\[\underset{n\to \infty }{\mathop{\lim }}\,\left[ \sum\limits_{r=1}^{n}{\frac{1}{2r}} \right],\] where [ ] denotes the greatest integer function, is equal to -
If \[f(x)=0\] be a quadratic equation such that \[f\,(-\,\pi )=f(\pi )=0\] and \[f\left( \frac{\pi }{2} \right)=-\frac{3{{\pi }^{2}}}{4},\] then\[\underset{n\to -\pi }{\mathop{\lim }}\,\frac{f(x)}{\sin (\sin x)}\] is equal to-
Let \[f(x)={{a}^{x}}(a>0)\] be written as \[f(x)=g(x)+h(x),\] where \[g(x)\] is an even function and \[h(x)\] is an odd function. Then the value of the \[g(x+y)+g(x-y)\] is -
\[f(x)={{(-1)}^{\left[ \frac{2x}{\pi } \right]}},\]\[g(x),\left| \,\sin x\, \right|-\left| \,\cos x\, \right|,\]\[\phi (x)=f(x)g(x),\] where [ ] denotes G.I.F. then fundamental period of \[f(x),\,\,g(x),\,\,\phi (x)\] are -
Two circles of radii 4 cm and 1 cm touch each other externally and \[\theta \] is the angle contained by their direct common tangents. Then sin \[\theta \] is equal to -
If \[\vec{a}\] and \[\vec{b}\] are two vectors such that \[\left| \vec{a}+\vec{b} \right|=2\] then value of \[\left[ \vec{a}\,\,\vec{b}\,\,\vec{a}\times \vec{b} \right]\] is equal to -
If a, b, c are in G.P., x and y be the arithmetic mean between a, b and b, c respectively, the \[\left( \frac{a}{x}+\frac{c}{y} \right)\,\,\left( \frac{b}{x}+\frac{b}{y} \right)\] is equal to-
If \[f=R\to R\] and for a fixed positive number c ; \[f(x+c)=1+[1-5f(x)+10{{\{f(x)\}}^{2}}\]\[-10\{{{(f(x)\}}^{3}}+5{{\{f(x)\}}^{4}}-\{f(x)\}5]\,\,1/5\,\,\forall \,\,\]\[x\in R,\] then f(x) is a periodic function whose period can be -
If positive numbers \[{{a}^{-1}},{{b}^{-1}},{{c}^{-1}}(a\ne b\ne c)\] Are in A.P., then product of roots of equation \[{{x}^{2}}-kx+2{{b}^{101}}-{{a}^{101}}-{{c}^{101}}=0(k\in R)-\]
If P(x) is a polynomial of the least degree that has a maximum equal to 6 at x = 1, and a minimum equal to 2 at x = 3, then \[\int\limits_{0}^{1}{p(x)dx}\] equals -
A tangent is drawn to the parabola \[{{y}^{2}}=4x\] at the point 'P' whose abscissa lies in the interval [1, 4]. The maximum possible area of the triangle formed by the tangent at 'P ' ordinates of the point 'P' and the x-axis is equal to -
\[f:R\to R\] such that \[f(x+2y)=f(x)+f(2y)+4xy,\]\[\forall \,\,x,\,\,y\,\,\in \,\,R.\] If \[{{I}_{1}}=\int\limits_{0}^{1}{f(x)dx,}\] \[{{I}_{2}}=\int\limits_{-1}^{0}{f(x)dx,}\] \[{{I}_{3}}=\int\limits_{1/2}^{2}{f(x)dx,}\]then-
A monkey of mass ' m' climbs up to a rope hung over a fixed pulley with an acceleration relative to the rope g/4. The opposite end of the rope is tied to a block of mass \[M\]lying on a rough horizontal plane. The coefficient of friction
Between the block and horizontal plane is \[\mu .\]Find the tension in the rope.
A)
\[\frac{M\left( 5m-4\mu M \right)g}{4\left( M+m \right)}+\mu Mg\]
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B)
\[\frac{m\left( 5m-4\mu M \right)g}{4\left( M+m \right)}+\mu Mg\]
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C)
\[\frac{M\left( 5m-5\mu M \right)g}{4\left( M+m \right)}+\mu Mg\]
Condenser \[A\] has a capacity of \[15\mu F\] which it is filled with a medium of dielectric constant 15. Another condenser B has a capacity of \[1\mu F\]with air between the plates. Both are charged separately by a battery of 100 V. After charging, both are connected in parallel without the battery and the dielectric medium being removed. The common potential now is
When an object is placed at a distance of 25 cm from a mirror, the magnification is \[{{m}_{1}}\]The object is moved 15 cm further away with respect to the earlier position, and the magnification becomes \[{{m}_{2}}.if\,{{m}_{1}}/{{m}_{2}}=4,\] the focal length of the mirror is:
Two wires are of same length and same area of cross-section. If first wire has resistivity \[{{\rho }_{1}}\]and Temperature coefficient of resistance \[{{\alpha }_{1}}\] but second wire has resistivity \[{{\rho }_{2}}\] and temperature coefficient of resistance\[{{\alpha }_{2}}\]. Their series Equivalent resistance is independent of small temperature changes. Then
Consider a system of three charges \[\frac{q}{3},\frac{q}{3}\] and \[\frac{2q}{3}\] placed at points A, B and C, respectively, as shown in figure. Take 0 to be the centre of circle of radius R and angle CAB = . Choose the correct one:
A)
The electric field at point \[{\mathrm O}\]is \[\frac{q}{8\pi {{\in }_{0}}{{R}^{2}}}\] directed along the negative x-axis.
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B)
The potential energy of the system is zero.
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C)
The magnitude of the force between the charges at \[C\]and \[B\]is \[\frac{{{q}^{2}}}{54\pi {{\in }_{0}}{{R}^{2}}}\]
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D)
The potential at points O is \[\frac{q}{12\pi {{\in }_{0}}{{R}^{{}}}}\]
A wedged shaped air film having an angle of 40 second is illuminated by a monochromatic light and the fringes are observed vertically down through a microscope. The fringe separation between two consecutive bright fringes is 0.12 cm. The wavelength of light is:
A bimetallic strip is formed out of two identical strips, one of copper and the other of brass. The Coefficients of linear expansions of two metals are \[{{\alpha }_{C}}\] and \[{{\alpha }_{B}}\] On heating, the temperature of the strip goes up by \[\Delta T\] and the strip bends to form an arc of radius \[R\] Find \[R\]
A rocket is launched normal to the surface of the Earth, away from the Sun, along the line joining the Sun and the Earth. The Sun is \[3\text{ }\times {{10}^{5}}\] times Heavier than the Earth and is at a distance \[2.5\times {{10}^{4}}\] times larger than the radius of the Earth. The escape velocity from Earth's gravitational field is \[{{v}_{e}}=11.2km{{s}^{-1}}\]. The minimum initial velocity \[({{v}_{s}})\] required for the rocket to be able to leave the Sun-Earth system is closest to (Ignore the rotation and revolution of the Earth and the presence of any other planet)
A particle of mass m and charge q enters a region of magnetic field (as shown) with speed v. There is a region in which the magnetic field is absent, as shown. The particle after entering the region collides elastically with a rigid wall. Time after which the velocity of particle becomes Antiparallel to its initial velocity is
\[{{S}_{1}}\]and \[{{S}_{2}}\] are two coherent current sources of radiations separated by distance \[100.25\lambda \],where \[\lambda \] Is the wavelength of radiation. \[{{S}_{1}}\] Leads \[{{S}_{2}}\]in phase by\[\pi \]/2. A and B are two points on the line joining \[{{S}_{1}}\]and \[{{S}_{2}}\] as shown in figure.
The ratio of amplitudes of source \[{{S}_{1}}\]and \[{{S}_{2}}\]are in the ratio \[~1:2\]then the ratio of intensity at A to that of \[\operatorname{B}\left( \frac{{{I}_{A}}}{{{I}_{B}}} \right)\]is
For the circuit shown in the figure the rms value of voltages across R and coil are \[{{E}_{1}}\]and \[{{E}_{2}}\]respectively. The power (thermal) developed
A stone is thrown from the top of a cliff 70m high at an angle of \[30{}^\circ \] below the horizontal and hits the sea 20m from the bottom of the cliff Find the initial speed of the stone and the direction in which it is moving when it hits the sea.
An electron of mass ?m? and charge ?e? initially at rest gets accelerated by a constant electric fields E. the rate of change of de-Broglie wavelength of this electron at time t, ignoring relativistic effects is:
A metal sphere of radius R and specific heat C is rotated about an axis passing through its Centre at a speed n rotation /second. It is suddenly Stopped and 50% of its energy is used in Increasing its temperature, then find the rise in Temperature of the sphere.
A sinusoidal voltage of amplitude 15 V is connected between the input terminals of the circuit shown in the figure. Assume that the diodes are ideal. In the output waveform
A)
Positive peaks of the input will be clipped to +12V and negative peaks will be clipped to -6V
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B)
Positive peaks of the input will be clipped to +6V and negative peaks will be clipped to -12V
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C)
Positive peaks of the input will be clipped to +12V and negative peaks will be clipped to -12V
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D)
Positive peaks of the input will be clipped to +6V and negative peaks will be clipped to-6V
The displacement of a particle is given by \[x\text{ }=\text{ }{{\left( t\text{ }-\text{ }2 \right)}^{2}}\], where x is in meters and t in seconds. The distance covered by the particle in first 4 s is
A ring is cut form a platinum tube 8.5 cm internal and 8.7 cm external diameter. It is supported horizontally from a pan of balance so that it comes in contact with the water in a glass vessel. What is the surface tension of water if an extra 3.97 g weight is required to pull it away from water? \[\left( g=980cm/{{s}^{2}} \right)\]
A 10 kW drilling machine is used to drill a bore in a small aluminum block of mass 8.0 kg. How much is the rise in temperature of the block in 2.5 minute? Assuming 50% of power is used up in heating the machine itself or lost the surroundings specific heat of aluminum \[=0.91J/g-{}^\circ C.\]
\[5.1\text{ }g\text{ }N{{H}_{4}}SH\] is introduced in \[3.0L\] evacuated flask at \[327{}^\circ C.\,\]\[30%\] of the solid \[N{{H}_{4}}SH\] decomposed to \[N{{H}_{3}}\] and \[{{H}_{2}}S\] as gases. The \[{{K}_{p}}\] of the reaction at \[327{}^\circ C\] is (\[R=0.082L\,atm\text{ }mo{{l}^{-1}}{{K}^{-1}},\] Molar mass of \[S=32\text{ }g\text{ }mo{{l}^{-1}},\] molar mass of \[N=14\text{ }g\text{ }mo{{l}^{-1}}\])
In the cell \[Pt(s)/{{H}_{2}}(g,1bar)/HCl(aq)/AgCl(s)/\] \[Ag(s)/pt(s)\] the cell potential is \[0.92V\] when a \[{{10}^{-6}}\]molal \[HCl\] solution is used. The standard electrode potential of \[(AgCl/Ag,C{{l}^{-}})\] electrode is: \[\left\{ Given,\frac{2.303RT}{F}=0.06V\,at298K \right\}\]
An aromatic compound 'A' having molecular formula \[{{C}_{7}}{{H}_{6}}{{O}_{2}}\] on treating with aqueous ammonia and heating forms compound 'B' The compound 'B' on reaction with molecular bromine and potassium hydroxide provides compound 'C' having molecular formula \[{{C}_{6}}{{H}_{7}}N.\] The structure of 'A' is:
An ideal gas undergoes isothermal compression from \[5{{m}^{3}}\] to \[1{{m}^{3}}\] against a constant external pressure of \[4N{{M}^{-2}}.\] Heat released in this process is used to increase the temperature of 1 mole of Al. If molar heat capacity of Al is \[24Jmo{{l}^{-1}}{{K}^{-1}},\] the temperature of Al is increases by:
Elevation in the boiling point for 1 molal solution of glucose is 2 K. The depression in the freezing point for 2 molal solution of glucose in the same solvent is 2 K. The relation between \[{{K}_{b}}\] and \[{{K}_{f}}\] is:
Insulin is a protein containing 51 amino acids. Thai include 17 of the 20 different amino acids commonly occurring in proteins. What is the minimum number of different kinds a: (RNA molecules involved in the synthesis of insulin?
Promoters and control elements work together to regulate transcription. What shows the possible locations of these in relation to a transcription start site on the DNA molecule?
During the light phase of photosynthesis, the photo activated pigments remove an electron from the hydroxylation derived from the water molecule. The fate of the free hydroxyl radical is that it
A)
is broken down into oxygen and a free radical of hydrogen
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B)
is used to raise the activation level of chlorophyll by donating a positive charge
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C)
is used to produce adenosine triphosphate from adenosine diphosphate
When mitochondria are extracted from cells for biochemical study, they are usually kept in a \[0.25mol\,d{{m}^{-\,3}}\]sucrose solution. Why is the sucrose solution used?
A)
To act as a solvent
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B)
To provide a source of food
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C)
To assist in the extraction of enzymes
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D)
To prevent the mitochondria from changing in structure
Four identical samples of plant tissue, each with a water potential \[(\psi )\]of sign\[-\,700\,kPa,\]are placed in four different solutions. Which solution induces full plasmolysis within the tissue?
Protein was the only food available to a mammal. If this supply of protein was only half of the minimum required to supply its energy needs, the mammal would show an increase in the
If \[({{x}_{1}},{{y}_{1}})\] & \[({{x}_{2}},{{y}_{2}})\] are the ends of diameter of a circle such that \[{{x}_{1}}\] & \[{{x}_{2}}\] are the roots of the equation \[a{{x}^{2}}+bx+c=0\] and \[{{y}_{1}}\] & \[{{y}_{2}}\] are the roots of the equation; \[p{{y}^{2}}+qy+c=0\] Then the co-ordinates of the centre of the circle is:
If \[\alpha ,\,\,\beta \] be the roots of the equation \[{{u}^{2}}-2u+2=0\] & if \[\cot \theta =x+1,\] then \[\frac{{{(x+\alpha )}^{n}}-{{(x+\beta )}^{n}}}{\alpha -\beta }\] is equal to:
A chord of the parabola \[y=-\,{{a}^{2}}{{x}^{2}}+5\,ax-\,4\] touches the curve \[y=\frac{1}{1-x}\] at the point \[x=2\]and is bisected by that point. If S is the sum of all possible values of a, then find 12 S:
A particle starts from rest at t = 0 and undergoes an acceleration a in m s~2 with time (in seconds which is shown in figure. Which one of the following plot represents velocity v in \[m{{s}^{-1}}\text{ }verus\]time t sec?
A solid sphere having uniform charge density p and radius \[R\] is show in figure. A Spherical cavity of radius \[\frac{R}{2}\] is hollowed out. So the potential at \[o\]Will be (Assuming potential of infinity to be Zero)
A solid cylinder of radius.?, made of a material of thermal conductivity K, is surrounded by a hollow cylinder of inner radius R and outer radius 2R made of material of thermal conductivity the two ends of the combined system are maintained at two different temperatures. There is no loss of heat across the cylindrical surface and the system is in steady state. The effective thermal conductivity of the system is
The counting rate observed from a radioactive Source at \[t=0\] second was \[1600\] counts per second and at \[t=8\] seconds it was \[100\]counts per second. The counting rate observed, as Counts per second at \[t=6\] seconds, will be
A balloon starting from the ground has been ascending vertically at a uniform velocity for 4.5 s and a stone left fall from it reaches the ground in 7 s. The velocity of the balloon when the stone was left fall is \[\left[ use\text{ }g\text{ }=\text{ }9.8\text{ }m/{{s}^{2}} \right].\]
Three bars of equal lengths x and equal area of cross-section \[A\] are connected in series. Their thermal conductivities are in the ratio of \[2:4:3.\] If the open ends of the first and the last bars are at temperature \[200{}^\circ C\] and \[18{}^\circ C\], respectively in the steady state, calculate the temperatures of Both the junctions.
The electric potential between a proton and an Electron is given by \[V={{V}_{0}}\ell n\frac{r}{{{r}_{0}}}\] , where r,, is a '0 constant. Assuming Bohr's model to be Applicable, write variation of r with n, n being the principal quantum number
A non conducting ring (of mass m, radius \[r\], Having charge\[Q\]) is placed on a rough horizontal Surface (in a cylindrical region with transverse magnetic field). The field is increasing with time at the rate \[R\] and coefficient of friction between The surface and the ring is \[\mu \]. For ring to remain in equilibrium u should be greater than equal to,
A bullet is fired vertically upwards with velocity \[v\]. From the surface of a spherical planet. When it Reaches its maximum height, its acceleration due to the planet's gravity is \[\frac{1}{4}th\] of its value of the Surface of the planet. If the escape velocity from The planet is \[{{V}_{esc}}=V\sqrt{N},\] then the value of N is (Ignore energy loss due to atmosphere)
An electric heater is used in a room of total wall Area 137 \[{{m}^{2}}\] to maintain a temperature of \[+20{}^\circ C\] inside it, when the outside temperature is \[-10{}^\circ C.\] The walls have three different layers materials. The innermost layer of wood of thickness \[2.5\] cm, the middle layer is of cement of thickness 1.0 cm and the outermost layer is of brick of thickness 25.0 cm. Find the power of the electric heater. Assume that there is no heat loss through the floor and the ceiling. The thermal conductivities of wood, cement and brick are \[0.125,1.5\,\]and \[1.0\operatorname{W}/m-{}^\circ C\] respectively.
A reaction of cobalt(III) chloride and ethylenediamine in a \[1:2\] mole ratio generates two isomeric products A (violet coloured) and B (green coloured). A can show optical activity, but, B is optically inactive. What type of isomers does A and B represent?
A compound of formula \[{{A}_{2}}{{B}_{3}}\]has the hcp lattice. Which atom forms the hcp lattice and what fraction of tetrahedral voids is occupied by the other atoms:
A)
hcp lattice - A, \[\frac{2}{3}\] Tetrahedral voids - B
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B)
hcp lattice - A, \[\frac{1}{3}\] Tetrahedral voids - B
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C)
hcp lattice - B, \[\frac{2}{3}\] Tetrahedral voids - A
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D)
hcp lattice - B, \[\frac{1}{3}\] Tetrahedral voids - A
In an experiment to determine the number of rats in a field, 80 rats were initially captured, marked and released. After one month, 100 rats were captured in the same field, of which 20 were previously marked ones. Based on the above observation, estimated population size of rats in the field will be
To understand prey-predator relationship, Didymium (predator) and Paramecium (prey) were used. Paramecium population was grown with and sediment as hiding place or refuge. To this population, Didymium was introduced only once. What could happen to the prey population in the course of time?
A)
The population will steadily decrease and vanish
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B)
The population will initially increase and then stabilize
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C)
The population will initially decrease, then increase and stabilize
A recessive inherited disease is expressed only in individuals of blood group O and not expressed in blood groups A, B and AB. Alleles controlling the disease and blood group are independently inherited. A normal woman with blood group A and her normal husband with blood group B already has one child with the disease. The woman is pregnant for second- time. What is the probability that the second child will also have the disease?
The region of the genome containing the RFLP used in this analysis is shown below.
Hind III indicates the restriction sites for this enzyme and * indicates the polymorphic site which is missing in the recessive allele q.
The black bar indicates the position of the probe used to detect the RFLP. DNA fragments from three different individuals, X, Y and Z, were subjected to restriction digestion by Hind. Ill and separate by gel electrophoresis. The following results were obtained.
In an investigation to determine the effect of temperature on the activity of an enzyme, the time for all the substrates to disappear from a standard solution was recorded. Which graph shows the result of this investigation?
In animals, four separate families of cell-cell adhesion proteins are listed in Column A and their functional characteristics are given in Column B.
Column A
Column B
A.
Integrin
1.
Lectins that mediate a variety of transient, cell-cell adhesion interactions in the bloodstream
B.
Cadherin
2.
Contains extracellular Ig-like domains and are mainly involved in the fine tuning of cell-cell adhesive interaction during development and regeneration
C.
Ig super family
3.
Mediates \[C{{a}^{2+}}\]dependent strong family hemophilic cell-cell adhesion
D.
Selecting
4.
Trans membrane cell adhesion protein that acts as extracellular matrix receptors.
Which one of the following is the correct combination?
The complete oxidation of one mole of glucose yields 2880 kJ of energy. The addition of one phosphate molecule to ADP requires 30.6 kJ of energy per mole. In aerobic respiration, 38 molecules of ATP are formed as a result of the breakdown of each glucose molecule. Which figure best represents the efficiency of aerobic respiration in trapping the energy released by the glucose molecule?