The equation \[\left( \cos p-1\, \right){{x}^{2}}+\left( \cos p \right)x\]+sin p=0 in the variable x, has real roots. Then p can take any value in the interval
Let \[f\left( x \right)={{\sin }^{3}}x+\lambda {{\sin }^{2}}x,-\frac{\pi }{2}<x<\frac{\pi }{2}\] in order that \[f\left( x \right)\] has exactly one minimum, \[\lambda \] should belong to
Let \[f:\left[ -2,3 \right]\to \left[ 0,\infty \right]\] be a continuous function such that \[f\left( 1-x \right)=f\left( x \right)\] for all \[x\in \left[ -2,3 \right]\] If \[{{R}_{1}}\]is the numerical value of area of the region bound by \[y=f\left( x \right),x=-2,x=3\] and the axis of x and \[{{R}_{2}}\]\[=\int\limits_{-2}^{3}{xf\left( x \right)dx\,then:}\]
Let \[f\] be a real valued function such that for any real \[x,f\left( \lambda +x \right)=f\left( \lambda -x \right)\,\,and\,\,f\left( 2\lambda +x \right)=-f\left( 2\lambda -x \right)\] for some\[\lambda >0.\] then
let \[f\left( x \right)=\left[ r+p\sin x \right],x\in \left( 0,\pi \right),r\in I\,and\,p\] is prime number ([.] denotes integer function).the number of points at which \[f\left( x \right)\] is non-differentiable is
Let \[f\] be a differentiable function satisfying \[{{\left[ f\left( x \right) \right]}^{n}}=f\left( nx \right)\] for all \[x\in R.\] Then, \[f'\left( x \right)f\left( nx \right)=\]
The number of all possible triplets \[\left( {{a}_{1}},{{a}_{2}},{{a}_{3}} \right)\] such that \[{{a}_{1}}+{{a}_{2}}\cos \left( 2x \right)+{{a}_{3}}{{\sin }^{2}}\left( x \right)=0\] for all x is
If \[U=\left\{ x:{{x}^{5}}-6{{x}^{4}}+11{{x}^{3}}-6{{x}^{2}}=0 \right\},\]\[A=\{x:{{x}^{2}}-5x+6=0)\] and \[B=\{x:{{x}^{2}}-3x+2=0\},then\,n(A\cap B)'\]is equal to:
Let P \[\left( 3\sec \theta ,2\tan \theta \right)\] and Q \[\left( 3\sec \phi ,2\tan \phi \right)\] where \[\theta +\phi =\frac{\pi }{2},\] be two distinct points on the hyperbola \[\frac{{{x}^{2}}}{9}-\frac{{{y}^{2}}}{4}=1.\]then the ordinate of the point of intersection of the normal at P and Q is:
The area of the rectangle formed by the perpendiculars from the Centre of ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=\]1 to the tangent and normal at a points whose eccentric angle is \[\frac{\pi }{4}is\left( a>b \right)\]
On each evening a boy either a boy watches DOORDARSHAN channel or TEN SPORTS. The probability that he watches TEN SPORTS is \[\frac{4}{5},\] if he watches DOORDARSHAN, there is a chance of \[\frac{3}{4},\] that he will be asleep, white it is \[\frac{1}{4}\] when he watches TEN SPORTS. On one day, the boy is found to be asleep. The probability that the boy watches DOORDARSHAN is
Let \[F\left( x \right)\] be a functions defined by \[F\left( x \right)=x-\left[ x \right],0\ne x\in R,\] where \[\left( x \right)\]is the greatest integer less than or equal to x, then the number of solutions of \[f\left( x \right)+F\left( \frac{1}{x} \right)=1\] is
Let \[A\equiv \left( -2,-2 \right)\] and \[B\equiv \left( 2,-2 \right)\] be two points and \[AB\] subtends an angle of \[45{}^\circ \] at any point \[P\] in the plane in such a way that area of \[\Delta APB\] is 8 square unit, then number of possible position (s) of P is
An object travels along a path shown with changing velocity as indicated by vectors A and B. Which vector best represents the net acceleration of the object from time \[{{t}_{A}}\] to \[{{t}_{B}}\]?
An object is dropped and accelerates downwards. As it falls it is affected by air friction, but never readies terminal velocity during the course of its fall. The graph that could indicate the magnitude of the object's acceleration as a function of time is:
An air track glider of mass M is built, consisting of two smaller connected gliders with a small explosive charge located between them. The glider is traveling along a frictionless rail at 2 m/s to the right when the charge is detonated, causing the smaller glider with mass \[\frac{1}{4}M,\] At, to move off to the right at 5 m/s. What is the final velocity of the second small glider?
A billiard ball hits the side of a pool table at an angle \[\theta \] as shown in the top view above, and bounces away at the same angle, and with the same speed. Which vector indicates the direction of the net change in momentum of the billiard ball?
A large cannon is mounted on a cart with frictionless wheels that is initially at rest on a horizontal surface. The cannon fires a large cannonball to the right with a speed \[{{v}_{cannonball}},\] which is then caught by a trap firmly attached to the cart. What is the filial speed v of the cannon-cart cannonball system?
The potential energy function U(x) is associated with a conservative force F and described by the graph given here. If a particle being acted upon by this force has a kinetic energy of 1.0 J at position\[{{x}_{0}},\] what is the particle's kinetic energy at position \[{{x}_{4}}?\]
A certain star, of mass m and radius r, is rotating with a rotational velocity\[\omega \]. After the star collapses, it has the same mass but with a much smaller radius. Which statement below is true?
A)
The star's moment of inertia I has decreased, and its angular momentum L has increased
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B)
The star's moment of inertia I has decreased, and its angular velocity \[\omega \] has decreased
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C)
The star's moment of inertia I remains constant, and its angular momentum L has increased
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D)
The star's angular momentum L remains constant, and its rotational kinetic energy has increased
A vertical straight conductor carries a current vertically upwards. A point P lies to the east of it at a small distance and another point Q lies to the west of it at the same distance. The magnetic field at P is (Consider earth magnetic field also) -
A)
Greater than at Q
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B)
Same as at Q
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C)
less than at Q
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D)
Greater or less than at Q depending upon the strength of the current
A wire of length \[\ell \] and carrying a current i is placed along X-axis in a magnetic field given by \[\overset{\to }{\mathop{B}}\,={{B}_{0}}(\hat{i}+\hat{j}+\hat{k}).\] magnitude of the force acting on the wire is -
A charged particle projected perpendicularly into a magnetic field of \[(7\hat{i}-3\hat{j})\times {{10}^{-\,3}}\] tesla acquires an acceleration of \[(x\hat{i}+7\hat{j})m{{s}^{-\,2}}.\] The value of x is-
The voltage and the current of an a.c. circuit are \[V=100\text{ }sin\,\,(\,100\,\,t\,)\] volt and \[i=100\text{ }sin\text{ }(100\,\,t+\pi /3)\text{ }mA\] respectively. The power dissipated in the circuit is -
An asteroid traveling through space collides with one end of a long, cylindrical satellite as shown and sticks to the satellite. Which of the following is true of the isolated asteroid-satellite system in this collision?
A)
Kinetic energy K is conserved
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B)
Total Energy E is conserved, but angular momentum L is not conserved
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C)
Angular momentum L is conserved, but linear momentum P is not conserved
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D)
Angular momentum L is conserved, and total energy E is conserved
Two ice skaters, of mass 30 kg and 80 kg, are skating across the surface of a frozen lake on a collision course, with respective velocities of 2.0 m/s in a general north direction, and 1.0 m/s generally west, as shown above. After they collide, the pair of skaters move off in a direction north of west with a momentum of approximately 100 kg m/s. How much kinetic energy was lost in the collision?
An ideal monoatomic gas is heated so that it expands at constant pressure. What percentage of the heat supplied to the gas is used to increase the internal energy of the gas?
When a tuning fork of frequency \[f\]is excited and held near me end of a straight pipe of length L open at both ends, the air column in the pipe vibrates in its fundamental mode and is in resonance with the tuning fork. The pipe is now kept vertical in a jar containing water so that half the length of the pipe is inside water. What should be the frequency of the tuning fork to be used to make the air column vibrate in its fundamental mode in resonance with the tuning fork now?
A wheel of radius R is rolling along a horizontal surface with a speed \[u.\] A pebble trapped on the wheel gets separated from the highest point of the wheel arrives at position P (figure). The horizontal range PQ of the pebble is:
A particle starts from the origin at time \[t=\text{ }0\]with velocity \[2\hat{j}\] and moves in the \[x-y\] plane with a constant acceleration of \[2\hat{i}+4\hat{j}\] where \[\hat{i}\] and \[\hat{j}\]are unit vectors along the x-direction and y- direction respectively. What will be the x- coordinate of the particle when its y-coordinate becomes 12 m -
Two solid bodies rotate about stationary mutually perpendicular intersecting axes with constant angular velocities \[{{\omega }_{1}}\] and \[{{\omega }_{2}}\]. What is the magnitude of angular velocity of one with respect to the other?
Equivalent conductance of \[NaCl,\]\[HCl\] and \[{{C}_{2}}{{H}_{5}}COONa,\] at infinite dilution are 126.45, 426.16 and \[91\,{{\Omega }^{-\,1}}c{{m}^{2}}\] respectively. The equivalent conductance of \[{{C}_{2}}{{H}_{5}}COOH\] is
For the following reaction, \[{{H}_{2}}(g)+C{{l}_{2}}(g)2HCl\,(g)\]. \[\Delta G{}^\circ \] is \[-\,262kJ.\] The equilibrium constant k for the reaction at 298 K is
Number of \[\alpha \] and \[\beta \text{-}\]particles emitted, when an atom of \[{}_{90}T{{h}^{232}}\] undergoes disintegration to produce an atom of \[{}_{82}P{{b}^{208}},\] are
Copper metal on treatment with dilute \[HN{{O}_{3}}\] produces a gas (X). X, when passed through acidic solution of stannous chloride, a nitrogen containing (Y) is obtained. Which on reaction with nitrous acid produces a gas (Z) is
Standard enthalpy of formation of \[{{C}_{3}}{{H}_{7}}N{{O}_{2}}(g),\]\[C{{O}_{2}}(g)\] and \[{{H}_{2}}O\,(l)\] are \[-133.57,-94.05\] and \[-\,68.32\,kcal/mol\] respectively. Standard enthalpy of combustion of \[C{{H}_{4}}\] at \[25{}^\circ C\] is\[-\,212.8\,kcal/mol\]. Calculate the enthalpy for combustion in kcal/mol of \[{{C}_{3}}{{H}_{7}}N{{O}_{2}}(s).\]
The rate constant, the activation energy and the Arrhenius parameter of a chemical reaction at \[25{}^\circ C\] are \[3\times {{10}^{-4}}{{s}^{-1}},\] \[104.4\,\,kJ\,mo{{l}^{-1}}\] and \[6.0\times {{10}^{14}}{{s}^{-\,1}}\] respectively. The value of rate constant at \[T\to \infty \] is closest to
In snapdragon, when a red flower plant is crossed with a white flower ones, The resultant hybrid plant have pink coloured flower if plant of \[{{F}_{1}}\]generation is crossed with a white flower ones, the progeny will be.
A red ball and a green ball are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin \[{{2}^{-k}}\] for \[k=1,\]\[2,3...\]what is the probability that the red ball is tossed into a higher numbered bin than green ball?
In a circle with centre 'O', PA and PB axe two chords PC is the chord that bisects the\[\angle APB\] the tangent to the circle at C is drawn meeting PA and PB extended at Q and R respectively. If \[QC=3,\]\[QA=2\] and \[RC=4,\]then length of RB equals
If \[\alpha =\int\limits_{0}^{1}{({{e}^{9x+3{{\tan }^{-1}}x}})\left( \frac{12+9{{x}^{2}}}{1+{{x}^{2}}} \right)}\,dx,\] where \[{{\tan }^{-1}}x\] takes only principal values, then the value of \[\left( {{\log }_{e}}\left| 1+\alpha \right|-\frac{3\pi }{4} \right)\]
Let \[{{n}_{1}}<{{n}_{2}}<{{n}_{3}}<{{n}_{4}}<{{n}_{5}}\]be positive integers such that \[{{n}_{1}}+{{n}_{2}}+{{n}_{3}}+{{n}_{4}}+{{n}_{5}}=20.\] Then, the number of such distinct arrangement \[({{n}_{1}},{{n}_{2}},{{n}_{3}},{{n}_{4}},{{n}_{5}})\] is
Let \[f:\left[ \frac{1}{2},1 \right]\]\[\to \]\[R\] (the set of all real numbers) be a positive non-constant and differentiable function such that \[f\,'(x)<2f(x)\] and \[f\,\left( \frac{1}{2} \right)=1.\] Then, the value of \[\int\limits_{1/2}^{1}{f(x)dx}\] lies in the interval
Tangent drawn from the point (1, 8) to the circle \[{{x}^{2}}+{{y}^{2}}-6x-4y-11=0\] touch the circle at the points A and B. The equation of the circumcircle of \[\Delta \,PAB\] is
A semicircular loop of radius R is rotated about its straight edge which divides the space into two regions one having a uniform magnetic field B and the other having no field. If initially the plane of loop is perpendicular to \[\overrightarrow{B}\](as shown), and if current flowing from O to A be taken as positive, the correct. plot of induced current v/s time for one time period is:
A ray hits the y-axis making an angle \[\theta \] with y-axis as shown in the figure. The variation of refractive index with x-coordinate is \[\mu ={{\mu }_{0}}\,\,\left( 1-\frac{x}{d} \right)\] for \[0\le x\le d\,\,\left( 1-\frac{1}{{{\mu }_{0}}} \right)\] and \[\mu ={{\mu }_{0}}\] for \[x<0,\]where d is a positive constant. The maximum x-coordinate of the path traced by the ray is:
A cobalt \[(\text{atomic}\,\,\text{no}\text{.}=27)\] target is bombarded with electrons, and the wavelengths of its characteristic x-ray spectrum are measured. A second weak characteristic spectrum is also found, due to an impurity in the target. The wavelengths of the \[{{K}_{\alpha }}\] lines are 225.0 pm (cobalt) and 100.0 pm (impurity). Atomic number of the impurity is \[(take\,\,b=1)\]
Two conducting spheres of radii r and 3 r initially have charges 3 q & q respectively. Their separation is much larger than their radii. If they are joined by a conductor of high resistance, the force between them will:
In an LRC series circuit at resonance, current in the circuit is \[10\sqrt{2}\,A.\] If now frequency of the source is changed such that now current lags by \[45{}^\circ \] than applied voltage in the circuit. Which of the following is correct?
A)
Frequency must be increased and current after the change is 10 A
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B)
Frequency must be decreased and current after the change is 10 A
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C)
Frequency must be decreased and current is same as that of initial value
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D)
The given information is insufficient to conclude anything
The oscillations represented by curve 1 in the graph are expressed by equation \[x=A\sin \omega \,t.\] The equation for the oscillations represented by curve 2 is expressed as:
A certain gas is taken to the five states represented by dots in the graph. The plotted lines are isotherms. Order of the most probable speed \[{{V}_{P}}\]of the molecules at these five states is:
An electron of the kinetic energy 10eV collides with a hydrogen atom in 1st excited state. Assuming loss of kinetic energy in the collision to be quantized which of the following statements is incorrect.
A uniform stick of mass M is placed in a frictionless well as shown. The stick makes an angle \[\theta \] with the horizontal. Then the force which the vertical wall exerts on right end of stick is:
In the radioactive decay \[_{z}{{X}^{A}}\to {{\,}_{Z+1}}{{Y}^{A}}\to \underset{high\,\,energy}{\mathop{_{Z-1}{{Z}^{A-4}}}}\,\to \underset{low\,\,energy}{\mathop{_{Z-1}{{Z}^{A-4}}}}\,\] sequence of the radiation emitted is-
The reaction, Sucrose \[\xrightarrow{{{H}^{+}}}\] Glucose + Frustose, takes place at certain temperature while the volume of solution is maintained at 1 litre. At time zero the initial rotation of the mixture is \[34{}^\circ .\]After 30 minutes the total rotation of solution is \[19{}^\circ \] and after a very long time, the total rotation is \[-\,11{}^\circ .\] Find the time when solution was optically inactive, (log 3 = 0.477, log 11 = 1.1)
Calculate the potential of a half cell having reaction:\[A{{g}_{2}}S\,(s)+2e-2\,Ag\,(s)+{{S}^{2\,-}}\,(aq)\] in a solution buffered at pH = 3 and which is also saturated with 0.1 M \[{{H}_{2}}S\,(aq).\] \[({{K}_{{{a}_{1}}}}.{{K}_{{{a}_{2}}}}{{H}_{2}}S={{10}^{-24}},\] \[{{K}_{sp}}(A{{g}_{2}}S)={{10}^{-49}}\]
Given the following molar conductivities at \[25{}^\circ C:,\] HCl, \[426\,\,{{\Omega }^{-\,1}}\,\,c{{m}^{2\text{ }}}mo{{l}^{-\,1}}\] NaCl, \[126\text{ }{{\Omega }^{-1}}c{{m}^{2\text{ }}}mo{{l}^{-1}};\] NaCl (sodium crotonate), \[\text{83 }{{\Omega }^{-1}}c{{m}^{2\text{ }}}mo{{l}^{-1}}.\] What is the ionization constant of crotonic acid? If the conductivity of a 0.001 M crotonic acid solution is \[3.83\text{ }\times \text{ 1}{{\text{0}}^{-5}}\text{ }{{\Omega }^{-1}}\text{ }c{{m}^{-1\text{ }}}?\]
A diploid somatic cell from a rat has a total of 42 chromosomes \[\left( 2n\text{ }=\text{ }42 \right).\]Read the following statements and select the option with correct statements.
[A] Rat has 21 different chromosomes.
[B] Rat has 42 DNA molecules during \[{{G}_{1}}.\]
[C] Rat has 21 bivalents in metaphase A of meiosis.
[D] Rat has 42 DNA molecules in metaphase B of meiosis.
The recognition site of the enzyme \[TaqI\] is TCGA. What would be the maximum number of recognition sites that this enzyme would have on a DNA molecule that is 5kb long?
E.coli has a circular DNA consisting of \[4.3\times {{10}^{6}}bp.\] There are 10 bp of DNA for each complete \[360{}^\circ \] turn of the double helix. Replication starts at one location in the genome and proceeds in both directions that is, there are two replication forks that move away from the origin of replication in opposite directions meeting when each has moved halfway around the genome. Under one particular set of growth conditions, 800 bp of DNA are formed per second at each replication fork. At how many revolutions per minute (rpm) is each replication fork rotating during replication?
If there are 34 amino acids and DNA contains only two nitrogenous based, what would be the minimum number of bases per codon that code for amino acids?