Three pieces of cakes of weights \[4\frac{1}{2}lbs,\,\,6\frac{3}{4}lbs\] and \[7\frac{1}{5}lbs\] respectively are to be divided into parts of equal weights. Further, each must be as heavy as possible. If one such part is served to each guest, then what is the maximum number of guests that could be entertained?
Number of solutions of the equation, \[[y+[y]]=2\text{ }cos\text{ }x\]is: \[(where\text{ }y=(1/3)\text{ }\!\![\!\!\text{ }sinx+[sinx+[sinx]]]\] and [ ] - greatest integer function)
If p and q are order and degree of differential equation \[{{y}^{2}}{{\left( \frac{{{d}^{2}}y}{d{{x}^{2}}} \right)}^{2}}+3x{{\left( \frac{dy}{dx} \right)}^{1/3}}+{{x}^{2}}{{y}^{2}}=\sin \,x\], then:
The sides of a rectangle are chosen at random, each less than 10 cm, all such lengths being equally likely. The chance that the diagonal of the rectangle is less than 10 cm is
In a triangle ABC, angle 8 < angle C and the values of B & C satisfy the equation \[2\tan x-k(1+{{\tan }^{2}}x)=0\] where \[(0<k<1)\]. Then the measure of angle A is:
The set of all values of the parameter 'a' for which the function; \[f(x)=8ax-a\text{ }\sin \text{ }6x-7x-\sin \text{ }5x\] increases & has no critical points for all \[x\in R\], is
The ratio of the roots of the equation \[a{{x}^{2}}+bx+c=0\] is same as the ratio of the roots of the equation \[A{{x}^{2}}+Bx+C=0\]. If \[{{D}_{1}}\] and \[{{D}_{2}}\] are the discriminants of \[a{{x}^{2}}+bx+c=0\] and \[A{{x}^{2}}+Bx+C=0\] respectively, then \[{{D}_{1}}:{{D}_{2}}=\]
If the tangents at the point P on the circle \[{{x}^{2}}+{{y}^{2}}+6x+6y=2\] meets the straight line \[5x-2y+6=0\] at a point Q on the y axis, then the length of PQ is
Two infinitely large charged planes having uniform surface charge density \[+\sigma \] and \[-\sigma \] are plane along x-y plane and \[yz\] plane respectively as shown in the figure. Then the nature of electric lines of forces in x-z plane is given by :
An \[\alpha \] particle is moving along a circle of radius R with a constant angular velocity \[\omega .\] Point A lies in the same plane at a distance 2R from the centre. Point A records magnetic field produced by \[\alpha \] particle. If the minimum time interval between two successive times at which A records zero magnetic field is \['t',\] the angular speed \[\omega ,\] in terms of t is :
Radius of a circular ring is changing with time and the coil is placed in uniform constant magnetic field perpendicular to its plane. The variation of \['r'\] with time \['t'\] is shown in the figure. Then induced e.m.f. \[\varepsilon \] with time will be best represented by.
Three identical capacitors are given a charge Q each and they are then allowed to discharge through resistance \[{{R}_{1}},\] \[{{R}_{2}}\] and \[{{R}_{3}}\] separately. Their charges, as a function of time are shown in the graph below. The smallest of the three resistances is:
AB is small object dipped in water at a depth of d. Its length is \[\ell .\] It is seen from air at near normal incidence. The length of the image is :
Two coherent light sources P and Q each of wavelength \[\lambda \] are separated by a distance \[3\lambda \] as shown. The maximum number of minima formed on line AB which runs from \[-\,\infty \] to \[\varphi \phi \] is:
A heavy nucleus having mass number 200 gets disintegrated into two small fragments of mass number 80 and 120. If binding energy per nucleon for parent atom is 6.5 MeV and for daughter nuclei is 7 MeV and 8 MeV respectively, then the energy released in the decay will be:
A rod of length \[\ell \] is in motion such that its ends A and B are moving along x-axis and y-axis respectively. It is given that \[\frac{d\theta }{dt}=2\] rad/s always. P is a fixed point on the rod. Let \[M\] be the projection of \[P\] on x-axis. For the time interval in which \[\theta \] changes from 0 to \[\frac{\pi }{2}\] choose the correct statement:
A)
The acceleration of \[M\] is always directed towards right
A small mass slides down an inclined plane of inclination \[\theta \] with the horizontal. The co-efficient of friction is \[\mu ={{\mu }_{0}}\,\,x\] where x is the distance through which the mass slides down and \[{{\mu }_{0}}\] a constant. Then the speed is maximum after the mass covers a distance of:
A source of frequency \['f'\] is stationary and an observer starts moving towards it at t = 0 with constant small acceleration. Then the variation of observed frequency f' registered by the observer with time is best represented as:
A tunnel is dug in the earth across one of its diameter. Two masses \['m'\] & \['2m'\] are dropped from the ends of the tunnel. The masses collide and stick to each other and perform S.H.M. Then amplitude of S.H.M. will be [R = radius of the earth]
A small uniform tube is bent into a circular tube of radius R and kept in the vertical plane. Equal volumes of two liquids of densities \[\rho \] and \[\sigma \,\,(\rho >\sigma )\] fill half of the tube as shown. \[\theta \] is the angle which the radius passing through the interface makes with the vertical:
All electrons ejected from a surface by incident light of wavelength 200 nm can be stopped before travelling 1 \[m\] in the direction of uniform electric field of 4 N/C. The work function of the surface is:
A point mass \['m'\] and charge \['q'\]is projected with a velocity v towards a stationary charge \[{{Q}_{0}}\] from a distance of \[2\text{ }m.\] The closest distance that q can approach is: \[\left[ k=\frac{1}{4\pi {{\varepsilon }_{0}}} \right]\]
The magnetic flux \[\phi \] through a metal ring varies with time t according to: \[\phi =3\,\,(a{{t}^{3}}-b{{t}^{2}})T{{m}^{2}},\] with \[a=2{{s}^{-3}}\] and \[b=6{{s}^{-2}}\]. The resistance of the ring is \[3\Omega .\]. The maximum current induced in the ring during the interval t = 0 to t = 2s, is:
A coin is released inside a lift at a height of 2 m from the floor of the lift. The height of the lift is 10 m. The lift is moving with an acceleration of \[9\text{ }m/{{s}^{2}}\] down wards. The time after which the coin will strike with the lift is: \[(g=10\text{ }m/{{s}^{2}}):\]
A solution of \[Ni{{(N{{O}_{3}})}_{2}}\] is electrolysed between platinum electrodes using 0.1 Faraday electricity. How many mole of Ni will be deposited at the cathode?
Molal depression constant for a solvent is 4.0 K kg \[mo{{l}^{-1}}.\] The depression in the freezing point of the solvent for\[0.03\text{ }mol\text{ }k{{g}^{-1}}\] solution of \[{{K}_{2}}OS{{O}_{4}}\] is: (Assume complete dissociation of the electrolyte)
In an acid-base titration, \[0.1\text{ }M\text{ }HCl\] solution was added to the\[NaOH\] solution of unknown strength. Which of the following correctly shows the change of pH of the titration mixture in this experiment?
10 mL of 1 mM surfactant solution forms a monolayer covering \[0.24\text{ }c{{m}^{2}}\] on a polar substrate. If the polar head is approximated as a cube, what is its edge length?
Emphysema damages the tissues of the lungs and slows pulmonary blood flow. This causes blood to back up, stretching and weakening the walls of the heart and blood vessels. Which of the following do you think would be most affected by this backup of blood from the lungs?
Centrifugation of a cell results in the rupture of the cell membrane and the contents compacting into a pellet in the bottom of the centrifuge tube. Bathing this pellet with a glucose solution yields metabolic activity including the production of ATP. One of the contents of this pellet is most likely which of the following?
Sita saw an old Tarzan movie on television. The movie supposedly took place in Africa, but Trina easily spotted that it was not really filmed there. Which of the following could have tipped her off?
A)
Chimps like cheetah do not live in Africa.
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B)
The monkeys in the jungle all had prehensile tails.
Patients with chronic lung disease and difficulty in breathing often adapt to the high concentration of\[{{\operatorname{CO}}_{2}}\], in their blood. The breathing centres stop responding to\[{{\operatorname{CO}}_{2}}\], level. If such a patient has difficulty in breathing, medical personnel are reluctant to give the patient pure oxygen. Based on what you know about control of breathing, why do you think this is the case?
A)
The patient's body would use the oxygen to make even more\[{{\operatorname{CO}}_{2}}\].
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B)
The oxygen would increase concentration of bicarbonate, altering pH.
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C)
Increased oxygen in the blood might slow or stop breathing.
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D)
The body is not used the oxygen, and the patient would overdose.
A zoologist examined an intestine cell from a crayfish and counted 200 chromosomes, each consisting of 2 chromatids, at prophase I of meiosis. What would he expect to see in each of the four cells at telophase II of meiosis if he looked in the crayfish ovary?
For approximately how long during the human female's menstrual cycle, progesterone concentrations are high enough to maintain the uterus in a proper condition for pregnancy?
Identification and arrangement of organisms on the basis of their cytological characteristics.
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B)
The classification of organisms based on broad morphological characters.
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C)
Delimiting various taxa of organisms and establishing their relationships.
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D)
The classification of organisms based on their evolutionary history and establishing their phylogeny on the totality of various parameters from all fields of studies.
If is a twice derivable function such that\[f'(2010)=f(2010)=\frac{1}{2010}\] and \[\int\limits_{0}^{2010}{f(x)dx=\frac{3}{2}}\], then \[\int\limits_{0}^{2010}{{{x}^{2}}}f''(x)dx=\]
Let f be a function defined from \[{{R}^{+}}\to {{R}^{+}}\]. 'If \[{{(f(xy))}^{2}}=x{{(f(y))}^{2}}\] for all positive numbers x and y and \[f(2)=6\], then \[f(50)\] is equal to
The vectors \[\overrightarrow{a}=-4\hat{i}+3\hat{k},\,\,\overrightarrow{b}=14\hat{i}+2\hat{j}-5\hat{k}\] are co-initial. The vector \[\overrightarrow{d}\] which is bisecting the angle between the vectors \[\overrightarrow{a}\] and \[\overrightarrow{b}\] and is having the magnitude \[\sqrt{6}\], is
(il) area bounded by\[y=f(x),\text{ }y={{x}^{4}}-4{{x}^{2}},\], y - axis and line \[x=t(0\,\,\le \,\,t\,\,\le \,\,2)\] is k times the area bounded by\[y=f(x),\,\,y=2{{x}^{2}}-{{x}^{3}}\], y - axis and line \[x=t(0\le t\le 2)\], is given as
If \[x=\frac{n\pi }{2}\], satisfies the equation \[\sin \frac{x}{2}-\cos \frac{x}{2}=1-\sin x\] and the inequality \[\left| \frac{x}{2}-\frac{\pi }{2} \right|\le \frac{3\pi }{4}\], then
Let \[{{d}_{1}}\text{, }{{d}_{2}}\,,.............\text{ }{{d}_{k}}\] be all the divisors of a positive Integer n including 1 and n. Suppose\[{{d}_{1}}+{{d}_{2}}+...+{{d}_{k}}=72\]. Then the value of \[\frac{1}{{{d}_{1}}}+\frac{1}{{{d}_{2}}}+......+\frac{1}{{{d}_{k}}}\] is
In the figure given below, the end B of the rod AB which makes angle \[\theta \] with the floor is pulled with a constant velocity \[{{v}_{0}}\] as shown. The length of rod is \[\ell .\] At an instant when \[\theta =37{}^\circ \]
A)
Velocity of end A is \[\frac{4{{v}_{0}}}{3}\]
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B)
Angular velocity of rod is \[\frac{5{{v}_{0}}}{6\ell }\]
A solid sphere of mass M and radius R is placed on a smooth horizontal surface. It is given a horizontal impulse J at a height h above the centre of mass and sphere starts rolling then, the value of h and speed of centre of mass are -
A)
\[h=\frac{2}{3}R\] and \[v=\frac{J}{M}\]
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B)
\[h=\frac{2}{5}R\] and \[v=\frac{2}{5}\frac{J}{M}\]
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C)
\[h=\frac{7}{5}R\] and \[v=\frac{7}{5}\frac{J}{M}\]
In the diagram shown, the charge +Q is fixed. Another charge +2q, is projected from a distance R from the fixed charge. Minimum separation between the two charges if the velocity becomes \[\frac{1}{\sqrt{3}}\] times of the projected velocity, at this moment is (Assume gravity to be absent) -
At t=0 the no of active nuclei in radioactive is number is \[{{N}_{0}}\] if decay constant is \[\lambda \] and rate of formation of isotope is K. Then number of active nuclei-
A container of dimension \[4m\,\,\times \,\,3m\,\,\times \,\,2m\] starts to move with uniform acceleration \[a=1.25\text{ }m/{{s}^{2}}\] at t = 0. The volume of liquid in vessel is \[18\text{ }{{m}^{3}}.\] The speed of liquid coming out from a very small orifice made at bottom of right side wall just after acceleration of container -
120 g of ice at \[0{}^\circ C\] is mixed with 100 g of water at \[80{}^\circ C.\] Latent heat of fusion is 80cal/g and specific heat of water is \[1\,\,cal/g{}^\circ C.\] The final temperature of the mixture is -
A stone of mass 1 kg is tied to a string 4 m long and is rotated at constant speed of \[40\text{ }m{{s}^{-1}}\] in a vertical circle. The ratio of the tension at the top and the bottom is-
A magnet is suspended in the magnetic meridian with an untwisted wire. The upper end of the wire is rotated through \[180{}^\circ \] to deflect the magnet by \[30{}^\circ \] from magnetic meridian. Now this magnet is replaced by another magnet and the upper end of the wire has to be rotated through \[270{}^\circ \] to deflect the magnet by \[30{}^\circ \] from magnetic meridian. The ratio of magnetic moments of the two magnet is -
During compression of a spring the work done is 10 kJ and 2 kJ escaped to the surroundings as heat. The change in internal energy, \[\Delta U\](in kJ) is:
At a given temperature T, gases Ne, Ar, Xe and Kr are found to deviate from ideal gas behaviour. Their equation of state is given as \[P=\frac{RT}{V-b}\] at T.
Which one of the following about an electron occupying the is orbital in a hydrogen atom is incorrect? (The Bohr radius is represented by \[{{a}_{0}}\]).
A)
The magnitude of the potential energy is double that of its kinetic energy on an average.
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B)
The probability density of finding the electron is maximum at the nucleus.
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C)
The total energy of the electron is maximum when it is at a distance\[{{a}_{0}}\]from the nucleus.
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D)
The electron can be found at a distance\[2{{a}_{0}}\]from the nucleus.
Brown \[({{y}^{+}})\] is dominant over yellow body colour (y) and red eyes \[({{w}^{+}})\] is dominant over white eyes (w). Both are carried on X chromosome. What is the genotype of a male Drosophila fly that has yellow body colour and red eyes?
A synthetically prepared mRNA contains repetitive AU sequences. The mRNA was incubated with mammalian cell extract which contains ribosomes, tRNAs and all the factors required for protein synthesis. Assuming no initiation codon is required for protein synthesis, which of the following peptides will most likely be synthesised?
A)
A single peptide composed of the same amino acid sequence.
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B)
A single peptide with alternating sequence of two amino acids.
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C)
A single peptide with alternating sequence of three amino acids.
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D)
Three different peptides, each sequence composed of a single amino acid.
The coding ratio in all known organisms is three, i.e., three nucleotides specify one amino acid. If DNA were to exclusively consist of only A-T base pairs, what would the minimum coding ratio be assuming that there are only 20 amino acids to be encoded?
A researcher was investigating the substrate specificity of two different enzymes, X and Y, on the same substrate. Both the enzymes were subjected to treatment with either heat or an inhibitor which inhibits the enzyme activity. Following are the results obtained where
a=inhibitor treatment, b=heat treatment and c=control.
Which of the following statements is correct?
A)
Only protein X is specific for the substrate, S.
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B)
Only protein Y is specific for the substrate, S.
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C)
Both X and Y are specific for the substrate, S.
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D)
Both X and Y are non-specific for the substrate, S.
In rabbits, two genes A and B are present on two different chromosomes. Products of both wild type A and B genes are essential for normal hearing. Homozygous recessive mutants either for A, B or both results in deafness. If a double heterozygous male (AaBb) is crossed with a double heterozygous female, the ratio of phenotypically normal and deaf rabbits will be:
The \[{{C}_{4}}\] carbon cycle is a \[C{{O}_{2}}\] concentrating mechanism evolved to reduce photorespiration. The followings are stated as important features of the \[{{C}_{4}}\] pathway:
(i) The leaves of \[{{C}_{4}}\] plants have Kranz anatomy that distinguishes mesophyll and bundle sheath cells.
(ii) In the peripheral mesophyll cells, atmospheric \[{{\operatorname{CO}}_{2}}\] is fixed by phosphoenol pyruvate carboxylase yielding a four-carbon acid.
(iii) In the inner layer of mesophyll, NAD-malic enzyme decarboxylates four-carbon acid and releases\[{{\operatorname{CO}}_{2}}\].
(iv) \[{{\operatorname{CO}}_{2}}\] is again re-fixed through Calvin cycle in the bundle sheath cells.
Which one of the following combinations is correct?
Choose the correct statement from the codes given below.
(i) Separation from extracellular medium allows the cells to maintain its chemical pool, orderliness of structure and reactions in contrast to disorderly distribution and randomly interacting molecules in the extracellular medium.
(ii) Cells are unable to recognise one another due to the presence of specific chemicals on their surface.
(iii) Cells of plant tissue are often connected with one another through cytoplasmic bridges called plasmodesmata.
(iv) Different cells of an organism communicate as well as exchange materials with one another.
A sample of enzyme lactase was isolated from the intestinal lining of a calf. Assays were undertaken to evaluate the activity of enzyme sample. The substrate of lactase is the disaccharide lactose. Lactase breaks a lactose molecule in two, producing a glucose molecule and a galactose molecule.
The results of two assays are:
Assay 1
Lactose Concentration (\[%\]w/v)
15
15
15
15
15
15
Concentration of enzyme sample (\[%\]v/v)
0
5
10
15
20
25
Rate of reaction (\[\mu \] mole glucose \[{{\sec }^{-1}}\]\[{{\operatorname{ml}}^{-1}}\])
0
25
50
75
100
125
Assay 2
lactone concentration (\[%\]w/v)
0
5
15
20
25
30
Concentration of enzyme sample (\[%\]v/v)
5
5
5
5
5
5
Rate of reaction (\[\mu \]mole glucose \[{{\sec }^{-1}}\]\[{{\operatorname{ml}}^{-1}}\])
0
15
25
35
40
40
Which of the following statements can be concluded from two assays?
A)
The reaction rate of lactase assay is always proportional to the amount of enzyme present.
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B)
The amount of lactose in an assay has no effect on the rate of reaction.
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C)
The reaction rate of the lactase assay is proportional to the amount of lactose present.
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D)
The reaction rate is proportional to the amount of enzyme present at a lactose concentration of \[15%\]w/v.