If \[{{e}^{\lambda }}+1,{{e}^{-\lambda }}+1\] are the roots of the equation \[{{x}^{2}}-2(p+1)x+5p-{{p}^{2}}=0\] where \[X,p\in R\] then number of integral values of 'p' is
If two sides of a triangle ABC are represented by vectors a and \[\left( \vec{a}\times \vec{b} \right)\times \vec{a}\] then maximum value of \[(\sin 2A+\sin B+\sin 2B+\sin 2C)\], is
If there are 10 stations on a route and the train has to be stopped at 4 of them, then the number of ways in which the train can be stopped so that atleast two stopping stations are consecutive is
Let \[P(3,3)\] and \[Q(2,1)\] be two points and the straight lines PQ and QR are equally inclined to the circle\[{{x}^{2}}+{{y}^{2}}=5,\] then equation of the line QR is
The number of integral values of 'a' for which the equation \[4\sin \left( x+\frac{\pi }{3} \right)\cos \left( x-\frac{\pi }{6} \right)={{a}^{2}}=\sqrt{3}\sin 2x-\cos 2x\]has a solution, is
If \[{{p}^{th}},{{q}^{th}},{{r}^{th}}\] terms of a G.P. are the positive numbers \[a,b,c\] respectively then angle between the vectors\[(\log {{a}^{2}})\hat{i}+(\log {{b}^{2}})\hat{j}+(\log {{c}^{2}})\hat{k}\] and \[(q-r)\hat{i}+(r-p)\hat{j}+(p-q)\hat{k}\]
The number of points (a, b), \[(a,b\in I)\] in the x-y plane from where two mutually perpendicular tangents can be drawn to the hyperbola\[\frac{{{x}^{2}}}{25}-\frac{{{y}^{2}}}{9}=1\], is
If each of 'n' readings are increased by 5, then the arithmetic mean of new 'n? readings is equal to 10. If each of the original readings are multiplied by 3, then the arithmetic mean of new readings will be
If the plane \[\frac{x}{2}+\frac{y}{3}+\frac{z}{4}=-\frac{1}{2}\] intersects \[x,y\] and z-axis at A, B and C respectively then volume of the tetrahedron OABC where '0' is the origin is
\[\left| \frac{{{z}_{1}}-2{{z}_{2}}}{2-{{z}_{1}}{{z}_{2}}} \right|=1\]and \[\left| {{z}_{2}} \right|\ne 1\] then the value of \[\left| {{z}_{1}} \right|\] is
The area bounded by the curve \[y={{x}^{2}}\] and \[y=\frac{2}{1+{{x}^{2}}}\] is \[\lambda \] sq. units, then the value of \[[\lambda ]\] is [Note: \[[k]\] denotes greatest integer less than or equal to k.]
If z satisfies the equation \[\left( \frac{z-2}{z+2} \right)\left( \frac{\overline{z}-2}{\overline{z}+2} \right)=1\], then minimum value of \[\left| z \right|\] is equal to
Let R be the real line. Consider the following subsets of the plane\[R\times R\]. \[S=\{(x,y):x-y+1=0\]and \[0<x<2\}\] \[T=\{(x,y):x-y\]is an integer} Which one of the following is true?
The angle of elevation of the top of pole from the point A on ground be \[\alpha \], where as angle of depression of the foot of pole from the point which is 'b' foot above point A is \[\beta \], then the length of pole is
If \[\left| {\vec{a}} \right|=\left| {\vec{b}} \right|=\left| {\vec{c}} \right|=2\] and \[\vec{a}.\vec{b}=\vec{b}.\vec{c}=\vec{c}.\vec{a}=2\], then \[[\vec{a}\,\vec{b}\,\vec{c}]\cos {{45}^{o}}\] is equal to
Let A be a square matrix of order 2 such that \[{{A}^{2}}-4A+4I=O\] where \[I\] is an identity matrix of order 2. If \[B={{A}^{5}}+4{{A}^{4}}+6{{A}^{3}}+4{{A}^{2}}+A\], then det.(B) is equal to
A \[2kg\] block is thrown on a rough horizontal surface with\[4m{{s}^{-1}}\]. Then work done by kinetic fraction on the horizontal surface and on the block respectively will be:
Two particles are executing SHM of the same amplitude A and frequency co along the x-axis. Their mean position is separated by \[{{X}_{0}}\] (where\[{{X}_{0}}>A\]). If maximum separation between them is \[{{X}_{0}}>2A\], then he phase difference between their motion is :
When three progressive wave : \[{{Y}_{1}}=4\sin (2x-6t),{{y}_{2}}=6\sin (2x-6t+\frac{\pi }{2})\] and \[{{Y}_{3}}=12\sin (2x-6t+\pi )\] are superimposed the amplitude of resultant wave is :
If acceleration and velocity of a particle are \[5\,m{{s}^{-2}}\hat{j}\] and \[3\hat{i}+4\hat{j}\,\,m{{s}^{-1}}\] respectively then centripetal acceleration and radius of current are at that point of the path where particle is present will be :
There is a spherical cavity of radius \[R/2\] in uniformly charged spherical region having charge density \[+\rho \] and radius R. If a small charge particle having charge \[{{q}_{0}}\] is released at point C, centre of the cavity then it will collide with the wall of the cavity with kinetic energy-
In spherical distribution, the charge density varies A as \[\rho (r)=\frac{A}{2}\] for \[a<r<b\] (as shown) where A is a constant A point charge Q lies at the centre of the sphere at \[r=0\]. The electric field in the region \[a<r<b\] has a constant magnitude for
A magnet of length 14 cm and magnetic moment M is broken into two parts of length 6 cm and 8 cm. They are put at right angles to each other with the opposite poles together. The magnetic dipole moment of the combination is
Two consider coils X and Y have equal number of turns and carry equal currents in the same sense and subtend same solid angle at point 0. The smaller coil \[X\] is midway between 0 and Y. the magnetic induction due to bigger coil Y at 0 is \[{{B}_{Y}}\] and that due to smaller coil \[X\] at 0 is \[{{B}_{X}}\]. Then
An induction coil has an impedance of \[10\,\Omega \]. When an AC signal of frequency \[1000\,Hz\] is applied to the coil, the voltage leads the current by \[{{45}^{o}}\]. The inductance of the coil is :
A ray of light makes normal incidence on the diagonal face of a right angled prism as shown in figure If\[\theta =37{}^\circ \], then the angle of deviation is \[(sin37{}^\circ \,3/5)\]
A hydrogen like atom (atomic number \[Z\]) is in a higher excited state a quantum number n. This excited atom can make a transition to the first excited state by successively emitting two photons of energies \[10.2\,\,eV\] and \[16.8\,\,eV\] respectively. Alternatively, the atom from the same excited state can make a transition to the second excited state by successively emitting two photons of energies \[4.25\,\,eV\] and \[5.95\,\,eV\] respectively. The values of n and \[Z\] are respectively (Ground state energy of hydrogen atom is \[-13.6\,\,eV\])
When photons of energy \[4.25\,\,eV\] strike the surface of a metal A, the ejected photoelectrons have maxi mum kinetic energy, \[{{T}_{A}}\] expressed in eV and de-Broglie wavelength \[{{\lambda }_{A}}\] . The maximum kinetic energy of photoelectrons liberated from another metal B by photons of energy \[4.70\,\,eV\] is \[{{T}_{B}}=({{T}_{A}}-1.50\,\,eV)\]. If the de-Broglie wavelength of these photoelectrons is \[{{\lambda }_{B}}=2{{\lambda }_{A}}\], then
If for a blackbody the graph of change is emissive power at different temperatures \[{{T}_{1}},{{T}_{2}}\] and \[T{{ }_{3}}\] with wavelength is according to the figure then.
The length of a needle floating, on the surface of water is \[2.25\,\,cm\]. The minimum force needed to lift the needle above the surface of water will be : \[(T=7.2\,N/cm)\]
The amount of work required for increasing the length of a given wire of length \[l\] by \[l\] will be : (A \[=\] Area, \[Y=\] Young's modulus of material of the wire)
If the terminal speed of a sphere of gold (density \[=19.5\,kg/{{m}^{2}}\]) is \[0.2\,m/s\] in a viscous liquid (density \[=1.5\,kg/{{m}^{2}}\]), find the terminal speed of a sphere of silver (density \[=10.5\,kg/{{m}^{3}}\]) of the same size in the same liquid:
The amplitude of a danyed harmonic oscillator become half in 3 second and will become \[1/x\] of the initial amplitude in next 6 second where \[x\] is:
\[0.04\,\,kg\] of nitrogen is enclosed in a vessel at a temperature of \[{{27}^{o}}C\]. How much heat has to be transferred to the gas to double the rms speed of its molecules \[[R=2\,cal/mol\,K]\]
Ideal gas 'A' is given by internal energy of\[U=2{{P}_{0}}{{V}_{0}}\]. If gas undergoes an expansion, as shown below, then heat absorbed in the process will be (\[{{P}_{0}}=\] external pressure)
In a compound element 'Q' crystallised in a hexagonal close packed array and element T' occupies two out of three octahedral void. Formula of compound would be:
\[ZnC{{O}_{3}}\xrightarrow{A}ZnO\xrightarrow{B}\underset{\operatorname{Im}\,pure}{\mathop{Zn}}\,\xrightarrow{C}\underset{Pure}{\mathop{Zn}}\,\] correct coding of A, B & C is
(A) Copper \[(I)\] compound are unstable in aqueous solution and undergo disproportionation,
(B) The stability of \[C{{u}^{2+}}\] rather than \[C{{u}^{1+}}\] is due to the much more negative \[\Delta \,{{H}_{Hyd.}}\] of \[C{{u}^{2+}}\] then\[C{{u}^{+}}\].
If at\[298\,K,\]the solubility of \[AgCl\] in \[0.05M\]\[BaC{{l}_{2}}\] (completely dissociated) is found to be very nearly \[{{10}^{-9}}M\]and\[E_{Ag/A{{g}^{+}}}^{0}=-\,0.80\,V\]. Then the value of \[E_{C{{l}^{-}}/AgCl/Ag}^{0}\] at the same temperature will be:
In solid state \[PC{{l}_{5}}\] exist as an ionic solid \[{{[PC{{l}_{4}}]}^{+}}{{[PC{{l}_{6}}]}^{-}}\] hybridisation state of action and anion respectively.
Decomposition of A follows first order kinetics by the following equation. \[4A(g)\xrightarrow{{}}B(g)+2C(g)\] If initially, total pressure was 800 mm of Hg and after 10 minutes it is found to be 650 mm of Hg What is half-life of A? (Assume only A is present initially)
Pure \[PC{{l}_{5}}(g)\] was placed in a flask at \[2atm\] and left for some time at a certain temperature. Where the following equilibrium was established \[PC{{l}_{5}}(g)\xrightarrow{{}}PC{{l}_{3}}(g)+C{{l}_{2}}(g)\].
The solubility product of \[A{{s}_{2}}{{O}_{3}}\] is \[10.8\times {{10}^{-9}}\]. It is \[50%\] dissociated in saturated solution. The solubility of salt is:
Compound (A) \[({{C}_{4}}{{H}_{8}}{{O}_{3}})\] reacts with \[NaHC{{O}_{3}}\] and evolves \[C{{O}_{2}}(g)\]. (A) reacts with \[LiA1{{H}_{4}}\] to give a compound (B) which is achiral. The structure of is:
Which of the following statements is incorrect about the reaction given below? \[\xrightarrow[Traces\,of\,KOH]{HCN}(B)\xrightarrow{{{H}_{2}}+Ni}(C)\xrightarrow{HN{{O}_{2}}}(D)\]
A)
In the formation of (D) from (C), ring expansion takes place
doneclear
B)
The product (D) is cyclopentanone
doneclear
C)
The product (D) is a, p-unsaturated cyclopentanone
doneclear
D)
Conversion of (B) to (C) can also be carried out with \[LiA1{{H}_{4}}\]
\[+C{{H}_{3}}-\underset{\begin{smallmatrix} || \\ O \end{smallmatrix}}{\mathop{C}}\,-C{{H}_{3}}\xrightarrow[EtOH]{EtONa}\xrightarrow[\Delta ]{{{H}_{2}}S{{O}_{4}}}[X]\]The product \[[X]\] is :