A body is projected up from the surface of the 3 earth with a velocity equal to \[\frac{3}{4}\] th of its escape velocity. If R be the radius of the earth, the height it reaches is -
The equation of SHM of a particle is given as \[2\frac{{{d}^{2}}x}{d{{t}^{2}}}+32x=0\] where x is the displacement from the mean position. The period of its oscillation (in seconds) is -
A long straight wire carrying a current of 30 A is placed in an external uniform magnetic field of induction\[4\times {{10}^{-4}}T\]. The magnetic field is acting parallel to the direction of current. The magnitude of the resultant magnetic induction in tesla at a point 2.0 cm away from the wire is
Rate of increment of energy in an inductor with time in series LR circuit getting charge with battery of e.m.f. E is best represented by: [inductor has initially zero current ]
A gas is found to obey the law \[{{P}^{2}}V\]= constant the initial temperature and volume are To and\[{{V}_{0}}\]. If the gas expands to volume\[2{{V}_{0}}\], its final temperature becomes-
Three objects colored black, grey and white are thrown into a furnace where each of them attains a temperature of\[2000{}^\circ C\]. Which object will glow the brightest?
The frequency of fundamental tone in an open organ pipe of length 0.48 m is 320 Hz. Speed of sound is 320 m/sec. Frequency of fundamental tone in closed organ pipe will be -
A box containing N molecules of a perfect gas at temperature \[{{T}_{1}}\] and pressure\[{{P}_{1}}\]. The number of molecules in the box is doubled keeping the total kinetic energy of the gas same as before. If the new pressure is \[{{P}_{2}}\] and temperature \[{{T}_{2}}\], then-
The Binding energy of \[_{17}^{35}Cl\] nucleus is 298 MeV. Find its atomic mass. \[\left[ {{m}_{p}}=1.007825\text{ }amu,\text{ }{{\text{m}}_{n}}=1.008665\text{ }amu \right]\]
The threshold frequency for a certain metal is \[{{V}_{0}}\]when light of frequency \[v=2{{v}_{0}}\] is incident on it. The maximum velocity of photoelectrons is \[4\times {{10}^{6}}m/s.\] If the frequency of incident radiation is increase to\[5\,{{v}_{0}}\], the maximum velocity of photo electrons in m/s will be.
An \[\alpha \]-particle and a proton are fired through the same magnetic fields which is perpendicular to their velocity vectors. The \[\alpha \]-particle and proton move such that radius of curvature of their path is same, then \[\frac{{{\lambda }_{\alpha }}}{{{\lambda }_{\rho }}}=\]
If the electric potential on the surface of inner most sphere is zero, then the relation between \[{{r}_{1}},{{r}_{2}}\] and \[{{r}_{3}}\] is (here \[\sigma \] is surface charge density)
A small sphere of mass m and carrying a charge q is attached to one end of an insulating thread of length a, the other end of which is fixed at (0, 0) as shown in figure. There exists a uniform electric field \[\vec{E}=-\text{ }{{\vec{E}}_{0}}\text{ }\hat{j}\] in the region. The minimum velocity which should be given to the sphere at (a. 0) in the direction shown so that it is able to complete the circle around the origin is (There is no gravity)
For a cell the graph between the potential difference (V) across the terminals of the cell and the current (I) drawn from the cell is shown in figure. The emf and the internet resistance of the cell are-
In the dimensional analysis of equation \[{{\left( velocity \right)}^{x}}={{\left( pressure\text{ }difference \right)}^{3/2}}\times {{\left( density \right)}^{-3/2}}.\]The value of x comes out to be equals to
A drunkard walking in a narrow lane takes 5 steps forward and 3 steps backward, followed again by 5 steps forward and 3 steps backward, and so on. Each step is 1 m long and required 1 second to cover. How long (in sec) the drunkard takes to fall in a pit 13 m away from the start?
An aeroplane was flying horizontally with a velocity of 720 km/h at an altitude of 490 m. When it is just vertically above the target a bomb is dropped from it. How far horizontally it missed the target (in m)?
The electronic configuration of an element is \[1{{s}^{2}}2{{s}^{2}}2{{p}^{6}}3{{s}^{2}}3{{p}^{3}}\]. What is the atomic number of the element, which is just below the above element in the periodic table?
Boron has two stable isotopes,\[^{10}B \left( 19\right)\] % and \[^{11}B \left( 81\right)\] %. Average atomic weight for boron in the periodic table is
A buffer solution is prepared by mixing 0.1 M ammonia and M ammonium chloride. At 298 K, the \[{{\operatorname{pK}}_{b}}\,of\,\,N{{H}_{4}}OH\] is 5.0. The pH of the buffer is
Given \[\operatorname{E}{{{}^\circ }_{C{{r}^{3+}}/Cr}}=-0.72V,\,\,E{}^\circ {{ }_{F{{e}^{2+}}/Fe}}=-\,0.42\,V\] The potential for the cell \[\operatorname{Cr}\left| C{{r}^{3+}}\left( 0.1\,M \right) \right|\left| F{{e}^{2+}}\,\left( 0.01\,M \right) \right|\] is
What is the energy (in eV) required by an electron in the \[L{{i}^{2+}}\] ion to be emitted from \[\,n = 2 state\]? (Given: Ground state ionization energy of hydrogen atom is 13.6 eV)
Diborane is a potential rocket fuel which undergoes combustion according to the equation \[{{B}_{2}}{{H}_{6}}(g)+3{{O}_{2}}\xrightarrow{{}}\,{{B}_{2}}{{O}_{3}}(s)+3{{H}_{2}}O(g)\]
Calculate the enthalpy change for the combustion of diborane in kJ/mol. Given
What amount of heat will be released (in kJ/mol) in formation of 35.2 g of \[C{{O}_{\text{2}}}\] from carbon and oxygen gas, if heat of combustion of carbon to \[C{{O}_{\text{2}}}\] is - 393.5 kJ/mol?
Let H be a set of hyperbolas. If a relation R on H is defined by \[R=\{({{h}_{1}},\,{{h}_{2}}):{{h}_{1}},{{h}_{2}}\]have same pair of asymptotes, \[{{h}_{1}},{{h}_{2}}\in H\},\] then the relation R is
A normal to the hyperbola \[\frac{{{x}^{2}}}{4}-\frac{{{y}^{2}}}{1}=1,\] has equal intercepts on positive x and y axes. If this normal touches the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1,\]then the value of \[({{a}^{2}}+{{b}^{2}})\] is
If matrix \[A=\left[ \begin{matrix} 1 & -3 \\ -1 & 1 \\ \end{matrix} \right]\] and \[S=A-\frac{1}{3}{{A}^{2}}+\frac{1}{9}{{A}^{3}}+.....\infty ,\]then the sum of elements of matrix S is
Let \[\alpha ,\beta \] be the roots of \[{{x}^{2}}-bx=b\,\,(b>0)\] and \[|\alpha |,\,|\beta |\] be the roots of \[{{x}^{2}}+px+q=0.\]The minimum value of \[({{p}^{2}}-8q)\] is equal to
If the plane \[\frac{x}{2}+\frac{y}{3}+\frac{z}{4}=\frac{1}{2}\] intersects x, y and z-axes at A, B and C, respectively, then volume of the tetrahedron OABC where O is the origin is
The area bounded by curves \[y=cosec\text{ }x,\]\[y=sec\text{ }x,\] \[y=cos\text{ }x\]and \[y=sin\text{ }x\]on interval \[\left( 0,\frac{\pi }{2} \right)\] ? is equal to
Let \[\vec{a},\vec{b},\vec{c}\]be three non-zero vectors such that projection of \[(\vec{b}+\vec{c})\]on a is \[2|\vec{a}|\], projection of \[|\vec{a}+\vec{c}|\]on \[\vec{b}\] is \[3|\vec{b}|\]and projection of \[|\vec{a}+\vec{b}|\]on \[\vec{c}\]is \[4|\vec{c}|,\] (where \[|\vec{a}|,|\vec{b}|,|\vec{c}|\in N\]). Then the minimum value of \[|\vec{a}+\vec{b}+\vec{c}|\] is equal to
A vertical pole stands at a point A on the boundary of a circular park of radius a and subtends an angle \[\alpha \] at another point B on the boundary. If the chord AB subtends an angle \[\alpha \] at the centre of the park, then the height of the pole is
Let there exist real numbers a and b such that for every positive number x, the identity \[{{\tan }^{-1}}\left( \frac{1}{x}-\frac{x}{8} \right)+{{\tan }^{-1}}(ax)+{{\tan }^{-1}}(bx)=\frac{\pi }{2}\] holds. Then the value of \[{{a}^{2}}+{{b}^{2}}\] is
If \[\alpha \] and \[\beta \] are acute angles such that \[\alpha +\beta =\lambda ,\] where \[\lambda \] is a constant, then maximum possible value of the expression \[\sin \alpha +sin\beta +cos\alpha +cos\beta \] is equal to
AB is a chord of the circle \[{{x}^{2}}+{{y}^{2}}=\frac{25}{2}.\]P is a point such that \[PA=4\]and\[PB=3\]. If \[AB=5,\]then distance of P from origin can be
If \[{{S}_{n}}{{=}^{n}}{{C}_{0}}{{.}^{n}}{{C}_{n}}{{+}^{n}}{{C}_{1}}{{.}^{n}}{{C}_{n-1}}{{+}^{n}}{{C}_{2}}{{.}^{n}}{{C}_{n-2}}+....{{+}^{n}}{{C}_{n}}{{.}^{n}}{{C}_{0}},\] then maximum value of \[\left[ \frac{{{S}_{n+1}}}{{{S}_{n}}} \right]\] is (where \[[.]\] denotes the greatest integer function)
The number of solutions of the equation \[{{\log }_{e}}({{\sec }^{2}}\theta +\cos e{{c}^{2}}\theta )={{\log }_{e}}4\sin \theta +{{\log }_{e}}16\cos \theta \]for \[\theta \in (0,2\pi )\] is
If \[2x+y=1\] is a tangent to \[y=f(x)\] at \[x=\frac{1}{3},\] then \[\underset{x\to 0}{\mathop{\lim }}\,\,\,\frac{\sin x-\sin 3x}{f\left( \frac{{{e}^{3x}}}{3} \right)-f\left( \frac{{{e}^{-3x}}}{3} \right)}\]is equal to _____.
Let \[f(x)=\left\{ \begin{matrix} x(x-1){{e}^{2x}}, & if\,\,x\le 0 \\ x(1-x){{e}^{-2x}}, & if\,\,x>0 \\ \end{matrix} \right.\] . Then \[f(x)\] attains its greatest value at _____.