A uniform thin rod AB of length L has linear mass density \[\mu \] \[(x)=a+\frac{bx}{L},\]where x is measured from A. If the CM of the rod lies at a distance of \[\left( \frac{7}{12} \right)L\] from A, then a and b are related as:
The radii of the two columns in U tube are \[{{r}_{1}}\] and \[{{r}_{2}}\]. When a liquid of density p (angle of contact is \[0{}^\circ \]) is fillied in it the level difference of liquid in two arms is h. The surface tension of liquid is (g = acceleration due to gravity)
Three objects A, B and C are kept in a straight line on a frictionless horizontal surface. These have masses m, 2m and m, respectively. The object A moves towards B with a speed 9 m/s and makes an elastic collision with it. Thereafter, B makes completely inelastic collision with C. All motions occur on the same straight line. Find the final speed (in m/s) of the object C.
A square surface of side L meter in the plane of the paper is placed in a uniform electric field E (volt/m) acting along the same plane at an angle Q with the horizontal side of the square as shown in Figure. The electric flux linked to the surface, in units of volt. w, is
Nine resistors each of \[1\,k\Omega \] are connected to a battery of 6 V as shown in the circuit given below. What is the total current flowing in the circuit.
The relation gives the value of P as \[P=\frac{\sqrt{abc}}{7d}.\]. Find the percentage error in P if the percentage error in a, b, c and d are \[2%,\text{ }2%,\text{ }2%\] and \[4%\] respectively
A block is dragged on a smooth plane with the help of a rope which moves with a velocity v as shown in figure. The horizontal velocity of the block is:
A projectile is thrown in the upward direction making an angle of \[60{}^\circ \] with the horizontal direction with a velocity of \[147\,m{{s}^{-1}}\]. Then the time after which its inclination with the horizontal is \[45{}^\circ \] is:
A source \[{{S}_{1}}\] is producing, \[{{10}^{15}}\]photons per second of avelength\[5000\overset{o}{\mathop{A}}\,\]. Another source \[{{S}_{2}}\] is producing \[1.02\times {{10}^{15}}\] photons per second of wavelength \[5100\overset{o}{\mathop{A}}\,\] Then, (power of \[{{S}_{2}}\]) to the (power of \[{{S}_{1}}\]) is equal to :
If in hydrogen atom, radius of \[{{n}^{th}}\] Bohr orbit is \[{{r}_{n}},\] frequency of revolution of electron in \[{{n}^{th}}\]orbit is \[{{f}_{n}},\] choose the correct option.
A sample of radioactive material decays simultaneously by two processes A and B with half-lives \[1/2\] and \[1/4\,hr.\]respectively. For first half hour it decays with the process A, next one hour with the process B and for further half an hour with both A and B. If originally there were \[{{N}_{0}}\] nuclei, the number of nuclei after 2 hour of such decay is \[{{N}_{0}}{{\left( \frac{1}{x} \right)}^{x}}\]then find the value of x.
Liquid oxygen at \[50\text{ }K\]is heated to \[300\text{ }K\]at constant pressure of 1 atm. The rate of heating is constant. Which one of the following graphs represents the variation of temperature with time?
A siren is fitted on a car going towards a vertical wall at a speed of\[36\text{ }km/hr\]. A person standing on the ground behind the car, listens to the siren sound coming directly from the source as well as that coming after reflection from the wall. The apparent frequency of the wave coming directly from the siren to the person and coming after reflection respectively are (Take the speed of sound to be\[340\text{ }m/s.\])
A \[2000\text{ }cc\]sample of \[{{O}_{2}}\] is confined to a cylinder. Initially, pressure of the gas is \[{{10}^{5}}\text{ }N/{{m}^{2}}\]and its temperature\[27{}^\circ C\]. The gas is subjected to a cyclic process. In the first step, its pressure is made twice at constant volume. In the second step, it expands to its initial pressure adiabatically and in the final step, the gas undergoes isobaric compression and finally attains its initial volume. Consider that the gas is ideal. Temperature of the gas at the ideal of the first step and volume of the gas at the end of the second step will be -
The density \[(\rho )\] versus pressure (P) of a given mass of an ideal gas is shown at two temperatures \[{{T}_{1}}\] and \[{{T}_{2}}\] Then relation between \[{{T}_{1}}\] and \[{{T}_{2}}\] maybe
A coil of resistance \[50\,\Omega \] is connected across a \[5.0\text{ }V\]battery. \[0.1\text{ }s\]after the battery is connected, the current in the coil is\[60\text{ }mA\]. Calculate the inductance of the coil.
A thin glass (refractive index\[1.5\]) lens has optical power of \[-5\text{ }D\]in air. Its optical power in a liquid medium with refractive index 1.6 will be
A beam of protons with velocity \[4\times {{10}^{5}}\text{ }m/s\]enters a uniform magnetic field of \[0.3\] tesla at an angle of \[60{}^\circ \] to the magnetic field. Find the radius (in cm) of the helical path taken by the proton beam. (mass of proton\[=1.67\times {{10}^{-27}}\text{ }kg\].)
Two blocks of masses 5 kg and 10 kg are connected by a metal wire going over a smooth pulley as shown in fig. The breaking stress of the metal wire is\[2\times 109\text{ }N/{{m}^{2}}\]. If \[g=10\text{ }m/{{s}^{2}},\]then find the minimum radius (in mm) of the wire which will not break.
In Young's double slit interference experiment, the distance between two sources is\[0.1/n\text{ }mm\]. The distance of the screen from the source is\[25\text{ }cm\]. Wavelength of light used is \[5000\overset{o}{\mathop{A}}\,\]. Determine the angular position of the first dark fringe.
A mass M attached to a spring oscillates with a period of 2s. If the mass is increased by 2 kg, then the period increases by 2s. Find the initial mass M (in kg) assuming that Hooke's law is obeyed.
A T.V. tower has a height 100m . What is the population density (in\[k{{m}^{-2}}\]) around the T.V. tower if the total population coverd is 50 lac? R (earth) \[=6.4\times {{10}^{6}}m\]
An oxide \[{{A}_{x}}{{O}_{y}}\] has molecular weight 288 u. Atomic weights of A and 0 respectively are 12 and 16. The formula of the compound, if A is 50% by weight is
Molar conductivity of 0.025 mol \[{{\text{L}}^{-1}}\]methanoic acid is\[46.1S\,\text{c}{{\text{m}}^{2}}\,\text{mo}{{\text{l}}^{-1}}\]. The degree of dissociation and dissociation constant will be (Given : \[\lambda _{{{H}^{+}}}^{o}=46.1S\,\text{c}{{\text{m}}^{2}}\,\text{mo}{{\text{l}}^{-1}}\]and \[\lambda _{HCO{{O}^{-}}}^{o}=54.6S\,\text{c}{{\text{m}}^{2}}\,\text{mo}{{\text{l}}^{-1}}\])
\[Mg\xrightarrow[Heat]{Air}X+Y\xrightarrow[Colourless\,gas]{{{H}_{2}}O}Z\] \[Z\xrightarrow[{}]{{{H}_{2}}O}Solution\xrightarrow[Blue\,coloured\,solution]{CuS{{O}_{4}}}(A)\] Substances X, Y, Z and A are respectively
At 1400 K, \[{{K}_{c}}=2.5\times {{10}^{-3}}\]for the reaction : \[C{{H}_{4(g)}}+2{{H}_{2}}{{S}_{(g)}}C{{S}_{2(g)}}+4{{H}_{2(g)}}\] A 10 L reaction vessel at 1400 K contains 2.0 mol of \[C{{H}_{4}},3.0\] mol of \[C{{S}_{2}},3.0\]mol of\[{{H}_{2}}\] and 4.0 mol of \[{{H}_{2}}S\]. In which direction does the reaction proceed to reach equilibrium?
Match the List I with List II for singly ionized helium atom if total energy of electron in first orbit in H-atom is -13.6 eV atom\[^{-1}\] and select the correct answer using the code given below the lists.
If an impurity in a metal has a greater affinity for oxygen and is more easily oxidised than the metal, then the purification of metal may be carried out by
The total number of molecules that do not follow octet rule among the following is___. \[CO,PC{{l}_{5}},PC{{l}_{3}},AlC{{l}_{3}},S{{F}_{6}},B{{F}_{3}},N{{H}_{3}}\]
Among the following, the number of underlined elements having +6 oxidation state is___. \[\underline{P}O_{4}^{3-},{{H}_{2}}{{\underline{S}}_{2}}{{O}_{8}},{{H}_{2}}\underline{S}{{O}_{5}},\underline{O}{{F}_{2}},{{\underline{Cr}}_{2}}O_{7}^{2-},\underline{Cr}{{O}_{5}}\]
If \[f\left( x \right)=g\left( x \right)|(x-1)\,(x-2)...(x-10)|-2\]is differentiable \[\forall x\in R,\] where \[g(x)\] is a polynomial of degree 9, then \[\frac{d}{dx}\,\left( f\left( {{x}^{2}}+g\left( x \right) \right) \right)\]at \[x=0\]is
Every tangent of a circle is perpendicular to exactly one member of the family of lines \[(x+y-2)+\lambda (7x-3y-4)=0\] at the point of contact of tangent. Also, the circle touches only one member of the family \[(2x-3y)+\mu (x-y-1)=0\]. The circle passes through the point
Quadratic equations \[{{x}^{2}}-6x+a=0\]and \[{{x}^{2}}-ex+6=0\]have one root \[\alpha \] in common. The other roots of the first and second equations are integers in the ratio\[4:3\]. Then which of the following expressions is negative for\[x=\alpha \]?
If f is a differentiable function satisfying \[f(x)=\int\limits_{0}^{x}{\sqrt{1-{{f}^{2}}(t)}}dt+\frac{1}{2},\] then the value of \[{{\sin }^{-1}}(f(\pi ))\] is equal to
For the curve defined parametrically as \[y=3\,\,\sin \theta .\cos \theta ,\] \[x={{e}^{\theta }}.\sin \theta ,\] where \[\theta \in [0,\pi ]\] the tangent is parallel to x-axis when \[\theta \] is
Let \[f(x)\] be a non-constant twice differentiable function on R such that \[f(2+x)=f(2-x)\] and \[f'\left( \frac{1}{2} \right)=f'(1)=0\] Then minimum number of roots of the equation \[f''(x)=0\]in \[(0,4)\]is
Let r, s, t be different prime numbers and a, b, c be positive integers. If LCM of a, b, c is \[{{r}^{2}}{{s}^{4}}{{t}^{2}}\]and HCF of a, b, c is \[r{{s}^{2}}t,\] then the number of ordered triplets (a, b, c) is
Locus of all the points which are at a distance of 3 units from the line \[\overset{\scriptscriptstyle\smile}{r}=\lambda (\hat{i}+\hat{j}+\hat{k})\] is given by \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-xy-yz-zx=k.\]The value of k is
If \[{{\tan }^{-1}}\left( -\alpha +\theta -\frac{{{\theta }^{3}}}{3!}+\frac{{{\theta }^{5}}}{5!}-\frac{{{\theta }^{7}}}{7!}+...\infty \right)+{{\cot }^{-1}}\]\[\left( \alpha -1+\frac{{{\theta }^{2}}}{2!}-\frac{{{\theta }^{4}}}{4!}+...\infty \right)=\frac{\pi }{2},\]then maximum value of \[\alpha \] equals to
If the system of equations \[px+2y-3=0,\]\[x+3y-4=0\] and \[p{{x}^{2}}+3{{y}^{2}}+3xy+(q-3)x-3y-1=0\left( p\ne \frac{2}{3} \right)\] has a unique solution, then the value of \[(p+q)\] is ______.
Let a and \[\beta (\alpha >3)\] be the roots of \[2{{x}^{2}}-ax+4=0\]and \[\sum\limits_{n=1}^{\infty }{{{\left( \frac{1}{\alpha }+\beta \right)}^{n}}=3.}\]Then the value of a is _______,