,$$ where $${\beta ^2} \in \left\{ { - 1,\,0,\,1} \right\}$$

STATEMENT - 1 : $$\left( {{p^2} - q} \right)\left( {{b^2} - ac} \right) \ge 0$$

and STATEMENT - 2 : $$b \ne pa$$ or $$c \ne qa$$

STATEMENT - 1 is True, STATEMENT - 2 is True;
STATEMENT - 2 is a correct explanation for
STATEMENT - 1

STATEMENT - 1 is True, STATEMENT - 2 is True;
STATEMENT - 2 is NOT a correct explanation for
STATEMENT - 1

STATEMENT - 1 is True, STATEMENT - 2 is False

STATEMENT - 1 is False, STATEMENT - 2 is True

IIT-JEE 2007 Paper 1 Offline

MCQ (Single Correct Answer)

-1

Let $$\alpha,\beta$$ be the roots of the equation $$x^2-px+r=0$$ and $$\frac{\alpha}{2},2\beta$$ be the roots of the equation $$x^2-qx+r=0$$. Then the value of r is

$$\frac{2}{9}(p-q)(2q-p)$$

$$\frac{2}{9}(q-p)(2p-q)$$

$$\frac{2}{9}(q-2p)(2q-p)$$

$$\frac{2}{9}(2p-q)(2q-p)$$

IIT-JEE 2006

MCQ (Single Correct Answer)

-1

Let $$a, b, c$$ be the sides of a triangle. No two of them are equal and $$\lambda \in R$$. If the roots of the equation $$x^{2}+2(a+b+c) x+3 \lambda(a b+b c+c a)=0$$ are real, then,

$$\lambda<\frac{4}{3}$$

$$\lambda>\frac{5}{3}$$

$$\lambda \in\left(\frac{1}{3}, \frac{5}{3}\right)$$

$$\lambda \in\left(\frac{4}{3}, \frac{5}{3}\right)$$

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