Let \[f(x):\frac{\tan \,[{{e}^{2}}]{{x}^{3}}-\tan \,[-\,{{e}^{2}}]{{x}^{3}}}{{{\sin }^{3}}x},x\ne 0\] ([.] represents greatest integer function). The value of \[f(0)\] for which \[f(x)\] is continuous, is
C is the centre of ellipse \[\frac{{{x}^{2}}}{16}+\frac{{{y}^{2}}}{9}=1\] and A and B are two points on the ellipse such that \[\angle ACB=90{}^\circ ,\] then is equal to
Let \[g(x)=\cos {{x}^{2}},\]\[f(x)=\sqrt{x}\] and \[\alpha ,\beta (\alpha <\beta )\] be the roots of the quadratic equation \[18{{x}^{2}}-9\pi x+{{\pi }^{2}}=0.\] Then, the area (in sq units) bounded by the curve \[y=(gof)(x)\] and the lines \[x=\alpha ,\]\[x=\beta \] and \[y=0\] is
If \[f(x)={{(x+2019)}^{n}},\] where x is a real variable and n is a positive integer, then the value of \[f(0)+f'(0)+\frac{f''(0)}{2!}+\frac{f'''(0)}{3!}+...+\frac{{{f}^{n-1}}(0)}{(n-1)!}\]
Two infinitely long charge wires with linear densities \[\lambda \] & \[3\lambda \] are placed along x & y axis respectively determined the \[\tan \theta \] where \[\theta \] is the angle that electric field at any point on the line \[y=\sqrt{3}\,\,x\] make with positive x-axis.
In the circuit shown the cell is ideal. The coil has an inductance of 4H and zero resistance. F is a fuse of zero resistance and will blow when the current through it reaches 5A. The switch is closed at t = 0. The fuse will blow-
A charged particle of mass m and charge q is projected into a uniform magnetic field of induction B with speed v which is perpendicular to B. The width of the magnetic field is d. The impulse imparted to fee particle by the field is \[\left( d<<\frac{mv}{qB} \right)\]
A ring of radius R having a linear charge density \[\lambda \] moves towards a solid imaginary sphere of radius 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 centers of the ring and the sphere. The maximum flux through the sphere in this process is -
Three point A, B & C are at a distance of 1 m, 2 m & 1 m from an infinitely long charged wire of linear charge density \[\lambda \,\,C/m.\] A charge q is taken from A to B, B to C and finally C to A. Which of the following is/are correct about the work done in the above process -
A particle of charge Q and of negligible initial speed is accelerated through a potential difference of U. The particle reaches a region of uniform magnetic field of induction B, where it undergoes circular motion. If potential difference is doubled and B is also doubled then magnetic moment of the circular current due to circular motion of charge Q will become -
A infinitely long line charge of charge density \[\lambda \] lies along the x axis and let the surface of zero potential passes through (0, 5, 12) m. The potential at point \[(2,3,-\,4)\] is -
A sample of radioactive material decays simultaneously by two processes A & B with half lives \[\frac{1}{2}\] & \[\frac{1}{4}hr\] respectively. For first half hr it decays with the process A, next one \[hr\] with the process B & for further half an hour with both A & B. If originally there were \[{{N}_{0}}\] nuclei the number of nuclei after 2 hr of such decay \[{{N}_{0}}{{\left( \frac{1}{2} \right)}^{a}}\] find the value of a -
Two mutually perpendicular infinitely long lines of charge having charge per unit length as \[{{\lambda }_{1}}\] and \[{{\lambda }_{2}}\] are located in air at a distance "a" from each other. The force of interaction between them is -
In figure \[{{C}_{1}}=2\mu F,\] \[{{C}_{2}}=6\mu F\] & \[{{C}_{3}}=3.5\mu F.\] If break down voltages of the individual capacitors are \[{{V}_{1}}\text{ }=\text{ }100V,\]\[{{V}_{2}}\text{ }=\text{ 5}0V,\] & \[{{V}_{3}}\text{ }=\text{ 40}0V,\]What maximum voltage can be placed across points a & b-
When light of intensity \[{{I}_{1}}\] & frequency \[{{v}_{1}}\] is incident on a substance. Photo electric emission take place and variation potential (V) is as shown by curve (1) in the figure. However, if light of intensity \[{{I}_{2}}\] & frequency \[{{v}_{2}}\] falls on the same substance, photo electric current varies with potential as shown by curve (2) which of the following is correct -
Two sources of emf 6V and internal resistance \[3\,\Omega \] and \[2\Omega \] are connected to an external resistance R as shown. If potential difference across battery A is zero then value of R is -
A wire PQRS carrying a current I run along three edges of a cube of side \[\ell \] as shown. There exists a uniform magnetic field of magnitude B along one of the sides of the cube. The magnitude of the force acting on the wire is ?
A particle is projected form the ground with an initial velocity of 20 m/s at an angle of \[30{}^\circ \]with horizontal. The magnitude of change in velocity in a time interval from t = 0 to t = 0.5 s is \[(g=10m/{{s}^{2}})\]
A smooth square platform ABCD is moving towards right with a uniform speed \[v.\] At what angle \[\theta \] must a particle be projected from A with speed \[u\]so that it strikes the point B ?
A particle moves in space along the path \[z=a{{x}^{3}}+6{{y}^{2}}\] in such a way that \[\frac{dx}{dt}=c=\frac{dy}{dt}\] where a, b and c are constants. The acceleration of the particle is -
A particle moves on a rough horizontal ground with some initial velocity say \[{{v}_{0}}.\] If 3/4th of its kinetic energy is lost in friction in time \[{{t}_{0}}.\] Then coefficient of friction between the particle and the ground is -
Two blocks of equal mass are tied with a light string, which passes over a massless pulley as shown in figure. The magnitude of acceleration of centre of mass of both the blocks is (neglect friction everywhere, inclined wedge is fixed at floor) -
Element 'B' forms \[ccp\] structure and 'A' occupies half the octahedral voids, while oxygen atoms occupy the tetrahedral voids. The structure of bimetallic oxide is:
Adorption of a gas follows Freundlich adsorption isotherm. \[x\] is the mass of the gas adsorbed on mass m of the adsorbent. The plot of \[\log \frac{x}{m}\] versus log p is in the given graph. \[\frac{x}{m}\] is proportional to:
In order to oxidise a mixture of one mole of each of \[Fe{{C}_{2}}{{O}_{4}},\] \[F{{e}_{2}}({{C}_{2}}{{O}_{4}})3,\] \[FeS{{O}_{4}}\] and \[F{{e}_{2}}{{(S{{O}_{4}})}_{3}}\] in acidic medium, the number of moles of \[KMn{{O}_{4}}\]required is :
Which one of the following equations does not correctly represent the first law of thermodynamics for the given processes involving an ideal gas? (Assume non-expansion work is zero)
The vapour pressures of pure liquids A and B are 400 and 600 mm Hg, respectively at 298 K. On mixing the two liquids, the sum of their initial volumes is equal to the volume of the final mixture. The mole fraction of liquid B is 0.5 in the mixture. The vapour pressure of the final solution, the mole fractions of components A and B in vapour phase, respectively are:
The correct order of the spin-only magnetic moment of metal ions in the following low spin complexes, \[{{[V{{\left( CN \right)}_{6}}]}^{4\,-}},\] \[{{[Fe{{\left( CN \right)}_{6}}]}^{4\,-}},\] \[{{[Ru{{(N{{H}_{3}})}_{6}}]}^{3\,+}},\] and \[[Cr{{\left( N{{H}_{3}} \right)}_{6}}^{2+}]\] is :
An organic compound neither reacts with neutral ferric chloride solution nor Fehling solution. It however, reacts with Grignard reagent and gives iodoform test. The compound is:
The area of the portion common to \[y={{\sin }^{-\,1}}(\sin x)\] and \[y=[{{\sin }^{-1}}(\sin x)]\] in \[[0,\pi ],\]here \[[\cdot ]\]denotes the greatest integer function is
If \[{{I}_{1}}=\int_{0}^{1}{\frac{{{e}^{x}}}{x+3}dx}\] and \[{{I}_{2}}=\int_{0}^{1}{\frac{{{x}^{3}}}{{{e}^{{{x}^{4}}}}(4-{{x}^{4}})}dx,}\] then \[\frac{{{I}_{1}}}{{{I}_{2}}}\] is
If \[\sin \alpha +cos\beta =\frac{1}{\sqrt{2}}\] and \[\cos \alpha +\sin \beta =\sqrt{\frac{2}{3}},\] then the value of \[\tan \left( \frac{\alpha -\beta }{2} \right)\] is
A die is so loaded that the probability of throwing a number is proportional to \[{{K}^{2}}.\] Then the probability that the number 3 appears given that when the die is rolled the number turned up is not even is equal to
The circle \[{{x}^{2}}+{{y}^{2}}-8x=0\] and hyperbola \[\frac{{{x}^{2}}}{9}-\frac{{{y}^{2}}}{4}=1.\] The equation of a common tangent with positive slope to the circle as well as to the hyperbola is
Which of the following intervals is a possible domain of the function \[f\,(x)={{\log }_{\{x\}}}[x]+{{\log }_{\{x\}}}\{x\}?\] Where [x] is the greatest integer function and {x} is the fractional part of x.
The spring block system as shown in figure is in equilibrium. The string connecting blocks A and B is cut. The mass of all the three blocks is m and spring constant of both the spring is k. The amplitude of resulting oscillation of block A is (string massless)
The plates of small size of a parallel plate capacitor are charged as shown. The force on the charged particle of 'q' at a distance \[\ell \] from the capacitor is : (Assume that the distance between the plates is \[d<<\ell )\]
A charged particle of specific charge (charge/mass) a is the projected from origin with a velocity \[\vec{u}={{v}_{0}}(\hat{i}+\hat{j})\] in a uniform and constant magnetic field \[\vec{B}={{B}_{0}}\hat{i}.\] The position co-ordinates of the particle at time \[t=\frac{\pi }{{{B}_{0}}\alpha }\] are -
In the circuit shown switch S is connected to position 2 for a long time and then joined to position 1. The total heat produced in resistance \[{{R}_{1}}\] is -
A parallel beam of monochromatic radiation of cross-section area \[A(<\pi {{a}^{2}}),\] intensity I and frequency v is incident on a solid conducting sphere of work function \[{{\phi }_{0}}[hv>{{\phi }_{0}}]\] and radius 'a'. The sphere is grounded by a conducting wire. Assume that for each incident photon one photoelectron is ejected. Just after this radiation is incident on initially uncharged sphere, the current through the conducting wire is -
In the figure ABC is the cross section of a right angled prism and BCDE is the cross section of a glass slab. The value of \[\theta \] so that light incident normally on the face AB does not cross the face BC is \[(given\,{{\sin }^{-1}}(3/5)=37{}^\circ )\]
Two particles having the same specific charge (q/m) enter a uniform magnetic field with the same speed but an angles of \[30{}^\circ \] and \[60{}^\circ \] with the field. Let a, b and c be ratios of their pitches, radii and periods of their paths respectively, then-
In the given electrical circuit, the potential difference between points A and B is (assume the battery is ideal and the conducting wires have almost zero resistance).
If solubility product of \[Z{{r}_{3}}{{\left( P{{O}_{4}} \right)}_{4}}\] is denoted by \[{{K}_{sp}}\] and its molar solubility is denoted by S, then which of the following relation between S and \[{{K}_{sp}}\] is correct ?
For silver \[{{C}_{p}}(J{{K}^{-\,1}}\,mo{{l}^{-\,1}})=23+0.01\,T.\] If the temperature (T) of 3 moles of silver is raised from 300 K to 1000 K at 1 atm pressure, the value of AH will be close to:
100 mL of a water sample contains 0.81 g of calcium bicarbonate and 0.73 g of magnesium bicarbonate. The hardness of this water sample expressed in terms of equivalents of \[CaC{{O}_{3}}\] is: (molar mass of calcium bicarbonate is \[162\,g\,\,mo{{l}^{-\,1}}\] and magnesium bicarbonate is \[146\text{ }g\text{ }mo{{l}^{-1}}\])
If an insect that eats plant seeds containing 100 jule of energy use 30 jule of that energy for respiration and excretes 20 jule in its faeces, what is the insect's net secondary production? what is its production efficiency
If two persons with 'AB' blood group marry and have sufficiently large number of children, these children could be classified as 'A' blood group: 'AB' blood group: 'B' blood group in 1 : 2 : 1 ratio. Modern technique of protein electrophoresis reveals presence of both' A' and 'B' type proteins in 'AB' blood group individuals. This is an exam pie of:
[a] Sustained high fever \[(30{}^\circ \,\,\text{to}\,\,40{}^\circ C),\] weakness, stomach pain constipation, headche & loss of appetite are some of the common symptoms of typhoid disease.
[b] The rupture of RBCs is associated with release of a toxic substance, haemozoin, which is responsible for the chill and high fever recurring every 3 - 4 days.
[c] Symptons of ascariasis disease include internal bleeding muscular pain, fever, anaemia & blockage of intestinal passage.
[d] Appearance of dry, scaly lesions on various part of body such as skin, nails and scalp are the main symptoms of viral disease.
[a] Treatment of AIDS with anti-retroviral drugs is partially effective
[b] ionising radiation like x-rays and gamma rays and non-ionising radiation like UV cause DNA damage leading to neoplastic transformation.
[c] The side effects of the use of anabolic steroids in females include masculisation (feature like males, increased aggressiveness, mood swings, depression etc.)
[d] Allergy is due to the release of chemicals like histamine & serotonin from the mast cells.
[a] Parturition is induced by a complex neuroendocrine mechanism.
[b] By end of 24 weeks (second trimester), the body is covered with fine hair, eye lids separate an eye lashes are formed.
[c] The blastomers in the blastocyst are arranged into an outer layer called trophoblast Sinner group of eel's attached to trophoblast called the inner cell mass.
[d] The excretion of the acrosome help the sperm inter into the cytoplasm of the ovum through the zone pellucida and the plasma membrane.