If \[\underset{x\to 0}{\mathop{\lim }}\,\xrightarrow[1+x-\cos x]{{{a}^{\sin x}}-1}=\underset{x\to 1}{\mathop{\lim }}\,\frac{x-1}{\ell n(x)}\] (where \[a>0\]), then a is equal to
Let \[f(x)=\left\{ \begin{matrix} 2-\left| {{x}^{2}}+5x+6 \right|, & x\ne -2 \\ {{b}^{2}}+1, & x=-2 \\ \end{matrix} \right.\] If \[f(x)\] has relative maximum at \[x=-2\], then the range of the b, is
The value of definite integral \[\int\limits_{0}^{\frac{\pi }{2}}{\sin \left( {{\sin }^{-1}}[x] \right)dx}\] is equal to [Note: \[[k]\] denote the largest integer less than or equal to k]
Tangent to the ellipse \[\frac{{{x}^{2}}}{32}+\frac{{{y}^{2}}}{18}=1\] having slope \[\frac{-3}{4}\] meet the coordinate axes in A and B. The area of \[\Delta AOB\] (0 is origin) equals
There are 10 children in a family. If the probability of a boy or a girl is equally likely, mutually exclusive and exhaustive then the chance that the family has at least three but atmost eight boys is
If \[{{z}_{1}}=\sqrt{2}\left( \cos \frac{\pi }{4}+i\sin \frac{\pi }{4} \right)\] and \[{{z}_{2}}=\sqrt{3}\left( \cos \frac{\pi }{3}+i\sin \frac{\pi }{3} \right)\], then \[\left| {{z}_{1}}{{z}_{2}} \right|\] is equal to [Note: \[i=\sqrt{-1}\]]
The figure show a \[\Delta AOB\] and the parabola\[y={{x}^{2}}\]. The ratio of the area of the \[\Delta AOB\] to the area of the region AOB of the parabola \[y={{x}^{2}}\], is equal to
Two circles with equations \[{{x}^{2}}+{{y}^{2}}+4x-10y+13=0\] and \[{{x}^{2}}+{{y}^{2}}-12x+2y+1=0\] share one common internal tangent. The y-intercept of the common tangent is
If the lines \[\frac{1-x}{3}=\frac{7y-14}{2p}=\frac{z-3}{2}\] and \[\frac{7-7x}{3p}=\frac{5-y}{1}=\frac{6-z}{5}\] are orthogonal to each other, then the value of p is
A house of height 100 m subtends a right angle at the window of an opposite house. If the height of the window is 64 m, then the distance between the two houses is
The curve represented by the differential equation \[xdy-ydx=ydy\], intersects the y-axis at A(0, 1) and the line \[y=e\] at (a, b), then the value of \[(a+b)\] is equal to (Here e denotes napier's constant)
Two light bulbs are connected in series in a circuit. The probability that an increase in voltage above its rated value will break the circuit if, under these assumptions, the chance that a bulb bums out is \[0.4\] for each of the two bulbs, is
If the area of triangle on the argand plane formed by the complex numbers \[-z,\,iz,\,z-iz\,(z\ne 0\,and\,i=\sqrt{-1})\] is 600, then \[\left| z \right|\] is equal to
The string just touch the block A in the figure as shown, such that A and B of equal mass are in equilibrium. All the surfaces have same coefficient of friction. Find it:
A particle is moving in a straight line under the action of variable force which supplies power proportional to the displacement, then displacement varies with time t as:
From a certain height h, a ball falls under gravity with some initial velocity to reach again the same height after losing \[50%\] of its energy in collision with ground. Find initial velocity if e is the coefficient of restitution.
A uniform disc of mass m and radius R is rolling up a rough inclined plane which makes an angle of \[{{30}^{o}}\] with the horizontal. If forces acting are gravitational and frictional, then value of \[\mu \] for maximum magnitude of the static fractional force acting on the disc is :
A particle moves along a straight line to follow the equation \[a{{x}^{2}}+b{{v}^{2}}=k\], where a, b and k are constant and \[x\] and \[v\] are \[x\]-coordinate and velocity of the particle respectively. Find the amplitude.
A satellite is orbiting at a height of \[h=10\,R\] where R is the radius of earth from the surface of the earth for which orbital velocity is\[\sqrt{\frac{11g{{R}^{2}}}{(R+h)}}\]. In which curve the satellite orbits the earth?
A wire of length L and cross-sectional area A is made of a material of Young's modulus Y. If the wire is stretched by an amount \[x\], the work done is:
Two unequal bars in length and cross-section with thermal conductivities \[{{k}_{1}}\] and \[{{k}_{2}}\] are connected in series. One end is maintained at \[{{\theta }_{1}}^{o}C\] and other end at \[{{\theta }_{2}}^{o}C\]. If the ratio of length of two bars is \[{{n}_{1}}\] and ratio of cross sectional area of two bars is \[{{n}_{2}}\], then find the value of \[n{{ }_{1}}/{{n}_{2}}\] for which temperature of the junction point is \[\frac{{{\theta }_{1}}+{{\theta }_{2}}}{2}\]
A police car moving at \[22\,m/s\] chases a motorcyclist. The police man sounds his horn at\[176\,Hz\], while both of them move towards a stationary siren of frequency \[165\,Hz\]. Calculate the speed of the motorcycle, if it is given that motor cyclist does not observe any beats: Police car Motor cycle Stationary siren
An object is placed at \[f/2\] away from first focus of a convex lens where \[f\] is the focal length of the lens. Its image is formed at a distance \[3f/2\] in a slab of refractive index \[3/2\], from the face of the slab facing the lens. Find the distance of this face of the slab from the second focus of the
A thin slice is cut out of a glass cylinder along a plane parallel to its axis. The slice is placed on a flat plate as shown, the observed interference fringes from this combination shall be:
The electric field \[E={{E}_{0}}y\hat{j}\] acts in teh space in which a cylinder of radius r and length \[l\] is placed with its axis parallel to y-axis. The charge inside the volume of cylinder is :
Seven capacitor each of capacitance \[2\mu F\] are connected in a configuration to obtain an effective 10 capacitance \[\frac{10}{11}\mu F\]. Which of the following combination will achieve the desired result.
At \[t=0\], switch S is closed. The charge on the capacitor is varying with time as \[Q={{Q}_{0}}(1-{{e}^{-\alpha t}})\] Find the value of \[{{Q}_{0}}\].
Two identical circular loops of metal wire are lying on a table without touching each other. Loop A carries a current which increases with time. In response, the loop B :
Initially only radioactive element A is present, which disintegrated into another stable element B. After 6 hrs the ratio of atoms of A and B is found to be \[1:7\]. Half-life of A is:
The potential difference applied to an X-ray tube is \[5\,KV\] and the current through it is \[3.2\,mA\]. Then number of electrons striking the target per second is:
The energy of a photon is equal to the Kinetic energy of a photon. The energy of photon is E. Let \[{{\lambda }_{1}}\] be the de-Broglie wavelength of the proton and \[{{\lambda }_{2}}\] be the wavelength of photon. The ratio \[{{\lambda }_{1}}/{{\lambda }_{2}}\] is proportional to:
There are four semiconductors A, B, C and D with respective Band gaps of \[3eV,\,2eV,\,1eV\] and \[0.6\] sec for use inphotodetector to detect \[14000\,\overset{0}{\mathop{A}}\,\] Select suitable semiconductor.
A particle, with restoring force proportional to displacement and resisting force proportional to velocity is subjected to a force F \[\sin \omega \,t\]. If the amplitude of the particle is maximum for \[\omega ={{\omega }_{1}}\] and the energy of the particle maximum for \[\omega ={{\omega }_{2}}\], then
A)
\[{{\omega }_{1}}={{\omega }_{0}}\] and \[{{\omega }_{2}}\ne {{\omega }_{0}}\]
doneclear
B)
\[{{\omega }_{\,1}}={{\omega }_{\,0}}\] and \[{{\omega }_{\,2}}={{\omega }_{\,0}}\]
doneclear
C)
\[{{\omega }_{\,1}}\ne {{\omega }_{\,0}}\] and \[{{\omega }_{\,2}}={{\omega }_{\,2}}\]
doneclear
D)
\[{{\omega }_{\,1}}\ne {{\omega }_{\,0}}\] and \[{{\omega }_{\,2}}\ne {{\omega }_{\,0}}\]
Consider the following sequence of reaction : \[Ph-COOH\xrightarrow{PC{{l}_{3}}}A\xrightarrow{N{{H}_{3}}}B\] \[\xrightarrow{POC{{l}_{3}}}C\xrightarrow{{{H}_{2}}/Ni}D\] The final product D is :
A buffer solution is prepared in which the concentration of \[N{{H}_{4}}^{+}\] is \[0.30\,M\] and the concentration of \[N{{H}_{4}}^{+}\] is \[0.20\,M\]. If the equation constant, \[{{K}_{b}}\] for \[N{{H}_{3}}\] equal to \[1.8\times {{10}^{-5}}\], what is the pH of this solution, \[(\log \,\,1.8=0.26)\]
1 mol \[C{{H}_{3}}COOH\] is added in \[250\,g\] benzene. Acetic acid dimerises in benzene due to hydrogen bond. \[{{K}_{b}}\] of benzene is \[2\,K\,kg\,mo{{l}^{-1}}\]. The boiling has increased by \[6.4\,K%\] dimerisation of acetic acid is:
If \[0.5\] mole \[{{H}_{2}}\], is reacted with \[0.5\] mole I, in a ten-litre container at \[{{444}^{o}}C\] and at same temperature value of equilibrium constant \[{{K}_{c}}\] is 49, the ratio of \[[HI]\] and \[[{{I}_{2}}]\] will be :
Potassium crystallizes in body centred cubic lattice with a unit cell length \[a=5.2\,\overset{o}{\mathop{\text{A}}}\,\]. What is the distance between nearest neighbours?
\[X\]ml of \[{{H}_{3}}\] gas effuse through a hole in a container in 5 seconds. The time taken for the effusion of the same volume of the gas specified below under identical conditions is:
Work out the possible structure for an alkylated resin formed from the following reactions. \[(Phthalicanhydride)+(Glycerol)\,\,\xrightarrow{Polymerisation}\]