Line \[{{L}_{1}}\] is parallel to vector \[\vec{\alpha }=-\,3\hat{i}+2\hat{j}+4\hat{k}\] and passes through a point A (7, 6, 2) and line \[{{L}_{2}}\] is parallel to a vector \[\vec{\beta }=2\hat{i}+\hat{j}+3\hat{k}\] and passes through a point B (5, 3, 4). Now a line \[{{L}_{3}}\] parallel to a vector \[\vec{r}=2\hat{i}-2\hat{j}-\hat{k}\] intersects the lines \[{{L}_{1}}\] and \[{{L}_{2}}\] at points C and D respectively, then \[\left| \overrightarrow{CD} \right|\] is equal to
The range of values of 'a' such that the angle \[\theta \] between the pair of tangents drawn from the point (a, 0) to the circle \[{{x}^{2}}+{{y}^{2}}=1\] satisfies \[\frac{\pi }{2}<\theta <\pi \] is:
Let \[f\,(x)=min\,(\{x\},\,\,\{-x\})\]\[\forall \]\[x\in R\] (where \[\{\cdot \}\] represents fractional part function) then \[\int\limits_{-100}^{100}{f\,(x)\,\,dx}\]
Two dice are rolled to get the coordinates of a point P(x, y) in the Cartesian plane. Find the probability that area of \[\Delta \,PAB\]is 1 sq. unit, where \[A\equiv (1,\,\,1),\]\[B\equiv (2,\,\,0).\] Given that P, A, B lie in anticlockwise order in the plane
The vector \[\vec{c}\] is perpendicular to the vectors \[\vec{a}=(2,\,\,-3,\,\,1),\,\,\vec{b}=(1,\,\,-2,\,\,3)\] and satisfies the condition \[\vec{c}.\,\,(\hat{i}+2\hat{j}+7\hat{k})=10.\] Then the vector \[\vec{c}=\]
The moment of inertia of a solid sphere, about an axis parallel to its diameter and at a distance of x from it, is 'I(x)'. Which one of the graphs represents the variation of l(x) with x correctly?
A load of mass M kg is suspended from a steel wire of length 2 m and radius 1.0 mm in Searle' experiment. The increase in length product wire is 4.0 mm. Now the load is fully immersed in a liquid of relative density 2. The relative density of the length of load is 8. The new value of increase in ength of the steel wire is:
In the above circuit, \[C=\sqrt{\frac{3}{2}}\mu F,\] \[R=20\Omega ,\] \[L=\frac{\sqrt{3}}{10}H\] and \[{{R}_{1}}=10\Omega .\]Current in \[L-{{R}_{1}}\]path is \[{{I}_{1}}\] and \[C-{{R}_{2}}\]Path is \[{{I}_{2}}.\] The voltage of A.C. source is given by, \[V=200\sqrt{2}\sin (100t)\] volts. The phase difference between \[{{I}_{1}}\] and \[{{I}_{2}}\]is:
An ideal gas is enclosed in a cylinder at pressure of 2 atm and temperature, 300 K. The mean time between two successive collisions is \[6\times {{10}^{-8}}s.\] If the pressure is doubled and temperature is increased to 500 K, the mean time between two successive collisions will be
In the figure, given that \[{{V}_{BB}}\] supply can vary from 0 to 5.0 V, \[{{V}_{cc}}=5V,\]\[{{\beta }_{dc}}=200,\] \[{{R}_{B}}=100K\Omega ,\] \[{{R}_{C}}=1k\Omega \] an \[{{V}_{BE}}=1.0V,\] The minimum base current and input voltage at which the transistor will go to saturation, will be respectively:
An alpha-particle of mass m suffers I-dimensional elastic collision with a nucleus at rest of unknown mass. It is scattered directly backwards losing, 64% of its initial kinetic energy. The mass of the nucleus is:
A 10 m long horizontal wire extends from North East to South West. It is falling, with a speed of \[5.0\,m{{s}^{-1}},\] at right angles to the horizontal component of the earth's magnetic field, of \[0.3\times {{10}^{-4}}\,\text{Wb/}{{m}^{2}}\]. The value of the induced emf in wire is:
A plano-convex lens (focal length \[{{f}_{2}},\]refractive index \[{{\mu }_{2}},\] radius of curvature R) fits exactly into a Plano concave lens (focal length \[{{f}_{1}},\]refractive index \[{{\mu }_{1}},\]radius of curvature R). Their plane surfaces are parallel to each other. Then, the focal length of the combination will be:
A vertical closed cylinder is separated into two parts by a frictionless piston of mass m and of negligible thickness. The piston is free to move along the length of the cylinder. The length of the cylinder above the piston is \[{{l}_{1}},\]and that below the piston is \[{{l}_{2}},\] such that \[{{l}_{1}}>{{l}_{2}}.\] Each part of the cylinder contains moles of an ideal gas at equal temperature T. If the piston is stationary, its mass, m, will be given by: (R is universal gas constant and g is the acceleration due to gravity)
Two satellites, A and B, have masses m and 2m respectively. A is in a circular orbit of radius R, B is in a circular orbit 2R around the earth. The ratio of their kinetic energies, \[{{\text{T}}_{\text{A}}}\text{/}{{T}_{B}},\]
A long cylindrical vessel is half filled with a liquid. When the vessel is rotated about its own vertical axis, the liquid rises up near the wall. If the radius of vessel is 5 cm and its rotational speed is 2 rotations per second, then the difference in the heights between the centre and the sides, in cm, will be:
A block kept on a rough inclined plane, as shown in the figure, remains at rest upto a maximum force 2 N down the inclined plane. The maximum external force up the inclined plane that does not move the block is 10 N. The coefficient of static friction between the block and the plane is: \[[Take\,g=10\,\text{m/}{{\text{s}}^{2}}]\]
In a Frank-hertz experiment, an electron of energy 5.6 eV passes through mercury vapour and emerges with an energy 0.7 eV. The minimum wavelength of photons emitted by mercury atoms is close to:
A particle of mass 20 g is released with an initial velocity 5 m/s along the curve from the point A, as shown in the figure. The point A is at height h from point B. The particle slides along the frictionless surface. When the particle reaches point B, its angular momentum about 0 will be: \[[Take\,g=10\,\text{m/}{{\text{s}}^{2}}]\]
A galvanometer, whose resistance is 50 ohm, has 25 divisions in it. When a current of \[4\times {{10}^{-4}}\]. A passes through it, its needle (pointer) deflects by one division. To use this galvanometer as a voltmeter of range 2.5 V, it should be connected to a resistance of:
A soap bubble, blown by a mechanical pump at the mouth of a tube, increases in volume, with time, at a constant rate. The graph that time dependence of pressure inside the bubble is given by:
In the given circuit diagram, the currents, \[{{I}_{1}}=-0.3A,\]\[{{I}_{4}}=0.8\,A\] and \[{{I}_{5}}=0.4\,A,\] are flowing as shown ' currents \[{{I}_{2}},{{I}_{3}}\] and \[{{I}_{6}},\]respectively are:
A resonance tube is old and has jagged end. It is still used in the laboratory to determine velocity in air. A tuning fork of frequency 512 HZ produces first resonance when the tube is filled with mark 11 cm below a reference mark, near end of the tube. The experiment is repeated fork of frequency 256 Hz which prod' resonance when water reaches a mark 27 the reference mark. The velocity of obtained in the experiment, is close to:
In the following sequence of reactions \[C{{H}_{3}}C{{H}_{2}}OH\xrightarrow{P+{{I}_{2}}}A\xrightarrow[ether]{Mg}B\xrightarrow{HCHO}C\xrightarrow{{{H}_{2}}O}D\]. The compound D is
A sample of 16 g charcoal was brought into contact with \[C{{H}_{4}}\] gas contained in a vessel of litre at \[27 {}^\circ C\]. The pressure of gas was found to fall from \[760\] to 608 torr. The density of charcoal sample is \[1.6\text{ }g/c{{m}^{3}}\]. What is the volume of the \[{{\operatorname{CH}}_{4}}\]Gas adsorbed per gram of the adsorbent at 608 torr and\[27 {}^\circ C\]?
Among the following series of transition metal ions, the one where all metal ions have \[3{{d}^{2}}\]electronic configuration is \[(At.nos.Ti=22;V=23;Cr=24;Mn=25)\]
A metal oxide has the formula \[Z{{ }_{2}}{{O}_{3}}\]. It can be reduced by hydrogen to give free metal and water. 0.1596 g of the metal oxide requires 6 mg of hydrogen for complete reduction. The atomic weight of the metal is
Two beaker A and B present in a closed vessel. Beaker A contains 152.4 g aqueous solution of urea, containing 12 g of urea beaker B contains 196.2 g glucose solution, containing 18 g of glucose. Both solutions allowed to attain the equilibrium. The mass % of glucose in its solution at equilibrium is
A certain buffer solution contains equal concentration of \[{{\operatorname{X}}^{-}}\]and HX. The \[{{K}_{b}}\] for \[{{X}^{-}}\] is \[{{10}^{-10}}\]. The pH of the buffer is:
Let f & g be two functions defined as follows; \[f\,(x)=\frac{x+|x|}{2}\] for all x & \[g\,(x)=\left[ \begin{matrix} x & \text{for} & x<0 \\ {{x}^{2}} & \text{for} & x\ge 0 \\ \end{matrix} \right.\] then:
A)
(gof)(x) & (fog)(x) are both cont. for all \[x\in R\]
A man wants to buy m mangoes in n different varieties, (where n > m) mangoes of the same variety being identical and they are available in abundance. Number of different ways he can plan his purchases, if he has to buy at least two mangoes of the same variety is:
A paramagnetic material has \[{{10}^{28}}\] atoms/ \[{{m}^{3}}.\] Its susceptibility at temperature 350 K is \[2.8\times {{10}^{-4}}.\] Its susceptibility at 300 K is:
In a radioactive decay chain, the initial nucleus is \[_{90}^{232}Th.\] At the end there are \[6\alpha -\]particles and \[4\beta \]particles with are emitted. If the end nucleus is\[_{Z}^{A}X,\] A and Z are given by:
When a certain photosensitive surface is illuminated with monochromatic light of frequency v, the stopping potential for the photo current is \[-{{\text{V}}_{\text{0}}}\text{/}2.\] When the surface illuminated by monochromatic light of frequency \[\text{V/2},\]the stopping potential is \[-{{V}_{0}}.\] The threshould Frequency for photoelectric emission is:
Two particles A, B are moving on two concentric circles of radii \[{{R}_{1}}\]and \[{{R}_{2}}\]with equal angular speed co. At t = 0, their positions and direction of motion are shown in the figure:
The relative velocity \[{{\overrightarrow{v}}_{A}}-{{\overrightarrow{v}}_{B}}\]at \[t=\frac{\pi }{2\omega }\]is given by:
The mean intensity of radiation on the surface of the Sun is about \[{{10}^{8}}W/{{m}^{2}}.\] The rms value of the corresponding magnetic field is closet to:
A parallel plate capacitor with plates of area \[1\,{{m}^{2}}\] each, are at a separation of 0.1 m. If the electric field between the plates is 100 N/C, the magnitude of chargen on each plate is:
An excess of liquid mercury is added to an acidified solution of \[1.0\times 1{{0}^{-3}}M\,F{{e}^{3+}}.\] It is found that 5% of \[{{\operatorname{Fe}}^{3+}}\]remains at equilibrium of the following reaction \[2Hg+2F{{e}^{3+}}\xrightarrow{{}}Hg_{2}^{2+}+2F{{e}^{2+}}\] The value of \[E{{{}^\circ }_{\operatorname{Hg}_{2}^{2+}/Hg}}\], is (given\[E{{{}^\circ }_{{{\operatorname{Fe}}^{3+}}/F{{e}^{2+}}}}=0.77V\])
Vapour pressure of solution containing 6 g of a non-volatile solute in \[180\]g water is 20.0 Torr. If 1 mole water is farther added vapour pressure increases by 0.02. The ratio of vapour pressure of water and molecular weight of non-volatile solute is
The density of gold is \[19 g/c{{m}^{3}}\]. If \[1.9\times {{10}^{-4}}g\]of gold is dispersed in one litre of water to give a Sol having spherical gold particles of radius 10 Nm, then the number of gold particles per mm3 of the sol will be:
At \[27{}^\circ C\], hydrogen is leaked through a tiny hole into a vessel for 20 minutes. Another unknown gas at the same temperature and pressure as that of I-L is leaked through the same hole for 20 minutes. After the effusion of the gases the mixture exerts a pressure of 6 atmosphere. The hydrogen content of the mixture is 0.7 mole. If the volume of the container is 3 litres, the molecular weight of the unknown gas is:
Given below is a matrix of possible interactions [beneficial \[(+),\]harmful \[(-),\]neutral (0)] between species 1 and 2. The names of interactions P, Q, R and S respectively, are
A gene that was 5055 base pairs long resulted in the expression of functional proteins that were 350, 450 and 1500 amino acids long. What could be most likely reason for this?
Which of the following statements about the life cycle of the HIV (Human Immunodeficiency Virus) are false?
I. The HIV particles recognise the host cell through the sialic acid containing proteins or lipids on the membrane of the host cell.
II. Upon entry of the HIV into the host cell, a drop of pH will result in a conformational change in the HIV structure, consequently releasing the viral genome into the host cell.
III. The viral DNA which enters the host cell's nucleus will be integrated into the genetic material of the host cell using the host cell's enzyme, integrase.
IV. The viruses are released from the host cell by exocytosis.
In cats, the genes controlling coat-colour are codominant (incompletely dominant) and are carried on the X-chromosomes. When a black female was mated with a ginger male, the resulting litter consisted of black male and tortoise shell female kittens. What phenotypic ratio would be expected in the \[{{F}_{2}}\text{-}\]generation?
A)
1 black male: 1 ginger male; 2 tortoise shell females
doneclear
B)
1 black male: 1 ginger male; 1 tortoise shell female: 1 black female
doneclear
C)
2 black males: 1 tortoise shell female: 1 ginger female
doneclear
D)
1 black male: 1 tortoise shell male: 1 ginger female: 1 black female
In an investigation into the properties of membranes, washed beet root cells were shaken for five minutes in different solvents. The intensity of the red colour of the solvent was then measured. Which of the following is possible?