1
IIT-JEE 2010 Paper 2 Offline
+4
-0
Match the statement in Column-$$I$$ with the values in Column-$$II$$

$$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ Column-$$I$$
(A)$$\,\,\,\,$$ A line from the origin meets the lines $$\,{{x - 2} \over 1} = {{y - 1} \over { - 2}} = {{z + 1} \over 1}$$
and $${{x - {8 \over 3}} \over 2} = {{y + 3} \over { - 1}} = {{z - 1} \over 1}$$ at $$P$$ and $$Q$$ respectively. If length $$PQ=d,$$ then $${d^2}$$ is
(B)$$\,\,\,\,$$ The values of $$x$$ satisfying $${\tan ^{ - 1}}\left( {x + 3} \right) - {\tan ^{ - 1}}\left( {x - 3} \right) = {\sin ^{ - 1}}\left( {{3 \over 5}} \right)$$ are
(C)$$\,\,\,\,$$ Non-zero vectors $$\overrightarrow a ,\overrightarrow b$$ and $$\overrightarrow c \,\,$$ satisfy $$\overrightarrow a \,.\,\overrightarrow b \, = 0.$$
$$\left( {\overrightarrow b - \overrightarrow a } \right).\left( {\overrightarrow b + \overrightarrow c } \right) = 0$$ and $$2\left| {\overrightarrow b + \overrightarrow c } \right| = \left| {\overrightarrow b - \overrightarrow a } \right|.$$
If $$\overrightarrow a = \mu \overrightarrow b + 4\overrightarrow c \,\,,$$ then the possible values of $$\mu$$ are
(D)$$\,\,\,\,$$ Let $$f$$ be the function on $$\left[ { - \pi ,\pi } \right]$$ given by $$f(0)=9$$
and $$f\left( x \right) = \sin \left( {{{9x} \over 2}} \right)/\sin \left( {{x \over 2}} \right)$$ for $$x \ne 0$$
The value of $${2 \over \pi }\int_{ - \pi }^\pi {f\left( x \right)dx}$$ is

$$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$Column-$$II$$
(p)$$\,\,\,\,$$ $$-4$$
(q)$$\,\,\,\,$$ $$0$$
(r)$$\,\,\,\,$$ $$4$$
(s)$$\,\,\,\,$$ $$5$$
(t)$$\,\,\,\,$$ $$6$$

A
$$\left( A \right) \to t;\,\,\left( B \right) \to p,r;\,\,\left( C \right) \to q,s;\,\,\left( D \right) \to r$$
B
$$\left( A \right) \to r;\,\,\left( B \right) \to p;\,\,\left( C \right) \to q,s;\,\,\left( D \right) \to r$$
C
$$\left( A \right) \to t;\,\,\left( B \right) \to p,r;\,\,\left( C \right) \to q;\,\,\left( D \right) \to r$$
D
$$\left( A \right) \to t;\,\,\left( B \right) \to r;\,\,\left( C \right) \to q,s;\,\,\left( D \right) \to r$$
2
IIT-JEE 2009 Paper 2 Offline
+3
-1

A line with positive direction cosines passes through the point P(2, $$-$$1, 2) and makes equal angles with the coordinate axes. The line meets the plane $$2x + y + z = 9$$ at point Q. The length of the line segment PQ equals

A
$$1$$
B
$${\sqrt 2 }$$
C
$${\sqrt 3 }$$
D
$$2$$
3
IIT-JEE 2009 Paper 2 Offline
+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)
4
IIT-JEE 2009 Paper 1 Offline
+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
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