If the chords of contact of tangents from two points \[(-4,2)\] and (2, 1) to the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] are at right angle, then the eccentricity of the hyperbola, is
The perpendicular bisector of the line segment with endpoints (2, 3, 2) and \[(-4,\,\,1,\,\,4)\] passes through the point \[(-3,\,\,6,\,\,1)\]) and has equation of, the form \[\frac{x+3}{a}=\frac{y-6}{b}=\frac{z-1}{c}\] where a, b and c are relatively prime integers with \[a>0\]. The value of abc \[-(a+b+c)\] is equal to
Let \[f(x)=\left\{ \begin{matrix} {{e}^{x}}, & x\le 0 \\ \left| 1-x \right|, & x>0 \\ \end{matrix} \right.\], then which of the following statement is INCORRECT?
A)
\[f\] is continuous and differentiable at \[x=0\].
Let \[{{A}_{i}}\], where \[i=\] 1, 2, 3,......n, be n independent event such that \[P(A)=\frac{1}{i+1}\], then probability that none of the event \[{{A}_{1}},{{A}_{2}},......,{{A}_{n}}\], occur is
A flagstaff 5m high is placed on a building 25m high. If the flag and building both subtend equal angles on the observer at a height 30m, the distance between the observer and the top of the flag is
Let \[\vec{a}=\hat{i}-\hat{j},\,\vec{b}=\hat{j}-\hat{k}\] and \[\vec{c}=\hat{k}-\hat{i}\]. If d is a unit vector such that \[\vec{a}.\vec{d}=0\] and \[[\vec{b}\vec{c}\vec{d}]=0\] then \[\vec{d}\] is
Let a solution \[y=y(x)\] of the differential equation \[{{e}^{y}}dy-(2+\cos x)dx=0\] satisfy \[y(0)=0\], then the value of \[y\left( \frac{\pi }{2} \right)\] is equal to
A plane meets coordinate axes at P, Q are R respectively such that position vector of centroid of \[\Delta PQR\] is \[2\hat{i}-5\hat{j}+8\hat{k}\], then the equation of plane is
A line L is common tangent to the circle \[{{x}^{2}}+{{y}^{2}}=1\] and the parabola\[{{y}^{2}}=4x\]. If \[\theta \] is the angle which it makes with the positive x-axis, then \[{{\tan }^{2}}\theta \] is equal to
The circumcentre of triangle whose vertices are\[\left( 2,\frac{\sqrt{3}-1}{2} \right),\left( \frac{1}{2},\frac{-1}{2} \right)\] and \[\left( 2,\frac{-1}{2} \right)\] is
A tower stands at the centre of a circular park. A and B are two points on the boundary of the park such that AB (= a) subtends an angle of \[{{60}^{o}}\] at the foot of the tower and the angle of elevation of the top of the tower from A or B is \[{{30}^{o}}\]. The height of tower is
The values of two resistors are\[{{R}_{1}}=(6\pm 0.3)k\Omega \] and \[{{R}_{2}}=(10\pm 0.2)k\Omega \]. The percentage error in the equivalent resistance when they are connected in parallel is :
A particle crossing the origin of coordinates at time \[t=0\]. moves in the xy-plane with a constant acceleration a in y-direction. If, its equation of motion is \[y=b{{x}^{2}}\] (where, is b is a constant), its velocity component in the x-direction is :
The instantaneous velocity of a point B of the given rod of length \[0.5\] m is 3 \[m{{s}^{-1}}\] in the represented direction. The angular velocity of the rod for minimum velocity of end A is :
A bullet of mass 10 g is fired horizontally with a velocity \[1000\,m{{s}^{-1}}\] from a rifle situated at a height 50 m above the ground. If the bullet reaches the ground with a velocity \[500\,m{{s}^{-1}}\], the work done against air resistance in the trajectory of the bullet is (take \[g=10\,m{{s}^{-2}}\])
The machine gun fires 240 bullets per minute. If the mass of each bullet is 10 g and the velocity of the bullets is \[600\,m{{s}^{-1}}\] , the power (in kW) of the gun is:
A particle of mass m is projected with a velocity v making an angle of \[{{30}^{o}}\] with the horizontal. The magnitude of angular momentum of the projectile about the point of projection when the particle is at its maximum height h is :
A timing fork produces 4 beats/s when sounded with a sonometer wire of vibrating length 48 cm. It produces 4 beats/s also when the vibrating length is 50 cm What is the frequency of the tuning fork?
Three objects coloured black, grey and white can with stand hostile conditions at \[{{2800}^{o}}C\]. These objects are thrown into furnace where each of them attains a temperature of \[{{2000}^{o}}C\]. Which object will glow brightest?
An air bubble of radius \[{{10}^{-2}}\] m is rising up at steady rate of \[2\times {{10}^{-3}}m{{s}^{-1}}\] through a liquid of density \[1.5\times {{10}^{3}}\] \[kg{{m}^{-3}}\], the coefficient of viscosity neglecting the density of air, will be (Take \[g=10\,m{{s}^{-2}}\])
Suppose, the gravitational force varies inversely as the \[{{n}^{th}}\] power of distance. Then, the time period of a planet in circular orbit of radius R around the sun will be proportional to:
The equation of trajectory of a projectile is \[y=10x-\left( \frac{5}{9} \right){{x}^{2}}\]. If we assume \[g=10\,m{{s}^{-2}}\], the range of projectile (in metre) is:
Sound waves of frequency \[f=600\] Hz fall normally on a perfectly reflecting wall. The shortest distance from the wall at which all particles will have maximum amplitude of vibration will be (speed of sound \[=300\,m{{s}^{-1}}\])
Two small spheres of masses \[{{M}_{1}}\] and \[{{M}_{2}}\] are suspended by weightless insulating threads of length \[{{L}_{1}}\] and \[{{L}_{2}}\]. The spheres carry charges \[{{Q}_{1}}\] and \[{{Q}_{2}}\] respectively. The spheres are suspended such that they are in level with one another and the threads are inclined to the vertical at angles of \[{{\theta }_{1}}\] and \[{{\theta }_{2}}\] as shown. Which one of the following y conditions is essential, if \[{{\theta }_{1}}={{\theta }_{2}}\]?
A)
\[{{M}_{1}}\ne {{M}_{2}}\] but \[{{Q}_{1}}={{Q}_{2}}\]
An immersion heater with electrical resistance \[7\,\Omega \] is immersed in \[0.1\] kg of water at \[{{20}^{o}}C\] for 3 min. If the flow of current is 4A. What is the final temperature of the water in ideal conditions? Specific heat capacity of water \[=4.2\times {{10}^{3}}Jk{{g}^{-1}}{{K}^{-1}}\]
In magnetic field of \[0.05\] T are of coil changes from 101 \[c{{m}^{2}}\] and 100 \[c{{m}^{2}}\] without changing the resistance which is \[2\,\Omega \]. The amount of charge that flow during this period is:
A rectangular loop of length \[l\] and breadth b is placed at distance of \[x\] from infinitely long wire carrying i such that the direction of current is parallel to breadth. If the loop moves away from the current wire in a direction perpendicular to it with a velocity v, the magnitude of the emf in the loop is (\[{{\mu }_{0}}=\] permeability of free space)
The figure shows three circuits with identical batteries inductors and resistance. Rank the circuit according to the currents through the battery just after the switch is closed, greatest first.
Let C be the capacitance of a capacitor discharging^ through a resister R. Suppose t is the time taken from the energy stored in the capacitor to reduce to half its initial value and \[{{t}_{2}}\] is the time taken for the charge to reduce to one-fourth its initial value. Then the ratio \[\frac{{{t}_{1}}}{{{t}_{2}}}\] will be:
An electromagnetic wave is propagating along x- axis. At \[x=1\] cm and \[t=10\] s, its electric vector \[\left| E \right|=6V/m\], then the magnitude of its magnetic vector is:
Following diffraction pattern was obtained using a diffraction grating using two different wavelengths \[{{\lambda }_{1}}\] and \[{{\lambda }_{2}}\] With the help of the figure identify which is the longer wavelength and their ratios?
A)
\[{{\lambda }_{2}}\] is longer than \[{{\lambda }_{1}}\] and the ratio of the longer to the shorter wavelength is \[1.5\]
doneclear
B)
\[{{\lambda }_{1}}\] is longer than \[{{\lambda }_{2}}\] and the ratio of the longer to the shorter wavelength is 1.5
doneclear
C)
\[{{\lambda }_{1}}\] and \[{{\lambda }_{2}}\] are equal and their ratio is \[1.0\]
doneclear
D)
\[{{\lambda }_{1}}\] is longer than \[{{\lambda }_{2}}\] and the ratio of the longer to the shorter wavelength is \[2.5\]
Maximum velocity of the photoelectrons emitted by a metal surface is \[1.2\times {{10}^{6}}m{{s}^{-1}}\] Assuming the specific charge of the electron to be \[1.8\times {{10}^{11}}C.k{{g}^{-1}}\], the value of the stopping potential in volt will be:
An electron is moving in an orbit of a hydrogen atom form which there can be a maximum of six transition. An electron is moving in an orbit of another hydrogen atom from which there can be a maximum of three transition. The ratio of the velocities of the electron in these two orbits is :
The maximum electron density in the ionosphere in the morning is \[{{10}^{10}}{{m}^{-3}}\]. At noon time, it increase to \[2\times {{10}^{10}}{{m}^{-3}}\]. Find the ratio of critical frequency at non and the critical frequency in the morning.
Two capillaries of same material but of different radii are dipped in a liquid. In one of the capillary the liquid rises to a height of 22 cm and in other to 66 cm The ratio of their radii is :
The electrode potentials for\[C{{u}^{2+}}_{(aq)}+{{e}^{-}}\xrightarrow{{}}C{{u}^{+}}_{(aq)}\]and \[C{{u}^{+}}_{(aq)}+{{e}^{-}}\xrightarrow{{}}C{{u}_{(s)}}\] are \[+0.15\] V and \[+0.50\] V respectively. The value of \[E_{C{{u}^{2+}}/Cu}^{o}\] Will be:
In the synthesis of sodium carbonate by Solvay (or ammonia soda) process the recovery of ammonia is done by treating \[N{{H}_{4}}C\ell \] with\[Ca{{(OH)}_{2}}\]. The by product obtained in this process is
The entropy change can be calculated by using the expression \[\Delta S=\frac{{{q}_{rev}}}{T}\] .When water freezes in a glass beaker, choose the correct statement amongst the following:
A)
\[\Delta S\] (system) decreases but \[\Delta S\] (surroundings) remains the same.
doneclear
B)
\[\Delta S\] (system) increases but \[\Delta S\] (surroundings) decreases.
doneclear
C)
\[\Delta S\] (system) decreases but \[\Delta S\] (surroundings) increases.
doneclear
D)
\[\Delta S\] (system) decreases and \[\Delta S\] (surroundings) also decreases.
A complex of certain metal has the magnetic moment of \[4.91\] BM, whereas another complex of the same metal with same oxidation state has zero magnetic moment. The metal ion could be :
Among the following graphs showing variation of rate (k) with temperature (T) for a reaction, the one that exhibits Arrhenius behavior over the entire temperature range is :
Two weak acids HA and HB with \[{{K}_{{{a}_{1}}}}\] and \[{{K}_{{{a}_{2}}}}\] as their dissociation constants are mixed to have equal concentration. Which of the following is incorrect, it \[{{K}_{{{a}_{1}}}}>{{K}_{{{a}_{2}}}}\]?