JEE Advance - Mathematics (2008 - Paper 1 Offline - No. 3)

Let $${L_1},$$ $${L_2},$$ $${L_3}$$ be the lines of intersection of the planes $${P_2}$$ and $${P_3},$$ $${P_3}$$ and $${P_1},$$ $${P_1}$$ and $${P_2},$$ respectively.

STATEMENT - 1Z: At least two of the lines $${L_1},$$ $${L_2}$$ and $${L_3}$$ are non-parallel and

STATEMENT - 2: The three planes doe not have a common point.

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

Explanation

We have,

$${P_1}:x - y + z = - 1$$

$${P_2}:x + y - z = - 1$$

$${P_3}:x - 3y + 3z = 2$$

Let dr's of the lines of L$$_1$$, L$$_2$$ and L$$_3$$ are $${a_1},{b_1},{c_1}:{a_2},{b_2},{c_2}$$ and $${a_3},{b_3},{c_3}$$ respectively.

Therefore,

$${a_1} + {b_1} - {c_1} = 0$$

$${a_1} - 3{b_1} + 3{c_1} = 0$$

$$ \Rightarrow {{{a_1}} \over 0} = {{{b_1}} \over { - 4}} = {{{c_1}} \over { - 4}}$$

$${a_1},{b_1},{c_1} = 0,1,1$$

again $${a_2} - {b_2} + {c_2} = 0$$

$${a_2} - 3{b_2} + 3{c_2} = 0$$

$${{{a_2}} \over 0} = {{{b_2}} \over { - 2}} = {{{c_2}} \over { - 2}}$$

$${a_2},{b_2},{c_2} = 0,1,1$$

Again $${a_3} - {b_3} + {c_3} = 0$$

$${a_3} + {b_3} - {c_3} = 0$$

$$ \Rightarrow {{{a_3}} \over 0} = {{{b_3}} \over 2} = {{{c_3}} \over 2} \Rightarrow {a_3},{b_3},{c_3} = 0,1,1$$

L$$_1$$, L$$_2$$ and L$$_3$$ are parallel.

JEE Advance - Mathematics (2008 - Paper 1 Offline - No. 3)

Explanation

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