1
IIT-JEE 2011 Paper 2 Offline
+4
-0
Match the statements given in Column -$$I$$ with the values given in Column-$$II.$$

$$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ Column-$$I$$
(A) $$\,\,\,\,$$If $$\overrightarrow a = \widehat j + \sqrt 3 \widehat k,\overrightarrow b = - \widehat j + \sqrt 3 \widehat k$$ and $$\overrightarrow c = 2\sqrt 3 \widehat k$$ form a triangle, then the internal angle of the triangle between $$\overrightarrow a$$ and $$\overrightarrow b$$ is
(B)$$\,\,\,\,$$ If $$\int\limits_a^b {\left( {f\left( x \right) - 3x} \right)dx = {a^2} - {b^2},}$$ then the value of $$f$$ $$\left( {{\pi \over 6}} \right)$$ is
(C)$$\,\,\,\,$$ The value of $${{{\pi ^2}} \over {\ell n3}}\int\limits_{7/6}^{5/6} {\sec \left( {\pi x} \right)dx}$$ is
(D)$$\,\,\,\,$$ The maximum value of $$\left| {Arg\left( {{1 \over {1 - z}}} \right)} \right|$$ for $$\left| z \right| = 1,\,z \ne 1$$ is given by

$$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ Column-$$II$$
(p)$$\,\,\,\,$$ $${{\pi \over 6}}$$
(q)$$\,\,\,\,$$ $${{2\pi \over 3}}$$
(r)$$\,\,\,\,$$ $${{\pi \over 3}}$$
(s)$$\,\,\,\,$$ $$\pi$$
(t) $$\,\,\,\,$$ $${{\pi \over 2}}$$

A
$$\left( A \right) \to q;\,\,\left( B \right) \to p;\,\,\left( C \right) \to s;\,\,\left( D \right) \to t$$
B
$$\left( A \right) \to q;\,\,\left( B \right) \to p;\,\,\left( C \right) \to t;\,\,\left( D \right) \to s$$
C
$$\left( A \right) \to p;\,\,\left( B \right) \to q;\,\,\left( C \right) \to s;\,\,\left( D \right) \to t$$
D
$$\left( A \right) \to q;\,\,\left( B \right) \to s;\,\,\left( C \right) \to p;\,\,\left( D \right) \to t$$
2
IIT-JEE 2011 Paper 2 Offline
Numerical
+4
-0
Let $$\overrightarrow a = - \widehat i - \widehat k,\overrightarrow b = - \widehat i + \widehat j$$ and $$\overrightarrow c = \widehat i + 2\widehat j + 3\widehat k$$ be three given vectors. If $$\overrightarrow r$$ is a vector such that $$\overrightarrow r \times \overrightarrow b = \overrightarrow c \times \overrightarrow b$$ and $$\overrightarrow r .\overrightarrow a = 0,$$ then the value of $$\overrightarrow r .\overrightarrow b$$ is
3
IIT-JEE 2011 Paper 2 Offline
+3
-1

If $$\mathop {\lim }\limits_{x \to 0} {[1 + x\ln (1 + {b^2})]^{1/x}} = 2b{\sin ^2}\theta$$, $$b > 0$$ and $$\theta \in ( - \pi ,\pi ]$$, then the value of $$\theta$$ is

A
$$\pm {\pi \over 4}$$
B
$$\pm {\pi \over 3}$$
C
$$\pm {\pi \over 6}$$
D
$$\pm {\pi \over 2}$$
4
IIT-JEE 2011 Paper 2 Offline
+3
-1

Let f(x) = x2 and g(x) = sin x for all x $$\in$$ R. Then the set of all x satisfying $$(f \circ g \circ g \circ f)(x) = (g \circ g \circ f)(x)$$, where $$(f \circ g)(x) = f(g(x))$$, is

A
$$\pm \sqrt {n\pi } ,\,n \in \{ 0,1,2,....\}$$
B
$$\pm \sqrt {n\pi } ,\,n \in \{ 1,2,....\}$$
C
$${\pi \over 2} + 2n\pi ,\,n \in \{ ....., - 2, - 1,0,1,2,....\}$$
D
$$2n\pi ,n \in \{ ....., - 2, - 1,0,1,2,....\}$$
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