The function \[f\,(x)=\left\{ \begin{matrix} {{x}^{2}}/a & 0\le x<1 \\ a & 1\le x<\sqrt{2} \\ (2{{b}^{2}}-4b)/{{x}^{2}} & \sqrt{2}\le x<\infty \\ \end{matrix} \right.\] is continuous for \[0\le x<\infty ,\] then the most suitable values of a and bare
Given \[A=\left| \begin{matrix} a & b & 2c \\ d & e & 2f \\ \ell & m & 2n \\ \end{matrix} \right|\] \[B=\left| \begin{matrix} f & 2d & e \\ 2n & 4\ell & 2m \\ c & 2a & b \\ \end{matrix} \right|,\] then
A point P lies on the hyperbola \[9{{x}^{2}}-16{{y}^{2}}=144\] such that \[P{{S}_{1}}:P{{S}_{2}}=3/2\] (where \[{{S}_{1}}\] and \[{{S}_{2}}\]are focii of hyperbola). Coordinates of point P is in the first quadrant are
An ellipse is inscribed in a circle and a point within the circle is chosen at random. If the probability that this point lies outside the ellipse is 2/3 then the eccentricity of the ellipse is:
If \[\vec{a}=\hat{i}+\hat{j}+\hat{k}\] & \[\vec{b}=\hat{i}-2\hat{j}+\hat{k},\] then the vector \[\vec{c}\]such that \[\vec{a}\,.\,\vec{c}=2\] & \[\vec{a}\times \vec{c}=\vec{b}\] is
A system with two 3 kg mass particles having velocities \[2\hat{i}+3\hat{j}\,m{{s}^{-1}}\]and \[4\hat{i}+6\hat{j}\,m{{s}^{-1}}\]is subjected to a constant force of \[24\hat{i}\,N\] for 6 s. Final velocity of centre of mass is
A block of mass 0.5 kg is placed over top of a wedge of mass 50 kg.
Angle of inclination \[\theta \]is \[37{}^\circ \]with horizontal. Block is released at \[t=0\] and it slide down the incline and reaches bottom with a speed of \[1.2625\,m{{s}^{-1}}.\]
There is no friction. Speed of the wedge with respect to ground, when block reaches at bottom is
A small block of mass 1 kg is pushed with a velocity \[2\,\hat{i}\,m{{s}^{-1}}\]over a horizontal plane which itself is moving with a velocity of \[-\,2\,\hat{j}\,m{{s}^{-1}}.\] Coefficient of friction between block and plane is 0.25. Force of friction acting on the block (in newton) nearly is
A pendulum bob is released from angular position \[\theta ,\]such that the magnitude of its initial acceleration and acceleration at lowest position are equal.
In steady state, heat conduction equations governing heat flow are identical to the electric current flowing through a conductor. What are the dimensions of quantity analogous to electrical resistance?
A rectangular glass wedge is partially dipped in water \[\left( {{n}_{\text{glass}}}=\frac{3}{2} \right).\]A beam of light entering face AB normally reaches entirely to AC.
A thick transparent slab is made such that its refractive index changes from \[{{n}_{1}}\] to \[{{n}_{2}}\] nearly linearly. This slab is used by a student to measure lateral displacement in lab. She draw following diagram using four pins. Angle p must be
The word "excellent" is written on a sheet of white paper with a red pencil and the word "good" is written with a green pencil, on a school notice board. A student look at this notice board through a green and red piece of glass.
A)
He can see "excellent" through red glass
doneclear
B)
He can see "excellent" through green glass
doneclear
C)
He can see "excellent" through both glasses
doneclear
D)
He cannot see "excellent" through any of the glasses
In an experiment of suspending spherical plastic balls in vertical turbulent jets of air and water it is found that if sphere is displaced slightly to left or right (from its suspended position), the air or water stream again pushes it back to its original position.
A piston moves upwards by 5 cm, when 200 J of heat is added.
Spring has a spring constant of \[50kN{{m}^{-1}}\]and is initially unstretched. Mass of piston is 60 kg and its diameter is 20 cm. Change of internal energy of vapour is
Let \[{{\lambda }_{1}}\] is the wavelength of an emitted photon in deexcitation of a Bohr's atom without considering recoil of atom. Also, \[{{\lambda }_{2}}\] is the wavelength of an emitted photon in deexcitation of a Bohr's atom considering recoil of atom. Then, correct statement is
One gram of activated carbon has a surface area of\[1000\text{ }{{m}^{2}}\]. Considering complete coverage as well as monomolecular adsorption, how much ammonia at 1 atm and 273 K would be adsorbed. On the surface of \[\frac{44}{7}g\]carbon if radius of a ammonia molecules is \[{{10}^{-8}}\]cm. \[\left[ Given:{{N}_{A}}\text{=6}\times \text{1}{{\text{0}}^{23}}\text{ } \right]\]
A gas present in a cylinder fitted with a frictionless piston expands against a constant pressure of 1 atm from a volume of 2 litre to a volume of 6 litre. In doing so, it absorbs \[800\text{ }\operatorname{J}\]heat from surroundings. Determine increase in internal energy of process.
In Duma's method of estimation of nitrogen, 0.35 g of an organic compound gave 55 mL of nitrogen at 300 K temperature and 715 mm pressure. The percentage composition of Nitrogen in the compound would be: (Aqueous Tension at 300 K = 15 mm)
The dipole moments of diatomic molecules AB and CD are 10.41Dand 10.27D, respectively while their bond distances are 2.82A and 2.67A, respectively. This indicates that
One mole of \[{{N}_{2}}{{O}_{4}}\](g) at 300 K is kept in a closed container under one atmosphere. It is heated to 600 K when \[20%\]by mass of \[{{N}_{2}}{{O}_{4}}\](g) decomposes to \[N{{O}_{2}}\](g). the resultant pressure is:
When a gas is bubbled through water at 298 K, a very dilute solution of the gas is obtained. Henry's law constant for the gas at 298 K is 100 kbar. If the gas exerts a partial pressure of 1 Bar, the number of millimoles of the gas dissolved in one litre of water is
A small particle of mass m moves in such a way that P.E.\[=-\frac{1}{2}mk{{r}^{2}}\], where k is a constant and r is the distance of the particle from origin. Assuming Bohr's model of quantization of Angular momentum and circular orbit, r is directly proportional to:
For the reaction \[\operatorname{C}\left( s \right)+C{{O}_{2}}(g)\to 2CO\left( g \right),\] \[{{K}_{P}}=63\] atm at 1000 K. If at equilibrium: \[{{P}_{C{{O}_{2}}}}\text{ }=\text{ }10\]then the total pressure of the gases at equilibrium is
\[S{{n}^{4+}}+2{{e}^{-}}\xrightarrow{{}}S{{n}^{2+}}E{}^\circ =0.13V\]\[B{{r}_{2}}+2{{e}^{-}}\xrightarrow{{}}2B{{r}^{-}}E{}^\circ =1.08V\] Calculate \[{{K}_{eq}}\]for the cell formed by two electrodes.
The electro negativities of four atoms labeled as D, E, F and G are as follows. D=3.8, E=3.3, F=2.8 and G=1.3. if the atoms from the molecules DE, DG, EG and DF, the order of arrangement of these molecules in the increasing order of covalent bond character is
In snapdragon, when a red flower plant is crossed with a white flower ones, the resultant hybrid plant have pink coloured flower if plant of \[{{F}_{1}}\]generation is crossed with a white flower ones, the progeny will be -
Equation of a straight line meeting the circle\[{{x}^{2}}+{{y}^{2}}=100\] in two points, each point at a distance of 4 from the point (8, 6) on the circle, is:
If in a rectangle ABCD with \[BC=3\,\,AB.\]Points P & Q are on BC such that \[\angle DBC={{\tan }^{-1}}(1/3);\] \[\angle \,\,DPC={{\tan }^{-1}}(1/2)\] & \[\angle \,\,DBC=\angle \,\,DQC-\angle \,\,DPC,\] then:
Messages are conveyed by arranging 4 white, 1 blue and 3 red flags on a pole. Flags of the same colour are alike. If a message is transmitted by the order in which the colours are arranged then the total number of messages that can be transmitted if exactly 6 flags are used is:
It is possible for a photon to materialize into an electron and a positron. In this way, electromagnetic energy is converted into matter. Both energy and linear momentum are conserved when an electron-positron pair is created near an atomic nucleus. In absence of nucleus, pair production cannot occur in empty space because
A)
it is impossible to conserve momentum without presence of nucleus
doneclear
B)
it is impossible to conserve energy without presence of nucleus
Particle deaccelerates on a straight line whose magnitude is \[|a|=\alpha \sqrt{v},\]where \[\alpha =a\] positive constant. If initial velocity of the particle is \[{{v}_{0}},\] then the distance travelled by particle before it stops is
Two candles of equal heights, 15 cm, each are placed in between vertical screens at a distance of 10 cm from each other and also from the nearer screen.
Candle A burns completely in \[1.5\text{ }h\]and candle B burns completely in 1 h. Shadows of A and B moves with speeds
Force of repulsion between two charges od \[+\,2\mu C\] and \[+\,3\mu C\]separated by a distance of \[10\,cm{{s}^{-\,1}}\]is \[{{F}_{1}}.\]If a dielectric slab of dielectric constant \[k=4\]is placed, such that it fills 5 cm distance (as shown), then force between charges will be
A diatomic ideal gas undergoes following two steps process, Step 1 Constant volume heating to triple pressure of gas. Step 2 Constant pressure heating to double volume Molar heat capacity of the gas for whole process is
A student measures magnitude of electric field due to a charged sphere S by two different ways. I. He places a very small similarly charged sphere near S and then measures force per unit charge. II. He places a large similarly charged sphere near S and then measures force per unit charge. Let his recordings are \[{{E}_{1}}\]and \[{{E}_{2}}.\]
A)
\[{{E}_{2}}>{{E}_{1}},\]when charge on S is negative
doneclear
B)
\[{{E}_{1}}={{E}_{2}}\]
doneclear
C)
\[{{E}_{1}}>{{E}_{2}}\]
doneclear
D)
\[{{E}_{2}}>{{E}_{1}},\]when charge on S is positive
An automatic rotating water sprinkler is mounted over top of a 5 m high pillar in a field.
Water is flowing at a rate of \[5\times {{0}^{-\,4}}{{m}^{3}}\]in 1 s through a nozzle of area\[1\text{ }c{{m}^{2}}\]. Maximum possible area of field that can be irrigated by the nozzle is (take, \[g=10m{{s}^{-\,2}}\])
A decinormal solution of potassium ferrocya- nide is \[50%\] dissociate at 300 K .the osmotic pressure of the solution is (Given\[\operatorname{R}=8.314J{{K}^{-1}}mo{{l}^{-1}}\] )
If the unit cell of a mineral has cubic close packed \[\left( ccp \right)\]Array of oxygen atoms with \[m\] fraction of Octahedral holes occupied by aliminium ions and \[n\]fraction of tetrahedral holes occupied by magnesium ions, m and n, respectively, are
\[100mL\]of tap water containing \[\operatorname{Ca}{{\left( HC{{O}_{3}} \right)}_{2}}\] Titrated with \[\operatorname{N}/50 HCl\] with methyl orange as Indicator. If \[30 mLof\,HCl\] were required, calculate the temporary hardness as parts of \[{{\operatorname{CaCO}}_{3}}\]per \[{{10}^{6}}\]Parts of water.
On treatment of \[100 mL\] of \[0.1\]M solution of \[{{\operatorname{CoCl}}_{3}}.6{{H}_{2}}O\]. With excess \[{{\operatorname{AgNO}}_{3}};1.2 \times 1{{0}^{22}}\] ions are precipitated. The complex is:
If x, y and z are total number of compounds in Which central atom used their all three p-orbitals, Only two p-orbitals and only one p-orbital in Hybridisation respectively. Then value of"\[''x + y - z'' is\]
The sodium salt of a carboxylic acid, was Produced by passing \[{{\operatorname{CO}}_{2}}\]into an aqueous Solution of caustic alkali at an elevated Temperature and pressure. On heating in presence of sodium hydroxide followed by Treatment with sulphuric acid gave a dibasic acid, C. A sample of \[0.4\]g of acid C, on combustion Gave \[0.08\]g of water and \[0.39\]g of carbon dioxide. The silver salt of the acid C weighing \[1.0\]g on ignition yielded 0.71 g of silver as residue. Identify C.
The following reaction is performed at 298 K.\[2NO\left( g \right)+{{O}_{2}}(g)2N{{O}_{2}}\left( g \right)\] The standard free energy of formation of \[\operatorname{NO}\left( g \right)\]is\[86.6 kJ/mol at 298 K\]. What is the standard free energy of formation of \[\operatorname{NO}\left( g \right) at 298 K?\] \[\left( {{K}_{P}} = 1.6 \times 1{{0}^{12}} \right)\]
When alleles of two contrasting characters are present together, one of the character expresses itself during the cross while the other remains hidden. This is the -