IIT-JEE 2010 Paper 1 Offline

Let \({S_k}\)= 1, 2,....., 100, denote the sum of the infinite geometric series whose first term is \(\,{{k - 1} \over {k\,!}}\) and the common ratio is \({1 \over k}\). Then the value of $${{{{100}^2}} \over {100!}}\,\, + \,\,\sum\limits_{...

The maximum value of the expression \({1 \over {{{\sin }^2}\theta + 3\sin \theta \cos \theta + 5{{\cos }^2}\theta }}\) is

The number of values of \(\theta\) in the interval, \(\left( { - {\pi \over 2},\,{\pi \over 2}} \right)\) such
that\(\,\theta \ne {{n\pi } \over 5}\) for \(n = 0,\, \pm 1,\, \pm 2\) and \(\tan \,\theta = \cot \,5\theta \,\) as well a...

The number of all possible values of \(\theta\) where \(0 < \theta < \pi ,\) for which the system of equations
\($\left( {y + z} \right)\cos {\mkern 1mu} 3\theta = \left( {xyz} \right){\mkern 1mu} \sin 3\theta\)$
$$$x\sin 3\theta = {{2...

Let \(P,Q,R\) and \(S\) be the points on the plane with position vectors \({ - 2\widehat i - \widehat j,4\widehat i,3\widehat i + 3\widehat j}\) and \({ - 3\widehat i + 2\widehat j}\) respectively. The quadrilateral \(PQRS\) must be a

If \(\overrightarrow a\) and \(\overrightarrow b\) are vectors in space given by \(\overrightarrow a = {{\widehat i - 2\widehat j} \over {\sqrt 5 }}\) and $$\overrightarrow b = {{2\widehat i + \widehat j + 3\widehat k} \over {\sqrt {14}...

An AC voltage source of variable angular frequency \(\omega\) and fixed amplitude V0 is connected in series with a capacitance C and an electric bulb of resistance R (inductance zero). When \(\omega\) is increased

Incandescent bulbs are designed by keeping in mind that the resistance of their filament increases with the increase in temperature. If at room temperature, 100, 60 and 40 W bulbs have filament resistances R100, R60 and R40 respectively, th...

To verify Ohm's law, a student is provided with a test resitor RT, a high resistance R1, a small resistance R2, two identical galvanometers G1 and G2, and a variable voltage source V. The correct circuit to carry out the experiment is

Consider a thin square sheet of side L and thickness, made of a material of resistivity \(\rho\). The resistance between two opposite faces, shown by the shaded areas in the figure is

When two identical batteries of internal resistance 1 \(\Omega\) each are connected in series across a resistor R, the rate of heat produced in R is J1. When the same batteries are connected in parallel across R, the rate is J2. If J1 = 2.2...

An \(\alpha\)-particle and a proton are accelerated from the rest by a potential difference of 100 V. After this, their de Broglie wavelengths are \(\lambda\)\(\alpha\) and \(\lambda\)p, respectively. The ratio $${{{\lambda _p}} \over {{\la...

A few electric field lines for a system of two charges \({Q_1}\) and \({Q_2}\) fixed at two different points on the \(x\)-axis are shown in the figure. These lines suggest that

A ray OP of monochromatic light is incident on the face AB of prism ABCD near vertex B at an incident angle of 60\(^\circ\) (see figure). If the refractive index of the material of the prism is \(\sqrt3\), which of the following is(are) cor...

The focal length of a thin biconvex lens is 20 cm. When an object is moved from a distance of 25 cm in front of it to 50 cm, the magnification of its image changes from m25 to m50. The ratio \({{{m_{25}}} \over {{m_{50}}}}\) is __________.

A thin uniform annular disc (see figure) of mass M has outer radius 4R and inner radius 3R. The work required to take a unit mass from point P on its axis to infinity is

Gravitational acceleration on the surface of a planet is \({{\sqrt 6 } \over {11}}g\), where \(g\) is the gravitational acceleration on
the surface of the earth. The average mass density of the planet is \({2 \over 3}\)
times that of the ea...

A binary star consists of two stars A (mass 2.2Ms) and B (mass 11Ms), where Ms is the mass of the sun. They are separated by distance d and are rotating about their centre of mass, which is stationary. The ratio
of the total angular momentu...

One mole of an ideal gas in initial state A undergoes a cyclic process ABCA, as shown in the figure. Its pressure at A is P0. Choose the correct option(s) from the following:

A real gas behaves like an ideal gas if its

A piece of ice (heat capacity = 2100 J kg-1 oC-1 and latent heat = 3.36 \(\times\) 105 J kg-1 ) of mass m grams is at
- 5 oC at atmospheric pressure. It is given 420 J of heat so that the ice starts melting. Finally when the
ice-water mix...

Two spherical bodies A (radius 6 cm ) and B (radius 18 cm ) are at temperature T1 and T2, respectively.
The maximum intensity in the emission spectrum of A is at 500 nm and in that of B is at 1500 nm.
Considering them to be black bodies, wh...

A point mass of 1 kg collides elastically with a stationary point mass of 5 kg. After their collision, the 1 kg
mass reverses its direction and moves with a speed of 2 ms−1. Which of the following statement(s) is (are) correct for the syste...

A block of mass m is on an inclined plane of angle θ. The coefficient of
friction between the block and the plane is μ and tan θ > μ. The block is held
stationary by applying a force P parallel to the plane. The direction of force
pointing ...

A thin flexible wire of length L is connected to two adjacent fixed points and carries a current I in the clockwise direction, as shown in the figure. When the system is put in a uniform magnetic field of strength B going into the plane of ...

In the graph below, the resistance R of a superconductor is shown as a friction of its temperature T for two different magnetic fields B1 (solid line) and B2 (dashed line). If B2 is larger than B1 which of the following graphs shows the cor...

A superconductor has Tc(0) = 100 K. When a magnetic field of 7.5 T is applied, its Tc decreases to 75 K. For this material, one can definitely say that when

If the total energy of the particle is E, it will perform periodic motion only if

For periodic motion of small amplitude A, the time period T of this particle is proportional to

The acceleration of this particle for \(|x| > {X_0}\) is

A 0.1 kg mass is suspended from a wire of negligible mass. The length of the wire is 1 m and its crosssectional area is 4.9 \(\times\) 10-7 m2. If the mass is pulled a little in the vertically downward direction and released, it performs ...

A student uses a simple pendulum of exactly 1m length to determine g, the acceleration due to gravity. He
uses a stop watch with the least count of 1 sec for this and records 40 seconds for 20 oscillations. For this
observation, which of th...

A stationary source is emitting sound at a fixed frequency f0, which is reflected by two cars approaching the source. The difference between the frequencies of sound reflected from the cars is 1.2% of f0. What is the difference in the speed...

When two progressive waves \({y_1} = 4\sin (2x - 6t)\) and \({y_2} = 3\sin \left( {2x - 6t - {\pi \over 2}} \right)\) are superimposed, the amplitude of the resultant wave is __________.