1
IIT-JEE 2009 Paper 2 Offline
MCQ (Single Correct Answer)
+3
-0

Match the statements/expressions in Column I with the values given in Column II:

Column I Column II
(A) Root(s) of the expression $$2{\sin ^2}\theta + {\sin ^2}2\theta = 2$$ (P) $${\pi \over 6}$$
(B) Points of discontinuity of the function $$f(x) = \left[ {{{6x} \over \pi }} \right]\cos \left[ {{{3x} \over \pi }} \right]$$, where $$[y]$$ denotes the largest integer less than or equal to y (Q) $${\pi \over 4}$$
(C) Volume of the parallelopiped with its edges represented by the vectors $$\widehat i + \widehat j + \widehat i + 2\widehat j$$ and $$\widehat i + \widehat j + \pi \widehat k$$ (R) $${\pi \over 3}$$
(D) Angle between vectors $$\overrightarrow a$$ and $$\overrightarrow b$$ where $$\overrightarrow a$$, $$\overrightarrow b$$ and $$\overrightarrow c$$ are unit vectors satisfying $$\overrightarrow a + \overrightarrow b + \sqrt 3 \overrightarrow c = \overrightarrow 0$$ (S) $${\pi \over 2}$$
(T) $$\pi$$

A
(A)$$\to$$(Q), (S); (B)$$\to$$(P), (R), (S), (T); (C)$$\to$$(Q); (D)$$\to$$(T)
B
(A)$$\to$$(R), (S); (B)$$\to$$(P), (R), (S), (T); (C)$$\to$$(T); (D)$$\to$$(P)
C
(A)$$\to$$(Q), (S); (B)$$\to$$(P), (R), (S), (T); (C)$$\to$$(T); (D)$$\to$$(R)
D
(A)$$\to$$(P), (S); (B)$$\to$$(Q), (R), (S), (T); (C)$$\to$$(T); (D)$$\to$$(R)
2
IIT-JEE 2009 Paper 1 Offline
MCQ (Single Correct Answer)
+3
-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

A
$$\overrightarrow a \,,\,\overrightarrow b ,\overrightarrow c$$ are non-coplanar
B
$$\overrightarrow b \,,\,\overrightarrow c ,\overrightarrow d$$ are non-coplanar
C
$$\overrightarrow b \,,\overrightarrow d$$ are non-parallel
D
$$\overrightarrow a ,\overrightarrow d$$ parallel and $$\overrightarrow b ,\overrightarrow c$$ are parallel
3
IIT-JEE 2009 Paper 1 Offline
MCQ (Single Correct Answer)
+4
-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 :

A
$${1 \over 4}$$
B
$$-{1 \over 4}$$
C
$${1 \over 8}$$
D
$$-{1 \over 8}$$
4
IIT-JEE 2008 Paper 2 Offline
MCQ (Single Correct Answer)
+3
-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 :
A
$$\widehat u = {{\widehat a + \widehat b} \over {\left| {\widehat a + \widehat b} \right|}}\,\,and\,\,M = {\left( {1 + \widehat a.\,\widehat b} \right)^{1/2}}$$
B
$$\widehat u = {{\widehat a - \widehat b} \over {\left| {\widehat a - \widehat b} \right|}}\,\,and\,\,M = {\left( {1 + \widehat a.\,\widehat b} \right)^{1/2}}$$
C
$$\widehat u = {{\widehat a + \widehat b} \over {\left| {\widehat a + \widehat b} \right|}}\,\,and\,\,M = {\left( {1 + 2\widehat a.\,\widehat b} \right)^{1/2}}$$
D
$$\widehat u = {{\widehat a - \widehat b} \over {\left| {\widehat a - \widehat b} \right|}}\,\,and\,\,M = {\left( {1 + 2\widehat a.\,\widehat b} \right)^{1/2}}$$
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