The number of values of 'r' satisfying the equation- \[^{39}{{C}_{3r}}-1-{{\,}^{39}}{{C}_{{{r}^{2}}}}={{\,}^{39}}{{C}_{{{r}^{2}}-1}}-{{\,}^{39}}{{C}_{3r}}\] is
The function\[f(x)=\left\{ \begin{matrix} {{x}^{2}}/a & 0\le x<1 \\ a & 1\le x<\sqrt{2} \\ (2{{b}^{2}}-4b){{x}^{2}} & \sqrt{2}\le x<\infty \\ \end{matrix} \right.\]is continuous for \[0\le x<\infty ,\] then the most suitable values of a and b are -
Given \[A=\left| \begin{matrix} a & b & 2c \\ d & e & 2f \\ \ell & m & 2n \\ \end{matrix} \right|,\] \[B=\left| \begin{matrix} f & 2d & e \\ 2n & 4\ell & 2m \\ c & 2a & b \\ \end{matrix} \right|,\] then-
A point P lies on the hyperbola \[9{{x}^{2}}-16{{y}^{2}}=144\] such that \[P{{S}_{1}}\text{ }:\text{ }P{{S}_{2}}\text{ }=\text{ }3/2\] (where \[{{S}_{1}}\] and \[{{S}_{2}}\] are foci of hyperbola). Coordinates of point P is in the first quadrant are-
An ellipse is inscribed in a circle and a point within the circle is chosen at random. If the probability that this point lies outside the ellipse is 2/3 then the eccentricity of the ellipse is -
If \[\vec{a}\text{ }=\hat{i}+\hat{j}+\hat{k}\] & \[\text{\vec{b} }=\hat{i}+-2\hat{j}+\hat{k},\] then the vector \[\vec{c}\] such that \[\vec{a}\,\,.\,\,\vec{c}=2\] & \[\vec{a}\,\,\times \,\,\vec{c}=\vec{b}\] is -
In an electron microscope, the resolution that can be achieved is of the order of the wavelength of electrons used. To resolve a width of \[7.5\times {{10}^{-12}}m,\] the minimum electron energy required is close to:
To mop-clean a floor, a cleaning machine presses a circular mop of radius R vertically down with a total force F and rotates it with a constant angular speed about its axis. If the force F is distributed uniformly over the mop and the floor is\[\mu ,\]the torque, applied the machine on the mop is:
A potentiometer wire AB having length L resistance 12 r is joined to a cell D of emf \[\varepsilon \]and internal resistance r. A cell C having emf \[\varepsilon \text{/2}\]and internal resistance 3r is connected. The length AJ galvanometer as shown in figure shows no deflection is:
A TV tower has a height of 140 m and of the receiving antenna is 40 m. What is maximurm distance upto which signals can be from this tower in LOS (Line of Sight) mode? (Given : radius of earth \[=6.4\times {{10}^{6}}m).\]
A uniform metallic wire has a resistance of \[18\,\Omega \]and is bent into an equilateral triangle. Then, the resistance between any two vertices of the triangle is:
In a Young's double slit experiment with slit separation 0.1 mm, one observes a bright fringe at angle \[\frac{1}{40}\,\,rad\] by using light of wavelength \[{{\lambda }_{1}}.\] When the light of wavelength \[{{\lambda }_{2}}\]is used a bright fringe is seen at the same angle in the same set up. Given that \[{{\lambda }_{1}}\] and\[{{\lambda }_{2}}\] are in visible range (380 nm to 740 nm), their values are:
Two guns A and B can fire bullets at speed 1 km/s and 2 km/s respectively. From a point on a horizontal ground, they are fired in all possible directions. The ratio of maximum areas covered by the bullets fired by the two guns, on the ground is:
The density of a material in SI units is \[128\,kg\,{{m}^{-3}}.\] In certain units in which the unit of length is 25 cm and the unit of mass 50 g, the numerical value of density of the material is:
A magnet of total magnetic moment \[{{10}^{-2}}\hat{i}\,A-{{m}^{2}}\] is placed in a time varying magnetic field, where B = 1 Tesia and \[\omega =0.125\] rad/s. The work done for reversing the direction of the magnetic moment at \[t=1\]second, is:
A heat source at \[T={{10}^{3}}\] K is connected to another heat reservoir at \[T={{10}^{2}}\] K by a copper slab which is 1 m thick. Given that the thermal conductivity of copper is \[0.1\,W{{K}^{-1}}\,{{m}^{-1}},\] the energy flux through it in the steady state is:
A parallel plate capacitor is of area \[6\,\,C{{m}^{2}}\] and a separation 3 mm. The gap is filled with three dielectric materials of equal thickness (see figure) with dielectric constants \[{{K}_{1}}=10,\] \[{{K}_{2}}=12\] and \[{{K}_{3}}=14.\]The dielectric constant of a material which when fully inserted in above capacitor, gives same capacitance would be:
'A charge Q is distributed over three concentric spherical shell of radii a, b, c (a < b < c) such that their surface charge densities are equal to one another. The total potential at a point at distance r from their common centre, where r < a, would be:
Three Carnot engines operate in series between a heat source at a temperature \[{{T}_{1}}\] and a heat sink at temperature \[{{T}_{4}}\] (see figure). There are two other reservoirs at temperature \[{{T}_{2}}\] and \[{{T}_{3}},\] as \[{{T}_{1}}>{{T}_{2}}>{{T}_{3}}>{{T}_{4}}.\] shown, with The three engines are equally efficient if:
A satellite is moving with a constant speed u in circular orbit around the earth. An object of mass \['m'\] is ejected from the satellite such that it just escapes from the gravitational pull of the earth. At the time of the earth. At the time of ejection. the kinetic energy of the object is:
Water flows into a large tank with flat bottom at the rate of \[{{10}^{-4}}{{m}^{3}}{{s}^{-1}}.\] Water is also leaking out of area 1 \[C{{m}^{2}}\] at its bottom. If the height of the water in the tank remains steady, then this height is:
A string of length 1 m and mass 5 g is fixed at both ends. The tension in the string is 8.0 N. The string is set into vibration using an external vibrator of frequency 100 Hz. The separation between successive nodes on the string is close to:
A train moves towards a stationary observers with 34 m/s. The train sounds a whistle and its registered by the observer is \[{{f}_{1}}.\]If the speed \[{{f}_{2}}.\] train is reduced to 17 m/s, the frequency registered is \[{{f}_{2}}.\]If speed of sound is 340 m/s, then the ratio \[{{f}_{\text{1}}}\text{/}{{f}_{2}}\] is:
A piano convex lens of refractive index \[{{\mu }_{1}}\] and length \[{{f}_{1.}}\] is kept in contact with another plano concave lens of refractive index \[{{\mu }_{2}}\] and focal length \[{{f}_{2}}.\] If the radius of curvature of their spherical faces is R each and \[{{f}_{1}}=2{{f}_{2}},\] then \[{{\mu }_{1}}\] and \[{{\mu }_{2}}\] are related as:
Conductivity of a saturated solution of a sparingly soluble salt AB at 298 K is \[1.85\times {{10}^{-6}}S{{m}^{-1}}\]. Solubility product of the salt AB at 298 K is \[\left[ gievn,\Delta _{m}^{o}\left( AB \right)=140\times {{10}^{-4}}S{{m}^{-2}}mo{{l}^{-1}} \right]\]
Vapour pressure of pure benzene is 119 torr and that of toluene is \[37.0\] torr at the same temperature. Mole fraction of toluene in vapour phase which is in equilibrium with a solution of benzene and toluene having a mole fraction of toluene \[0.50,\]will be
\[{{C}_{3}}{{H}_{6}}C{{l}_{2}}\] on reaction with \[NaOH\] forms \[{{C}_{3}}{{H}_{6}}O\] which gives yellow precipitate on heating with \[NaOH\] and \[{{I}_{2}}\]. Thus, \[{{C}_{3}}{{H}_{6}}C{{l}_{2}}\] is
If you needed to take a blood sample during the time in a woman's menstrual cycle when the concentration of her gonadotropic hormones would be at their lowest levels, which of the following days on average would be the best choices for sampling?
During the early stages of alcoholic fermentation, there is a high rate of growth of yeast. After some time, the rate decreases. Which of the following conditions in the culture medium is least likely to have caused this?
The curve given below shows enzymatic activity with relation to three conditions (pH, temperature and substrate concentration). What do the two axis (x and y) represent?
For a substance passing through a membrane by simple diffusion, select the curve in the following figure that best shows the relationship between its concentration outside the membrane and its rate of movement through the membrane.
Suppose that in sheep, a dominant allele [B] produces black hair and a recessive allele [b] produces white hair. If you saw a black sheep, you would be able to identify
If \[f(x)={{\left( [\{x\}]{{\tan }^{-1}}\left( \frac{{{x}^{2}}-3x-1}{{{x}^{2}}-3x+5} \right)+3-{{x}^{7}} \right)}^{1/7}},\]where \[[k]\] and \[\{k\}\]denotes greatest integer and fractional part functions of k respectively, then the value of \[{{f}^{-1}}(50)+f\,(f\,(100))-f\,(50),\]
The sum to n terms of the series, \[1+\left( 1+\frac{1}{2}+\frac{1}{{{2}^{2}}} \right)+\left( 1+\frac{1}{2}+\frac{1}{{{2}^{2}}}+\frac{1}{{{2}^{3}}}+\frac{1}{{{2}^{4}}} \right)+......\] is-
If \[cos\text{ }\alpha ,\] \[cos\text{ }\beta \] and \[cos\text{ }\gamma \] are the roots of the equation \[9{{x}^{3}}-9{{x}^{2}}-x+1=0,\] \[\alpha ,\beta ,\gamma ,\]\[\in [0,\pi ],\] then radius of the circle whose centre is \[\left( \sum{a},\sum{\cos \alpha } \right)\] and passing through \[\left( 2{{\sin }^{-1}}\left( \tan \frac{\pi }{4} \right),4 \right)\] is-
Two circles \[{{C}_{1}}\] and \[{{C}_{2}}\] intersect at two distinct points P & Q in a plane. Let a line passing through P meets circle \[{{C}_{1}}\] and \[{{C}_{2}}\] in A and B respectively. Let Y is mid-point of AB and QY meets circle \[{{C}_{1}}\] and \[{{C}_{2}}\] in X and Z respectively, then-
The most general values of x for which \[\sqrt{3}\sin x-\cos x=\underset{\lambda \varepsilon R}{\mathop{\min }}\,\{2,{{e}^{2}},\pi ,{{\lambda }^{2}}-4\lambda +7\}\]
Given that the vectors \[\vec{a}\] and \[\overrightarrow{b}\] are non collinear, the values of x and y for which the vector equality \[2\,\,\vec{u}-\vec{v}=\vec{w}\] holds true if \[\vec{u}-x\,\vec{a}=2y\vec{b},\]\[\vec{v}=-\,2y\vec{a}+3x\vec{b},\] \[\vec{w}=4\vec{a}-2\vec{b}\]are-
If the magnetic field of a plane electromagnetic wave is given by (The speed of light \[=3\times {{10}^{8}}m\text{/s}\]\[B=100\times {{10}^{-6}}\sin \left[ 2\pi \times 2\times {{10}^{15}}\left( t-\frac{x}{c} \right) \right]\]then the maximum electric field associated with it is:
Using a nuclear counter the count rate of en particles from a radioactive source is measured. At \[t=0\] it was 1600 counts per second and \[t=8\] seconds it was 100 counts per second. The observed, as counts per second, at. \[t=6\] second is close to:
A solid metal cube of edge length 2 cm is moving in positive direction at a constant speed There is a uniform magnetic field of positive z-direction. The potential difference the two faces of the cube perpendicular to is:
Two electric dipoles A, B with respective dipole moments \[\overset{\xrightarrow{{}}}{\mathop{{{d}_{A}}}}\,=-4qa\hat{i}\]and \[\overset{\xrightarrow{{}}}{\mathop{{{d}_{B}}}}\,=2qa\hat{i}\] placed on the x-axis with a separation R, as shown in the figure The distance from A at which both of them produce is:
A piece of wood of mass 0.03 kg is dropped from the top of a 100 m height building. At the same time, a bullet of mass 0.02 kg is fired vertically upward, with a velocity 100 \[m{{s}^{-1}},\] from the ground. The bullet gets embedded in the wood. Then the maximum height to which the combined system reaches above the top of the building before falling below is: \[(g=10\,m{{s}^{-2}})\]
An insulating thin rod of length i has a linear charge density \[\rho (x)={{\rho }_{0}}\frac{x}{\ell }\]on it. The rod is rotated about an axis passing through the origin \[(x=0)\]and perpendicular to the rod. If the rod makes n rotations per second, then the time averaged magnetic moment of the rod is:
A homogeneous solid cylindrical roller of radius R and mass M is pulled on a cricket pitch by a horizontal force. Assuming rolling without slipping, angular acceleration of the cylinder is:
The heat of atomisation of \[P{{H}_{3}}\left( g \right)\] and \[{{P}_{2}}{{H}_{4}}\left( g \right)\]are 953 and \[1485\,kJ\,mo{{l}^{-1}}\] respectively. The \[P-P\] bond energy in kJ \[mo{{l}^{-1}}\] is
An organic compound undergoes first order decomposition. The time taken for its decomposition to 1/8 and 1/10 of its initial concentration are \[{{t}_{1/8}}\] and \[{{t}_{1/10}}\] respectively. What is the value of \[\frac{\left[ {{t}_{1/8}} \right]}{\left[ {{t}_{1/10}} \right]}\times 10?\]\[({{\log }_{10}}2=0.3)\]
The given graph represents the variation of compressibility factor\[(Z)=\frac{pV}{nRT},\] for three real gases A, B and C. Identify the only incorrect statement.
A)
For the gas A, \[a=0\] and its dependence on p is linear at all pressure
doneclear
B)
For the gas B, \[b=0\] and its dependence on p is linear at all pressure
doneclear
C)
For the gas C, which is typical real gas for which neither a nor\[b=0\]. By knowing the minima and point of the intersection, with \[Z=1,\]a and b can be calculated
doneclear
D)
At high pressure, the slope is positive for all real gases
\[E{{{}^\circ }_{F{{e}^{3+}}/Fe}}=-0.336V,\]\[E{{{}^\circ }_{F{{e}^{2+}}/Fe}}=-0.439V\]The value of standard electrode potential for the charge, \[F{{e}^{3+}}\left( aq \right)+{{e}^{-}}\xrightarrow{{}}F{{e}^{2+}}\left( aq \right)\]will be
Among \[Ga\left( Z=64 \right)\], \[Lu\left( Z=71 \right)\],\[La\left( Z=103 \right)\]\[Ta\left( Z=73 \right)\] the elements having half-filled f-orbital is
The correct statement with respect to the complexes \[\left[ Ni{{\left( CO \right)}_{4}} \right]\] and\[{{\left[ Ni{{\left( CN \right)}_{4}} \right]}^{2-}}\] is
A)
nickel is in the same oxidation state in both
doneclear
B)
both are paramagnetic in nature
doneclear
C)
have square planar and tetrahedral geometry respectively
doneclear
D)
have tetrahedral and square planar geometry Respectively
Compound \[A,{{C}_{5}}{{H}_{10}}{{O}_{5}}\]gives a tetraacetate with \[A{{c}_{2}}O\] and oxidation of 'A' with \[B{{r}_{2}}/{{H}_{2}}O\] gives an acid, \[{{C}_{5}}{{H}_{10}}{{O}_{6}}\]. Reduction of 'A' with \[HI\]gives iso-pentane. What is the possible structure of A?
Consider a chemical reaction in which substrate A is enzymatically converted to product. The rate of change of substrate to product with increasing concentration of substrate is shown by broken line. The rate of reaction with increasing concentration of substrate A with a fixed amount of substance B is shown by unbroken line.
If the same reaction is carried out with fixed quantity of substrate and increasing concentration of B, the expected result will be:
Phenylketonuria (PKU) results due to absence of phenylalanine hydroxylase and Alkaptonuria (AKU) results due to the absence of homogentisic acid oxidase. The following pathway shows where these enzymes function.
If a person is homozygous for recessive alleles of both PKU and AKU, he will show symptoms of:
'A' is an inhibitor of chloroplast function. The Production of \[{{o}_{2}}\] and the synthesis of ATP are measured in illuminated chloroplasts before and after addition of' A' as shown below.
Which statement is correct?
A)
'A' inhibits the reduction of \[NAD{{P}^{+}}\]
doneclear
B)
'A' inhibits the proton gradient and the reduction of \[NAD{{P}^{+}}\]
doneclear
C)
'A' inhibits the proton gradient but not the reduction of \[NAD{{P}^{+}}\]
doneclear
D)
'A' inhibits neither the proton gradient nor the reduction of \[NAD{{P}^{+}}\]
Red hair is a recessive trait in humans, in a randomly mating population in Hardy-Weinberg equilibrium, approximately 9% of individuals are red-haired. What is the frequency of heterozygotes?
In an experiment, the aleurone layer of oat seeds is destroyed chemically. It is observed that such seeds fail to germinate. Which of the following treatments will be useful to trigger the germination?
A)
Soaking the seeds in water containing glucose for long time.
doneclear
B)
Soaking the seeds in low concentration of abscisic acid.
Consider a gene 25.5kb in length. The regulatory region is 500bp long. The number of exons and introns in a gene are 9 and 8 respectively with the mean size of each being 145bp and 2960 bp respectively. What will be the length of a polypeptide chain synthesised by this gene?
A sample of DNA and mRNA is purified to analyse the base composition of each DNA strand and of the mRNA. The data is shown in the table below. Which strand of DNA serves as a template for mRNA synthesis?