From an external point P, pair of tangent lines are drawn to the parabola \[{{y}^{2}}=4x.\] If \[{{\theta }_{1}}\] and \[{{\theta }_{2}}\] be the inclinations of these tangents with the axis of x such that \[{{\theta }_{1}}+{{\theta }_{2}}=\frac{\pi }{4}\] then the locus of P is -
If \[{{\overline{X}}_{1}}\] and \[{{\overline{X}}_{2}}\] are the means of two distributions such that \[{{\overline{X}}_{1}}<{{\overline{X}}_{2}}\] and \[\overline{X}\] is the mean of the combined distribution, then
Let A, B, C be three events in a probability space. Suppose that \[P(A)=0.5,\text{ }P(B)=0.3,\text{ }P(C)=0.2,\]\[P(A\cap B)=0.15,\] \[P(A\cap C)=0.1\] and \[P(B\cap C)=0.06,\] the greatest possible value of \[P({{A}^{c}}\cap {{B}^{c}}\cap {{C}^{c}})\] is [Note: \[{{A}^{c}}\]denotes compliment of event A]
If the chords of contact of tangents from two points \[(-\,4,\,\,2)\] and (2, 1) to the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] are at right angle, then the eccentricity of the hyperbola is -
Let \[\alpha \] and \[\beta \] be the solutions of the quadratic equation \[{{x}^{2\text{ }}}-1154x+1=0,\] then the value of \[\sqrt[4]{\alpha }+\sqrt[4]{\beta }\] is equal to -
A point Q is selected at random inside the equilateral triangle. If sum of lengths of perpendicular dropped on sides from Q is P, then length of altitude of \[\Delta \] is-
If \[\vec{a}\] & \[\vec{b}\] are any two vectors of magnitudes 1 and 2 respectively, and \[{{(1-3\overrightarrow{a}\,\,.\overrightarrow{b})}^{2}}+|2\overrightarrow{a}+\overrightarrow{b}+3(\overrightarrow{a}\,\,\times \,\,\overrightarrow{b}){{|}^{2}}=47,\] then the angle between \[\vec{a}\] and \[\vec{b}\] is -
Let \[\{{{a}_{n}}\}\,\,(n\ge 1)\] be a sequence such that \[{{a}_{1}}=1,\] and \[3{{a}_{n+1}}-3{{a}_{n}}=1\] for all \[n\ge 1.\] Then \[{{a}_{2002}}\] is equal to
A current loop, having two circular arcs joined by two radial lines shown in the figure. It carries a current of 10 A. The magnetic field at point O will be close to:
A gas can be taken from A to B via two different processes ACB and ADB.
When the 'path ACB is used 60 J if heat flows into the system and 30 J of work is done by the system. If path ADB is used work done by the system is 10 J. The heat flows into the system in path ADB is:
Plane electromagnetic wave of frequency 50 MHz "travels in free space along the positive x direction. At a particular point in space and time, \[\vec{E}=6.3\hat{j}\] V/m. The corresponding magnetic field B, at that point will be:
Two coherent source- produce waves of different intensities which interfere. After interference, the ratio of to the minimum intensity is of the waves are in the ratio:
An L-shaped object, made of thin rods of uniform mass density, is suspended with a string as shown in figure. If \[AB=BC,\] and the angle made by AB with downward vertical is \[\theta ,\] then:
A mixture of 2 moles of helium gas (atomic mas\[=4u\]) and I mole of argon gas (atomic mas\[=40u\]) is kept at 300 K in a container. The ration of their rms \[\text{speeds}\,\,\left[ \frac{{{\text{V}}_{rms}}\,(\text{helium})}{{{\text{V}}_{rms}}(\text{argon})} \right],\,\text{is}\,\text{close}\,\text{to}:\]
A bar magnet is demagnetized by inserting it inside a solenoid of length 0.2 m, 100 turns, and carrying a current of 5.2 A. The coercivity of the "bar magnet is:
A rod, of length L at room temperature and uniform area of cross section A. is made of a metal having coefficient of linear expansion a/ \[{}^\circ \text{C}.\] It is observed that an external compressive force F, is applied on each of its ends, prevents any change in the length of the rod, when its temperature rises by AT K. Young's modulus, Y, for this metal is:
A block of mass m, lying on a smooth horizontal surface, is attached to a spring (of negligible mass) of spring constant k. Trie other end of the spring is fixed, as shown in the figure. The block is initially at rest in its equilibrium position. If now the block is pulled with a constant force F, the maximum speed of the block is:
Three charges +Q, q, +Q are placed respectively, at distance. 0. d/2 and d from the origin, on the x-axis. If the net force experienced by +Q, placed at \[x=0,\]is zero. then value of q is:
A conducting circular loop made of a thin wire, has area \[3.5\times {{10}^{-3}}\,{{\text{m}}^{2}}\] and resistance 10 Ω. It is placed perpendicular to a time dependent magnetic field \[B(t)~=(0.4T)\sin (50\pi t).\] The field is uniform in space. Then the net charge flowing through the loop during \[t=0s\] and \[t=10ms\] is close to:
Two masses m and \[\frac{m}{2}\] are connected at the two ends of a massless rigid rod of length \[l.\] The rod is suspended by a thin wire of torsional constant k at the Centre of mass of the rod-mass system (see figure). Because of torsional constant k, the restoring torque is \[\tau \,=k\theta \] for angular displacement 0. If the rod is rotated by \[{{\theta }_{0}}\]and released, the tension in it when it passes through its mean position will be:
A parallel plate capacitor is made of two square plates of side 'a', separated by a distance d (d<<a). The lower triangular portion is filled with a dielectric of dielectric constant K, as shown in the figure. Capacitance of this capacitor is:
Mobility of electrons in a semiconductor is defined as the ratio of their drift velocity to the applied electric field. If, for an n-type semiconductor, the density of electrons is \[{{10}^{19}}{{\text{m}}^{-3}}\] and their mobility is \[\text{1}\text{.6}\,\text{,}{{\text{m}}^{\text{2}}}\]/(V.s) then the resistivity of the semiconductor (since it is an n-type semiconductor contribution of holes is ignored) is close to:
A block of mass 10 kg is kept on a rough inclined plane as shown in the figure. A force of 3 N is applied on the block. The coefficient of static friction between the plane and the block is 0.6. What should be the minimum value of force P, such that the block does not move downward?
Temperature difference of \[120{}^\circ C\] is maintained between two ends of a uniform rod AB of length 2 L. Another bent rod PQ, of same cross section as AB and length 3L/2, is connected across AB (See figure). In steady state, temperature difference between P and Q will be close to:
The rate constant of the reaction \[A\to B\] is \[0.6\times {{10}^{-\,3}}mol{{e}^{-\,1}}\]per second. If the concentration of A is 5 M then concentration of B after 20 min is
The elevation in boiling point, when 13.44 g of freshly prepared \[CuC{{l}_{2}}\]are added to one kilogram of water, is [Some useful data, \[{{K}_{b}}=0.52K\,kg\,mo{{l}^{-1}},\]molecular weight of \[CuC{{l}_{2}}=134.4\,g].\]
A metal crystallises into two cubic phases, face centred cubic (fee) and body centred cubic (bcc), whose unit cell lengths are 3.5 and \[3.0\overset{{}^\circ }{\mathop{A}}\,\] respectively. Calculate the ratio of densities of fee and bcc.
The standard reduction potential values of three metallic cations, X, Y, Z are \[0.52,\]\[-3.03\] and \[-1.18V\]respectively. The order of reducing power of the corresponding metals is
Given that the standard states for iodine and chlorine are \[{{I}_{2}}(s)\] and \[C{{l}_{2}}\,(g),\] the standard enthalpy for the formation of \[ICl\left( g \right)\] is
The solubility of \[Pb{{(OH)}_{2}}\]in water is \[6.7\times {{10}^{-\,6}}M.\] Calculate the solubility of \[Pb\,{{(OH)}_{2}}\] in a buffer solution of \[pH=8.\]
The intermediate lobe of the pituitary gland produces a secretion which causes a dramatic darkening of the skin of many fishes, amphibians and reptiles. It is
If \[ax+by-5=0\] is the equation of the shortest chord of the circle \[{{\left( x-3 \right)}^{2}}+{{\left( y-4 \right)}^{2}}=4\] passing through the point (2, 3), the \[\left| a+b \right|\] is-
If the quadratic equation \[f(x)=p{{x}^{2}}-qx+r=0\] has two distinct roots in (0, 2) where p, q, \[r\in N\] and \[f(1)=-\,1\] then the minimum value of p is-
If the line \[y=\sqrt{3}x\] intersects the curve \[{{x}^{3}}+{{y}^{3}}+3xy+5{{x}^{2}}+\text{4}x+5y-1=0\] at the points A, B, C then \[OA.\,\,OB.\,\,OC\]is (Here \['O'\] is origin)
Three glass cylinders of equal height \[H=30cm\] and me refractive index \[n=1.5\] are placed on a horizontal surface as shown in figure. Cylinder I has a flat top, cylinder II has a convex top and cylinder III has a concave top. The radii of curvature of the two curved tops are same \[(R=3m).\] If \[{{\text{H}}_{\text{1}}}\text{,}\]\[{{\text{H}}_{2}}\] and \[{{\text{H}}_{\text{3}}}\] are the apparent depths of a point X on the bottom of the three cylinder, respectively, the correct statement(s) is/ are:
An electric dipole with dipole moment \[\frac{{{p}_{0}}}{\sqrt{2}}\,(\hat{i}+\hat{j})\] is held fixed at the origin O in the presence of an uniform electric field of magnitude \[{{\text{E}}_{0}}.\]If the potential is constant on a circle of radius R centered at the origin as shown in figure, then the correct statement(s) is/are: \[({{\varepsilon }_{0}}\] is permittivity of free space, R >> dipole size)
A)
Total electric field at point A is \[{{\vec{E}}_{A}}\,=\,\sqrt{2}{{E}_{0}}\,(\hat{i}+\hat{j})\]
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B)
Total electric field at point B is \[{{\vec{E}}_{B}}\,=\,0\,\]
A free hydrogen atom after absorbing a photon of wavelength \[{{\lambda }_{a}}\] gets excited from the state \[n=1\] to the state \[n=4.\] Immediately after that the electron jumps to \[n=m\] state by emitting a photon of wavelength \[{{\lambda }_{e}}.\] Let the change in momentum of atom due to the absorption and the emission are \[\vartriangle {{P}_{a}}\] and \[\vartriangle {{P}_{e}},\] respectively. If \[{{\lambda }_{a}}\text{/}{{\lambda }_{e}}\,=\,\frac{1}{5}\] which of the option(s) is/ are correct?
[Use \[hc=1242\text{ }eVnm;\]\[1nm=~{{10}^{-9}}m,h\] and c are Plank's constant and speed of light, respectively]
In a Young's double slit experiment, the slit separation d is 0.3 mm and the screen distance D is 1 m. A parallel beam of light of wavelength 600 nm is incident on the slits at angle a as shown in figure. On the screen, the point O. is equidistant from the slits and distance PO is 11.0 mm. Which of the following statement(s) is/are correct?
A)
Fringe spacing depends on a
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B)
For \[\alpha =0,\] there will be constructive interference at point P
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C)
for \[\alpha \,=\,\frac{0.36}{\pi }\]degree, there will be destructive interference at point P.
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D)
for \[\alpha \,=\,\frac{0.36}{\pi }\] degree, there will be destructive interference at point 0.
A thin and uniform rod of mass M and length L is held vertical on floor with large friction. The rod is released from rest so that it falls by rotating about is contact-point with the floor without slipping. Which of the following statement(s) is/are correct. When the rod makes an angle \[60{}^\circ \] with vertical? [g is the acceleration due to gravity]
A)
The normal reaction force from the floor on the mg rod will be \[\frac{mg}{16}\]
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B)
The angular acceleration of the rod will be \[\frac{2g}{L}\]
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C)
The angular speed of the rod will be \[\sqrt{\frac{3g}{2L}}\]
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D)
The radial acceleration of the rod's center of mass will be \[\frac{3g}{4}\]
A mixture of ideal gas containing 5 moles of monoatomic gas and 1 mole of rigid diatomic gas is initially at pressure \[{{\text{P}}_{0.}}\]volume \[{{\text{V}}_{0}}\] and temperature \[{{T}_{0.}}\] If the gas mixture is adiabatically compressed to a volume \[{{\text{V}}_{\text{0}}}\text{/}4,\] then the correct statement(s) is/are: (\[\text{given}\,{{2}^{1.2}}\,=\,2.3;\,{{2}^{3.2}}=\,9.2;\,\text{R}\,\]is gas constant)
A)
Adiabatic constant of the gas mixture is 1.6
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B)
The average kinetic energy of the gas mixture after compression is in between \[18\text{R}{{\text{T}}_{0}}\] and \[19\text{R}{{\text{T}}_{0.}}\]
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C)
The final pressure of the gas mixture after compression is in between \[9{{\text{P}}_{0}}\] and \[10{{\text{P}}_{0}}.\]
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D)
The work \[\left| \,\text{W}\, \right|\]done during the process is \[13\text{R}{{\text{T}}_{0}}.\]
A block of mass 2 M is attached to a massless spring with spring-constant K. This block is connected to two other blocks of masses M and 2 M using two massless pulleys and strings. The accelerations of the blocks are \[{{a}_{1}},\]\[{{a}_{2}}\]and \[{{a}_{3}}\] as shown in the figure. The system is released from rest with the spring in its unstretched state. The maximum extension of the spring is \[{{x}_{0}}.\] Which of the following option(s) is/are correct? [g is the acceleration due to gravity. Neglect friction]
A)
\[{{x}_{0}}=\frac{4\text{Mg}}{k}\]
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B)
When spring achieves an extension of \[\frac{{{x}^{0}}}{2}\]for the first time, the speed of the block connected to the spring is \[3g\sqrt{\frac{\text{M}}{5k}}\]
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C)
At an extension of \[\frac{{{x}_{0}}}{4}\] of the spring, the magnitude of acceleration of the block connected to the spring is \[\frac{3g}{10}\]
A small particle of mass m moving inside a heavy hollow and straight tube along the tube axis undergoes elastic collision at two ends. The tube has no friction and it is closed at one end by a flat surface while the other end is fitted with a heavy movable flat piston as shown in figure. When the distance of the piston from closed end is the particle speed is \[v\,=\,{{v}_{0}}.\] The piston is moved inward at a very low speed V such that \[\text{V}\,<<\,\frac{\text{dL}}{\text{L}}{{\text{V}}_{0}}.\] Where \[dL\] the infinitesimal displacement of the piston. Which of the following statements) is/are correct?
A)
After each collision with the piston, the particle speed increases by 2V
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B)
If the piston moves inward by dL, the particle speed increases by \[2v\frac{d\text{L}}{\text{L}}\]
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C)
The rate at which the particle strikes the piston is v/L.
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D)
The particle's kinetic energy increases by a factor of 4 when the piston is moved inward from \[{{\text{L}}_{0}}\] to \[\frac{1}{2}{{\text{L}}_{0}}.\]
A monochromatic light is incident from air on a refracting surface of a prism of angle \[75{}^\circ \] and refractive index \[{{n}_{0}}=\,\sqrt{3.}\]The other refracting surface of the prism is coated by a thin film of material of refractive index n as shown in figure. The light suffers total internal reflection at the coated prism surface for an incidence angle of \[\theta \,\le \,{{60}^{0}}.\]The value of \[{{n}^{2}}\] is _______ .
A perfectly reflecting mirror of mass M mounted on a spring constitutes a spring-mass system of angular frequency Ω such that \[\frac{4\pi M\Omega }{h}\,=\,{{10}^{24}}{{m}^{-2}}\] with h as h Planck's constant. N photons of wavelength \[\lambda \,=\,8\pi \,\times {{10}^{-6}}\] m strike the mirror simultaneously at normal incidence such that the mirror gets displaced by 1 \[\mu m.\] If the value of N is \[x\times {{10}^{12}},\] then the value of x is ___ [Consider the spring as massless]
On treatment of 100 mL of 0.1 M solution of \[CoC{{l}_{3}}.6{{H}_{2}}O\] with excess of \[AgN{{O}_{3}};1.2\times {{10}^{22}}\] ions are precipitated. The complex is
For one mole of a van der Waals' gas when \[b=0\]and T = 300 K, the pV vs 1/V plot is shown below. The value of the van der Waals' constant a \[(\text{atm}\,\text{Lmo}{{\text{l}}^{-2}})\]
Ahexapeptide has the composition Ala, Gly, Phe, Val. Both the N-terminal and C-terminal units are Val. Cleavage of the hexapeptide by chemotrypsin gives two different tripeptides, both having Val as the N-terminal group. Among the products of randod hydrolysis is a Ala-Val dipeptide fragment. What is the primary structure of the hexapeptide?
One mole of an organic compound A with the formula \[{{C}_{3}}{{H}_{8}}O\] reacts completely with two moles of \[HI\] to from X and Y. When Y is boiled with aqueous alkali it forms Z. Z answers the iodoform test. The compound A is
[a] The female of many primates, including human's have_____(i)______cycle, in which the _____(ii)_____is shed, [b] Whereas other anammals have.....(iii) cycle. [c] During the first 14 days of the menstrucal cycle growth of the follicle is promoted by, ____ (iv)
A married couple adopted a male child. A few years later, twin boys were born to them. The blod group of the couple is AB positive and O negative. The blood group of the three sons is A positive, B positive, and O positive. The blood group of the adopted son is
A)
O positive
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B)
A positive
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C)
B positive
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D)
Cannot be determined on the basis of the given data
Match the names of disease listed under column I with meanings given under column II, choose the answer which gives the correct combination of the alphabets of the columns.
ABC dominant genes a are required for production of purple colour of flower in a plant A purple plant with genotype AABBCC crossed with a colourless plant with genotype aabbcc produces purple colour in \[{{F}_{1}}\] hybrid. On selfing of \[{{F}_{1}}\]-what proportion of coloured offspring in \[{{F}_{2}}\]