The sum of the common terms of the following three arithmetic progressions\(3,7,11,15, \ldots , 399\), \(2,5,8,11, \ldots , 359\)and \(2,7,12,17, \ldots , 197,\) is equal to _____
If the x-intercept of a focal chord of the parabola \(y^2=8 x+4 y+4\) is 3 , then the length of this chord is equal to ___
Let \(\alpha x+\beta y+y z=1\) be the equation of a plane passing through the point\((3,-2,5)\)and perpendicular to the line joining the points \((1,2,3)\) and \((-2,3,5)\) Then the value of \(\alpha \beta y\)is equal to ____
The line \(x=8\) is the directrix of the ellipse \(E : \frac{x^2}{ a ^2}+\frac{y^2}{b^2}=1\)with the corresponding focus \((2,0)\) If the tangent to \(E\)at the point \(P\) in the first quadrant passes through the point \((0,4 \sqrt{3})\)and intersects the\(x\)-axis at \(Q\), then \((3 PQ )^2\)is equal to ____
$0.3\, g$ of ethane undergoes combustion at $27^{\circ} C$ in a bomb calorimeter The temperature of calorimeter system (including the water) is found to rise by $05^{\circ} C$ The heat evolved during combustion of ethane at constant pressure is ___$kJ \, mol ^{-1}$ (Nearest integer) [Given : The heat capacity of the calorimeter system is $20 \, kJ\, K ^{-1}, R =83 \, JK ^{-1}\, mol ^{-1}$ Assume ideal gas behaviour Atomic mass of $C$ and $H$ are 12 and $1\, g\, mol ^{-1}$ respectively]
In the figure given below, a block of mass $M=490 g$ placed on a frictionless table is connected with two springs having same spring constant $\left( K =2 N m ^{-1}\right)$ If the block is horizontally displaced through ' $X$ ' $m$ then the number of complete oscillations it will make in $14 \pi$ seconds will be
A solid sphere of mass $1 \,kg$ rolls without slipping on a plane surface Its kinetic energy is $7 \times 10^{-3} J$. The speed of the centre of mass of the sphere is ___$cm s ^{-1}$.
Expression for an electric field is given by $\overrightarrow{ E }=4000 x^2 i \frac{ V }{ m }$ The electric flux through the cube of side $20 cm$ when placed in electric field (as shown in the figure) is ___$V cm$
In a medium the speed of light wave decreases to $0.2$ times to its speed in free space The ratio of relative permittivity to the refractive index of the medium is $x: 1$ The value of $x$ is(Given speed of light in free space $=3 \times 10^8 m s ^{-1}$ and for the given medium \(\mu =1\))
Let the line $L: \frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-3}{1}$ intersect the plane $2 x+y+3 z=16$ at the point $P$ Let the point $Q$ be the foot of perpendicular from the point $R(1,-1,-3)$ on the line $L$ If $\alpha$ is the area of triangle $P Q R$, then $\alpha^2$ is equal to
Number of 4-digit numbers that are less than or equal to 2800 and either divisible by 3 or by 11 , is equal to
Let for $x \in R$ $f(x)=\frac{x+|x|}{2} \text { and } g(x)=\begin{cases}x, & x<0 \\x^2, & x \geq 0\end{cases} $
Then area bounded by the curve $y=(f \circ g)(x)$ and the lines $y=0,2 y-x=15$ is equal to
For the given circuit, in the steady state, $\left| V _{ B ^{-}}- V _{ D }\right|=$___$V$
Two parallel plate capacitors $C_1$ and $C_2$ each having capacitance of $10 \mu F$ are individually charged by a $100 V D C$ source Capacitor $C_1$ is kept connected to the source and a dielectric slab is inserted between it plates Capacitor $C_2$ is disconnected from the source and then a dielectric slab is inserted in it Afterwards the capacitor $C_1$ is also disconnected from the source and the two capacitors are finally connected in parallel combination The common potential of the combination will be ___$V$(Assuming Dielectric constant $=10$ )
A ball is dropped from a height of $20 m$ If the coefficient of restitution for the collision between ball and floor is $0.5$, after hitting the floor, the ball rebounds to a height of ___$m$
A series $LCR$ circuit consists of $R =80 \Omega, X _{ L }=100 \Omega$, and $X _{ C }=40 \Omega$. The input voltage is 2500 $\cos (100 \pi t ) V$. The amplitude of current, in the circuit, is ___$A$
Two discs of same mass and different radii are made of different materials such that their thicknesses are 1 cm and 0.5 cm respectively. The densities of materials are in the ratio 3:5. The moment of inertia of these discs respectively about their diameters will be in the ratio of \(\frac{x}{6}\). The value of x is ____.
The displacement equations of two interfering waves are given by\(y_1=10 \sin \left(\omega t+\frac{\pi}{3}\right)\) cm , \(y_2=5[\sin \omega t+\sqrt{3} \cos \omega t]\) cm respectively.The amplitude of the resultant wave is ___ cm
If the binding energy of ground state electron in a hydrogen atom is $136 eV$, then, the energy required to remove the electron from the second excited state of $Li ^{2+}$ will be : $x \times 10^{-1} eV$. The value of $x$ is
A water heater of power $2000 W$ is used to heat water. The specific heat capacity of water is $4200 J$ $kg ^{-1} K ^{-1}$ .The efficiency of heater is $70 \%$ .Time required to heat $2 kg$ of water from $10^{\circ} C$ to $60^{\circ} C$ is___$ s$(Assume that the specific heat capacity of water remains constant over the temperature range of the water)
Two bodies are projected from ground with same speeds $40\, ms ^{-1}$ at two different angles with respect to horizontal The bodies were found to have same range If one of the body was projected at an angle of $60^{\circ}$, with horizontal then sum of the maximum heights, attained by the two projectiles, is ____$m$ (Given $g=10 ms ^{-2}$ )
Enthalpies of formation of $CCl _4( g ), H _2 O ( g ), CO _2( g )$ and $HCl ( g )$ are $-105,-242,-394$ and $-92\, kJ$ mol $^{-1}$ respectively The magnitude of enthalpy of the reaction given below is ___$kJ \, mol ^{-1}$ (nearest integer) $CCl _4( g )+2 H _2 O ( g ) \rightarrow CO _2( g )+4 HCl ( g )$
If the CFSE of $\left[ Ti \left( H _2 O \right)_6\right]^{3+}$ is $-960 kJ / mol$, this complex will absorb maximum at wavelength ___$nm$ (nearest integer) Assume Planck's constant $( h )=64 \times 10^{-34} Js$, Speed of light $( c )=30 \times 10^8 m / s$ and Avogadro's Constant $\left( N _{ A }\right)=6 \times 10^{23} / mol$
