Let \[(x)={{\cos }^{-1}}\left( \frac{1-{{x}^{2}}}{1+{{x}^{2}}} \right)+{{\sin }^{-1}}\left( \frac{2x}{1+{{x}^{2}}} \right),x\in [-1,0]\] Then the number of points where g(x) is non-differentiable in [-1,0] is
If the lines \[\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-k}\]and \[\frac{x-1}{k}=\frac{y-4}{2}=\frac{z-5}{1}\]are coplanar, then the sum of all possible values of k is
Suppose that co and z are complex numbers such that both (1 + 2i) \[\omega \] and (1 + 2i)z are different real numbers. The slope of the line connecting \[\omega \] and z in the complex plane is
A test is made up of 5 questions, for each question there are 4 possible answers and only one is correct. For every right choice you gain 1 mark while for each wrong choice there is a penalty of 1 mark. The probability of getting at least 2 marks answering to every question in a random way is
Four students from University \[{{U}_{1}}\], one of them is Mr. A, and five students from University \[{{U}_{2}}\], one of them is Mr. B, are going to see an Inter University Cricket tournament. However, they found, that only 5 ticket remaining, so 4 of them must go back. Suppose that at least one student from each University must go to see the game, and at least one of Mr. A and Mr. B must go to see the game. Number of ways of selecting 5 students satisfying the above constraint, is equal to
A ray with the starting point at the origin in the complex plane passes through the point\[\left( 2+2i \right)\]. Another ray with the same starting point passes through the point \[(2-2\sqrt{3}i).\]The angle formed by these two
\[=2\hat{i}-\hat{j}+\hat{k},\overrightarrow{q}=\hat{i}+2\hat{j}-\hat{k}\]and \[=\hat{i}+\hat{j}-2\hat{k}.\]. If \[=\overrightarrow{p}+\lambda \overrightarrow{q}\] and projection of \[\]on \[\overrightarrow{r}\] is \[\frac{4}{\sqrt{6}},\] then \[\lambda \] equals
If a curve passes through the point \[M\left( -1,\text{ }1 \right)\]and has slope \[\left( 2x-\frac{1}{{{x}^{2}}} \right)\] at any point P(x, y) on it, then the ordinate of the point on the curve whose abscissa is \[-2\], is
Consider the conic \[=\frac{{{(x+1)}^{2}}}{\pi }+\frac{{{y}^{2}}}{3}=1.\] Suppose P is any point on the conic and \[{{S}_{1}},{{S}_{2}}\] are the foci of the conic, then the maximum value of \[(P{{S}_{1}}+P{{S}_{2}})\] is
If tangents drawn to circle \[\left| z \right|=4\] at points \[A({{z}_{1}})\] and \[B({{z}_{2}})\] intersect at P such that arg \[\left( \frac{{{z}_{2}}}{{{z}_{1}}} \right)=\frac{\pi }{2},\], then locus of P is
Let S be the sum of the first n terms of the arithmetic sequence 8, 12, 16,....., and T be the sum of first n terms of arithmetic sequence 17,19,21......... If S-T=0, then n is equal to
Let \[y'(x)+\frac{g'(x)}{g(x)}y(x)=\frac{g'(x)}{1+{{g}^{2}}(x)}\]where f(x) denotes \[\frac{df(x)}{dx}\]and g(x) is a given non-constant differentiable function an R If \[g\left( 1 \right)=y\left( 1 \right)=1\]and \[g(e)=\sqrt{(2e-1)}\] then y(e) equals (Here e denotes napier's constant)
Let P be arbitrary point on the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}-1=0,a>b>0.\]Suppose \[{{F}_{1}}\] and \[{{F}_{2}}\] are the foci of the ellipse. The locus of the centroid of the triangle \[P{{F}_{1}}{{F}_{2}}\] as P moves on the 1 ellipse, is
Let p and q be any two logical statements and \[r:p\to \left( \sim \text{ }p\vee \text{ }q \right)\]. If r has a truth value F, then the truth values of p and q are respectively
A flag is hoisted on a car which is moving towards east with velocity 60 km/h and wind is blowing with 60 km/h south to north .The direction of flutter of the flag is:
A particle is projected at an angle \[{{\tan }^{-1}}\left( \frac{1}{\sqrt{3}} \right)\]At a height, velocity of the particle is \[9\hat{i}+3\hat{j}\] (m/s). Find the height \[\left( g=10m/{{s}^{2}} \right)\]
A block of mass 0.1 kg is held against a wall applying a horizontal force of 5N on the block. If the coefficient of friction between the block and the wall is 0.50, the magnitude of frictional force acting on the block is:
A hemispherical vessel of radius R moving with a constant velocity \[{{v}_{0}}\] and containing a ball, is suddenly halted. Find the height by which ball will rise in the vessel. The surface is smooth.
Four particles A, B, C and D of equal masses are placed at the comers of a square. They move with equal uniform speed V towards the intersection of the diagonals. After collision A comes to rest, B traces its path back with same speed and C and D move with equal velocities. What is the velocity of C after collision.
A equilateral triangle ABC formed from a uniform wire has two small indentical beads initially located at A. The triangle is set rotating about the vertical axis AO. Then the beads are released from rest simultaneously and allowed to slide down one along AB and the other along AC, as shown, neglect friction. The quantities that are conserved as beads slies down are:
A)
angular velocity and total energy.
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B)
angular velocity and moment of inertia about axis of rotation
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C)
total angular momentum and moment of inertia about axis of rotation.
A particle is executing SHM. At a point x = A/3, kinetic energy of the particle is K, where A is the amplitude. At a point x = 2A/3, kinetic energy of the particle will be :
The orbital velocity of a satellite at point B with radius \[{{r}_{B}}\] is v. The radius of point A is \[{{r}_{A}}.{{r}_{A}}\]and \[{{r}_{B}}\] are semi major and semi minor axis respectively. If the orbit is increased in radial direction so that \[{{r}_{A}}\] becomes 1\[1.2{{r}_{A}}\]. find the orbital velocity at \[(1.2{{r}_{A}})\]
Wire A and B are connected with blocks P are as shown. The ratio of lengths, radii and Young?s modulus of wires A and B are r, 2r and 3r respectively (r is a constant). Find the mass of block P if ratio of increase in their corresponding lengths is \[1/6{{r}^{2}}\]. The mass of block Q is 3M.
Earth's receives \[1400W/{{m}^{2}}\] of solar power. If all the solar energy falling on a lens of area \[0.2{{m}^{2}}\]is focused onto a block of ice of mass 280 g, the time taken to melt the ice will be (in min.)
A bubble rises from bed of a lake to its surface. The depth of lake is 103 times the value by which bubble increases in size. If the atmospheric pressure is equal to pressure due to a column of water of 10 m, find the value by which bubble increases in size (approximately)
Three charges \[+Q,+q\] and \[+q\] are placed at the vertices of a right angle triangle (isosceles triangle) as shown. The net electrostatic energy of the configuration is zero, if Q is equal to:
A parallel place capacitor of capacitance C is connected to a battery and is charged to a potential difference V. Another capacitor of capacitance 2C is similarly charged to a potential difference 2V. The charging battery is now disconnected and the capacitor are connected in parallel to each other in such a way that the positive terminal of one is connected to the negative terminal of the other. The final energy of the configuration is:
A battery of internal resistance 4Q-is connected to the network of resistances as shown. In order that the maximum power can be delivered to the network, the value of R in \[\Omega \] should be
The current is uniformly distributed over the cross-section of a straight cylindrical conductor of radius r. The variation of magnetic field B along a distance x from axis of conductor is shown in the curves. Find the correct option:
A small square loop of wire of side \[l\] is placed inside a large square loop of wire of side L \[\left( L>>l \right)\]. The loops are coplanar and their centre coincides. The mutual inductance of the system is proportional to:
LCR circuit is connected to a 200 V, AC source L= 10 H, \[C=160\mu F\] and \[R=80\Omega \] at resonance. Let \[{{i}_{1}},{{i}_{2}}\] and \[{{i}_{3}}\] be rms current through L, C and R respectively then:
A)
\[{{i}_{1}}={{i}_{2}}\] and \[{{i}_{1}}>{{i}_{3}}\]
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B)
\[{{i}_{1}}={{i}_{2}}\] and \[{{i}_{1}}<{{i}_{3}}\]
After 280 days, the activity of a radioactive sample is 600 dps. The activity reduces to 300 dps after another 140 days. The initial activity of the sample in dps is:
\[{{K}_{\alpha }}\]wavelength emitted by an atom of atomic number \[Z=11\]is \[\lambda \]. Find the atomic number for an atom that emits \[{{K}_{\alpha }}\]radiation with wavelength \[4\lambda \].
A proton colloids with a stationary hydrogen atom in ground state elastically. Energy of colliding photon is 10.2eV. After a time interval of the order microsecond another photon collides with same hydrogen atom in elastically with an energy of 15 eV. What will be observed by the detector:
A)
2 photon of energy 10.2 eV
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B)
2 photon of energy 1.4 eV
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C)
One photon of energy 10.2 eV and an electron of energy 1.4 Ev
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D)
One photon of energy 10.2 eV and another photon of energy 1.4 eV
The circuit sown in figure contains two diodes each with a forward resistance of 500 and with infinite backward resistance. If the battery voltage is 6V, the current through 100W resistance (in amperes) is:
A radar has power of 1kW and is operating at a frequency 10 GHz. It is located on a steep mountain of height 600 m. The maximum distance up to which it can detect an object on the surrounding earth surface is :
The time period of a simple pendulum is 2 sec. Initial amplitude is 10 degree becomes 5degree in 100 oscillations .The quality factor of the weakly damped oscillation will be:
Two simple harmonic motions are represented by the equations \[{{y}_{1}}=0.1\,\sin \,(100\pi t+\pi /3)\] and \[\,(100\pi t+\pi /3)\]. The phase difference of the velocity of particle 1 with respect to the velocity of particle 2is:
1 mol \[C{{H}_{3}}OOH\] is added in 250 g benzene. Acetic acid diereses in benzene due to hydrogen bond K of benzene is \[2\text{ }K\text{ }kgmo{{l}^{-1}}\]. The boiling point has increased by 6.4K. % dimerisation of acetic acid is:
\[{{H}_{2}}S\]reacts with lead acetate forming a black compound which reacts with \[{{H}_{2}}{{O}_{2}}\] to form another compound. The colour of the compound is:
(I) \[{{\{MnC{{l}_{6}}\}}^{3-}},{{[Fe{{F}_{6}}]}^{3-}}\], and \[{{[Co{{F}_{6}}]}^{3-}}\]are paramagnetic having four, five and four unpaired electrons respectively.
(II) Valence bond theory gives a quantitative interpretation of the thermodynamic stabilities of coordination compounds.
(III) The crystal field spliting \[{{\Delta }_{o}}\], depends upon the field produced by the ligand and charge on the metal ion.
A \[5.0\text{ }c{{m}^{3}}\] solution of \[{{H}_{2}}{{O}_{2}}\] liberates 0.508 g of \[{{1}_{2}}\], from an acidified KI solution. The strength of \[{{H}_{2}}{{O}_{2}}\], solution in terms of volume strength at STP is:
For the equilibrium, \[CuS{{O}_{4}}.5{{H}_{2}}O(s)\rightleftharpoons CuS{{O}_{4}}.3{{H}_{2}}O(s)+2{{H}_{2}}O(g),{{K}_{p}}=1.086\times {{10}^{-4}}\] at \[{{25}^{o}}C\]. The efflorescent nature of \[CuS{{O}_{4}}.5{{H}_{2}}O(s)\] can be noticed when the vapour pressure of water in the atmosphere is (in mm Hg):
In FCC lattice A, B, C, D atoms are arranged at corners, face centres, octahedral voids and tetrahedral voids respectively then the body diagonal contains:
The vander Waals' constant 'a' for the gases\[C{{H}_{4}},{{N}_{2}},N{{H}_{2}}\], and \[{{O}_{2}}\] are \[2.25,1.39,4.17\]and at \[1.3\text{ }{{L}^{2}}\] atai- \[mo{{l}^{-2}}\] respectively. The gas which shows highest critical temperature is:
[3]\[H\centerdot +B{{r}_{2}}\xrightarrow{{}}HBr+Br\centerdot \] Fast step
Calculate the rate of reaction, if concentration of hydrogen is twice that of bromine and the rate constant is equal to \[I{{M}^{-1/2}}Se{{c}^{-1}}\]. Concentration of bromine is 1M.
\[Zn\] Amalgam is prepared by electrolysis of aqueous \[ZnC{{l}_{2}}\] using Hg cathode (9gm). How much current is to be passed through \[ZnC{{l}_{2}}\] solution for 1000 seconds to prepare a Zn Amalgam with 25% Zn by wt.(Zn=65.4)
A certain acid-base indicator is red in acid solution and blue in basic solution. At pH = 5.75% of the indicator is present in the solution in its blue form. Calculate the pH at which the indicator shows 90% red form? \[\left( Given\text{ }{{10}^{-4.523}}=3\times {{10}^{-5}} \right)\]