If direction of two sides of a triangle are fixed and length of third side is constant and is sliding between these sides, then locus of the orthocenter of the triangle is
Let \[{{S}_{n}}=\sum\limits_{k=1}^{n}{\frac{n}{{{n}^{2}}+kn+{{k}^{2}}}}\] and \[{{T}_{n}}=\sum\limits_{k=0}^{n-1}{\frac{n}{{{n}^{2}}+kn+{{k}^{2}}}}\] \[n=1,2,3,.........Then,\]
Two friends visit a restaurant randomly during 5 pm to 6 pm. Among the two, whoever comes first waits for 15 min. and then leaves. The probability that they meet is:
A trapezium is inscribed in the parabola \[{{y}^{2}}=4x\] such that its diagonal pass through the point \[(1,0)\] and each has length\[\frac{25}{4}\]. Then the area of trapezium is equal to (in sq. units)
A person goes to office either by car, scooter, bus or train whose probabilities are respectively, \[\frac{1}{7},\frac{3}{7},\frac{2}{7}\] and\[\,\frac{1}{7}\]. The probability that he travelled by car is
A variable point \[P\]on the ellipse of eccentricity \[e\] is joined to the foci \[S\]and \[S'.\]if the locus of the incentre of the triangle \[PSS'\] is a conic of eccentricity \[{{e}_{1}},\]then\[\frac{2}{e_{1}^{2}}-\frac{1}{e}\]equals to
Let \[z,{{z}_{0}}\]be two complex numbers \[{{\bar{z}}_{0}}\] being the conjugate of \[{{z}_{0}}\]. The numbers \[z,{{z}_{0}},z\,{{\bar{z}}_{0}}\] 1 and 0 \[P,{{P}_{0}},QA\] and the origin respectively if \[\left| z \right|=1,\] consider the following statement:
(I) \[PO{{P}_{0}}\] and \[AOQ\]are congruent
(II) \[\left| z-{{z}_{0}} \right|=\left| z{{{\bar{z}}}_{0}}-1 \right|\]
A person throws two dice, one the common cube and the other regular tetrahedron with numbers \[1,2,3,4\]on its faces, the number on the lowest face being taken in the case of a tetrahedron. The chance that the sum of numbers throws is not less than 5 is
Let \[{{A}_{1}},{{A}_{2}}....{{A}_{n}}\]be the vertices of a regular polygon of \[n\] sides inscribed in a circle of radius unity and \[a={{\left| {{A}_{1}}{{A}_{2}} \right|}^{2}}+{{\left| {{A}_{1}}{{A}_{3}} \right|}^{2}}+....{{\left| {{A}_{1}}{{A}_{n}} \right|}^{2}}\]\[b=\left| {{A}_{1}}{{A}_{2}} \right|\left| {{A}_{1}}{{A}_{3}} \right|...\left| {{A}_{1}}{{A}_{n}} \right|,\operatorname{then}\frac{a}{b}=\]
If \[p,q,r,s\] are in arithmetic progression and \[f\left( x \right)=\left| \begin{align} & p+\sin x\,\,\,\,\,\,\,q+\sin x\,\,\,\,\,\,\,p-r+\sin x \\ & q+\sin x\,\,\,\,\,\,\,\,\,r+\sin x\,\,\,\,\,\,\,-1+\sin x \\ & r+\sin x\,\,\,\,\,\,\,\,\,\,s+\sin x\,\,\,\,\,\,\,s-q+\sin x \\ \end{align} \right|\]. Such that \[\int\limits_{0}^{2}{f\left( x \right)dx=-4,}\] then the common difference of the progession is
A thin circular plate of mass M and radius R has its density varying as \[P(r)={{P}_{0}}r\] with \[{{P}_{0}}\] as Constant and \[r\] is the distance from its center. The moment of inertia of the circular plate about an axis perpendicular to the plate and passing through its edge is \[I=aM{{R}^{2}}.\]The value of the coefficient a is:
A solid conducting sphere, having a charge Q is surrounded by an uncharged conducting hollow spherical shell. Let the potential difference between the surface of the hollow shell be V. If the shell given a charge of \[-\,4Q\]the new potential difference between the same two surface is:
In figure, the optical fiber is \[l=2m\] long and has a diameter of \[d=20\mu m.\]If a ray of light is incident on one end of the fiber at angle \[{{\theta }_{1}}=40{}^\circ ,\]the number of reflections it makes before emerging from the other end is close to (refractive index of fiber is 1.31 and \[\sin 40{}^\circ =0.64)\]
An alternating voltage \[v(t)=220\,\sin \,100\pi t\]volt is applied to a purely resistive load of \[50\,\,\Omega .\] The time taken for the current to rise from half of the peak value to the peak value is:
Radiation coming from transitions \[n=2\] to \[n=1\]of ydrogen atoms fall on \[H{{e}^{+}}\]ions in n = 1 and \[n=2\] states. The possible transition of helium ions as they absorb energy from the radiation is:
A plane electromagnetic wave travels in free space along the x-direction. The electric field component of the wave at a particular point of space and time is \[E=6\,\,V{{m}^{-\,1}}\]along y-direction. Its corresponding field component, B would be:
Form particles A, B, C and D with masses \[{{m}_{A}}=m,\] \[{{m}_{B}}=2m,\]\[{{m}_{C}}=3m\]and \[{{m}_{D}}=4m\] are at the corners of a square. They have accelerations of equal magnitude with directions as shown. The acceleration of the centre of the centre of mass of particles is:
Two identical beakers A and B contain equal volumes of two different liquids at \[60{}^\circ C\] each and left to cool down. Liquid in A has density of \[8\times {{10}^{2}}\,\,\text{kg/}{{\text{m}}^{3}}\]and specific heat of \[2000\,J\,k{{g}^{-\,1}}{{K}^{-\,1}}\]while liquid in B has density of \[{{10}^{3}}\,kg\,\,{{m}^{-\,3}}\]and specific heat of \[4000\,J\,k{{g}^{-1}}{{K}^{-1}}.\] Which of the following best describes their temperature versus time graph schematically? (assume the emissivity of both the beakers to be the same)
For the circuit shown, with \[{{R}_{1}}=1.0\,\,\Omega ,\]\[{{R}_{2}}=2.0\,\Omega ,\]\[{{E}_{1}}=2V\] and \[{{E}_{2}}={{E}_{3}}=4V,\] the potential difference between the points 'a' and ?b? is approximately (in V):
A 20 Henry inductor and coil is connected to a 10 ohm resistance in series as shown in figure. The time at which rated of dissipation of energy (Joule's heat) across resistance is equal to the rate at which magnetic energy is stored in the inductor, is:
Two particles move at right angle to each other. Their de-Broglie wavelengths are\[{{\lambda }_{1}}\]and \[{{\lambda }_{2}}\] respectively. The particles suffer perfectly inelastic collision. The de-Broglie wavelength \[\lambda ,\]of the final particle, is given by:
In an interference experiment the ratio of amplitudes of coherent waves is \[\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{1}{3}.\]The ratio of maximum and minimum intensities of fringes will be:
A steel wire having a radius of 2.0 mm, carrying a load of 4 kg, is hanging from a ceiling. Given that \[g=3.1\pi \,m{{s}^{-\,2}},\]what will be the tensile stress that would be developed in the wire?
Water from a pipe is coming at a rate of 100 liters per minute. If the radius of the pipe is 5 cm, the Reynolds number for the flow is of the order of: (density of water \[=1000\,\text{kg/}{{\text{m}}^{3}},\]coefficient of viscosity of water\[=1\,mPa\,s\])
The bob of a simple pendulum has mass 2g and charge of \[5.0\mu C.\] It is at rest in a uniform horizontal electric field of intensity 2000 V/m. At equilibrium, the angle that the pendulum makes with the vertical \[(take\,\,g=10\text{m/}{{\text{s}}^{2}})\]
A wire of length 2L, is made by joining two wires A and B of same length but different radii r and 2r and made of the same material. It is vibrating at a frequency such that the joint of the two wires forms a node. the number of antinodes in wire A is p and that in is q then the ratio p : q is:
An upright object is placed at a distance of 40 cm in front of a convergent lens of focal length 2 cm. : convergent mirror of focal length 10 cm is placed a: distance of 60 cm on the other side of the lens. The position and size of the final image will be:
A)
40 cm from the convergent lens, twice the size the object
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B)
20 cm from the convergent mirror, twice the of the object
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C)
40 cm from the convergent mirror, same size the object
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D)
20 cm from the convergent mirror, same size as the object
[a] A circular coil having N turns and radius r carries a current I. It is held in the XZ plane in a magnetic \[B\hat{i}.\]The torque on the coil due to the magnetic field is :
The wurtzite structure is described as hcp of \[{{S}^{2-}}\] ions with the alternate tetrahedral voids occupied by \[Z{{n}^{2+}}\] ions. A compound containing A, B and X atoms had the A and B atoms arranged as \[Z{{n}^{2+}}\] and \[{{S}^{2-}}\] respectively in the hcp lattice. The X - atoms occupied alternate octahedral voids. Which of the following is incorrect?
A)
Formula of the compound is \[{{A}_{2}}{{B}_{2}}X\]
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B)
Coordination number of A can be 4.
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C)
Coordination number of X can be 6.
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D)
Centres of 12A atoms lie on the edges of one unit cell.
\[{{S}_{1}}\]: Melting of Cu(s) is favourable at high temperature and high pressure.
\[{{S}_{2}}\]: Equilibrium \[{{H}_{2}}(g)+{{I}_{2}}(g)\,\,\,\,2HI(g)\] is stabilished in V(L). If complete equilibrium mixture is transferred to 2V(L) container then the partial pressure of HI will remain same in the new container.
\[{{S}_{3}}\]: Formation of diamond is favourable at very high temperature and very high pressure.
Which of the following is INCORRECT regarding entropy?
A)
The entropy of solution formed is higher if the detergent concentration is slightly less than CMC as compared to when its concentration is slightly more than CMC.
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B)
The vapour pressure of a solution of 10 g protein molecules is expected to be more than that of 10 g urea molecules as entropy of urea solution is more.
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C)
The entropy of pure solid solvent is lesser than entropy of pure liquid solvent but equal to entropy of liquid solution of non-volatile solute.
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D)
The entropy of colloid molecules decrease on coagulation
In the reaction : \[A+2B\to product\]\[Rate=k\,[A]\,\,{{[B]}^{0}}\] The initial concentration of both A and B are a M. The graph of concentration of B vs time is: \[({{t}_{1}}={{t}_{1/2\,}}for\,A)\]
Equal volume of two solution having pH = 2 and pH = 10 are mixed together at \[90{}^\circ C.\] Then pH of resulting solution is: \[(Take\,\,{{K}_{w}}\,at\,\,90{}^\circ C={{10}^{-12}})\]
The correct \[C-C\] bond length order is: \[\begin{matrix} \underset{(x)}{\mathop{{{H}_{3}}C-C{{H}_{3}}}}\, & \underset{(y)}{\mathop{{{H}_{3}}C-C{{H}_{4}}}}\,-Cl \\ \end{matrix}\]
If sexual reproduction takes place between the filaments of Rhizopus of different strains, one with 80 nuclei and another with 24 nuclei, what would be the total number of spores of different strains put together?
Farmers in a particular region were concerned that premature yellowing of leaves of a pulse crop might cause decrease in the yield. Which treatment could be most beneficial to obtain maximum seed yield?
A)
Frequent irrigation of the crop
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B)
Treatment of the plants with cytokinins along with a small dose of nitrogenous fertilizer
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C)
Removal of all yellow leaves and spraying the remaining green leaves with 2, 4, 5-trichlorophenoxy acetic acid
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D)
Application of iron and magnesium to promote synthesis of chlorophyll
If there are 120 adenine molecules in a B-DNA double helical structure showing 20 coils, what is the number of pyrimidine nucleotides forming three hydrogen bonds in it?
Palaeontologists unearthed a human skull during excavation. A small fragment of the scalp tissue was still attached to it. Only little DNA could be extracted from it. If the genes of the ancient man need to be analysed, the best way of getting sufficient amount of DNA from this extract is by
The number of integral values of a for which the point \[(a-1,\text{ }a+1)\] lies in the larger segment of the circle \[{{x}^{2}}+{{y}^{2}}-xy6=0\] cut by the chord whose equation is \[x+y-2=0\] is equal to
Let \[1\le m<n\le p\]. The number of subsets of the Set \[A=\text{ }\left\{ 1,2,3,\text{ }.......p \right\}\]having in, n as the least and the greatest elements respectively, is
Let \[f:\mathbf{R}\to \mathbf{R}\] be a continuous function which satisfies \[f(x)=\int\limits_{0}^{x}{f(t)dt.}\] Then the value of \[f\left( \ln 5 \right)\]is
Two numbers x and y are chosen at random without replacement from amongst the numbers\[1,2,\text{ }3,..........\text{ }3n\]. The probability that \[{{x}^{3}}+{{y}^{3}}\] is Divisible by 3 is
Let \[\frac{dF(x)}{dx}={{x}^{\sqrt{1-{{x}^{2}}}}},0<x\le 1\] and \[I=\int\limits_{\pi /6}^{\pi /2}{{{\left( \sin x \right)}^{\cos x}}\cos x\,dx=F\left( n \right)-F\left( \frac{1}{2} \right)}\]
Given the following statements
(i) domain of \[F'(x)\]is \[(0,1]\]
(ii) Possible value of n is \[\frac{{{\pi }^{2}}}{4}\]
(iii) Possible value of n is 1
(iv) The value of I can be evaluated by substitution.
Four identical particles of mass M are located at the corners of square of side 'a'. What should be their speed if each of them revolves under the influence of others gravitational field in a circular orbit circumscribing the square?
A thermally insulated vessel contains 150 g of water \[0{}^\circ C.\] Then the air from the vessel is pumped out adiabatically. A fraction of water turns into ice and the rest evaporates at \[0{}^\circ C.\] itself. The mass of evaporated will be closest to: (Latent heat of vaporization of water \[=\,2.10\times {{10}^{6}}Jk{{g}^{-1}}\] and Latent heat of fusion of water \[=3.36\times {{10}^{5}}jk{{g}^{-\,1}})\]
A particle moves in one dimension from rest under the influence of a force that varies with the distance travelled by the particle as shown in the figure. The kinetic energy of the particle after it has travelled 3 m is :
Ship is sailing towards north-east with velocity \[\vec{v}=30\hat{i}+50\hat{j}\] where \[\hat{i}\] points east and \[\hat{j},\] north. Ship B distance of 80 km east and 150 km north of Ship A and is sailing towards west at 10 km/hr. will be at minimum distance from B in:
A boy ?s catapult is made of rubber cord which is 42 cm long, with 6 mm diameter of cross-section and The boy keeps a stone weighing 0.02 kg on it and stretches the cord by 20 cm by applying a constant force. When released, the stone flies off with a velocity of \[20\,m{{s}^{-1}}.\] Neglect the change in the area of cross-section of the cord while stretched. The Young's modulus of rubber is closest to:
A thin strip 10 cm long is on a U shaped wire of negligible resistance and it is connected to a spring of spring constant \[0.5N{{m}^{-1}}.\](see figure). The assembly is kept in a uniform magnetic field of 0.1 T. If the strip is pulled from its equilibrium position and released, the number of oscillations it performs before its amplitude decreases by a factor of e is N. If the mass of the strip is 50 grams, its resistance \[10\,\Omega \] and air drag negligible, N will be close to:
If \[{{10}^{22}}\]gas molecules each of mass \[{{10}^{-\,\,26}}\] collide with a surface (perpendicular to it) elastically per second over an area \[1\,{{m}^{2}}\]with a speed \[{{10}^{4}}\,\text{m/s,}\]the pressure exerted by the gas molecules will be of the order of:
Voltage rating of a parallel plate capacitor is 500 V. Its dielectric can withstand a maximum electric field of \[{{10}^{6}}\,\text{V/m}.\]The plate area is \[{{10}^{-4}}{{m}^{2}}.\] What is the dielectric constant if the capacitance is 15 pF? \[\text{given}\,{{\in }_{0}}=8.86\times {{10}^{-12}}{{\text{C}}^{\text{2}}}\text{/N}{{\text{m}}^{\text{2}}}\]
A compound with molecular formula \[{{C}_{4}}{{H}_{10}}{{O}_{3}}\] is converted by the action of acetyl chloride to a compound of molecular mass 190 the original compound \[({{C}_{4}}{{H}_{10}}{{O}_{3}})\] has -
The 'brown ring' formed at the junction of two layers in the test of nitrate is due to the formation of a complex ion, \[{{[Fe{{({{H}_{2}}O)}_{5}}NO]}^{2}}.\] Which of the following statements are correct for this complex \[[\mu =3.87B.M.]-\]
(I) Oxidation state of Fe is +1 and NO exists as \[N{{O}^{+}}\]
(II) The complex ion is in octahedral geometry as attained by \[s{{p}^{3}}{{d}^{2}}\]hybridisation
(III) The complex is paramagnetic and has three unpaired electrons due to transfer of electron from NO to \[F{{e}^{2+}}\]
(IV) The complex is octahedral geometry as attained by \[{{d}^{2}}s{{p}^{3}}\] hybridisation
(V) The brown colour of the complex is attributed to d-d-transition of electron
Strips of plant tissue were immersed in a range of sucrose solutions of different concentrations. Their lengths were measured before immersion and after 30 minutes in the different solutions. The graph shows the ratio of initial length to final length.
Which concentration of sucrose solution has the same water potential as the cell sap?
A plant is heterozygous for a pair of alleles that are codominant. This plant is self-pollinated and the resulting seeds are germinated and allowed to grow. Which ratios are expected in the offspring?
In fruitflies, a sex-linked gene controls the development of eye colour. The eyes are either red or white. The male is the heterogametic sex. What will be the expected percentage of eye colours in the progeny when a heterozygous red-eyed female is crossed with a white-eyed male?
The mRNA triplet UGA acts as a stop codon to terminate the synthesis of a polypeptide. The diagram shows a strand of DNA coding for 4 amino acids. Where would a mutation, involving the insertion of a thymine nucleotide, result in the termination of translation?
Albinism in humans is controlled by a recessive allele. How many copies of this allele will be found at one of the poles of a cell at telophase-I of meiosis in an albino person?