1
IIT-JEE 2011 Paper 1 Offline
+3
-1

Let a, b and c be three real numbers satisfying

$$[\matrix{ a & b & c \cr } ]\left[ {\matrix{ 1 & 9 & 7 \cr 8 & 2 & 7 \cr 7 & 3 & 7 \cr } } \right] = [\matrix{ 0 & 0 & 0 \cr } ]$$ ........(E)

Let $$\omega$$ be a solution of $${x^3} - 1 = 0$$ with $${\mathop{\rm Im}\nolimits} (\omega ) > 0$$. If a = 2 with b and c satisfying (E), then the value of $${3 \over {{\omega ^a}}} + {1 \over {{\omega ^b}}} + {3 \over {{\omega ^c}}}$$ is equal to

A
$$-$$2
B
2
C
3
D
$$-$$3
2
IIT-JEE 2011 Paper 1 Offline
+3
-1

Let a, b and c be three real numbers satisfying

$$[\matrix{ a & b & c \cr } ]\left[ {\matrix{ 1 & 9 & 7 \cr 8 & 2 & 7 \cr 7 & 3 & 7 \cr } } \right] = [\matrix{ 0 & 0 & 0 \cr } ]$$ ........ (E)

Let b = 6, with a and c satisfying (E). If $$\alpha$$ and $$\beta$$ are the roots of the quadratic equation ax2 + bx + c = 0, then $$\sum\limits_{n = 0}^\infty {{{\left( {{1 \over \alpha } + {1 \over \beta }} \right)}^n}}$$ is

A
6
B
7
C
$${6 \over 7}$$
D
$$\infty$$
3
IIT-JEE 2011 Paper 1 Offline
Numerical
+3
-1

Let $$f:[1,\infty ) \to [2,\infty )$$ be a differentiable function such that $$f(1) = 2$$. If $$6\int\limits_1^x {f(t)dt = 3xf(x) - {x^3} - 5}$$ for all $$x \ge 1$$, then the value of f(2) is ___________.

4
IIT-JEE 2011 Paper 1 Offline
+3
-0.75
A dense collection of equal number of electrons and positive ions is called neutral plasma. Certain solids containing fixed positive ions surrounded by free electrons can be treated as neutral plasma. Let 'N' be the number density of free electrons, each of mass 'm'. When the electrons are subjected to an electric field, they are displaced relatively away from the heavy positive ions. If the electric field becomes zero, the electrons begin to oscillate about the positive ions with a natural angular frequency '$${\omega _p}$$' which is called the plasma frequency. To sustain the oscillations, a time varying electric field needs to be applied that has an angular frequency $$\omega$$, where a part of the energy is absorbed and a part of it is reflected. As $$\omega$$ approaches $${\omega _p}$$ all the free electrons are set to resonance together and all the energy is reflected. This is the explanation of high reflectivity of metals.

Taking the electronic charge as 'e' and the permittivity as $$'{\varepsilon _0}'$$. Use dimensional analysis to determine the correct expression for $${\omega _p}$$.

A
$$\sqrt {{{Ne} \over {m{\varepsilon _0}}}}$$
B
$$\sqrt {{{m{\varepsilon _0}} \over {Ne}}}$$
C
$$\sqrt {{{N{e^2}} \over {m{\varepsilon _0}}}}$$
D
$$\sqrt {{{m{\varepsilon _0}} \over {N{e^2}}}}$$
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