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1

### AIEEE 2002

If $$a,\,b,\,c$$ are distinct $$+ ve$$ real numbers and $${a^2} + {b^2} + {c^2} = 1$$ then $$ab + bc + ca$$ is
A
less than 1
B
equal to 1
C
greater than 1
D
any real no.

## Explanation

As $$\,\,\,\,\,{\left( {a - b} \right)^2} + {\left( {b - c} \right)^2} + {\left( {c - a} \right)^2} > 0$$

$$\Rightarrow 2\left( {{a^2} + {b^2} + {c^2} - ab - bc - ca} \right) > 0$$

$$\Rightarrow 2 > 2\left( {ab + bc + ca} \right)$$

$$\Rightarrow ab + bc + ca < 1$$
2

### AIEEE 2002

If $$p$$ and $$q$$ are the roots of the equation $${x^2} + px + q = 0,$$ then
A
$$p = 1,\,\,q = - 2$$
B
$$p = 0,\,\,q = 1$$
C
$$p = - 2,\,\,q = 0$$
D
$$p = - 2,\,\,q = 1$$

## Explanation

$$p + q = - p$$ and $$pq = q \Rightarrow q\left( {p - 1} \right) = 0$$

$$\Rightarrow q = 0$$ or $$p=1.$$

If $$q = 0,$$ then $$p=0.$$ i.e.$$p=q$$

$$\therefore$$ $$p=1$$ and $$q=-2.$$
3

### AIEEE 2002

Difference between the corresponding roots of $${x^2} + ax + b = 0$$ and $${x^2} + bx + a = 0$$ is same and $$a \ne b,$$ then
A
$$a + b + 4 = 0$$
B
$$a + b - 4 = 0$$
C
$$a - b - 4 = 0$$
D
$$a - b + 4 = 0$$

## Explanation

Let $$\alpha ,\beta$$ and $$\gamma ,\delta$$ be the roots of the equations $${x^2} + ax + b = 0$$

and $${x^2} + bx + a = 0$$ respectively.

$$\therefore$$ $$\alpha + \beta = - a,\alpha \beta = b$$

and $$\gamma + \delta = - b,\gamma \delta = a.$$

Given $$\left| {\alpha - \beta } \right| = \left| {\gamma - \delta } \right|$$

$$\Rightarrow {\left( {\alpha - \beta } \right)^2} = {\left( {\gamma - \delta } \right)^2}$$

$$\Rightarrow {\left( {\alpha + \beta } \right)^2} - 4\alpha \beta = {\left( {\gamma + \delta } \right)^2} - 4\gamma \delta$$

$$\Rightarrow {a^2} - 4b = {b^2} - 4a$$

$$\Rightarrow \left( {{a^2} - {b^2}} \right) + 4\left( {a - b} \right) = 0$$

$$\Rightarrow a + b + 4 = 0$$

( as $$a \ne b$$ )
4

### AIEEE 2002

Product of real roots of equation $${t^2}{x^2} + \left| x \right| + 9 = 0$$
A
is always positive
B
is always negative
C
does not exist
D
none of these

## Explanation

Product of real roots $$= {9 \over {{t^2}}} > 0,\forall \,t\, \in R$$

$$\therefore$$ Product of real roots is always positive.

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