1
IIT-JEE 2011 Paper 1 Offline
+4
-1

Let $${U_1}$$ and $${U_2}$$ be two urns such that $${U_1}$$ contains $$3$$ white and $$2$$ red balls, and $${U_2}$$ contains only $$1$$ white ball. A fair coin is tossed. If head appears then $$1$$ ball is drawn at random from $${U_1}$$ and put into $${U_2}$$. However, if tail appears then $$2$$ balls are drawn at random from $${U_1}$$ and put into $${U_2}$$. Now $$1$$ ball is drawn at random from $${U_2}$$ being white is

Given that the drawn ball from $${U_2}$$ is white, the probability that head appeared on the coin is

A
$${{17} \over {23}}$$
B
$${{11} \over {23}}$$
C
$${{15} \over {23}}$$
D
$${{12} \over {23}}$$
2
IIT-JEE 2011 Paper 1 Offline
+4
-1
Let $$\overrightarrow a = \widehat i + \widehat j + \widehat k,\,\overrightarrow b = \widehat i - \widehat j + \widehat k$$ and $$\overrightarrow c = \widehat i - \widehat j - \widehat k$$ be three vectors. A vector $$\overrightarrow v$$ in the plane of $$\overrightarrow a$$ and $$\overrightarrow b ,$$ whose projection on $$\overrightarrow c$$ is $${{1 \over {\sqrt 3 }}}$$ , is given by
A
$$\widehat i - 3\widehat j + 3\widehat k$$
B
$$-3\widehat i - 3\widehat j - \widehat k$$
C
$$3\widehat i - \widehat j + 3\widehat k$$
D
$$\widehat i + 3\widehat j - 3\widehat k$$
3
IIT-JEE 2011 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-1
The vector (s) which is/are coplanar with vectors $${\widehat i + \widehat j + 2\widehat k}$$ and $${\widehat i + 2\widehat j + \widehat k,}$$ and perpendicular to the vector $${\widehat i + \widehat j + \widehat k}$$ is/are
A
$$\widehat j - \widehat k$$
B
$$-\widehat i + \widehat j$$
C
$$\widehat i - \widehat j$$
D
$$-\widehat j + \widehat k$$
4
IIT-JEE 2011 Paper 1 Offline
+3
-1

Let $$P = \{ \theta :\sin \theta - \cos \theta = \sqrt 2 \cos \theta \}$$ and $$Q = \{ \theta :\sin \theta + \cos \theta = \sqrt 2 \sin \theta \}$$ be two sets. Then

A
$$P \subset Q$$ and $$Q - P \ne \emptyset$$
B
$$Q \not\subset P$$
C
$$P \not\subset Q$$
D
$$P = Q$$
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