1
JEE Advanced 2016 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-2
Change Language
Let $$\widehat u = {u_1} \widehat i + {u_2}\widehat j + {u_3}\widehat k$$ be a unit vector in $${{R^3}}$$ and
$$\widehat w = {1 \over {\sqrt 6 }}\left( {\widehat i + \widehat j + 2\widehat k} \right).$$ Given that there exists a vector $${\overrightarrow v }$$ in $${{R^3}}$$ such that $$\left| {\widehat u \times \overrightarrow v } \right| = 1$$ and $$\widehat w.\left( {\widehat u \times \overrightarrow v } \right) = 1.$$ Which of the following statement(s) is (are) correct?
A
There is exactly one choice for such $${\overrightarrow v }$$
B
There are infinitely many choices for such $${\overrightarrow v }$$
C
If $$\widehat u$$ lies in the $$xy$$-plane then $$\left| {{u_1}} \right| = \left| {{u_2}} \right|$$
D
If $$\widehat u$$ lies in the $$xz$$-plane then $$2\left| {{u_1}} \right| = \left| {{u_3}} \right|$$
2
JEE Advanced 2016 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-2
Change Language
Consider a pyramid $$OPQRS$$ located in the first octant $$\left( {x \ge 0,y \ge 0,z \ge 0} \right)$$ with $$O$$ as origin, and $$OP$$ and $$OR$$ along the $$x$$-axis and the $$y$$-axis, respectively. The base $$OPQR$$ of the pyramid is a square with $$OP=3.$$ The point $$S$$ is directly above the mid-point, $$T$$ of diagonal $$OQ$$ such that $$TS=3.$$ Then
A
the acute angle between $$OQ$$ and $$OS$$ is $${\pi \over 3}$$
B
the equation of the plane containing the triangle $$OQS$$ is $$x-y=0$$
C
the length of the perpendicular from $$P$$ to the plane containing the triangle $$OQS$$ is $${3 \over {\sqrt 2 }}$$
D
the perpendicular distance from $$O$$ to the straight line containing $$RS$$ is $$\sqrt {{{15} \over 2}} $$
3
JEE Advanced 2015 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-1
In $${R^3},$$ consider the planes $$\,{P_1}:y = 0$$ and $${P_2}:x + z = 1.$$ Let $${P_3}$$ be the plane, different from $${P_1}$$ and $${P_2}$$, which passes through the intersection of $${P_1}$$ and $${P_2}.$$ If the distance of the point $$(0,1, 0)$$ from $${P_3}$$ is $$1$$ and the distance of a point $$\left( {\alpha ,\beta ,\gamma } \right)$$ from $${P_3}$$ is $$2,$$ then which of the following relations is (are) true?
A
$$2\alpha + \beta + 2\gamma + 2 = 0$$
B
$$2\alpha - \beta + 2\gamma + 4 = 0$$
C
$$2\alpha + \beta - 2\gamma - 10 = 0$$
D
$$2\alpha - \beta + 2\gamma - 8 = 0$$
4
JEE Advanced 2015 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-1
In $${R^3},$$ let $$L$$ be a straight lines passing through the origin. Suppose that all the points on $$L$$ are at a constant distance from the two planes $${P_1}:x + 2y - z + 1 = 0$$ and $${P_2}:2x - y + z - 1 = 0.$$ Let $$M$$ be the locus of the feet of the perpendiculars drawn from the points on $$L$$ to the plane $${P_1}.$$ Which of the following points lie (s) on $$M$$?
A
$$\left( {0, - {5 \over 6}, - {2 \over 3}} \right)$$
B
$$\left( { - {1 \over 6}, - {1 \over 3},{1 \over 6}} \right)$$
C
$$\left( { - {5 \over 6},0,{1 \over 6}} \right)$$
D
$$\left( { - {1 \over 3},0,{2 \over 3}} \right)$$
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