JEE Main 2014 (Offline)

If \(a \in R\) and the equation \(- 3{\left( {x - \left[ x \right]} \right)^2} + 2\left( {x - \left[ x \right]} \right) + {a^2} = 0\) (where [\(x\)] denotes the greater integer \(\le x\)) has no integral solution, then all possible values...

If \({(10)^9} + 2{(11)^1}\,({10^8}) + 3{(11)^2}\,{(10)^7} + ......... + 10{(11)^9} = k{(10)^9},\), then k is equal to :

Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. then the common ratio of the G.P. is :

The variance of first 50 even natural numbers is

Let \(PS\) be the median of the triangle with vertices \(P(2, 2)\), \(Q(6, -1)\) and \(R(7, 3)\). The equation of the line passing through \((1, -1)\) band parallel to PS is :

Let \(a, b, c\) and \(d\) be non-zero numbers. If the point of intersection of the lines \(4ax + 2ay + c = 0\) and \(5bx + 2by + d = 0\) lies in the fourth quadrant and is equidistant from the two axes then :

Let \(f_k\left( x \right) = {1 \over k}\left( {{{\sin }^k}x + {{\cos }^k}x} \right)\) where \(x \in R\) and \(k \ge \,1.\)
Then \({f_4}\left( x \right) - {f_6}\left( x \right)\,\,\) equals :

If $$\left[ {\overrightarrow a \times \overrightarrow b \,\,\,\,\overrightarrow b \times \overrightarrow c \,\,\,\,\overrightarrow c \times \overrightarrow a } \right] = \lambda {\left[ {\overrightarrow a\,\,\,\,\,\,\,\, \overrightarrow ...

In the circuit shown here, the point \('C'\) is kept connected to point \('A'\) till the current flowing through the circuit becomes constant. Afterward, suddenly, point \('C'\) is disconnected from point \('A'\) and connected to point $$'...

The radiation corresponding to \(3 \to 2\) transition of hydrogen atom falls on a metal surface to produce photoelectrons. These electrons are made to enter a magnetic field \(3 \times {10^{ - 4}}\,T.\) If the radius of the larger circular ...

Hydrogen \(\left( {{}_1{H^1}} \right)\), Deuterium \(\left( {{}_1{H^2}} \right)\), singly ionised Helium \({\left( {{}_2H{e^4}} \right)^ + }\) and doubly ionised lithium \({\left( {{}_3L{i^6}} \right)^{ + + }}\) all have one electron aroun...

A parallel plate capacitor is made of two circular plates separated by a distance \(5\) \(mm\) and with a dielectric of dielectric constant \(2.2\) between them. When the electric field in the dielectric is \(3 \times {10^4}\,V/m\) the char...

In a large building, three are \(15\) bulbs of \(40\) \(W\), \(5\) bulbs of \(100\) \(W\), \(5\) fans of \(80\) \(W\) and \(1\) heater of \(1\) \(kW.\) The voltage of electric mains is \(220\) \(V.\) The minimum capacity of the main fuse of...

During the propagation of electromagnetic waves in a medium :

Match List - \(1\) (Electromagnetic wave type ) with List - \(2\) (Its association/application) and select the correct option from the choices given below the lists:

The forward biased diode connection is:

The current voltage relation of diode is given by \({\rm I} = \left( {{e^{100V/T}} - 1} \right)mA,\) where the applied voltage \(V\) is in volts and the temperature \(T\) is in degree kelvin. If a student makes an error measuring $$ \pm 0.0...

Assume that an electric field \(\overrightarrow E = 30{x^2}\widehat i\) exists in space. Then the potential difference \({V_A} - {V_O},\) where \({V_O}\) is the potential at the origin and \({V_A}\) the potential at \(x=2\) \(m\) is :

A green light is incident from the water to the air - water interface at the critical angle \(\left( \theta \right)\). Select the correct statement.

A thin convex lens made from crown glass \(\left( {\mu = {3 \over 2}} \right)\) has focal length \(f\). When it is measured in two different liquids having refractive indices \({4 \over 3}\) and \({5 \over 3},\) it has the focal lengths $...

Four particles, each of mass \(M\) and equidistant from each other, move along a circle of radius \(R\) under the action of their mutual gravitational attraction. The speed of each particle is :

Three rods of Copper, Brass and Steel are welded together to form a \(Y\) shaped structure. Area of cross - section of each rod \(= 4c{m^2}.\) End of copper rod is maintained at \({100^ \circ }C\) where as ends of brass and steel are kept ...

One mole of a diatomic ideal gas undergoes a cyclic process \(ABC\) as shown in figure. The process \(BC\) is adiabatic. The temperatures at \(A, B\) and \(C\) are \(400\) \(K\), \(800\) \(K\) and \(600\) \(K\) respectively. Choose the corr...

A block of mass \(m\) is placed on a surface with a vertical cross section given by \(y = {{{x^3}} \over 6}.\) If the coefficient of friction is \(0.5,\) the maximum height above the ground at which the block can be placed without slipping...

The coercivity of a small magnet where the ferromagnet gets demagnetized is \(3 \times {10^3}\,A{m^{ - 1}}.\) The current required to be passed in a solenoid of length \(10\) \(cm\) and number of turns \(100,\) so that the magnet gets demag...

A conductor lies along the \(z\)-axis at \(- 1.5 \le z < 1.5\,m\) and carries a fixed current of \(10.0\) \(A\) in \(- {\widehat a_z}\) direction (see figure). For a field $$\overrightarrow B = 3.0 \times {10^{ - 4}}\,{e^{ - 0.2x}}\,\,{\...

From a tower of height H, a particle is thrown vertically upwards with a speed u. The time taken by the
particle, to hit the ground, is n times that taken by it to reach the highest point of its path. The relation
between H, u and n is:

On heating water, bubbles being formed at the bottom of the vessel detach and rise. Take the bubbles to be spheres of radius \(R\) and making a circular contact of radius \(r\) with the bottom \(R\) and making a circular contact of radius $...

An open glass tube is immersed in mercury in such a way that a length of \(8\) \(cm\) extends above the mercury level. The open end of the tube is then closed and scaled and the tube is raised vertically up by additional \(46\) \(cm\). What...

There is a circular tube in a vertical plane. Two liquids which do not mix and of densities \({d_1}\) and \({d_2}\) are filled in the tube. Each liquid subtends \({90^ \circ }\) angle at center. Radius joining their interface makes an angle...

The pressure that has to be applied to the ends of a steel wire of length \(10\) \(cm\) to keep its length constant when its temperature is raised by \({100^ \circ }C\) is:
(For steel Young's modulus is $$2 \times {10^{11}}\,\,N{m^{ - 2}}$...

A bob of mass \(m\) attached to an inextensible string of length \(l\) is suspended from a vertical support. The bob rotates in a horizontal circle with an angular speed \(\omega \,rad/s\) about the vertical. About the point of suspension:

A mass \('m'\) is supported by a massless string wound around a uniform hollow cylinder of mass \(m\) and radius \(R.\) If the string does not slip on the cylinder, with what acceleration will the mass fall or release?

A particle moves with simple harmonic motion in a straight line. In first \(\tau s,\) after starting from rest it travels a distance \(a,\) and in next \(\tau s\) it travels \(2a,\) in same direction, then:

A student measured the length of a rod and wrote it as 3.50 cm. Which instrument did he use to measure it?

Two beams, \(A\) and \(B\), of plane polarized light with mutually perpendicular planes of polarization are seen through a polaroid. From the position when the beam \(A\) has maximum intensity (and beam \(B\) has zero intensity), a rotatio...

A pipe of length \(85\) \(cm\) is closed from one end. Find the number of possible natural oscillations of air column in the pipe whose frequencies lie below \(1250\) \(Hz\). The velocity of sound in air is \(340\) \(m/s\).

When a rubber-band is stretched by a distance \(x\), it exerts restoring force of magnitude \(F = ax + b{x^2}\) where \(a\) and \(b\) are constants. The work done in stretching the unstretched rubber-band by \(L\) is :