1
JEE Advanced 2016 Paper 1 Offline
Numerical
+3
-0

Let $$z = {{ - 1 + \sqrt 3 i} \over 2}$$, where $$i = \sqrt { - 1} $$, and r, s $$\in$$ {1, 2, 3}. Let $$P = \left[ {\matrix{ {{{( - z)}^r}} & {{z^{2s}}} \cr {{z^{2s}}} & {{z^r}} \cr } } \right]$$ and I be the identity matrix of order 2. Then the total number of ordered pairs (r, s) for which P2 = $$-$$I is ____________.

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2
JEE Advanced 2016 Paper 1 Offline
Numerical
+3
-0

Let $$\alpha$$, $$\beta$$ $$\in$$ R be such that $$\mathop {\lim }\limits_{x \to 0} {{{x^2}\sin (\beta x)} \over {\alpha x - \sin x}} = 1$$. Then 6($$\alpha$$ + $$\beta$$) equals _________.

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3
JEE Advanced 2016 Paper 1 Offline
MCQ (More than One Correct Answer)
+3
-1
A length-scale (l) depends on the permittivity ($$\varepsilon $$) of a dielectric material, Boltzmann constant (kB), the absolute temperature (T), the number per unit volume (n) of certain charged particles, and the charge (q) carried by each of the particles. Which of the following expression(s) for I is(are) dimensionally correct?
A
$$l = \sqrt {\left( {{{n{q^2}} \over {\varepsilon {k_b}T}}} \right)} $$
B
$$l = \sqrt {\left( {{{\varepsilon {k_b}T} \over {n{q^2}}}} \right)} $$
C
$$l = \sqrt {\left( {{{{q^2}} \over {\varepsilon {n^{2/3}}{k_B}T}}} \right)} $$
D
$$l = \sqrt {\left( {{{{q^2}} \over {\varepsilon {n^{1/3}}{k_B}T}}} \right)} $$
4
JEE Advanced 2016 Paper 1 Offline
MCQ (Single Correct Answer)
+3
-1
A uniform wooden stick of mass 1.6 kg and length $$l$$ rests in an inclined manner on a smooth, vertical wall of height h ( < $$l$$ ) such that a small portion of the stick extends beyond the wall. The reaction force of the wall on the stick is perpendicular to the stick. The stick makes an angle of $$30^\circ $$ with the wall and the bottom of the stick is on a rough floor. The reaction of the wall on the stick is equal in magnitude to the reaction of the floor on the stick. The ratio $${h \over l}$$ and the frictional force f at the bottom of the stick are ( g =10 ms-2 )
A
$${h \over l} = {{\sqrt 3 } \over {16}},f = {{16\sqrt 3 } \over 3}N$$
B
$${h \over l} = {3 \over {16}},f = {{16\sqrt 3 } \over 3}N$$
C
$${h \over l} = {{3\sqrt 3 } \over {16}},f = {{8\sqrt 3 } \over 3}N$$
D
$${h \over l} = {{3\sqrt 3 } \over {16}},f = {{16\sqrt 3 } \over 3}N$$
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