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IIT-JEE 2010 Paper 1 Offline

MCQ (Single Correct Answer)
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
A
$$\left( {{p^3} + q} \right){x^2} - \left( {{p^3} + 2q} \right)x + \left( {{p^3} + q} \right) = 0$$
B
$$\left( {{p^3} + q} \right){x^2} - \left( {{p^3} - 2q} \right)x + \left( {{p^3} + q} \right) = 0$$
C
$$\left( {{p^3} - q} \right){x^2} - \left( {5{p^3} - 2q} \right)x + \left( {{p^3} - q} \right) = 0$$
D
$$\left( {{p^3} - q} \right){x^2} - \left( {5{p^3} + 2q} \right)x + \left( {{p^3} - q} \right) = 0$$

Explanation

Sum of roots = $${{{\alpha ^2} + {\beta ^2}} \over {\alpha \beta }}$$ and product = 1

Given, $$\alpha$$ + $$\beta$$ = $$-$$ p and $$\alpha$$3 + $$\beta$$3 = q

$$ \Rightarrow (\alpha + \beta )({\alpha ^2} - \alpha \beta + {\beta ^2}) = q$$

$$\therefore$$ $${\alpha ^2} + {\beta ^2} - \alpha \beta = {{ - q} \over p}$$ ..... (i)

and $${(\alpha + \beta )^2} = {p^2}$$

$$ \Rightarrow {\alpha ^2} + {\beta ^2} + 2\alpha \beta = {p^2}$$ ..... (ii)

From Eq. (i) and (ii), we get

$${\alpha ^2} + {\beta ^2} = {{{p^3} - 2q} \over {3p}}$$

and $$\alpha \beta = {{{p^3} + q} \over {3p}}$$

$$\therefore$$ Required equation is

$${x^2} - {{({p^2} - 2q)x} \over {({p^3} + q)}} + 1 = 0$$

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

2

IIT-JEE 2008

MCQ (Single Correct Answer)
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 $$c \ne qa$$

A
STATEMENT - 1 is True, STATEMENT - 2 is True;
STATEMENT - 2 is a correct explanation for
STATEMENT - 1
B
STATEMENT - 1 is True, STATEMENT - 2 is True;
STATEMENT - 2 is NOT a correct explanation for
STATEMENT - 1
C
STATEMENT - 1 is True, STATEMENT - 2 is False
D
STATEMENT - 1 is False, STATEMENT - 2 is True
3

IIT-JEE 2007

MCQ (Single Correct Answer)
Let $$\alpha ,\,\beta $$ be the roots of the equation $${x^2} - px + r = 0$$ and $${\alpha \over 2},\,2\beta $$ be the roots of the equation $${x^2} - qx + r = 0$$. Then the value of $$r$$
A
$${2 \over 9}\left( {p - q} \right)\left( {2q - p} \right)$$
B
$${2 \over 9}\left( {q - p} \right)\left( {2p - q} \right)$$
C
$${2 \over 9}\left( {q - 2p} \right)\left( {2q - p} \right)$$
D
$${2 \over 9}\left( {2p - q} \right)\left( {2q - p} \right)$$
4

IIT-JEE 2006

MCQ (Single Correct Answer)
Let $$a,\,b,\,c$$ be the sides of triangle where $$a \ne b \ne c$$ and $$\lambda \in R$$.
If the roots of the equation $${x^2} + 2\left( {a + b + c} \right)x + 3\lambda \left( {ab + bc + ca} \right) = 0$$ are real, then
A
$$\lambda < {4 \over 3}$$
B
$$\lambda > {5 \over 3}$$
C
$$\lambda \in \left( {{1 \over 3},\,{5 \over 3}} \right)$$
D
$$\lambda \in \left( {{4 \over 3},\,{5 \over 3}} \right)$$

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