WB JEE 2022

The value of a for which the sum of the squares of the roots of the equation \({x^2} - (a - 2)x - a - 1 = 0\) assumes the least value is

If a, b, c are in G.P. and log a \(-\) log 2b, log 2b \(-\) log 3c, log 3c \(-\) log a are in A.P., then a, b, c are the lengths of the sides of a triangle which is

Let \({a_n} = {({1^2} + {2^2} + .....\,{n^2})^n}\) and \({b_n} = {n^n}(n!)\). Then

A is a set containing n elements. P and Q are two subsets of A. Then the number of ways of choosing P and Q so that P \(\cap\) Q = \(\varphi\) is

Let S, T, U be three non-void sets and f : S \(\to\) T, g : T \(\to\) U and composed mapping g . f : S \(\to\) U be defined. Let g . f be injective mapping. Then

For the mapping \(f:R - \{ 1\} \to R - \{ 2\}\), given by \(f(x) = {{2x} \over {x - 1}}\), which of the following is correct?

Let R and S be two equivalence relations on a non-void set A. Then

If the algebraic sum of the distances from the points (2, 0), (0, 2) and (1, 1) to a variable straight line be zero, then the line passes through the fixed point

If the sum of the distances of a point from two perpendicular lines in a plane is 1 unit, then its locus is

Consider the equation \(y - {y_1} = m(x - {x_1})\). If m and x1 are fixed and different lines are drawn for different values of y1, then

The equation of the plane through the intersection of the planes x + y + z = 1 and 2x + 3y \(-\) z + 4 = 0 and parallel to the x-axis is

The line \(x - 2y + 4z + 4 = 0\), \(x + y + z - 8 = 0\) intersect the plane \(x - y + 2z + 1 = 0\) at the point

If \((\cot {\alpha _1})(\cot {\alpha _2})\,......\,(\cot {\alpha _n}) = 1,0 < {\alpha _1},{\alpha _2},....\,{\alpha _n} < \pi /2\), then the maximum value of \((\cos {\alpha _1})(\cos {\alpha _2}).....(\cos {\alpha _n})\) is given by

If \(\overrightarrow a = \widehat i + \widehat j - \widehat k\), \(\overrightarrow b = \widehat i - \widehat j + \widehat k\) and \(\overrightarrow c\) is unit vector perpendicular to \(\overrightarrow a\) and coplanar with $$\overright...

If \({\overrightarrow \alpha }\) is a unit vector, \(\overrightarrow \beta = \widehat i + \widehat j - \widehat k\), \(\overrightarrow \gamma = \widehat i + \widehat k\) then the maximum value of $$\left[ {\overrightarrow \alpha \over...

When an AC source of emf E with frequency \(\omega\) = 100 Hz is connected across a circuit, the phase difference between E and current I in the circuit is observed to be \({\pi \over 4}\) as shown in the figure. If the circuit consist of ...

If the kinetic energies of an electron, an alpha particle and a proton having same de-Broglie wavelength are \({\varepsilon _1},{\varepsilon _2}\) and \({\varepsilon _3}\) respectively, then

Suppose in a hypothetical world the angular momentum is quantized to be even integral multiples of \({h \over {2\pi }}\). The largest possible wavelength emitted by hydrogen atoms in visible range in a world according to Bohr's model will b...

A sample of hydrogen atom in its ground state is radiated with photons of 10.2 eV energies. The radiation from the sample is absorbed by excited ionized He+. Thenwhich of the following statement/s is/are true?

Find the equivalent capacitance between A and B of the following arrangement :

Three concentric metallic shells A, B and C of radii a, b and c (a < b < c) have surface charge densities +\(\sigma\), \(-\)\(\sigma\) and +\(\sigma\) respectively. The potential of shell B is

A particle is moving in an elliptical orbit as shown in figure. If \(\overrightarrow p\), \(\overrightarrow L\) and \(\overrightarrow r\) denote the linear momentum, angular momentum and position vector of the particle (from focus O) re...

A body of mass m is thrown with velocity u from the origin of a co-ordinate axes at an angle \(\theta\) with the horizon. The magnitude of the angular momentum of the particle about the origin at time t when it is at the maximum height of t...

Three particles, each of mass 'm' grams situated at the vertices of an equilateral \(\Delta\)ABC of side 'a' cm (as shown in the figure). The moment of inertia of the system about a line AX perpendicular to AB and in the plane of ABC in g-c...

A battery of emf E and internal resistance r is connected with an external resistance R as shown in the figure. The battery will act as a constant voltage source if

In a closed circuit there is only a coil of inductance L and resistance 100 \(\Omega\). The coil is situated in a uniform magnetic field. All on a sudden, the magnetic flux linked with the circuit changes by 5 Weber. What amount of charge w...

A Zener diode having break down voltage Vz = 6V is used in a voltage regulator circuit as shown in the figure. The minimum current required to pass through the Zener to act as a voltage regulator is 10 mA and maximum allowed current through...

The expression \(\overline A (A + B) + (B + AA)(A + \overline B )\) simplifies to

The electric potential for an electric field directed parallel to X-axis is shown in the figure. Choose the correct plot of electric field strength.

Two charges, each equal to \(-\)q are kept at (\(-\)a, 0) and (a, 0). A charge q is placed at the origin. If q is given a small displacement along y direction, the force acting on q is proportional to

A neutral conducting solid sphere of radius R has two spherical cavities of radius a and b as shown in the figure. Centre to centre distance between two cavities is c. qa and qb charges are placed at the centres of cavities respectively. T...

Consider two concentric conducting sphere of radii R and 2R respectively. The inner sphere is given a charge +Q. The other sphere is grounded. The potential at \(r = {{3R} \over 2}\) is

A hemisphere of radius R is placed in a uniform electric field E so that its axis is parallel to the field. Which of the following statement(s) is/are true?

The human eye has an approximate angular resolution of \(\theta\) = 5.8 \(\times\) 10\(-\)4 rad and typical photo printer prints a minimum of 300 dpi (dots per inch, 1 inch = 2.54 cm). At what minimal distance d should a printed page be hel...

A body of mass m is thrown vertically upward with speed \(\sqrt3\) ve, where ve is the escape velocity of a body from earth surface. The final velocity of the body is

Certain amount of an ideal gas is taken from its initial state 1 to final state 4 through the paths 1 \(\to\) 2 \(\to\) 3 \(\to\) 4 as shown in figure. AB, CD, EF are all isotherms. If vp is the most probable speed of the molecules, then

Consider a thermodynamic process where integral energy \(U = A{P^2}V\) (A = constant). If the process is performed adiabatically, then

One mole of a diatomic ideal gas undergoes a process shown in P-V diagram. The total heat given to the gas (ln 2 = 0.7) is

One mole of an ideal monoatomic gas expands along the polytrope PV3 = constant from V1 to V2 at a constant pressure P1. The temperature during the process is such that molar specific heat \({C_V} = {{3R} \over 2}\). The total heat absorbed ...

If a string, suspended from the ceiling is given a downward force F1, its length becomes L1. Its length is L2, if the downward force is F2. What is its actual length?

A golf ball of mass 50 gm placed on a tee, is struck by a golf-club. The speed of the golf ball as it leaves the tee is 100 m/s, the time of contact on the ball is 0.02 s. If the force decreases to zero linearly with time, then the force at...

Two infinite line-charges parallel to each other are moving with a constant velocity v in the same direction as shown in the figure. The separation between two line-charges is d. The magnetic attraction balances the electric repulsion when...

An electron revolves around the nucleus in a circular path with angular momentum \(\overrightarrow L\). A uniform magnetic field \(\overrightarrow B\) is applied perpendicular to the plane of its orbit. If the electron experiences a torqu...

A straight wire is placed in a magnetic field that varies with distance x from origin as \(\overrightarrow B = {B_0}\left( {2 - {x \over a}} \right)\widehat k\). Ends of wire are at (a, 0) and (2a, 0) and it carries a current I. If force o...

A horizontal semi-circular wire of radius r is connected to a battery through two similar springs X and Y to an electric cell, which sends current I through it. A vertically downward uniform magnetic field B is applied on the wire, as shown...

As shown in figure, a rectangular loop of length 'a' and width 'b' and made of a conducting material of uniform cross-section is kept in a horizontal plane where a uniform magnetic field of intensity B is acting vertically downward. Resist...

27 drops of mercury coalesce to form a bigger drop. What is the relative increase in surface energy?

Two wires A and B of same length are made of same material. Load (F) vs. elongation (x) graph for these two wires is shown in the figure. Then which of the following statement(s) is/are true?

A particle is subjected to two simple harmonic motions in the same direction having equal amplitudes and equal frequency. If the resultant amplitude is equal to the amplitude of the individual motion, the phase difference (\(\delta\)) betwe...

Given : The percentage error in the measurements of A, B, C and D are respectively, 4%, 2%, 3% and 1%. The relative error in \(Z = {{{A^4}{B^{{1 \over 3}}}} \over {C{D^{{3 \over 2}}}}}\) is