If in \[\Delta ABC,\]\[A=\,(1,10)\], circumcentre \[\left( -\frac{1}{3},\frac{2}{3} \right)\]and orthocentre \[\left( \frac{11}{3},\frac{4}{3} \right)\] The coordinates of the mid-point of the sides opposite to A is
Given a, b, c are positive integers forming an increasing GP, \[b-a\] is a perfect square of a natural number and \[{{\log }_{6}}a+{{\log }_{6}}b+{{\log }_{6}}c=6,\] then \[a+b+c\] is equal to
Let set 'A' has 7 elements and set B has 5 elements. If one function is selected from all possible defined function from A to B, then the probability that it is onto
If \[\frac{{{a}_{0}}}{n+1}+\frac{{{a}_{1}}}{n}+\frac{{{a}_{2}}}{n-1}+...+\frac{{{a}_{n}}-1}{2}{{a}_{n}}=0,\]then the equation \[{{a}_{0}}{{x}^{n}}+{{a}_{1}}{{x}^{n-1}}+{{a}_{2}}{{x}^{n-2}}+...+{{a}_{n-1}}x+{{a}_{n}}=0\] has
Two mutually perpendicular tangents of the parabola \[{{y}^{2}}=4ax\]meet its axis in \[{{P}_{1}}\]and \[{{P}_{2}}\]If S is the focus of the parabola, then \[\frac{1}{S{{P}_{1}}}+\frac{1}{S{{P}_{2}}}\] is equal to
Let \[S=\left\{ x\in (-\pi ,\pi ):x\ne 0,\pm \frac{\pi }{2} \right\}.\] The sum of all distinct solution of the equation\[\sqrt{3}\sec x+\text{cosec}\,x+2\,(\tan x-\cot x)=0\]in the set S is equal to
Let C be the circle with centre at \[(1,1)\] and radius = 1. If T is the circle centred at \[(0,y)\] passing through origin and touching the circle C externally, then the radius of T is equal to
Two infinitely large charged planes having uniform surface charge density \[+\,\sigma \] and \[-\,\sigma \] are placed along x-y plane and yz plane respectively as shown in the figure. Then the nature of electric lines of forces in x-z plane is given by:
An \[\alpha \] particle is moving along a circle of radius R with a constant angular velocity \[\omega .\] Point A lies in the same plane at a distance 2R from the centre. Point A records magnetic field produced by \[\alpha \] particle. If the minimum time interval between two successive times at which A records zero magnetic field is 't', the angular speed \[\omega ,\] in terms of t is:
Radius of a circular ring is changing with time and the coil is placed in uniform constant magnetic field perpendicular to its plane. The variation of 'r' with time 't' is shown in the figure. Then induced e.m.f. \[\varepsilon \] with time will be best represented by:
Three identical capacitors are given a charge Q each and they are then allowed to discharge through resistance \[{{R}_{1}},\]\[{{R}_{2}}\] and \[{{R}_{3}}\] separately. Their charges, as a function of time are shown in the graph below. The smallest of the three resistances is
Two coherent light sources P and Q each of wavelength \[\lambda \] are separated by a distance \[3\,\lambda \]as shown. The maximum number of minima formed on line AB which runs from \[-\,\infty \] to \[+\,\infty \]is:
A heavy nucleus having mass number 200 gets disintegrated into two small fragments of mass number 80. and 120. If binding energy per nucleon for parent atom is 6.5 MeV and for daughter nuclei is 7 MeV and 8 MeV respectively, then the energy released in the decay will be:
A rod of length \[\ell \] is in motion such that its ends A and B are moving along x-axis and y-axis respectively. It is given that \[\frac{d\theta }{dt}=2\,\,rad/s\] always. P is a fixed point on the rod. Let M be the projection of P on x-axis. For the time interval in which \[\theta \] changes from 0 to \[\frac{\pi }{2},\] choose the correct statement,
A)
The acceleration of M is always directed towards right
A small mass slides down an inclined plane of inclination \[\theta \] with the horizontal. The co-efficient of friction is \[\mu ={{\mu }_{0}}\,x\]where x is the distance through which the mass slides down and \[{{\mu }_{0}}.\] a constant. Then the speed is maximum after the mass covers a distance of
A source of frequency 'f' is stationary and an observer starts moving towards it at \[t=0\] with constant small acceleration. Then the variation of observed frequency f' registered by the observer with time is best represented as:
A tunnel is dug in the earth across one of its diameter. Two masses' m' &' 2 m' are dropped from the ends of the tunnel, The masses collide and stick to each other and perform S.H.M. Then amplitude of S.H.M. will be: [\[R=\]radius of the earth]
A small uniform tube is bent into a circular tube of radius R and kept in the vertical plane. Equal volumes of two liquids of densities \[\rho \] and \[\sigma \,(\rho >\sigma )\] fill half of the tube as shown. \[\theta \] is the angle which the radius passing through the interface makes with the vertical.
All electrons ejected from a surface by incident light of wavelength 200 nm can be stopped before travelling 1 m in the direction of uniform electric field of 4 N/C. The work function of the surface is:
A point mass 'm' and charge 'q' is projected with a velocity v towards a stationary charge \[{{Q}_{0}}\] from a distance of 2 m. The closest distance that q can approach is:
The magnetic flux \[\phi \] through a metal ring varies with time t according to: \[\phi =3\,(a{{t}^{3}}-b{{t}^{2}})T{{m}^{2}},\] with \[a=2{{s}^{-\,3}}\]and \[b=6\,{{s}^{-\,2}}.\] The resistance of the ring is \[3\,\Omega \,.\]The maximum current induced in the ring during the interval \[t=0\] to \[t=2s,\] is
A coin is released inside a lift at a height of 2 m from the floor of the lift. The height of the lift is 10 m. The lift is moving with an acceleration of \[9\,m/{{s}^{2}}\] downwards. The time after which the coin will strike with the lift is:\[(g=10m/{{s}^{2}})\]
The half-life of a radioactive isotope is 3 hours. If the initial mass of the isotope were 256 gm, the mass of it remaining undecayed after 18 hours would be -
Dinucleotide is obtained by joining two nucleotides together by phosphodiester linkage. Between which carbon atoms of pentoe sugars of nucleotides are these linkages present?
In antirrhinum (Snapdragon), a red flower was crossed with a white flower and in \[{{F}_{1}}\]generation pink flowers were obtained. When pink flowers were selfed, the \[{{F}_{2}}\]generation showed white, red and pink flowers. Choose the incorrect statement from the following:
A)
This experiment does not follow the Principle of Dominance.
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B)
Pink colour in \[{{F}_{1}}\] is due to incomplete dominance.
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C)
Ratio of \[{{F}_{2}}\] is \[\frac{1}{4}\] (Red) : \[\frac{2}{4}\](Pink) : \[\frac{1}{4}\](White)
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D)
Law of Segregation does not apply in this Experiment
The tangent to the curve \[y={{e}^{x}}\] drawn at the point \[(c,{{e}^{c}})\] intersects the line joining the points \[(c-1,{{e}^{c-1}})\] and \[(c+1,{{e}^{c+1}})\] is
Let \[g\,(x)=\frac{{{(x-1)}^{n}}}{\log {{\cos }^{m}}(x-1)};0<x<2,m\] and n are integers, \[m\ne 0,\]\[n>0,\] and let P be the left hand derivatives of \[\left| x-1 \right|\] at \[x=1.\] If \[\lim g(x)=P,\] then
The number of \[3\times 3\] matrices A whose entries are either 0 or 1 and for which the system
\[A\,\left[ \begin{matrix}
x \\
y \\
z \\
\end{matrix} \right]=\left[ \begin{matrix}
1 \\
0 \\
0 \\
\end{matrix} \right]\] has exactly two distinct solutions, is
Let w be a complex cube root of unity with \[w\ne 1.\] A fair die is thrown three times. If \[{{r}_{1}},{{r}_{2}}\] and \[{{r}_{3}}\] are the numbers obtained on the die, then the probability \[{{w}^{{{r}_{1}}}}+{{w}^{{{r}_{2}}}}+{{w}^{{{r}_{3}}}}=0\]
Let P, Q, R and S be the points on the plane with position vectors \[-\,2i-j,\]\[4i,\]\[3i+3j\] and \[-\,3i+2j\] respectively. The quadrilateral PQRS must be
A)
parallelogram, which is neither a rhombus nor a rectangle
Let \[({{x}_{0}},{{y}_{0}})\] be the solution of the following equations \[{{(2x)}^{\ln 2}}={{(3y)}^{\ln 3}},{{3}^{\ln x}}={{2}^{\ln y}},\] then \[{{x}_{0}}\]
For a real number x let [x] denote the largest integer less or equal to x and \[\{x\}=x-[x].\]The smallest integer value of x for which \[\int\limits_{1}^{n}{[x]}\{x\}dx\] exceeds 2020 is
A cylindrical optical fibre (quarter circular shape) of refractive index n = 2 and diameter d = 4mm is surrounded by air. A light beam is sent into the fibre along its axis as shown in figure. Then the smallest outer radius R (as shown in figure) for which no light escapes during first refraction from curved surface of fibre is:
A charge passing through a resistor is varying with time as shown in the figure. The amount of heat generated in time 't' is best represented (as a function of time) by:
In a Young's double slit experiment, \[d\,=\,1\,\,mm\], \[\,=\,6000\,\overset{o}{\mathop{A}}\,\] and \[D=1\,m\] (where d, \[\] and D have usual meaning). Each of slit individually produces same intensity on the screen. The minimum distance between two points on the screen having 75 % intensity of the maximum intensity is :
An ac voltage source \[V\,=\,{{V}_{0}}\,\sin \,\,\omega t\] is connected across resistance R and capacitance C as shown in figure. It is given that \[R\,=\,\frac{1}{\omega \,C}\]. The peak current is \[{{I}_{0}}\]. If the angular frequency of the voltage source is changed to \[\frac{\omega }{\sqrt{3}}\] then the new peak current in the circuit is :
A capillary tube is made of glass having index of refraction n and is surrounded by air. The outer radius of the tube is R. The tube is filled with a liquid having index of refraction n' (n' < n). For any ray that hits the outer surface of tube from air as shown to also enter the liquid, the minimum internal radius r of the tube is given by:
A stone is thrown horizontally under gravity with a speed of 10m/sec. Find the radius of curvature of it's trajectory at the end of 3 sec after motion began.
A particle of mass m initially at rest, is acted upon by a variable force F for a brief interval of time T. It attains a velocity u after the force stops acting. F is shown in the graph as a function of time. The curve is a semicircle, find u.
The electric potential in a region is given by \[V\,=\,(2{{x}^{2}}\,-\,3y)\] volt where x and y are in meters. The electric field intensity at a point (0, 3m, 5m) is
A solid conducting sphere of radius R is moved with a velocity V in a uniform magnetic field of strength B such that \[\overrightarrow{B}\] is perpendicular to \[\overrightarrow{V}\]. The maximum e.m.f. induced between two points of the sphere is:
A parallel plate capacitor is charged to a potential difference of 100 V and disconnected from the source of emf. A slab of dielectric is then inserted between the plates. Which of the following three quantities change?
Triplet state of Reaction intermediates like carbene & nitrene are more stable than their singlet state. This fact can be explained on the basis of -
A)
Triplet intermediates can show syn addition whereas singlet intermediates can also show the ant addition
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B)
State of minimum multiplicity is most stable state due to least energy
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C)
The paired electronic configuration provides maximum multiplicity, which leads to more stabilization for intermediate
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D)
The electron is separate orbitals bring the state of maximum multiplicity according to formula 2S + 1, where 5 = sum of spin quantum numbers for all the electrons. This leads to net decrease in energy according to hunds rule of maximum multiplicity
Hyper conjugation (HC) and reverse hyper conjugation (RHC) both are the example of \[\sigma -\pi \] conjugation. Which one of following statement is correct about the above two electronic effect -
A)
HC directing the \[{{E}^{\otimes }}\] at meta position where as RHC activates the \[{{E}^{\otimes }}\] at ortho & para position
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B)
HC directing the \[{{E}^{\otimes }}\] at ortho position RHC directing the \[{{E}^{\otimes }}\] at meta & para position
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C)
HC directing the \[{{E}^{\otimes }}\] at ortho & para position, RHC directing \[N{{u}^{\otimes }}\] at ortho & para position
A reaction \[2{{A}_{(g)}}\,\,\xrightarrow{k}\,\,{{B}_{(g)}}+3{{C}_{(g)}}\] has \[k=0.98\,\,h{{r}^{-1}}\] initial pressure of A is \[3{{P}_{0}}\] and that of B and C is zero. After time ?t? the total pressure due to products is \[{{P}_{0}}.\] The value of kt for the reaction is:
From evolutionary point of view, retention of the female gametophyte with developing young embryo on the parent sporophyte for some time, is first observed in: