If [x] denotes the greatest integer \[\le x,\] then \[\underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{{{n}^{3}}}\{[{{1}^{2}}x]+[{{2}^{2}}x]+[{{3}^{2}}x]+...+[{{n}^{2}}x]\}\]equals
If the complex number is \[{{\left( 1+ri \right)}^{3}}=\lambda \left( 1+i \right),\] when \[i=\sqrt{-1},\] for some real \[\lambda \], the value of \[r\] can be
A, B are C are contesting the election for the post of secretary of a club which does not allow ladies to become members. The probabilities of A, B and C winning the election are \[\frac{1}{3},\frac{2}{9}\] and \[\frac{4}{9}\] respectively. The probabilities of introducing the clause of admitting lady member to the club by A, B and C are 0.6, 0.7 and 0.5, respectively. The probability that ladies will be taken as members in the club after the election is
The number of solutions of the equation \[{{\sin }^{3}}x\cos x+{{\sin }^{2}}x{{\cos }^{2}}x+\sin x{{\cos }^{3}}x=1,\] in the interval \[[0,2\pi ],\] is
Let \[f:\text{ }R\to R\] be such that \[f\left( 1 \right)=3\] and \[f'\left( 1 \right)=6.\] Then \[\underset{x\to 0}{\mathop{\lim }}\,{{\left( \frac{f\left( 1+x \right)}{f\left( 1 \right)} \right)}^{1/x}}\] equals
If the integers m and n are chosen at random between 1 and 100, then the probability that a number of the form \[{{7}^{m}}+{{7}^{n}}\] is divisible by 5 equals
The circumcentre of a triangle lies at the origin and its centroid is the mid-point of the line segment joining the points \[\left( {{a}^{2}}+1, {{a}^{2}}+1 \right)\] and \[(2a,-2a),a\ne 0.\] Then for any a, the orthocentre of this triangle lies on the line:
The set of equations: \[\lambda x-y+\left( \cos \theta \right)z=0;3x+y+2z=0;\] \[\left( cos\theta \right)x+y+2z=0; 0\le \theta <2\pi ,\] has non -trivial solutions
A)
For no value of \[\lambda \operatorname{and}\theta \]
doneclear
B)
For all values of \[\lambda \]and\[\theta \]
doneclear
C)
For all values of\[\lambda \], and only two values of\[\theta \]
doneclear
D)
For only one value of \[\lambda \] and all values of \[\theta \]
If \[a\in \left[ -\text{ }20,0 \right]\], then probability that the graph of the function \[y=16{{x}^{2}}+8\left( a+5 \right)\text{ }x-7a-5\] is strictly above the \[x-\]axis is
Let \[f:R\to R\] and \[g:R\to R\] be two one-one and onto functions such that they are the mirror images of each other about the line \[y=a.\] If \[h(x)=f(x)+\text{ }g(x),\] then \[h(x)\] is
If \[\int\limits_{0}^{\pi /2}{\frac{dx}{{{a}^{2}}{{\cos }^{2}}x+{{b}^{2}}{{\sin }^{2}}x}=\frac{\pi }{16}.}\] Then minimum value of \[a\cos x+b\sin \text{ }x\] is
Which of the following graphs best represents the force acting on a charged particle kept at distance x from the centre of a square and on the axis of the square whose corners have equal charges.
A charged particle moves with a constant velocity \[(\hat{i}+\hat{j})\,\,m/s\] in a magnetic field \[\vec{B}=(2\hat{i}+3\hat{k})\,T\]and uniform electric field \[\vec{E}=(a\hat{i}+b\hat{j}+c\hat{k})\,N/C,\] then (assuming all quantities in S.I. unit):
Wire bent as ABOCD as shown, carries current I entering at A and leaving at D. Three uniform magnetic fields each \[{{B}_{0}}\] exist in the region as shown. The force on the wire is:
A metal rod of length moving with an angular velocity and velocity of its centre is v. Find potential difference between points A and B at the instant shown in figure. A uniform magnetic field of strength B exist perpendicular to plane of paper:
An uncharged capacitor of capacitance C is connected with an ideal cell. The emf of the cell is slowly increased from 0 to V (by some mechanism). The total energy taken from the cell in the process of charging of the capacitor is (assume the resistance of the circuit is very small):
The capacitor shown in figure 1 is charged completely by connecting switch S to contact a. If switch S is thrown to contact b at time \[t=0,\]which of the curves in figure 2 above represents the magnitude of the current through the resistor R as function of time t?
A beam of light parallel to central line AB is incident on the plane of slits in a YDSE experiment as shown. The number of minima obtained on the large screen is Now if the beam is tilted by some angle as shown in the figure, then the number of minima obtained is Then:
A charge particle\[{{q}_{0}}\] of mass \[{{m}_{0}}\] is projected along the y-axis at \[t=0\] from origin with a velocity \[{{V}_{0}}.\] If a uniform electric field \[{{E}_{0}}\] also exists along the x-axis, then the time at which debroglie wavelength of the particle becomes half of the initial value is:
In an x-ray tube, if the accelerating potential difference is changed, then:
A)
the frequency of characteristic x-rays of a material will get changed
doneclear
B)
number of electrons emitted will change
doneclear
C)
the difference between \[{{\lambda }_{0}}\] (minimum wavelength) and\[{{\lambda }_{\,k\,\alpha }}\](wavelength of \[{{k}_{\alpha }}\] x-ray) will get changed
doneclear
D)
difference between \[{{\lambda }_{\,k\,\alpha }}\] and \[{{\lambda }_{\,k\,\beta }}\] will get changed.
A lens is placed between a source of light and a wall, It forms images of area \[{{A}_{1}}\]and \[{{A}_{2}}\] on the wall, for its two different positions. The area of the source of light is (source and wall are fixed)-
The binding energies of the atom of elements A & B are \[{{E}_{a}}\]& \[{{E}_{b}}\] respectively. Three atoms of the element 5 fuse to give one atom of element A. This fusion process is accompanied by release of energy e. Then \[{{E}_{a}},\] \[{{E}_{b}}\] are related to each other as
Two solid bodies of equal mass m initially at \[T=0{}^\circ C\] are heated at a uniform and same rate under identical conditions. The temperature of the first object with latent heat \[{{L}_{1}}\] and specific heat capacity in solid state \[{{C}_{1}}\] changes according to graph 1 on the diagram. The temperature of the second object with latent heat \[{{L}_{2}}\] and specific heat capacity in solid state \[{{C}_{2}}\] changes according to graph 2 on the diagram.
Based on what is shown on the graph, the latent heats \[{{L}_{1}}\] and \[{{L}_{2}},\] and the specific heat capacities \[{{C}_{1}}\] and \[{{C}_{2}}\] in solid state obey which of the following relationships:
A sphere of radius R is in contact with a wedge. The point of contact is R/5 from the ground as shown in the figure. Wedge is moving with velocity 20 m/s, then the velocity of the sphere at this instant will be:
As shown in figure, S is a point on a uniform disc rolling with uniform angular velocity on a fixed rough horizontal surface. The only forces acting on the disc are its weight and contact forces exerted by horizontal surface. Which graph best represents the magnitude of the acceleration of point S as a function of time
When a ball is released from rest in a very long column of viscous liquid, its downward acceleration is 'a' (just after release). Then its acceleration when it has acquired two third of the maximum velocity:
The work done by the force \[\vec{F}=A\,({{y}^{2}}\hat{i}+2{{x}^{2}}\hat{j}),\] where A is a constant and x & y are in meters around the path shown is:
Two strings X and Y of a sitar produces a beat of frequency 4Hz. When the tension of string Y is slightly increased, the beat frequency is found to be 2HZ. If the frequency of X is 300 HZ, then the original frequency of Y was
Thermal coefficient of volume expansion at constant pressure for an ideal gas sample of n moles having pressure \[{{P}_{0}}\] volume \[{{V}_{0}}\] and temperature \[{{T}_{0}}\] is
A process that has \[\Delta H=200\text{ }J\]\[mo{{l}^{-1}}.\] and \[\Delta S=40\,J{{K}^{-1}}\]\[40\text{ }J{{K}^{-1}}\]\[mo{{l}^{-1}}.\] Out of the values given below, choose the minimum temperature above which the process will be spontaneous:
Consider the given plots for a reaction obeying Arthenius equation \[({{0}^{{}^\circ }}C<T<300{}^\circ C):\](k and \[{{E}_{a}}\]are constant and activation energy respectively)
The correct structure of product 'P' in the following reaction is: \[Asn-Ser+\underset{(excess)}{\mathop{{{(C{{H}_{3}}C{{O}_{3}})}_{2}}}}\,\xrightarrow{NE{{t}_{3}}}P\]
If \[A\]denotes the arithmetic mean of the real numbers \[{{a}_{1}},{{a}_{2}},...{{a}_{n}},\] then \[\sum\limits_{i=1}^{n}{{{\left( x-{{a}_{i}} \right)}^{2}}}\]has a minimum at.
If the function \[f\left( x \right)=\frac{x-1}{c-{{x}^{2}}+1}\] does not take any value in the internal \[\left[ -1,-\frac{1}{3}, \right]\] then the largest integral value that \[c\]can attain is equal to
Let \[f\left( x \right)=\left[ x \right]+\left| 1-x \right|,-1\le x<3,\] (here [.] denotes greatest integer function). The number of points, where \[f\left( x \right)\] is non-differentiable is
If \[\left| \overrightarrow{a}+\overrightarrow{b} \right|=\left| \overrightarrow{a}-\overrightarrow{b} \right|\] then the vectors \[\vec{a}\]and \[\vec{b}\]are adjacent sides of
ABCD is a convex quadrilateral. 3, 4, 5 and 6 points are marked on the sides AB, BC, CD and DA respectively. Find the number of triangles with vertices on different sides.
Figure shows a metal ball of mass 50 kg and radius \[\frac{2}{\sqrt{\pi }}m\] is placed on an insulating uncharged stand. In space an upward electric field \[5\times {{10}^{5}}N/C\] is switched on. A stream of light ions is incident on the ball from left side at a speed \[2\times {{10}^{6}}m/s\] m/s as shown in figure. If charge on ball at \[t=0\] was zero, find the time in seconds at which ball will be lifted from the stand. The charge density of ion beam is \[5\times {{10}^{-12}}\text{coul}/{{m}^{3}}.\] Assume that all charge incident on the ball is absorbed.
Two smooth spherical non conducting shells each of radius R having uniformly distributed charge \[Q\And -\,Q\] on their surfaces are released on a smooth non-conducting surface when the distance between their centres is 5 R. The mass of A is m and that of B is 2 m. The speed of A just before A and B collide is:
The xz plane separates two media A and B with refractive indices \[{{\mu }_{1}}\] & \[{{\mu }_{2}}\] respectively. A ray of light travels from A to B. Its directions in the two media are given by the unit vectors, \[{{\vec{r}}_{A}}=a\,\hat{i}+b\,\hat{j}\] & \[{{\vec{r}}_{B}}=\alpha \,\hat{i}+\beta \,\hat{j}\] respectively where \[\hat{i}\] & \[\hat{j}\] are unit vectors in the x & y directions. Then:
A ring of radius R having a linear charge density \[\lambda \] moves towards a solid imaginary sphere of radius \[\frac{R}{2},\] so that the centre of ring passes through the centre of sphere. The axis of the ring is perpendicular to the line joining the centres of the ring and the sphere. The maximum flux through the sphere in this process is:
Figure shows a conducting horizontal rod of resistance r is made to oscillate simple harmonically with a fixed amplitude in a uniform and constant magnetic field B, directed inwards. The ends of rod always touch two parallel fixed vertical conducting rails. The ends of rails are joined by an inductor and a capacitor having self inductance and capacitance \[\frac{1}{\pi }\] Henry and \[\frac{1}{\pi }\] farad respectively. The amplitude of current in the circuit depends on the frequency of oscillation of rod. The amplitude of the current will be maximum when the time period of rod is: (do not consider self inductance anywhere other than in the inductor)
The distance between two slits in a Young's double slit experiment is 3 mm. The distance of the screen from the slits is 1 m. Microwaves of wavelength 1 mm are incident on the plane of the slits normally. The distance of the first maxima on the screen from the central maxima will be:
Half lives of two isotopes X and Y of a material are known to be \[2\times {{10}^{9}}\]years and \[4\times {{10}^{9}}\] respectively. If a planet was formed with equal number of these isotopes, then the current age of planet, given that currently the material has 20% of X and 80% of Y by number, will be:
When a metallic surface is illuminated with monochromatic light of wavelength \[\lambda ,\] the stopping potential is \[5\,{{V}_{0}}.\] When the same surface is illuminated with light of wavelength \[3\lambda ,\] the stopping potential is \[{{V}_{0}}.\] Then the work function of the metallic surface is:
An isolated and charged spherical soap bubble has a radius 'r' and the pressure inside is atmospheric. If 'T' is the surface tension of soap solution, then charge on drop is:
A mixture of 100 m mol of \[Ca{{(OH)}_{2}}\]sodium sulphate dissolved in water and the volume was made up to 100 mL. The mass of calcium sulphate formed and .the concentration of OH~ in resulting solution, respectively are (Molar mass of \[Ca{{(OH)}_{2}},\]\[N{{a}_{2}}S{{O}_{4}}\]and \[CaS{{O}_{4}}\]are 74, 143 and 136 g \[mo{{l}^{-1}}\] respectively \[{{K}_{sp}}\] of \[Ca{{(OH)}_{2}}\]is \[5.5\times {{10}^{-6}}\])
Liquids A and B form an ideal solution in the entire composition range. At 350 K, the vapour pressures of pure A and pure B are \[7\times {{10}^{3}}\] and \[12\times {{10}^{3}}\] Pa, respectively. The composition of the vapour in equilibrium with a solution containing 40 mole per cent of A at this temperature is:
In Antirrhinum majus, the Red (RR) flowered plant crossed with white flowered (rr) plant & in \[{{F}_{1}}\] generation pink (Rr) flowered plants obtained on selfing of \[{{F}_{1}}\] generation, \[{{F}_{2}}\] generation obtained. In which the ratio of Red & white flowered plants will -