For \[{{x}^{2}}\ne n\pi 1,\]\[n\in N\] (the set natural numbers). The integral \[\int{x\sqrt{\frac{2\sin ({{x}^{2}}-1)-\sin 2({{x}^{2}}-1)}{2\sin ({{x}^{2}}-1)+\sin 2({{x}^{2}}-1)}}dx}\] is equal to; (where c is a constant of integration)
If \[y=y(x)\] is the solution of the differential equation, \[x\frac{dy}{dx}+2y={{x}^{2}}\], \[y(1)=1,\] then \[y\left( \frac{1}{2} \right)\] is equal to:
Consider a class of 5 girls and 7 boys. The number of different teams consisting of 2 girls and 3 boys that can be formed from this class, if there are two specific boys A and B, who refuse to be members of the same team, is:
Axis of a parabola lies along x-axis. If its vertex and focus are at distance 2 and 4 respectively from the origin, on the positive \[x-\]axis then which of the following points does not lie on it?
Let \[\vec{a}=\hat{i}-\hat{j},\]\[\vec{b}=\hat{i}+\hat{j}+\hat{k}\] and \[\vec{c}\] be a vector such that \[\vec{a}\times \vec{c}+\vec{b}=0\]and \[\vec{a},\vec{c}=4,\] then \[{{\left| {\vec{c}} \right|}^{2}}\] is equal
Let \[{{a}_{1}},\]\[{{a}_{2,}}\]...\[{{a}_{30}}\] be an A.P., \[S=\sum\limits_{i=1}^{30}{ai}\] and \[\sum\limits_{i=1}^{15}{{{a}_{{{(2i-1)}^{.}}}}}\] If \[{{a}_{5}}=27\] and \[\text{S}-2T=75,\] then \[{{a}_{10}}\] is equal to:
5 students of a class have an average height 150 cm and variance \[18\text{ }c{{m}^{2}}\]. A new student, whose height is 156 cm, joined them. The variance \[(\text{in}\,\,\text{c}{{\text{m}}^{2}})\] of the height of these six students is:
Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Let X denote the random variable of number of aces obtained in the two drawn cards. Then \[\text{P}(X=1)+\text{P}\,(X=2)\] equals:
For \[x\in R-\{0,1\},\] let \[{{f}_{1}}(x)=\frac{1}{x},\]\[{{f}_{2}}(x)=1-x\] and \[{{f}_{3}}(x)=\frac{1}{1-x}\] be three-given functions. If a function, \[J(x)\] is equal to:
Let \[\text{A}=\left\{ \theta \,\in \left( \frac{\pi }{2},\pi \right):\frac{3+2i\text{sin}\theta }{1-2i\text{sin}\theta } \right\}\] purely imaginary. Then the sum of the element in A is:
The particle of mass m and charge q will touch the infinitely large plate of uniform charge density \[\sigma \] if its velocity v is more than: \[\{\text{Give}\,\,\text{that }\sigma \,\,q\,\,>0\}\]
A negative charge is given to a loop and the loop is rotated in the plane of paper about its centre as shown, the magnetic field produced by the ring affects a small magnet placed above the ring in the same plane of paper.
A)
the magnet does not rotate
doneclear
B)
the magnet rotates clockwise as seen by observer from below
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C)
the magnet rotates anti-clockwise as seen form below
An inelastic ball of mass m has been thrown vertically upwards (positive z - direction) from the ground at \[z=0.\] Linear momentum of ball is \[{{P}_{z}}.\] The phase trajectory (graph between \[{{P}_{z}}\] versus z) of the ball after successive bouncing on the ground is:
In a cylindrical region uniform magnetic field which is perpendicular to the plane of the figure is increasing with time and a conducting rod PQ is placed in the region. Then
A)
P will be at higher potential than Q.
doneclear
B)
Q will be at higher potential than P.
doneclear
C)
Both P and Q will be equipotential.
doneclear
D)
no emf will be developed across rod as it is not crossing / putting any line of force.
In the plane mirror, the co-ordinates of image after two and half time periods are (initial velocity \[{{V}_{0}}\] is in the xy-plane and the plane mirror is perpendicular to the x-axis. A uniform magnetic field \[B\widehat{i}\] exists in the whole space. \[{{P}_{0}}\] is pitch of helix, \[{{R}_{0}}\] is radius of helix).
In a YDSE both slits produce equal intensities on the screen. A 100% transparent thin film is placed in front of one of the slits. Now the intensity of the geometrical centre of system on the screen becomes 75% of the previous intensity. The wavelength of the light is \[6000\,\overset{{}^\circ }{\mathop{A}}\,\] and \[{{\mu }_{film}}=1.5.\] The thickness of the film cannot be:
A beam of electrons striking a copper target produces X-rays. Its spectrum is as shown. Keeping the voltage same if the copper target relative is replaced with a different metal, the cut-off wavelength and characteristic lines of the new spectrum will change in comparison with old as:
A)
Cut-off wavelength will remain unchanged while characteristic lines will be different.
doneclear
B)
Both cut-off wavelength and characteristic lines will remain unchanged.
doneclear
C)
Both cut-off wavelength and characteristic lines will be different.
doneclear
D)
Cut-off wavelength will be different while characteristic lines will remain unchanged.
Two point charges \[+\,q\] and \[-\,4\,q\] are placed at \[(-\,a,\,\,0).\] and \[(+\,a,\,\,0).\] Take electric field intensity to be positive if it is along positive x-direction. The variation of the electric field intensity as one moves along the x-axis is
If at \[t=0\] the switch \[{{S}_{w}}\] is closed, then the charge on capacitor in the given circuit (initially uncharged) when the current through battery becomes 50% of its maximum value is (assume battery is ideal):
Two rods of same length and areas of cross section \[{{A}_{1}}\] and \[{{A}_{2}}\] have their ends at same temperature. If \[{{K}_{1}}\] and \[{{K}_{2}}\] are their thermal conductivities, \[{{C}_{1}}\] and \[{{C}_{2}}\] their specific heats and \[{{\rho }_{1}}\] and \[{{\rho }_{2}}\] are their densities, then the condition that rate of flow of heat is same in both the rods is
A chord attached about an end to a vibrating fork divides it into 6 loops, when its tension is 36 N. The tension at which it will vibrate in 4 loops is:
Water flows through a frictionless horizontal duct with a cross-section varying as shown in figure. Pressure p at points along the axis is represented by:
A pebble is thrown horizontally from the top of a 20 m high tower with an initial velocity of 10 m/s. The air drag is negligible. The speed of the pebble when it is at the same distance from top as well as base of the tower \[(g=10\,m/{{s}^{2}})\]
Two bodies A and B have emissivities 0.5 and 0.8 respectively. At some temperatures the two bodies have maximum spectral emissive powers at wavelength \[8000\,\overset{{}^\circ }{\mathop{A}}\,\] and \[4000\,\overset{{}^\circ }{\mathop{A}}\,\] respectively. The ratio of their emissive powers at these temperatures are:
\[{{S}_{1}}:\] An uncharged conductor kept near a charged body is attracted by that body whether the charge on the body is positive or negative.
\[{{S}_{2}}:\] A solid conductor is placed in a uniform electric field. The electric field due to conductor will be same at every point inside the conductor.
\[{{S}_{3}}:\] The electric field produced by an infinitely large sheet is same on both sides.
\[{{S}_{4}}:\] Intensity of electric field decreases as you go away from the centre of a uniformly charged solid sphere (having uniform volume charge density).
State, in order, whether \[{{S}_{1}},\]\[{{S}_{2}},\]\[{{S}_{3}},\]\[{{S}_{4}}\] are true or false
The two particles A and B have de-Broglie wavelengths 2 nm and 5 nm respectively. If mass of A is twice the mass of B, the ratio of kinetic energies of A and B would be -
For a general \[{{n}^{th}}\] order process \[A\to ~P\] with initial concentration of reactant \[''a''\] and rate constant k, the expression for time for 75% completion of reaction is -
Liquids and vapours curves show the compositions of the solution and vapour in equilibrium at the boiling point of the former. Pick out the correct diagram of the following representing the variation of boiling point of solution of two liquid \[\,A\] and B (where A is more volatile) with composition.
The percentage of ammonia obtainable, if equilibrium were to be established during the Haber process, is plotted against the operating pressure for two temperatures, \[400{}^\circ C\] and \[500{}^\circ C.\] Which of the following graph correctly represent the two process?
The gold numbers of protective colloids A, B, C and D are 0.16, 0.06, 0.01 and 0.1 respectively. The protective powers of A, B, C and D are in the order -
The specific conductance of a 0.1 N KCl solution at \[23{}^\circ C\] is \[0.0112\text{ }oh{{m}^{-1}}\text{ }c{{m}^{-1}}.\] The resistance of the cell containing the solution at the same temperature was found to be 55 ohm. The cell constant will be:
The process by which ATP is produced in the inner membrane of a mitochondrion, the electron transport system transfer protons from the inner compartment to the outer, as the protons flow back to the inner compartment, the energy of their movement is used to add phosphate to ADP, forming ATP is:
Imagine a gel through which DNA fragments have moved in response to an applied electrical current the band on this gel that is farthest from the top (that is, from the place where the DNA fragments were added to the ?well?) represents the
A)
Shortest fragments of DNA.
doneclear
B)
Longest fragments of DNA.
doneclear
C)
Restriction enzyme used to cut the DNA into fragments.
Which of the following events would likely occur if a plants tissue culture is treated with a solution containing a relatively high concentration of cytokinin and a relatively low concentration of auxin?
A)
Bud and shoot formation; rapid cell expansion; increased cell division.
doneclear
B)
Bud and shoot formation; slow cell expansion; increased cell division
doneclear
C)
Bud and shoot formation; rapid cell expansion; increased cell division.
If there are 999 bases in an RNA that codes for a protein with 333 amino acids, and the base at position 901 is deleted such that the length of the RNA becomes 998 based, how many codons will be altered?
Which of the following statement (s) is /are incorrect?
[A] light is essential for life to exist on the earth.
[B] Many species of small plant under the canopy of tall trees in forest show optimal use of available light due to having large sized antenna and higher number of thylakoids.
[C] UV-rays is not harmful for many organisms.
[D] Photoperiodic requirement is not essential for many plants for flowering.
[E] Red algae can live in deeper water of sea because of having pigments, phycoerythrin.
Insulin is a protein. It includes 17 of the 20 different amino acids. What is the minimum number of \[tRNA\] molecules involved in the synthesis of insulin?
[A] The first recombinant DNA was constructed by using a piece of DNA from a plasmid carrying antibiotic resistance gene in the bacterium Salmo \[typhimurium\] and linked it to the plasmid of E.coli.
[B] Cohen and Boyer are known as Father of genetic engineering.
[C] When cut by the same restriction enzyme, the resultant DNA fragments have the same kind of sticky ends and these can be joined together using DNA ligase.
[D] Endonucleases remove nucleotides from the ends of the DNA whereas exonucleases make cuts at specific positions within the DNA.
[E] Presence of more than one recognition sites within the vector will generate several fragments which will complicate the gene cloning.
[F] Humulin was the first recombinant DNA based product, produced and marketed
[G] YAC vectors contain the telomeric sequence, the centromere and autonomously replicating sequence yeast chromosomes.
[H] Alkaline phosphatase is used to prevent unwanted self-ligation of the vector DNA molecules in procedures of DNA technology
[I] \[pBR322\] vector was the first artificial ideal vector constructed by Bolivar and Rodriguez
[J] Plasmid DNA is coated with histone proteins and can act as genetic factor.
When a physician elicits the knee-jerk reflex by tapping deep tendons in the knee, the normal response is for the lower leg to swing forward. When this happens,
A)
Muscles in the back of the lower leg are contracting but that in the front are relaxing.
doneclear
B)
Muscles in the front of the lower leg are contracting but that in the back are relaxing.
doneclear
C)
Muscles in the back of the thigh are contracting but that in the front are relaxing.
doneclear
D)
Muscles in the front of the thigh are contracting but that in the back are relaxing.
A line \[y=mx+1\] intersects the circle \[{{(x-3)}^{2}}+\,{{(y+2)}^{2}}=25\] at the points P and Q. If the midpoint of the line segment PQ has x-coordinate \[\frac{-\,3}{5}\] then which one of the following options is correct?
Let \[\,M\,=\,\left[ \begin{matrix} {{\sin }^{4}}\theta & -1-{{\sin }^{2}}\theta \\ 1+{{\cos }^{2}}\theta & {{\cos }^{4}}\theta \\ \end{matrix} \right]=\alpha I+\beta {{M}^{-1}}\] where \[\alpha \,=\alpha \,(\theta )\] and \[\beta =\,\beta \,(\theta )\] are real numbers, and I is the \[2\times 2\]identity matrix. If \[{{\alpha }^{*}}\] is the minimum of set and \[\{a\,(\theta ):\theta \in [0,2\pi )\}\]\[\beta *\] is the minimum of the set \[\{\beta \,(\theta ):\,\theta \,\in [0,2\pi )\}\] then the value of \[\alpha *+\beta *\]
Let S be the set of all complex numbers z satisfying \[\left| z\,-\,2+i \right|\,\le \,\sqrt{5}.\] If the complex number \[{{z}_{0}}\] is such that \[\frac{1}{\left| {{z}_{0}}\,-\,1 \right|}\] is the maximum of the set \[\left\{ \frac{1}{\left| z-1 \right|}:z\,\in \,\text{S} \right\},\] then the principal argument of \[\frac{4\,\,-\,\,{{z}_{0}}-\,\bar{z}}{{{z}_{0}}\,-\,{{{\bar{z}}}_{0}}+2i}\] is:
Let \[\left| \!{\overline {\, {} \,}} \right. \]denotes a curve \[y=y(x)\] which is in the first quadrant and let the point (1, 0) lie on it. Let the tangent to \[\left| \!{\overline {\, {} \,}} \right. \]at a point P intersect the v-axis at \[{{\text{Y}}_{{{\text{p}}^{\text{.}}}}}\]. If \[\text{P}{{\text{Y}}_{\text{p}}}\] has length 1 for each point P on \[\left| \!{\overline {\, {} \,}} \right. \], then which of the following options is/are correct?
Define the collections \[\{{{\text{E}}_{1}},{{\text{E}}_{2}},\,{{\text{E}}_{3}},\,....\}\]of ellipse and \[\{{{\text{R}}_{1}},\,{{\text{R}}_{2}},\,{{\text{R}}_{3}},\,.....\}\]of rectangles as follows:
\[\text{Let}\,\,\text{M}\,=\,\left[ \begin{matrix} 0 & 1 & a \\ 1 & 2 & 3 \\ 3 & b & 1 \\ \end{matrix} \right]\,\,\text{and}\,\,\,\text{adj}\,\,\text{M}\,=\,\left[ \begin{matrix} -1 & 1 & -1 \\ 8 & -6 & 2 \\ -5 & 3 & -1 \\ \end{matrix} \right]\] Where a and b are real numbers. Which of the following .options is/are correct?
Let \[\alpha \] and \[\beta \] be the roots of \[{{x}^{2}}-x-1=0,\] with \[\alpha \,>\,\beta .\] For all positive integer n, define \[{{a}_{n}}\,=\,\frac{{{\alpha }^{n}}\,-\,{{\beta }^{n}}}{\alpha \,-\,\beta },\,n\ge \,1\], \[{{b}_{1}}\,=\,1\] and \[{{b}_{n}}\,=\,{{a}_{n-1}}\,+\,{{a}_{n}}+{{1}^{,\,}}\,n\,\,\le \,2\]. Then which of the following options is/are correct?
A)
\[{{a}_{1}}+{{a}_{2}}+{{a}_{3}}+{{a}_{n}}={{a}_{n+2}}-\,1\] for all \[n\,\,\ge \,\,1\]
doneclear
B)
\[{{b}_{n}}={{a}^{n}}+{{\beta }^{n}}\] for all \[n\,\ge \,1\]
There are three bags \[{{\text{B}}_{\text{1}}}\text{,}\]\[{{\text{B}}_{\text{2}}}\] and \[{{\text{B}}_{\text{3}}}\text{.}\]The bag \[{{\text{B}}_{\text{1}}}\] contains 5 red and 5 green balls, contains 3 red and 5 green balls, and \[{{\text{B}}_{\text{2}}}\] contains 5 red \[{{\text{B}}_{3}}\] and 3 green balls. Bags \[{{\text{B}}_{\text{1}}}\text{,}\]\[{{\text{B}}_{\text{2}}}\] and \[{{\text{B}}_{\text{3}}}\] have probabilities \[\frac{3}{10},\,\frac{3}{10}\] and \[\frac{4}{10}\] respectively of being chosen. A bag is selected at random and a ball is chosen at random from the bag. Then which of the following options is/are correct?
A)
Probability that the chosen ball is green, given that the selected bag is \[{{\text{B}}_{\text{3}}}\] equals \[\frac{3}{8}\]
doneclear
B)
Probability that the selected bag is \[{{\text{B}}_{\text{3}}}\] and the chosen ball is green equals \[\frac{3}{10}\]
doneclear
C)
Probability that the selected bag is \[{{\text{B}}_{\text{3}}}\] and the chosen ball is green equals \[\frac{5}{13}\]
doneclear
D)
Probability that the chosen ball is green equals \[\frac{39}{80}\]
The spring block system as shown in figure is in equilibrium. The string connecting blocks A and B is cut. The mass of all the three blocks is m and spring constant of both the spring is k. The amplitude of resulting oscillation of block A is (string massless)
The plates of small size of a parallel plate capacitor are charged as shown. The force on the charged particle of 'q' at a distance \['\ell '\] from the capacitor is: (Assume that the distance between the plates is \[d<<\ell \])
A charged particle of specific charge (charge/mass) \[\alpha \] is projected from origin with a velocity \[\vec{u}={{v}_{0}}\,(\hat{i}+\hat{j})\] in a uniform and constant magnetic field \[\vec{B}={{B}_{0}}\hat{i}.\] The position co-ordinates of the particle at time \[t=\frac{\pi }{{{B}_{0}}\alpha }\] are
In the circuit shown switch S is connected to position 2 for a long time and then joined to position 1. The total heat produced in resistance \[{{R}_{1}}\] is:
A parallel beam of monochromatic radiation of cross-section area \[A\,(<\pi {{a}^{2}}),\] intensity I and frequency \[\nu \] is incident on a solid conducting sphere of work function \[{{\phi }_{0}}\,[h\nu >{{\phi }_{0}}]\] and radius 'a'. The sphere is grounded by a conducting wire Assume that for each incident photon one photoelectron is ejected. Just after this radiation is incident on initially uncharged sphere, the current through the conducting wire is:
In the figure ABC is the cross section of a right angled prism and BCDE is the cross section of a glass slab. The value of \[\theta \] so that light incident normally on the face AB does not cross the face BC is \[(given\,\,{{\sin }^{-1}}(3/5)=37{}^\circ )\]
Two particles having the same specific charge (q/m) enter a uniform magnetic field with the same speed but an angles of \[30{}^\circ \] and \[60{}^\circ \] with the field. Let a, b and c be ratios of their pitches, radii and periods of their paths respectively, then:
In the given electrical circuit, the potential difference between points A and B is (assume the battery is ideal and the conducting wires have almost zero resistance):
The cell \[Pt({{H}_{2}})(1\,atm)|{{H}^{+}}(pH=?)||.\]\[(a=1)|Ag\] has emf, \[{{E}_{298}}=0.\] The standard electrode potential for the reaction \[AgL+{{e}^{-}}\to \text{ }Ag+\overset{\Theta }{\mathop{I}}\,\] is \[-\,0.151\,A\] volt. Calculate the pH value-
The freezing point of aqueous solution that contains 3 % urea, 7.45 % KCl and 9 % of glucose is (given \[{{K}_{f}}\] of water =1.86 and assume molarity = molarity)
The activation energy is lowered by \[8.314\text{ }kJ\text{ }mo{{l}^{-1}}\] for the catalysed reaction. How many times the rate of this catalysed reaction greater than that of uncatalysed reaction? \[(Given\,\,{{e}^{3.33}}=28)\]
Compounds X and Z in the following reaction are-\[C{{H}_{3}}CHO\xrightarrow[(ii)\,\,{{H}_{2}}O]{(i)\,\,C{{H}_{3}}MgBr}(X)\xrightarrow{{{H}_{2}}S{{O}_{4}},\,\Delta }(Y)\]\[\xrightarrow{Hydroboration\,\,oxidation}(Z)\]
A person has \[400\] million alveoli per lung with an average radius of \[0.1\]mm for each alveolus. Considering the alveoli are spherical in shape, The total respiratory surface of that person is closest to:
A gene that was \[5055\] bp long resulted in three Polypeptides-350 amino acids long, \[450\] amino acids long and \[1500\] amino acids long. The most Likely reason for this is:
Minisatellites are used as marker for identifying Individuals via DNA fingerprinting as the alleles May differ in the number of repeats. From the Southern blot shown below identify the progeny (A, B, C and D) for the given parents
Dividing chromosomes can be labelled with a thymine analogue, bromodeoxy - uridine. After differential staining, the chromosomes can be seen as darkly stained (old) strands and lightly stained (new) strands. The following chromosomes were observed and photographed while studying division of human blood cells.
From the picture, which of the following can be deduced?
[A] The chromosomes belong to metaphase stage.
[B] The cell division was taking place in mature red blood cells.
[C] Parts of the chromatids were exchanged by crossing over.
[D] The different colours of the two sister chromatids confirm that DNA replication is semi-conservative.
\[Igf2\] Gene of mouse encodes for insulin like growth factor II. A mouse that carries two normal wild type alleles of this gene is normal in size, whereas a mouse that carries two mutant alleles is dwarf. One heterozygous male was crossed with a heterozygous female, both being normal sized. The ratio of normal to dwarf was \[1:1\] instead of the expected \[3:1\] when some of these \[{{\operatorname{F}}_{1}}\] male dwarfs were mated with homozygous mutant female dwarfs, normal mice appeared in the following progeny. This can happen if the normal \[Igf2\] gene got imprinted in parent during