If A (n) represents the area bounded by the curve \[y=n\]\[\ell nx,\] where \[n\in N\] and \[n>1,\] the x-axis and the lines \[x=1\] and \[x=e,\] then the value of \[A\,(n)+n\,A\,(n-1)\] is equal to:
If \[P\,({{x}_{1}},\,\,{{y}_{1}}),\]\[Q\,({{x}_{2}},\,\,{{y}_{2}}),\]\[R\,({{x}_{3}},\,\,{{y}_{3}})\] and \[S\,({{x}_{4}},\,\,{{y}_{4}})\] are four cyclic points on a rectangular hyperbola \[xy={{c}^{2}}\] the coordinate of the orthocenter of the \[\Delta \,PQR\] are
If \[\underset{x\to \,0}{\mathop{lim}}\,\frac{{{729}^{x}}-{{243}^{x}}-{{81}^{x}}+{{9}^{x}}+{{3}^{x}}-1}{{{x}^{3}}}=K\,{{(\ell n\,3)}^{3}}\] then k is equal to
If \[{{\vec{e}}_{1}}\] & \[{{\vec{e}}_{2}}\] are two unit vectors and \[\theta \] is the angle between them, then \[\sin \left( \frac{\theta }{2} \right)\] is:
A fixed container is fitted with a piston which is attached to a spring of spring constant k. The other end of the spring is attached to a rigid wall. Initially the spring is in its natural length and the length of container between the piston and its side wall is L. Now an ideal diatomic gas is slowly filled in the container so that the piston moves quasistatically. If pushed the piston by x so that the spring now is compressed by x. The total rotational kinetic energy of the gas molecules in terms of the displacement x of the piston is (there is vacuum outside the container)
A battery of internal resistance \[2\Omega \] is connected to a variable resistor whose value can vary from \[4\Omega \] to \[10\Omega \,.\] The resistance is initially set at\[4\Omega \,\,.\] If the resistance is now increased then -
Two identical spheres of same mass and specific gravity (which is the ratio of density of a substance and density of water) 2.4 have different charges of Q and \[-\,3Q.\] They are suspended from two strings of same length \[\ell \] fixed to points at the same horizontal level, hut distant \[\ell \] from each other. When the entire set up is transferred inside a liquid of specific gravity 0.8. It is observed that the inclination of each string in equilibrium remains unchanged. Then the dielectric constant of the liquid is -
The element which has a \[{{k}_{\alpha }}\] x-rays line of wavelength 1.8 \[\overset{\text{o}}{\mathop{\text{A}}}\,\] is - \[(R=1.1\times {{10}^{7}}{{m}^{-1}},b=1\,and\,\sqrt{5/33}=0.39)\]
\[{{M}_{1}}\text{ }and\text{ }{{M}_{2}}\] are plane mirrors and kept parallel to each other. At point O there will be maxima for wavelength. Light from monochromatic source S of wavelength \[\lambda \] is not reaching directly on the screen. The \[\lambda \] is \[-\,[D>>d,d>>\lambda ]\]
A parallel plate capacitor without any dielectric has capacitance\[{{C}_{0}}.\] A dielectric slab is made up of two dielectric slabs of dielectric constants K and 2K and is of same dimensions as that of capacitor plates and both the parts are of equal dimensions arranged serially as shown. If this dielectric slab is introduced (dielectric K enters first) in between the plates at constant speed, then variation of capacitance with time will be best represented by
A uniform rod of length L is charged uniformly with charge q and is rotating about an axis passing through its centre and perpendicular to rod. Magnetic moment of the rod is -
A object is moving with velocity v (w.r.t. earth) parallel to plane mirror\[{{M}_{2}}\]. Another plane mirror \[{{M}_{1}}\] makes an angle \[\beta \] with the vertical as shown in figure. Then velocity of image in mirror \[{{M}_{2}}\] w.r.t. the image in \[{{M}_{1}}\] is -
The graph between photo electric current and cathode potential when the anode is kept at zero potential, for light of two different intensities out of the same frequency looks like the one -
There are three concentric thin spheres of radius \[a,\text{ }b,\text{ }c\,\text{(}a>b>c).\] The total surface charge densities on their surfaces are \[\sigma ,\]\[-\sigma \] and a respectively. The magnitude of electric field at r (distance from center) such that \[a>r>b\]is -
The direction of field B at a point P symmetric to P with respect to the vertex, i.e., along the axis and the same distance d, but inside the V is along
A thin prism of glass is placed in air and water successively. If \[_{a}{{\mu }_{g}}=3/2\] and \[_{a}{{\mu }_{g}}=4/3,\]then the ratio of deviation produced by the prism for a small angle of incidence when placed in air and water is -
A wire of fixed length is wound in such a way that it forms a solenoid of length \['\ell '\]and radius 'r'. Its self-inductance is found to be L. Now if same wire is wound in such a way that it forms a solenoid of length \[\frac{\ell }{2}\] and radius \[\frac{r}{2}\] then the self-inductance will be -
Two identical samples (same material and same amount) P and Q of a radioactive substances having mean life T are observed to have activities \[{{A}_{p}}\]& \[{{A}_{Q}}\]respectively at the time of observation. If P is older than Q, then the difference in their ages is -
Consider the two hypothetical reactions given below:
I. \[aA\to products,k=xmo{{l}^{-1}}L{{\min }^{-1}}\]
II. \[bB\to products,k=y{{\min }^{-1}}\]
The half-lives of both the reactions are the same, equal to 1 hr when molar concentration of the reactant is 1.0 M in each case. If these reactions are started at the same time taking 1 M of the reactant in each case, the ratio \[\left[ A \right]/\left[ B \right]\] after 3hr will be:
\[28g\,{{N}^{2}}\] and \[6.0\]g of \[{{H}_{2}}\] are heated over catalyst in a closed one litre flask of \[450{}^\circ \]C. the entire equilibrium mixture required 500 ml of 1.0 \[M\,{{H}_{2}}S{{O}_{4}}\] for neutralization. The value of \[{{K}_{c}}\] for the reaction \[{{N}_{2}}(g)+3{{H}_{2}}(g)2N{{H}_{3}}(g)\,is\]
1g of charcoal adsorbs \[100\text{ }mL\text{ }0.5\text{ }M\]\[\,C{{H}_{3}}COOH\] to form a monolayer, and thereby the molarity of \[C{{H}_{3}}COOH\] reduces to 0.49. Calculate the surface area of the charcoal covered by each molecule of acetic acid. Surface area of charcoal \[=3.10\times {{10}^{2}}{{m}^{2}}/g.\]
When \[KMn{{O}_{{{4}^{{}}}}}\] acts as an oxidising agents and ultimately forms \[Mn{{O}_{{{4}^{2-}}}},M{{n}_{2}}{{O}_{3}},M{{n}^{2+}}\] then the number of electrons transferred in each case respectively is
An organic compound whose empirical and molecular formula are same, contains \[~20%\]% carbon, \[6.7%\]% hydrogen, \[46.7%\]% nitrogen and the rest oxygen. On heating it yields ammonia, leaving a solid residue. The solid residue. a violet colour with dilute solution of alkaline copper sulphate. The organic compound is
An electron in the ground state of hydrogen was excited to a higher energy level using monochromatic radiations of wave length \[(\lambda )975\]\[\overset{{}^\circ }{\mathop{\text{A}}}\,\]. The longest wave length that appears in the resulting spectrum is due to transition from:
\[0.02\] mole of \[\left[ Co{{(N{{H}_{3}})}_{5}}Br \right]C{{l}_{2}}\] and \[0.02\] mole of \[\left[ Co{{(N{{H}_{3}})}_{5}}Cl \right]S{{O}_{4}}\] are present in 200cc of a solution X. the number of moles of the precipitated Y and Z that are formed when the solution X is treated with excess silver nitrate and excess barium chloride respectively are
In the following sequence of reactions, \[\begin{align} & C{{H}_{3}}-\underset{\underset{N{{H}_{2}}}{\mathop{|}}\,}{\mathop{CH}}\,-C{{H}_{3}}\xrightarrow{HN{{O}_{2}}}A\xrightarrow{oxidation} \\ & B\xrightarrow[ii){{H}_{3}}{{O}^{+}}]{i)C{{H}_{3}}MgI}C \\ & \\ \end{align}\] The compound C formed will be
In the given conformation, if \[{{C}_{2}}\] is rotated about \[{{C}_{2}}-{{C}_{3}}\] bond anticlockwise by an angle of \[120{}^\circ \] then the conformation obtained is
Work done for the conversion of 0.5 mole of water at \[100{}^\circ \]C to steam at 1 atm pressure is (heat of vaporision of water\[at\,100{}^\circ C\text{ }is\text{ }40670\text{ }J\text{ }mo{{l}^{-1}}~~\])
The nuclei of elements X, Y and Z have same number of protons, but different numbers of neutrons. According to mendeleef periodic table, the elements X,Y and Z
A)
Belong to same group and same period
doneclear
B)
Belong to different group and different periods
doneclear
C)
Belong to same group and different periods
doneclear
D)
Are isotopes, which do not have different positions
On being placed in water, sodium peroxide not only produces an alkaline solution but also some bubbles. If we assume that the peroxide ion picks up two protons from water to produce a compound that can be seen as the disproportion. Using the above information complete the following equation. \[\begin{align} & N{{a}_{2}}{{O}_{2}}(s)\,+{{H}_{2}}O(1)\xrightarrow{{}}(A)+(B) \\ & (A)\,and\,(B)\,are \\ \end{align}\]
For AB bond if percent ionic character is plotted against electronegativity difference \[({{X}_{A}}-{{X}_{B}}),\]the shape of the curve would look like The correct curve is
Certain nerve gases developed for military purposes work by producing convulsive muscular contractions upon the slightest stimulation. This suggests that their function is to inhibit the action of
Two cells A and B are contiguous. Cell A has osmotic pressure 10 atm, turgor pressure 7 atm and diffusion pressure deficit 3 atm. Cell B has osmotic pressure 8 atm, turgor pressure 3 atm and diffusion deficit 5 atm. The result will be
A mutation in one of the replication enzyme causes DNA polymerase to be unable to add nucleotides at the origin of replication, hence no daughter strands of DNA can be synthesised. Which of the following is the defective enzyme?
The range of \['\alpha '\] for which the point, \[(\alpha ,\,\,\alpha )\] lies inside the region bounded by the curves \[y=\sqrt{1-{{x}^{2}}}\] and \[x+y=1\] is
The solution of the differential equation \[{{x}^{2}}\frac{dy}{dx}.\cos \,\,\left( \frac{1}{x} \right)-y\sin \,\,\left( \frac{1}{x} \right)=-1,\] where \[y\to -1\] as \[x\to \infty \]is.
A variable chord PQ of parabola \[{{y}^{2}}=4ax\] subtends a right angle at the vertex. Find the locus of point of intersection of the tangents at P and Q.
A dumbbell consists of two masses connected by a rigid rod of negligible mass and length d. A physics student takes the dumbbell and rotates it about its center of mass with an angular velocity\[\omega \], giving it an angular momentum\[{{L}_{1}}\]. The student then takes a second dumbbell, with masses 2m and length 2d, and rotates them with the same angular velocity\[\omega \]. What is the angular momentum \[{{L}_{2}}\] of this second dumbbell?
In Young's double slit experiment, the introduction of a thin transparent film reduces the intensity at centre of screen by 75%. Then (\[\mu \]= refractive index of film, t = thickness of film and \[\lambda \] = wavelength of light used)
A certain mass of gas undergoes a process given by \[dU=\frac{dW}{2}.\] If the molar heat capacity of the gas for this process is \[\frac{15}{2}R,\] then the gas is:
There is rectangular wire frame having a thin film of soap solution. A massless thin wire of radius R and area of cross section A is placed on the surface of film and inside portion of the film is pricked. If surface tension of soap solution is S and Young's modulus of wire is Y then change in radius of the wire is:
Two coils have self-inductances \[{{L}_{1}}=8\,\,mH\] and \[{{L}_{2}}=2\,\,mH.\] In both of them currents are increased at the same constant rate. At a certain instant the power given to the two coils is the same. If at that instant, \[{{i}_{1}},\] \[{{V}_{1}},\] \[{{U}_{1}}\] and \[{{i}_{2}},\] \[{{V}_{2}},\] \[{{U}_{2}}\] be the currents, induced voltages and energies stored in the two coils respectively, then-
A thin semi-circular conducting ring of radius -R is falling its plane vertical in a horizontal magnetic induction \[\overrightarrow{B}.\] At the position MNQ the speed of the ring is y and potential difference developed across the ring is -
A)
Zero
doneclear
B)
\[B\text{ v }\pi {{R}^{2}}/2\] and M is at higher potential
A piece of granite floats at the interface of mercury and water contained in a beaker (Fig.). If the densities of granite, water and mercury are \[\rho ,\]\[{{\rho }_{1}}\]and \[{{\rho }_{2}}\] respectively, the ratio of the volume of granite in water to the volume in mercury is -
A simple pendulum is constructed by attaching a mass m to a thin rod of length \[\ell \]. The pendulum is pulled back to some angle \[\theta >30{}^\circ \] from the vertical and released. Which of the following techniques could be used to change the frequency f of this pendulum?
I. Changing the mass m on the end of the pendulum
II. Changing the length \[\ell \] for the pendulum
III. Changing the angle \[\theta \] from which the pendulum is released
Two identical trucks each of mass M (excluding sacks of rice) move on national highways with speeds \[{{v}_{1}}\] and \[{{v}_{2}}\] towards each other. When they meet each other, a sack of rice of mass \[m\]is thrown from one track to the other and an identical sack of rice is thrown from the second to the first. Calculate their velocities \[{{v}_{1}}'\] and \[{{v}_{2}}'\] after the exchange of sacks, given m = 50 kg, M=200kg, \[\left| \overrightarrow{\,{{v}_{1}}}\, \right|=50m/s\] and \[\left| \overrightarrow{\,{{v}_{2}}}\, \right|=200m/s\] Friction of the road may be neglected-
A particle moves in the x-y plane with velocity \[\vec{v}=a\hat{i}+bx\hat{j},\]where a and b are constants. Initially, the particle was at the origin. The trajectory of the particle is -
The octahedral complex of a metal ion \[{{M}^{3+}}\]with four monodentate ligands \[{{L}_{1}},{{L}_{2}},{{L}_{3}}\]and \[{{L}_{4}}\]absorb wavelength in the region of red, green, yellow and blue, respectively. The increasing order of ligand strength of the four ligands is:
Consider a class room of dimension \[5\times 10\times 3\,{{m}^{3}}\] at temperature \[20{}^\circ C\] and pressure 1 atm. There are 50 peoples in the room, each losing energy at the average of 150 e=watt. Assuming that the walls, ceiling, floor and furniture perfectly insulated and none of them absorbing heat: the time needed for rising the temperature of air in the room to body temperature,\[i.e.,37{}^\circ C\] will be (for air Cp=7/2 R. loss of air to the outside as the temperature rises may be neglected)
\[AgNO{{~}_{3}}\](aq.) was added to an aqueous KCl solution gradually and the conductivity of the solution was measured. The plot of conductance \[\left( \wedge \right)\]versus the volume of \[AgNO{{~}_{3}}\] is
Uncertainty in position of an electron \[(mass\text{ }=9.1\times {{10}^{-28}}g)\] moving with a velocity \[3\times {{10}^{4}}\,cm/s,\] accurate up to \[0.001%,\] will be: \[(h\text{ }=\text{ }6.626\times \text{ 1}{{\text{0}}^{-27}}\])
An organic compound A upon reacting with \[~N{{H}_{3}}\] gives B. On heating B gives C. C in presence of KOH reacts with \[B{{r}_{2}}\] to give \[C{{H}_{3}}C{{H}_{2}}N{{H}_{2}}.\] A is:
The thickness of a piece of paper is \[0.0036\] inch Suppose a certain book has an Avogadro's number of pages calculate the thickness of the book in light-years. (1 light-year equal to \[~5.88\text{ }\times \text{1}{{\text{0}}^{12}}\]miles)
X mL of \[{{H}_{2}}\] gas effuses through a hole in a container in 5 seconds. The time taken for the effusion of the same volume of the gas specified below under identical conditions is
A man has normal red-green colour vision. His blood group is rhesus negative (homozygous recessive). His wife also has normal colour vision but is rhesus positive. She is heterozygous at both the red-green colour vision locus and the blood group locus. What is the probability that their first child will be rhesus negative, red green colourblind boy?
A biologist studied the population of rats in a barn. He found that the average natality was 250, average mortality 240, immigration 20 and emigration 30. The net increase in population is
The gene map of a plasmid, which has been used for genetic engineering is shown below. It contains two antibiotic resistance genes. The positions of several restriction endonuclease binding sites are also shown below
Which enzyme(s) would be the most suitable for the cleavage of the plasmid and the DNA containing the gene of interest?
If there were 34 amino acids and DNA only contained two types of nitrogenous bases, what would be the minimum number of bases per codon that could code for proteins?
Each graph shows the rate of reaction of an uninhibited enzyme and that of the same enzyme in the presence of a constant amount of either a competitive or a non-competitive inhibitor. Which graph is correctly labelled?