The curve amongst the family of curves represented by the differential equation, \[({{x}^{2}}-{{y}^{2}})dx+2xy\]\[dy=0\] which passes through \[(1,1),\] is:
A)
a circle with centre on the \[x\]-axis
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B)
an ellipse with major axis along the\[y\]-axis
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C)
a circle with centre on the \[y\]-axis
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D)
a hyperbola with transverse axis along the\[x\]-axis
Let \[f:(-1,1)\to R\] be a function defined by\[f(x)=max\left\{ -|x|,-\sqrt{1-{{x}^{2}}} \right\}.\] If K be the set of all points at which / is not differentiable, then K has exactly:
The positive value of \[\lambda \] for which the co-efficient of \[{{x}^{2}}\] in the expression \[{{x}^{2}}{{\left( \sqrt{x}+\frac{\lambda }{{{x}^{2}}} \right)}^{10}}\] is \[720,\] is:
Let \[N\] be the set of natural numbers and two functions\[f\] and \[g\] be defined as \[f,g:N\to N\] such that: \[f(n)=\left\{ \begin{matrix} \frac{n+1}{2} & if\,n\,is\,odd \\ \frac{n}{2} & if\,n\,is\,even \\ \end{matrix} \right.\] and\[g(n)=n-{{(-1)}^{n}}.\]Then\[fog\]is:
The number of values of \[\theta \in (0,\pi )\] for which the system of linear equations \[x+3y+7z=0\], \[-x+4y+7z=0\], \[(sin3\theta )x+(cos2\theta )y+2z=0\] has a non-trivial solution, is:
Let \[\overrightarrow{a}=(\lambda -2)\overrightarrow{a}+\overrightarrow{b}\] and \[\overrightarrow{\beta }=(4\lambda -2)\overrightarrow{a}+3\overrightarrow{b}\] be two given vectors where a and b are non collinear. The value of \[\lambda \] for which vectors \[\vec{\alpha }\] and \[\vec{\beta }\] are collinear, is:
Two sides of a parallelogram are along the lines, \[x+y=3\] and \[x-y+3=0.\] If its diagonals intersect at \[\left( 2,4 \right),\] then one of its vertex is:
Let \[z={{\left( \frac{\sqrt{3}}{2}+\frac{i}{2} \right)}^{5}}+{{\left( \frac{\sqrt{3}}{2}+\frac{i}{2} \right)}^{5}}.\] If \[R(z)\] and \[I(z)\] respectively denote the real and imaginary parts of\[z,\] then:
If the probability of hitting a target by a shooter, in any shot, is \[\frac{1}{3},\] then the minimum number of independent shots at the target required by him so that the probability of hitting the target at least once is greater than \[\frac{5}{6}\] is:
If\[\int_{{}}^{{}}{{{x}^{5}}{{e}^{-4{{x}^{3}}}}dx=\frac{1}{48}{{e}^{-4{{x}^{3}}}}}f(x)+C,\] where C is a constant of integration, then\[f(x)\] is equal to:
Let \[A=\left[ \begin{matrix} 2 & b & 1 \\ b & {{b}^{2}}+1 & b \\ 1 & b & 2 \\ \end{matrix} \right]\] where\[b>0.\] Then the minimum value of \[\frac{\det (A)}{b}\] is:
A body slides down an inclined plane of inclination 6. The coefficient of friction down the plane varies in direct proportion to the distance moved down the plane \[(\mu =k\,x)\]. The body will move down the plane with a
A spherical steel ball released at the top of a long column of glycerine of length L, falls through a distance L/2 with accelerated motion and the remaining distance L/2 with a uniform velocity. If \[{{t}_{1}}\] and \[{{t}_{2}}\] denote the times taken to cover the first and second half and \[{{W}_{1}}\] and \[{{W}_{2}}\] the work done against gravity in the two halves, then
A plane mirror is placed at x = 0 with its plane parallel to the y-axis. An object starts from x = 3 m and moves with a velocity of \[(2\hat{i}+2\hat{j})m{{s}^{-1}}\] away from the mirror The relative velocity of the image with respect to the object is
A)
\[2\sqrt{2}\,m{{s}^{-1}}\] making an angle of \[45{}^\circ \] with the \[+x\] axis
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B)
\[2\sqrt{2}\,m{{s}^{-1}}\] making an angle of \[135{}^\circ \] with the \[+x\] axis
Which of the following graphs best represents the force acting on a charged particle kept at distance x from the centre of a square and on the axis of the square whose corners have equal charges.
Wire bent as ABOCD as shown, carries current I entering at A and leaving at D. Three uniform magnetic fields each \[{{B}_{0}}\] exist in the region as shown. The force on the wire is
In an x-ray tube, if the accelerating potential difference is changed, then:
A)
the frequency of characteristic x - rays of a material will get changed
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B)
number of electrons emitted will change
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C)
the difference between \[{{\lambda }_{0}}\] (minimum wavelength) and \[{{\lambda }_{k\alpha }}\] (wavelength of \[{{k}_{\alpha }}x-ray\]) will get changed
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D)
difference between \[{{\lambda }_{k\alpha }}\] and \[{{\lambda }_{k\beta }}\] will get changed.
A charge particle \[{{q}_{0}}\]of mass \[{{m}_{0}}\] is projected along the y-axis at t = 0 from origin with a velocity\[{{V}_{0}}\]. If a uniform electric field \[{{E}_{0}}\] also exists along the x-axis, then the time at which debroglie wavelength of the particle becomes half of the initial value is:
A lens is placed between a source of light and a wall. It forms images of area \[{{A}_{1}}\] and \[{{A}_{2}}\] on the wall, for its two different positions. The area of the source of light is (source and wall are fixed)-
Two solid bodies of equal mass m initially at \[T=0{}^\circ C\] are heated at a uniform and same rate under identical conditions. The temperature of the first object with latent heat \[{{L}_{1}}\] and specific heat capacity in solid state \[{{C}_{1}}\] changes according to graph 1 on the diagram. The temperature of the second object with latent heat \[{{L}_{2}}\] and specific heat capacity in solid state \[{{C}_{2}}\] changes according to graph 2 on the diagram. Based on what is shown on the graph, the latent heats \[{{L}_{1}}\] and \[{{L}_{2}}\], and the specific heat capacities \[{{C}_{1}}\] and \[{{C}_{2}}\] in solid state obey which of the following relationships :
Thermal coefficient of volume expansion at constant pressure for an ideal gas sample of n moles having pressure \[{{P}_{0}}\] volume \[{{V}_{0}}\] and temperature \[{{T}_{0}}\] is
Two scales on voltmeter measure voltages up to 20V and 30V. The resistance connected in series with galvanometer is \[1680\Omega \] for the 20V scale and \[2930\Omega \] for the 30V scale in the same galvanometer. The resistance of the galvanometer and the full scale current are:
A container of a large uniform cross-sectional area A resting on a horizontal surface holds two immiscible, non-viscous and incompressible liquids of densities 'd' and '2d' each of height (1/2)H as shown. The smaller density liquid is open to atmosphere. A homogeneous solid cylinder of length \[L\left( <\frac{1}{2}H \right)\] cross-sectional area (1/5) A is immersed such that it floats with its axis vertical to the liquid-liquid interface with length (1/4) L in denser liquid. if D is the density of the solid cylinder then:
A satellite with mass 2000 kg and angular momentum magnitude \[2\times {{10}^{12}}kg.{{m}^{2}}/s\] is moving in an elliptical orbit around a planet. The rate at which area is being swept out by the satellite around the planet, is equal to
A uniform triangular wedge of mass M is kept on smooth horizontal surface. A horizontal force F is applied on the wedge, then net force on the upper portion ABC is (AB = l/2 and neglect any effect of rotation)
A uniform rod of mass m, length S is placed over a smooth horizontal surface along y-axis and is at rest as shown in figure- An impulsive force F is applied for a small time At along positive x-direction at end A of the rod. The x-coordinate of end A of the rod when the rod becomes parallel to x-axis for the first time is (initially the coordinate of centre of mass of the rod is (0,0)):
A particle executes SHM in a straight line. In the first second starting from rest it travels 'a' distance a and in the next second a distance 'b' in the same direction. The amplitude of S.H.M will be
An ether \[(A),{{C}_{5}}{{H}_{12}}O,\] when heated with excess of hot concentrated HI produced two alkyl halides which when treated with \[NaOH\] yielded compounds [B] and [C]. Oxidation of [B] and [C] gave a propanone and an ethanoic acid respectively. The IUPAC name of the ether [A] is:
For a homogeneous gaseous reaction \[\operatorname{A}\xrightarrow{{}}3B,\], if pressure after time t was \[{{\operatorname{P}}_{t}}\] and after completion of reaction, pressure was \[{{\operatorname{P}}_{\infty }}\] then select correct relation
In \[\operatorname{O}_{2}^{-},{{\operatorname{O}}_{2}}\] and \[\operatorname{O}_{2}^{2-}\] molecular species, that total number of antibonding electrons respectively are
(ii) \[\operatorname{A}\xrightarrow[in\,absence\,of\,peroxide]{HBr,dark}\underset{\left( major \right)}{\mathop{\operatorname{C}}}\,+\underset{\left( minoor \right)}{\mathop{\operatorname{D}}}\,\]
Pottasium has a bcc structure with nearest neighbour distance \[4.52\overset{\text{o}}{\mathop{\,\text{A}}}\,\]. Its atomic weight is 39. Its density (in kg \[{{m}^{-3}}\]) will be
If \[{{k}_{1}}\], is much smaller than \[{{\operatorname{k}}_{2}}\] the most suitable qualitative plot of potential energy (P. E.) versus reaction co-ordinate (R.C.) for the above reaction
A 0.60 g sample consisting of only \[{{\operatorname{CaC}}_{2}}{{O}_{4}}\]and \[Mg{{C}_{2}}{{O}_{4}}\] is heated at \[500{}^\circ C\], converting the two salts to \[{{\operatorname{CaCO}}_{3}}\], and \[MgC{{O}_{3}}\]. The sample then weighs 0.465 g. If the sample had been heated to\[900{}^\circ C\], where the products are \[CaO\] and \[MgO\]. What would the mixtures of oxides have weighed?
What is the freezing point of a solution containing 8.1 g HBr in 100 g water assuming the acid to be 90% ionised? \[({{K}_{f}}. for water = 1.86 K kg mo{{l}^{-1}}):\]
Consider the following reaction: \[M{{X}_{4}}+X{{'}_{2}}\xrightarrow{{}}M{{X}_{4}}X{{'}_{2}}\]
If atomic number of M is 52 and X and X' are halogens and X' is more electronegative than X. Then choose correct statement regarding given information:
A)
Both X' atoms occupy axial positions which are formed by overlapping of p and \[d\]-orbitals only
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B)
All M - X bond lengths are identical in both \[{{\operatorname{MX}}_{4}}\] and \[{{\operatorname{MX}}_{4}}X'{{ }_{2}}\] compounds
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C)
Central atom 'M' does not use non-axial set of \[d\]-orbital in hybridization of final product.
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D)
Hybridization of central atom 'M' remains same in both reactant and final product.
The ammonia evolved from the treatment of 0.30 g of an organic compound for the estimation of nitrogen was passed in 100 mL of 0.1 M sulphuric acid. The excess of acid required 20 mL of 0.5 M sodium hydroxide solution for complete neutralization. The organic compound is
In an electrolysis experiment current was passed for 5 hours through two cells connected in series. The first cell contains a solution of gold and the second contains copper sulphate solution. 9.85 g of gold was deposited in the first cell. If the oxidation number of gold is +3, the amount of copper deposited on the cathode of the second cell and magnitude of the current in amperes is. (1 Faraday = 96,500 Coulombs)
The plane which bisects the line segment joining the points \[(-3,-3,4)\] and \[(3,7,6)\] at right angles, passes through which one of the following points?
If mean and standard deviation of 5 observations \[{{x}_{1}},{{x}_{2}},{{x}_{3}},{{x}_{4}},{{x}_{5}}\] are 10 and 3, respectively, then the variance of 6 observations \[{{x}_{1}},{{x}_{2}},.....{{x}_{3}}\] and \[-50\] is equal to:
Let\[f\]be a differential function such that \[f'(x)=7-\frac{3}{4}\frac{f(x)}{x},(x>0)\]and\[f(1)\ne 4.\]Then\[\underset{x\to {{0}^{+}}}{\mathop{\lim }}\,xf\left( \frac{1}{x} \right):\]
The value of \[\int\limits_{{\scriptstyle{}^{-\pi }/{}_{2}}}^{{\scriptstyle{}^{\pi }/{}_{2}}}{\frac{dx}{[x]+[sinx]+4},}\] where \[[t]\] denotes the greatest integer less than or equal to \[t,\] is:
Let \[{{a}_{1}},{{a}_{2}},{{a}_{3}},...{{a}_{10}}\] be in G.P. with\[{{a}_{i}}>0\]for\[i=1,\]\[2,....,10\] and S be the set of pairs \[(r,k),r,k\in N\] (the set of natural numbers) for which \[\left| \begin{matrix} {{\log }_{e}}a_{1}^{r}a_{2}^{k} & {{\log }_{e}}a_{2}^{r}a_{3}^{k} & {{\log }_{e}}a_{3}^{r}a_{4}^{k} \\ {{\log }_{e}}a_{4}^{r}a_{5}^{k} & {{\log }_{e}}a_{5}^{r}a_{6}^{k} & {{\log }_{e}}a_{6}^{r}a_{7}^{k} \\ {{\log }_{e}}a_{7}^{r}a_{8}^{k} & {{\log }_{e}}a_{8}^{r}a_{9}^{k} & {{\log }_{e}}a_{9}^{r}a_{10}^{k} \\ \end{matrix} \right|=0\] Then the number of elements in S, is:
A helicopter is flying along the curve given by \[y-{{x}^{3/2}}=7,(x\ge 0).\] A soldier positioned at the point \[\left( \frac{1}{2},7 \right)\] wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is:
A solid metallic sphere of radius r is enclosed by a thin metallic shell of radius 2r, A charge q is given to the inner sphere- When the inner sphere is connected to the shell by a metal wire, the heat energy generated in it is given by
A proton, when accelerated through a potential difference of V volts, has a wavelength \[\lambda \] associated with it. If an alpha particle is to have the same wavelength \[\lambda \], it must be accelerated through a potential difference of
A solid sphere released from rest from the top of an inclined plane of inclination\[{{\theta }_{1}}\], rolls without sliding and reaches the bottom with speed \[{{v}_{1}}\] and its time of descent is\[{{t}_{1}}\]. The same sphere is then released from rest from the top of another inclined plane of inclination\[{{\theta }_{2}}\] but of the same height, rolls without sliding and reaches the bottom with speed \[{{v}_{2}}\] and its time of descent is \[{{t}_{2}}\]. If \[{{\theta }_{2}}>{{\theta }_{1}}\], then
Figure shows a metal ball of mass 50 kg and radius \[\frac{2}{\sqrt{\pi }}m\] is placed on an insulating uncharged stand, in space an upward electric field \[5\times {{10}^{5}}N/C\] is switched on. A stream of light ions is incident on the ball from left side at a speed \[2\times {{10}^{6}}m/s\] shown in figure. If charge on ball at \[t=0\] was zero, find the time in seconds at which ball will be lifted from the stand. The charge density of ion beam is \[5\times {{10}^{-12}}coul/{{m}^{3}}\]. Assume that all charge incident on the ball is absorbed.
A particle is projected from point A, that is at a distance 4R from the centre of the earth, with speed \[{{V}_{1}}\] in a direction making \[30{}^\circ \] with the line joining the centre of the earth and point A, as shown. Find the speed \[{{V}_{1}}\] if particle grazes the surface of the earth as shown in figure. Consider gravitational interaction only between these two. \[(use\,\,\frac{GM}{R}=6.4\times {{10}^{7}}{{m}^{2}}/{{s}^{2}})\]
Half lives of two isotopes X and Y of a material are known to be \[2\times {{10}^{9}}\] years and \[4\times {{10}^{9}}\] years respectively. If a planet was formed with equal number of these isotopes, then the current age of planet, given that currently the material has 20% of X and 80% of Y by number, will be
An isolated and charged spherical soap bubble has a radius 'r' and the pressure inside is atmospheric. If 'T' is the surface tension of soap solution, then charge on drop is:
A rod of length 1000 mm and co-efficient of linear expansion \[a={{10}^{-4}}\] per degree is placed symmetrically between fixed walls separated by 1001 mm. The Young's modulus of the rod is \[{{10}^{11}}\text{ }N/{{m}^{2}}\]. If the temperature is increased by \[20{}^\circ C\,\], then the stress developed in the rod is (in\[N/{{m}^{2}}\]):
A ring of radius R having a linear charge density \[\lambda \] moves towards a solid imaginary sphere of radius \[\frac{R}{2}\], so that the centre of ring passes through the centre of sphere. The axis of the ring is perpendicular to the line joining the centres of the ring and the sphere. The maximum flux through the sphere in this process is:
Based on the given diagram, the correct statements regarding the homogenous solutions of two volatile liquids is
(1) Plots AD and BC show that Raoult's law is obeyed for the solution in which B is a solvent and A is the solute and as well as for that in which A is solvent and B is solute.
(2) Plot CD shows that Dalton's law of partial pressures is obeyed by the binary solution of components A and B.
(3) EF + EG = EH; and AC and BD correspond to the vapour pressures of the pure solvents A and B respectively.
The rate constant for the decomposition of a certain substance is \[2.80\times {{10}^{-3}}{{M}^{-1}}{{s}^{-1}}\]at \[30{}^\circ C\]and \[1.38\times {{10}^{-2}}{{M}^{-1}}{{s}^{-1}}\] at \[50{}^\circ C.\] The Arrhenius parameters [A] of the reaction is: \[(R= 8.314 \times {{10}^{-3}}kJmo{{l}^{-1}}{{K}^{-1}}).\]
Two metals X and Y form covalent halides. Both halides can act as Lewis acids and a catalyst in Friedel Crafts reaction. Halide of X is polymer in the solid state and a dimer in the vapour state, which decomposes to monomer at 1200 K. However, halide of Y is a dimer in vapour state and becomes ionic in polar solvent. X and Y are respectively
Among \[[Ni{{(CO)}_{4}}],{{[NiC{{l}_{4}}]}^{2-}},[Co{{(N{{H}_{3}})}_{4}}C{{l}_{2}}]Cl,\] \[N{{a}_{3}}[Co{{F}_{6}}],N{{a}_{2}}{{O}_{2}}\] and \[Cs{{O}_{2}}\] the total number of paramagnetic compounds is
A mixture of \[{{\operatorname{CuSO}}_{4}}.5{{H}_{2}}\operatorname{O}\] and \[{{\operatorname{MgSO}}_{4}}.7{{H}_{2}}O\] is heated until all the water is lost. If 5.020 g of the mixture gives 2.988 g of the anhydrous salts, what is the percent by mass of\[{{\operatorname{CuSO}}_{4}}.5{{H}_{2}}\operatorname{O}\] in the mixture?
For the reaction:\[\operatorname{Ag}\left( CN \right)_{2}^{-}A{{g}^{+}}+2C{{N}^{-}},\]the \[{{\operatorname{K}}_{c}}\] at \[25{}^\circ C\] is \[4\times {{10}^{-14}}.\] What will be the \[[A{{g}^{+}}]\] in solution which was originally 0.1 M in KCN and 0.03M in \[{{\operatorname{AgNO}}_{3}}.\]
The accessibility of the promoter regions of prokaryotic DNA is (in many cases) regulated by the interaction of proteins with the sequences termed as -