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1

WB JEE 2009

MCQ (Single Correct Answer)

The sum of all real roots of the equation $$|x - 2{|^2} + |x - 2| - 2 = 0$$ is

A
7
B
4
C
1
D
5

Explanation

$$|x - 2{|^2} + |x - 2| - 2 = 0$$

Let $$|x - 2| = t$$

$$\therefore$$ $${t^2} + t - 2 = 0 \Rightarrow (t + 2)(t - 1) = 0$$

$$\Rightarrow$$ t = $$-$$2 or 1

If t = $$-$$2 then $$|x - 2| = - 2$$, but modulus of any value is non negative, so it has no solution.

If t = 1, then $$|x - 2| = 1 \Rightarrow x - 2 = \pm 1 \Rightarrow x = 3$$ or 1

$$\therefore$$ Sum of roots = 3 + 1 = 4

2

WB JEE 2009

MCQ (Single Correct Answer)

If $$\alpha$$, $$\beta$$ be the roots of $${x^2} - a(x - 1) + b = 0$$, then the value of $${1 \over {{\alpha ^2} - a\alpha }} + {1 \over {{\beta ^2} - a\beta }} + {2 \over {a + b}}$$ is

A
$${4 \over {a + b}}$$
B
$${1 \over {a + b}}$$
C
0
D
$$-$$1

Explanation

$${\alpha ^2} - a\alpha = {\beta ^2} - a\beta = - (a + b)$$

$$ \Rightarrow {1 \over {{\alpha ^2} - a\alpha }} + {1 \over {{\beta ^2} - a\beta }} + {2 \over {a + b}} = 0$$

3

WB JEE 2008

MCQ (Single Correct Answer)

The equation $${x^2} - 3|x| + 2 = 0$$ has

A
No real root
B
One real root
C
Two real roots
D
Four real roots

Explanation

$${x^2} - 3|x| + 2 = 0$$

$$ \Rightarrow |x{|^2} - 3|x| - |x| + 2 = 0$$ ($$\because$$ $${x^2} = |x{|^2} = |x{|^2}$$)

$$ \Rightarrow |x|(|x| - 2) - 1(|x| - 2) = 0 \Rightarrow (|x| - 1)(|x| - 2) = 0$$

$$ \Rightarrow |x| - 1 = 0$$ or $$|x| - 2 = 0$$

If $$|x| - 1 = 0 \Rightarrow |x| = 1 \Rightarrow x = \pm 1$$

If $$|x| - 2 = 0 \Rightarrow |x| = 2 \Rightarrow x = \pm 2$$

4

WB JEE 2008

MCQ (Single Correct Answer)

If one root of the equation $${x^2} + (1 - 3i)x - 2(1 + i) = 0$$ is $$-$$1 + i, then the other root is

A
$$-$$ 1 $$-$$ i
B
$${{( - 1 - i)} \over 2}$$
C
i
D
2i

Explanation

$${x^2} + (1 - 3i)x - 2(1 + i) = 0$$

sum of roots $$ = - {{coefficient\,of\,x} \over {coefficient\,of\,{x^2}}} = {{ - (1 - 3i)} \over 1}$$

$$\therefore$$ $$( - 1 + i) + \alpha = - 1 + 3i$$ ($$\because$$ one root is $$ - 1 + i$$)

$$ \Rightarrow \alpha = 2i$$ $$\therefore$$ other root is 2i.

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