IIT-JEE 2009 Paper 1 Offline | Vector Algebra and 3D Geometry Question 60 | Mathematics

IIT-JEE 2009 Paper 1 Offline

MCQ (Single Correct Answer)

-1

If $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c $$ and $$\overrightarrow d $$ are unit vectors such that $$(\overrightarrow a \times \overrightarrow b )\,.\,(\overrightarrow c \times \overrightarrow d ) = 1$$ and $$\overrightarrow a \,.\,\overrightarrow c = {1 \over 2}$$, then

$$\overrightarrow a \,,\,\overrightarrow b ,\overrightarrow c $$ are non-coplanar

$$\overrightarrow b \,,\,\overrightarrow c ,\overrightarrow d $$ are non-coplanar

$$\overrightarrow b \,,\overrightarrow d $$ are non-parallel

$$\overrightarrow a ,\overrightarrow d $$ parallel and $$\overrightarrow b ,\overrightarrow c $$ are parallel

IIT-JEE 2009 Paper 1 Offline

MCQ (Single Correct Answer)

-1

Let $$P(3,2,6)$$ be a point in space and $$Q$$ be a point on the line $$$\widehat r = \left( {\widehat i - \widehat j + 2\widehat k} \right) + \mu \left( { - 3\widehat i + \widehat j + 5\widehat k} \right)$$$

Then the value of $$\mu $$ for which the vector $${\overrightarrow {PQ} }$$ is parallel to the plane $$x - 4y + 3z = 1$$ is :

$${1 \over 4}$$

$$-{1 \over 4}$$

$${1 \over 8}$$

$$-{1 \over 8}$$

IIT-JEE 2008 Paper 2 Offline

MCQ (Single Correct Answer)

-1

Let two non-collinear unit vectors $$\widehat a$$ and $$\widehat b$$ form an acute angle. A point $$P$$ moves so that at any time $$t$$ the position vector $$\overrightarrow {OP} $$ (where $$O$$ is the origin) is given by $$\widehat a\cos t + \widehat b\sin t.$$ When $$P$$ is farthest from origin $$O,$$ let $$M$$ be the length of $$\overrightarrow {OP} $$ and $$\widehat u$$ be the unit vector along $$\overrightarrow {OP} $$. Then :

$$\widehat u = {{\widehat a + \widehat b} \over {\left| {\widehat a + \widehat b} \right|}}\,\,and\,\,M = {\left( {1 + \widehat a.\,\widehat b} \right)^{1/2}}$$

$$\widehat u = {{\widehat a - \widehat b} \over {\left| {\widehat a - \widehat b} \right|}}\,\,and\,\,M = {\left( {1 + \widehat a.\,\widehat b} \right)^{1/2}}$$

$$\widehat u = {{\widehat a + \widehat b} \over {\left| {\widehat a + \widehat b} \right|}}\,\,and\,\,M = {\left( {1 + 2\widehat a.\,\widehat b} \right)^{1/2}}$$

$$\widehat u = {{\widehat a - \widehat b} \over {\left| {\widehat a - \widehat b} \right|}}\,\,and\,\,M = {\left( {1 + 2\widehat a.\,\widehat b} \right)^{1/2}}$$

IIT-JEE 2008 Paper 2 Offline

MCQ (Single Correct Answer)

-1

Consider the lines

$${L_1}:{{x + 1} \over 3} = {{y + 2} \over 1} = {{z + 1} \over 2}$$

$${L_2}:{{x - 2} \over 1} = {{y + 2} \over 2} = {{z - 3} \over 3}$$

The unit vector perpendicular to both $${L_1}$$ and $${L_2}$$ is :

$${{ - \widehat i + 7\widehat j + 7\widehat k} \over {\sqrt {99} }}$$

$${{ - \widehat i - 7\widehat j + 5\widehat k} \over {5\sqrt 3 }}$$

$${{ - \widehat i + 7\widehat j + 5\widehat k} \over {5\sqrt 3 }}$$

$${{7\widehat i - 7\widehat j - \widehat k} \over {\sqrt {99} }}$$

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