A particle of mass \[m=5\] is moving with a uniform speed \[\text{v=3}\sqrt{2}\] in the XOY plane along the line\[y=x+4\]. The magnitude of the angular momentum of the particle about the origin is
The depth d at which the value of acceleration due to gravity becomes \[\frac{1}{n}\] times the value at the surface of the earth, is [R = radius of the earth]
A stress of \[3.18\times {{10}^{8}}N{{m}^{-2}}\] is appleid to a steel rod of length 1 m along its length. Its Young's modulus is \[2\times {{10}^{11}}N{{m}^{-2}}\]. Then the elongation produced in the rod in mm is
A parallel plate air capacitor has a capacitance C. When it is half filled with a dielectric of dielectric constant 5, the percentage increase in the capacitance will be
The length and breadth of a rectangular sheet are \[16.2\text{ }cm\]and \[10.1\text{ }cm,\] respectively. The area of the sheet in appropriate significant figures and error is
Two trains A and B each of length 400 m are moving on two parallel tracks in the same direction (with A ahead of B) with same speed\[72\,km/h\]. The driver of B decides to overtake A and accelerates by\[1\,m/{{s}^{2}}\]. If after 50s. B just brushes past A, calculate the original distance between A and B.
The resultant of two vectors \[\vec{A}\] and \[\vec{B}\] is perpendicular to the vector \[\vec{A}\] and its magnitude is equal to half the magnitude of vector \[\vec{B}\]. The angle between \[\vec{A}\] and \[\vec{B}\] is
Surface of certain metal is first illuminated with light of wavelength \[{{\lambda }_{1}}=350nm\]and then, by light of wavelength\[{{\lambda }_{2}}=540nm\]. It is found that the maximum speed of the photo electrons in the two cases differ by a factor of (2) The work function of the metal (in eV) is close to: (Energy of photon \[=\frac{1240}{\lambda \left( in\,nm \right)}eV\])
One of the lines in the emission spectrum of \[L{{i}^{2+}}\] has the same wavelength as that of the 2nd line of Balmer series in hydrogen spectrum. The electronic transition corresponding to this line is \[=12\to n=x.\]. Find the value of x.
A radioactive element X converts into another stable element Y. Half-life of X is 2 hrs. Initially only X is present. After time t, the ratio of atoms of X and Y is found to be \[1:4,\] then t in hours is
The change in the entropy of a 1 mole of an ideal gas which went through an isothermal process from an initial state \[({{P}_{1}},{{V}_{1}},T)\] to the final state \[({{P}_{2}},{{V}_{2}},T)\] is equal to
Two springs, of force constants \[{{k}_{1}}\] and \[{{k}_{2}}\] are connected to a mass m as shown. The frequency of oscillation of the mass is f If both \[{{k}_{1}}\] and \[{{k}_{2}}\] are made four times their original values, the frequency of oscillation becomes
Wave pulse on a string shown in figure is moving to the right without changing shape. Consider two particles at positions \[{{x}_{1}}=1.5m\] and \[{{x}_{2}}=2.5m\]. Their transverse velocities at the moment shown in figure are along directions
In a coil of resistance \[100\,\Omega \] a current is induced by changing the magnetic flux through it as shown in the figure. The magnitude of change in flux through the coil is
In an electromagnetic wave in free space the root mean square value of the electric field is \[{{E}_{rms}}=6V/m.\]. The peak value of the magnetic field is :-
A point source S is placed at the bottom of a transparent block of height \[10\text{ }mm\]and refractive index\[2.72\]. It is immersed in a lower refractive index liquid as shown in the figure. It is found that the light emerging from the block to the liquid forms a circular bright spot of diameter \[11.54\text{ }mm\]on the top of the block. The refractive index of the liquid is
Two Polaroids \[{{P}_{1}}\] and \[{{P}_{2}}\] are placed with their axis perpendicular to each other. Unpolarised light \[{{I}_{0}}\] is incident on \[{{P}_{1}}\]. A third polaroid \[{{P}_{3}}\] is kept in between \[{{P}_{1}}\] and \[{{P}_{2}}\] such that its axis makes an angle \[45{}^\circ \] with that of \[{{P}_{1}}\]. The intensity of transmitted light through \[{{P}_{2}}\] is
A solid sphere of radius R has a charge Q distributed in its volume with a charge density \[\rho =k\,{{r}^{a}},\] where k and a are constants and r is the distance from its centre. If the electric field at \[r=\frac{R}{2}\] is \[\frac{1}{8}\] times that at \[r=R,\] find the value of a.
The potential energy of a 1 kg particle free to move along tile x-axis is given by \[V(x)=\left( \frac{{{x}^{4}}}{4}-\frac{{{x}^{2}}}{2} \right)J.\] The total mechanical energy of the particle is 2 J. Then, the maximum speed (in m/s) is
An ac source of angular frequency co is fed across a resistor R and a capacitor C in series. The current registered is I. If now the frequency of source is changed to \[\omega /3\] (but maintaining the same voltage), the current in the circuit is found to be halved. The ratio of reactance to resistance at the original frequency \[\omega \]is
An aluminum rod and a copper rod of equal length 1 m and cross-sectional area \[1\text{ }c{{m}^{2}}\]are welded together as shown in the figure. One end is kept at a temperature of \[20{}^\circ C\]and other at \[60{}^\circ C\]. Calculate the amount of heat (in joule/sec) taken out per second from the hot end. Thermal conductivity of aluminum is \[200\text{ }W/m{}^\circ C\]and of copper is\[390\text{ }W/m{}^\circ C\].
The wavelength of the first spectral line in the Banner series of hydrogen atom is\[6561\text{ }!\overset{o}{\mathop{A}}\,\]. The wavelength (in \[\overset{o}{\mathop{A}}\,\]) of the second spectral line in the Balmer series of singly-ionized helium atom is
Consider the following compounds I to IV with respect to their Su2 reactivity with a given nucleophile \[\underset{I}{\mathop{C{{H}_{3}}C{{H}_{2}}Br}}\,\] \[\underset{II}{\mathop{C{{H}_{3}}C{{H}_{2}}C{{H}_{2}}Br}}\,\] \[\underset{III}{\mathop{{{\left( C{{H}_{3}} \right)}_{2}}CHC{{H}_{2}}Br}}\,\] \[\underset{IV}{\mathop{{{\left( C{{H}_{3}} \right)}_{3}}CC{{H}_{2}}Br}}\,\] The correct order of decreasing reactivity is -
The standard free energy change for the following reaction is - 210 KJ. What is me standard cell potential? \[2{{H}_{2}}{{O}_{2}}\left( aq \right)\xrightarrow{{}}2{{H}_{2}}O(\ell )+{{O}_{2}}\left( g \right)\]
\[100\text{ }c{{m}^{3}}\] of a solution of an acid (Molar Mass = 98) containing 29.4 g of the acid per litre were completely neutralized by 90.0 cm3 of aq. \[NaOH\]containing 20 g of \[NaOH\] per \[500\text{ }c{{m}^{3}}\]. The basicity of the acid is:
In an atom, an electron is moving with a speed of 600 m/s with an accuracy of 0.005%. Certainty with swish the position of the electron can be located is \[(h=6.6\times {{10}^{-~34}}kg\,{{m}^{2}}{{s}^{-1}},{{e}_{m}}=9.1\times {{10}^{-31}}kg)\]-
\[S{{O}_{3}}\left( g \right)\] is heated in a closed vessel. An equilibrium \[2S{{O}_{3}}\left( g \right)2S{{O}_{2}}\left( g \right)+{{O}_{2}}\left( g \right)\] is established. The vapour density of the mixture in which \[S{{O}_{3}}\] is 50% dissociated is
A gas expands adiabatically at content pressure such that\[T\propto \frac{1}{\sqrt{V}}\]. The value of V i.e. \[\frac{{{C}_{p}}}{{{C}_{V}}}\] of the gas will be
The total no. of stereoisomers possible for the compound \[C{{H}_{3}}C{{H}_{2}}CH\underset{Br}{\mathop{\underset{|}{\mathop{C}}\,}}\,{{H}_{2}}CH=CHC{{H}_{3}}\] is-
If A ,B and C have positive values such that \[A+B+C=\pi ;\] then the minimum value of \[\frac{\cos \left( \frac{A-B}{2} \right)}{\cos \left( \frac{A+B}{2} \right)}+\frac{\cos \left( \frac{B-C}{2} \right)}{\cos \left( \frac{B+C}{2} \right)}+\frac{\cos \left( \frac{C-A}{2} \right)}{\cos \left( \frac{C+A}{2} \right)}A\]is
The real numbers c, b and a form an arithmetic sequence with\[a\ge b\ge c\ge 0\]. If the quadratic equation \[a{{x}^{2}}+bx+c=0\]has exactly one root, then the root is
If m is the slope of a line which is a tangent to the hyperbola \[\frac{{{x}^{2}}}{{{\alpha }^{2}}}-\frac{{{y}^{2}}}{{{({{\alpha }^{3}}+{{\alpha }^{2}}+\alpha )}^{2}}}=1,\] then
If \[f(x)=\left\{ \begin{matrix} \sqrt{\left\{ x \right\}} & for & x\in /Z \\ 1 & for & x\in /Z \\ \end{matrix} \right.\]where \[\{.\}\]denotes the fractional part of x, then the area bounded by \[f(x)\]and \[g(x)\] for \[x\in \left[ 0,6 \right]\]is
If f is a continuous function in \[[0,1]\] such that \[f\left( \frac{1}{k} \right)=k\,\,\forall k\in N,\] then \[\underset{n\to \infty }{\mathop{\lim }}\,f\left( \sqrt[3]{{{n}^{2}}-{{n}^{3}}}+n \right)\] is equal to
If \[f(x)\] is a continuous function \[\forall \,x\in R-\{-2\}\] and satisfies \[{{x}^{3}}-{{x}^{2}}(f(x)+2)-x+2(2f(x)+1)=0\forall \,x\in R-\{-2\},\] then \[f(2)\]is equal to
If the curves \[y=\frac{{{\log }_{e}}x}{x}\] and \[y=\lambda {{x}^{2}}\] (where \[\lambda \] is constant) touch each other, then the value of \[\lambda \]is
Let \[f(x)={{x}^{3}}+5x+8\] and \[x=\alpha \] be a point such that \[f'(\alpha )\ne \frac{f(b)-f(a)}{b-a}\] for any values of \[a,b,\in R\]. Then the number of such points is
Equation of line in the plane \[P:2x-y+z-4=0\] which is perpendicular to the line \[L:\frac{x-2}{1}=\frac{y-2}{1}=\frac{z-3}{-2}\] and which passes through the point of intersection of L and P is
If \[{{n}_{1}},{{n}_{2}},{{n}_{3}},....,{{n}_{100}}\] are positive real numbers such that \[{{n}_{1}}+{{n}_{2}}+{{n}_{3}}+.....+{{n}_{100}}=20\]and \[k={{n}_{1}}({{n}_{2}}+{{n}_{3}}+{{n}_{4}})\,({{n}_{5}}+{{n}_{6}}+....+{{n}_{9}})({{n}_{10}}+....+{{n}_{16}})\]\[...(...+{{n}_{100}}),\] then \[k\in \]
Let AB be the chord of contact of the point \[(3,-3)\] w.r.t. the circle\[{{x}^{2}}+{{y}^{2}}=9\]. Then the locus of the orthocentre of \[\Delta PAB,\]where P be any point moving on the circle, is
Let W denote the words in the English dictionary. Define the relation R = { \[(x,y)\in W\times W:\] as the words x and y have at least one letter in common}. Then R is
Let \[{{A}_{i}},\] where \[i=1,2,3,...,10,\]be an independent event such that\[P({{A}_{i}})=\frac{1}{i+1}\]. Then probability that none of the events \[{{A}_{1}},{{A}_{2}},.....{{A}_{10}}\] occurs is____.
A point \[P(x,y)\] moves such that the sum of its distances from the lines \[2x-y-3=0\]and \[x+3y+4=0\]is 7. The area bounded by locus of P is (in sq. units) _____.
For a regular polygon, let r and R be the respective radii of the inscribed and the circumscribed circles. Then the value of \[\frac{r}{R}\]CANNOT be____.