If the value of the definite integral\[\int\limits_{-4\pi \sqrt{2}}^{4\pi \sqrt{2}}{\left( \frac{\sin x}{1+{{x}^{4}}}+1 \right)dx}\] is equal to\[k\pi \], then the value of \[k\]is
An aeroplane flying with uniform speed horizontally 1 kilometre above the ground is observed at an elevation of\[{{60}^{o}}\]. After 10 seconds if elevation is observed to be \[{{30}^{o}}\], then the speed of the plane (in kilometre /hour) is
A house of height 100m subtends a right angle at the window of an opposite house. If the height of the window is 64m, then the distance between the two houses is
In a class of 100, students there are 70 boys whose average marks in a subject are 75. If the average marks of the complete class is 72, then what is the average of the girls?
The median of a set of 9 distinct observations is\[20.5\]. If each of the largest 4 observations of the set is increased by 2, then the median of the new set
Let \[f(t)={{t}^{2}}\] for \[0\le t\le l\] and \[g(t)={{t}^{3}}\] for\[0\le t\le l\]. The value of c with \[0<c<l\] at which \[\frac{f(l)-f(0)}{g(l)-g(0)}=\frac{f'(c)}{g'(c)}\], is
Let A and B be two events such that\[P(\overline{A\cup B})=\frac{1}{6}\], and \[P(\overline{A})=\frac{1}{4}\], where \[\overline{A}\] stands for complement of event A. Then, events A and B are
If the area of auxiliary circle of the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=l\,;\,(a>b)\] is twice the area of the ellipse, then the eccentricity of the ellipse is
If \[{{x}_{1}},{{x}_{2}},\,..........\,,{{x}_{18}}\] are observations such that \[\sum\limits_{j=l}^{18}{({{x}_{j}}-8)=9}\] and \[\sum\limits_{j=l}^{18}{{{({{x}_{j}}-8)}^{2}}=45}\], then the standard deviation of these observations is
Sum of the abscissa and ordinate of the centre of the circle touching the line \[3x+y+2=0\] at the point \[M(-1,\,\,1)\] and passing through the point N(3, 5) is
The function \[f(x)=\frac{l-\sin x+\cos x}{l+\sin x+\cos x}\] in not defined at \[x=\pi \]. The value off \[f(\pi )\], so that \[f(x)\] is continuous at \[x=n\], is
Hat W contains two white balls and one black ball. Hat B contains two black balls and one white ball. At random, one of the following strategies is selected:
Two different balls are drawn from hat W.
Two different balls are drawn from hat B.
One ball is drawn from each hat.
The probability of getting at least one white ball, is
Let \[y=f(x)\] satisfies the differential equation \[(1+{{x}^{2}})\frac{dy}{dx}+2xy=2x\] and \[f(0)=2\]. Then the number of integers in the range of \[f(x)\] is
Let the term independent of x in the expansion of \[{{\left( {{x}^{2}}-\frac{l}{x} \right)}^{9}}\] has the value p and q be the sum of the coefficients of its middle terms, then \[(p-q)\] equals
Consider the ten numbers\[ar,a{{r}^{2}}a{{r}^{3}},\,.......\,,\,a{{r}^{10}}\]. If their sum is 18 and the sum of their reciprocals is 6 then the product of these ten numbers, is
Let matrix A of order 3 is such that \[{{A}^{2}}=2A-I\] where \[I\] is an identity matrix of order 3. Then for \[n\in N\] and \[n\ge 2,\,{{A}^{n}}\] is equal to
If the derivative of the function \[f(x)\] is everywhere continuous and is given by \[f(x)=\left\{ \begin{matrix} b{{x}^{2}}+ax+4; & x\ge -l \\ a{{x}^{2}}+b; & x<-l \\ \end{matrix} \right.\] , then
Statement 1: Consider two curves \[{{C}_{l}}:z\overline{z}+i\overline{z}+b=0\] and \[(b\in R,z=x+iy\] and \[i=\sqrt{-1})\]. where\[(b\in R,\,z=x+iy\] and \[i=\sqrt{-1}\]). If \[{{C}_{1}}\] and \[{{C}_{2}}\] intersects orthogonally then\[b=-2\].
Statement 2: If two curves intersects orthogonally then the angle between the tangents at all their points of intersection is\[\frac{\pi }{2}\].
A)
Statement-1 is true, Statement-2 is true and Statement-2 is correct explanation for Statement-1.
doneclear
B)
Statement-1 is true, Statement-2 is true and Statement-2 is NOT the correct explanation for Statement-1.
Statement 1: The circles \[{{x}^{2}}+{{y}^{2}}=9\] and \[(2x-3)(x-1)+2y(y-\sqrt{6})=0\] touches each other internally.
Statement 2: Circle described on the focal distance as diameter of the ellipse \[8{{x}^{2}}+9{{y}^{2}}=72\] touches the auxiliary circle \[{{x}^{2}}+{{y}^{2}}=9\] internally.
A)
Statement-1 is true, Statement-2 is true and Statement-2 is correct explanation for Statement-1.
doneclear
B)
Statement-1 is true, Statement-2 is true and Statement-2 is NOT the correct explanation for Statement-1.
In the arrangement shown in the figure ends P and Q of an unstretchable string move downwards with uniform speed U. Pulleys A and B are fixed. Mass M moves upwards with a speed.
Two blocks of masses M and m are connected with either end of a massless string which pass over a smooth pulley\[(M>m)\]. The pulley is connected with the roof of an elevator moving with acceleration 'a' upwards. The relative acceleration of block M relative to block m is 2a. Ratio \[\frac{M}{m}\] is:
A thin wire of length L and uniform linear mass density p is bent into a circular loop with centre at The moment of inertia of the loop about the tangent to the loop in its plane \[x-x'\] is:
Time period (T) and amplitude are same for two particles which undergo SHM along the same line. At one particular instant, one particle is at phase \[\frac{3\pi }{2}\] and other is at phase zero, while moving in the same direction. Find the time at which they will cross each other.
A solid sphere of mass M and of uniform density and radius 4R is located with its centre at origin, O. Two spheres of equal radii R with their centres at A (2R, 0, 0) and B(0, 2R, 0) respectively are taken out of the solid sphere leaving spherical cavities. The gravitational force at the origin will be (\[R=1\] Unit)
A block of ice at \[(-{{10}^{o}}C)\] is slowly heated and converted to steam at \[{{100}^{o}}C\]. Which of the following curves represent the phenomena qualitatively [\[T=\] temperature, \[H=\] Heat applied]
Two cylinders fitted with pistons and placed as shown, connected with string through a small tube of negligible volume, are filled with a gas at pressure \[{{P}_{0}}\] and temperature \[{{T}_{0}}\]. The radius of smaller cylinder is half of the other. If the temperature is increased to \[2{{T}_{0}}\], find the pressure, if the piston of bigger cylinder moves towards left by 1 metre?
The displacement y of a particle executing periodic motion is given by:\[y=4{{\cos }^{2}}\left( \frac{1}{2}t \right)\sin 1000\,t)\] How many independent harmonic motions may be considered to be a result of the superposition by this expression?
Yellow light is used in a single slit diffraction experiment with slit width of \[0.6\] mm If yellow light is replaced by x-rays, then the observed pattern will reveal.
A particle combination of \[0.1\] MO resistor and a \[10\mu F\] capacitor is connected across a \[1.5\] V source of negligible resistance. The time required for the capacitor to get charged upto \[0.75\] V is approximately (in sec.)
Three spherical conductors \[{{S}_{1}},{{S}_{2}}\] and \[{{S}_{4}}\] of radii a, b and d respectively \[(a<b<d)\] are arranged concentrically. \[{{S}_{1}}\] and \[{{S}_{2}}\] are connected and \[{{S}_{4}}\] is grounded. Find the equivalent capacitance of the system
In hydrogen atom, electron moves in an orbit of radius \[0.5\,\overset{0}{\mathop{A}}\,\] making \[{{10}^{16}}\] revolutions per second. The magnetic moment associated with orbital motion of the electron is :
A thin semicircular conducting ring of radius R is falling with its plane vertical in a horizontal magnetic induction\[\vec{B}\]. At the position MNQ the speed of the ring is v and the potential difference developed across the ring is:
A)
zero
doneclear
B)
\[B\pi v{{R}^{2}}/2\] and M is at higher potential
A series circuit is having resistance of \[22\,\Omega \] and impedance of \[44\,\Omega \] 220 V(rms) is applied across the circuit. The power consumed is:
The time during which three-fourth of a sample will decay if decaying both by \[\alpha \] and \[\beta \] -emission simultaneously is 312 year. The mean life of this sample is 900 years for \[\alpha \] - emission. Find the mean life of this sample for \[\beta \] - emission.
Consider \[\alpha \] - particle, \[\beta \] - particle and \[\gamma \]- rays each having an energy of \[0.5\] MeV. In increasing order of penetrating powers, the radiations are:
A stationary hydrogen atom of mass m in the ground state achieve minimum excitation energy after head-on, inelastic collision with a moving hydrogen atom. Find the velocity of moving hydrogen atom.
A transistor has current amplification factor (current gain) of 50. In a CE - amplifier circuit the collector resistance is\[1\,k\Omega \]. The output voltage if input voltage is 0.01 V
The time period of a simple pendulum is 2 sec and its amplitude is 10 degree. Its amplitude decreases to 5degree after 10 oscillations. The ?relaxation time? (in sec.) will be :
The magnetic moment of a bar magnet is \[5\times {{10}^{-5}}\] web-m and it is being suspended in a magnetic field of \[8\times {{10}^{-4}}\]tesla. The time period of oscillation of magnet is 15 sec, the M.I. of the bar magnet (in kg-m2) will be:
In a solid AB having NaCI structure, atom A occupy the comers of cubic unit cell. If all the atoms along one of the body diagonal axis are removed then resultant stoichiometry of the solid is
For the cell, \[Hg\left| H{{g}_{2}}C{{l}_{2}} \right|C{{l}^{-}}(0.1\,M)||\,C{{l}^{-}}(0.01\,M)\]\[\left| C{{l}_{2}},Pt\,{{E}^{o}}_{cell}=1.10\,V \right.\], Hence \[{{E}_{cell}}\] is
The ammonia gas evolved from the treatment of \[0.30\] g of an organic compound for the estimation of \[{{N}_{2}}\], was passed in 100 ml of \[0.1\] M \[{{H}_{2}}S{{O}_{4}}\]. The excess of acid required 20 ml \[0.5\] NaOH, solution for complete neutralization. The organic compound is:
\[PC{{l}_{5}}(g)\overset{{}}{leftrightarrows}PC{{l}_{3}}(g)+C{{l}_{2}}(g)\]. In above reaction, at equilibrium condition mole fraction of \[PC{{l}_{5}}\] is \[0.4\] and mole fraction of\[C{{l}_{2}}\], is\[0.3\]. Then find out mole fraction of \[PC{{l}_{3}}\]
First nucleophilic attack in the reaction.\[\phi -C\underset{A}{\mathop{H}}\,O+\underset{B}{\mathop{HCHO}}\,\xrightarrow{conc.O{{H}^{-}}}\] should occur on