The locus of centroid of the triangle whose vertices are \[\left( acos\text{ }t,\text{ }asin\text{ }t \right),\] (b sin t, - bcos t) and (1,0) where t is a parameter, is
The number of possible tangents which can be drawn to the curve \[4{{x}^{2}}-9{{y}^{2}}=36\], which are perpendicular to the straight line \[5x+2y-10=0\] is
If \[x\frac{{{d}^{2}}y}{d{{x}^{2}}}+\frac{dy}{dx}=\ln x,\,y(l)=\text{l }\]and \[y'(\text{l})=-\text{l}\] then \[y''\left( \frac{1}{3} \right)\], equals
Let \[S\left( 5,\text{ }12 \right)\]and \[S'\left( -12,\text{ }5 \right)\]are the foci of an ellipse passing through the origin. The eccentricity of ellipse equals
The values of a for which the points of extremism of the function \[f\left( x \right)={{x}^{3}}-3\alpha {{x}^{2}}+3\left( {{\alpha }^{2}}-1 \right)\text{ }x+1\]lie in the interval \[\left( -2,\text{ }4 \right)\]will be equal to
Let \[A=\left[ _{0}^{\sin \theta }\,\,_{-\sin \theta }^{0} \right]\]. If \[A\text{ + }{{A}^{T}}\] is a null matrix, then the number of values of \[\theta \] in (0, 6), is
A person standing on the bank of a river observes that the angle of elevation of the top of the tree on the opposite bank of the river is \[60{}^\circ \] and when he retires 40 m away from the tree, the angle of elevation becomes \[30{}^\circ \]. The breadth of the river is
Let \[\overrightarrow{a},\overrightarrow{b}\] are unit vectors and c is a vector of magnitude 2. If these three vectors are such each one of them is perpendicular to the sum of the other two, then the value of \[|a+b+c|\] is
Of all the five-digit numbers that can be formed with the digits 0, 1, 3, 5, 7 and 9 without repetition of the digits, a number is randomly selected. The probability that it is divisible by 10, is
If a quadrilateral is formed by four tangents to the ellipse \[\frac{{{x}^{2}}}{9}+\frac{{{y}^{2}}}{4}=1\] is a square, then the area of the square is equal to
Let \[f(x)=\left\{ _{a{{x}^{2}}+bx+c\,\,\,\,\,\,\,\,\,\,\,\,x>1}^{{{x}^{3}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\le 1} \right.\,\] If \[f''\left( x \right)\]is continuous everywhere, then which one of the following is correct?
Let \[{{x}_{n}}\] be the sequence of numbers denoted by\[{{x}_{n}}=\frac{195}{4{{P}_{n}}}-\frac{^{n+3}{{P}_{3}}}{{{P}_{n+1}}}(n\in N)\] where \[{{P}_{n}}\] denotes the number of ways in which n distinct things can be arranged on n different places in a definite order. The sum of all possible values of \[n\in N\] for which \[{{x}_{n}}>0\], is
The number of ordered triplets (x, y, z) of real numbers that satisfy the equation \[{{({{\sin }^{-1}}x)}^{2}}=\frac{{{\pi }^{2}}}{4}+{{({{\sec }^{-1}}y)}^{2}}+{{({{\tan }^{-1}}z)}^{2}}\]is
Let \[{{L}_{1}}:\overrightarrow{r}=\hat{i}-\hat{j}-10\hat{k}+\lambda (2\hat{i}-3\hat{j}+8\hat{k})\] and \[{{L}_{2}}:\overrightarrow{r}=4\hat{i}-3\hat{j}-\hat{k}+\mu (\hat{i}-4\hat{j}+7\hat{k})\] represent two lines in R3, then which one of the following is incorrect?
A)
\[{{L}_{1}}\] is parallel to the vector \[4\hat{i}-6\hat{j}+16\hat{k}.\]
doneclear
B)
\[{{L}_{2}}\]is parallel to the vector \[-\hat{i}+4\hat{j}-7\hat{k}.\].
doneclear
C)
\[{{L}_{1}}\] and \[{{L}_{2}}\] are coplanar.
doneclear
D)
Angle between the lines \[{{L}_{1}}\] and \[{{L}_{2}}\] is \[{{\cos }^{-1}}\left( \frac{70}{11\sqrt{7}} \right)\].
\[\underset{x\to 0}{\mathop{Lim}}\,\frac{\int\limits_{0}^{x}{{{\left( {{t}^{2}}+{{e}^{{{t}^{2}}}} \right)}^{\frac{1}{1-\cos \,t}}}dt}}{({{e}^{x}}-1)}\] is equal to
The length of a given cylindrical wire is increased by 100% due to consequent change in diameter The percentage change in the resistance of the wire will be:
Choose current statement for a straight line motion
A)
if \[a=-3\text{ }m{{s}^{-2}}\]then the motion must be retarded motion.
doneclear
B)
If a particle is project vertically up they towards positive y-axis than acceleration will be negative at the time of ascent and positive at the time of its decent.
doneclear
C)
When velocity is zero acceleration must be zero.
doneclear
D)
motion is said to be retarded motion when acceleration and velocity have apposite sign.
What is the minimum energy required to launch a satellite of mass 'm? from surface of the planet of mass 'M' and radius 'R' in a circular orbit at an altitude '3R'.
The amplitude of a clamped oscillator decreases to half of its original length in 2 sec. then in next 4 sec. it becomes \[\alpha \] times of original amplitude where a will be:
A wall is moving with velocity u and a source of sound moves with velocity \[\frac{u}{2}\] in the same direction as shown in the figure. Assuming that the sound travels with velocity 10 u. The ratio of incident sound wavelength on the wall to the reflected sound wavelength by the wall, is equal to:
Four charges each equal to - Q are placed at the four comers of a square and a charge 'q? is at its centre. If the system is in equilibrium the value of q is:
\[30\Omega \]and \[12\Omega \] resistance are connected with a 84V ideal battery in series. A voltmeter of \[20\Omega \] resistance is used to measure the potential difference across \[30\Omega \] resistance. The reading of the voltmeter is:
Proton, deuteron and alpha particle of same kinetic energy are moving in circular trajectories in a constant magnetic field. The radii of proton deuteron and alpha particle are respectively \[{{\gamma }_{p}},{{\gamma }_{d}}\]and \[{{\gamma }_{\alpha }}\] then their relation will be:
In a region of space magnetic field exists in a cylindrical region of radius 'a' centered at origin. The field is given by: \[\overrightarrow{B}={{B}_{0}}t\hat{k}\]. There for experienced by a stationary charge q placed at (r,0,0) where r > a, is:
A metallic rod of length is tied to a suing of length \[(3\ell )\] and made to rotate with constant angular velocity \[(\omega )\] on horizontal table with one end of the string is fixed. There is a vertical magnetic field B in the region, the emf induced accross the ends of the rod is:
An unpolarised beam of light of intensity I is incident on a set of four polarising plates such that each plate makes an angle of \[\pi /3\] with preceding sheet. The intensity of light transmitted through the combination is:
In Young's experiment the source is red light of wave length \[7\times {{10}^{-7}}m\]. When a thin glass plate of refractive index 1.5 at this wavelength is put in the path of one of the interfering beams, the central bright shifts by \[{{10}^{-3\text{ }}}m\] to the position previously occupied by the 5th bright fringe then the thickness of the plate is :
A radioactive sample P having activity \[10\mu C\] has twice the number of nuclei as another sample Q which has an activity \[20\mu C\], The half lives of P and Q can be:
A proton when accelerated through 24000 V has wavelength associated with it. An a particle in order to have the same wave length must be accelerated through:
In a hydrogen atom, the binding energy of electron in the ground state is \[{{E}_{1}}\] then the frequency of revolution of the electron in the nth orbit is:
A transistor is connected in common emitter configuration. The collector supply is 8V and the voltage drop accross a resistor of \[800\Omega \] in the collector circuit is 0.5 V. If the current gain factor a is 0.96, then the base currents is :
A light emitting diode has a voltage drop of 2 volt across it and passes a current of 10 mA when it operates with 6 volt battery through a limiting resistor R. The value of R is:
If the TV - Telecast is to cover a radius of 120 km then the height of the transmitting antenna is (Use radius of the earth \[{{R}_{e}}=\text{ }6400\]km)
An emf of 15 volt is applied in a circuit containing 5 Henry inductance and \[10\Omega \] resistance. The ratio of the current at time \[t=\infty \] and \[t=1\text{ }sec.\]is :
An LCR series circuit with a resistance of \[100\sqrt{5}\Omega \] is connected to an AC-source of 200 V when the capacitor is removed from the circuit, current lags behind emf by \[{{45}^{o}}\]. When the inductor is removed from the circuit, keeping capacitor and resistor in the circuit current leads by an angle of \[{{\tan }^{-1}}\left( \frac{1}{2} \right)\]. Then the current in the LCR circuit will be:
Compound 'X' \[({{C}_{5}}{{H}_{8}}{{O}_{2}})\] forms a dioxime with \[N{{H}_{2}}OH\]. On Clemmensen reduction it forms a compound which forms only one monochloro derivative 'X' can be:
A mixture of [a] and [b], which are two miscible liquids, is under equilibrium conditions at atmospheric pressure. The mole fraction of [a] in solution is 0.3 and in vapour phase is 0.6. If the solution behaves ideally the ratio of \[P_{a}^{o}\] to \[P_{b}^{o}\] is:
A zero-order reaction, A\[\to \] Product, with an initial concentration \[2{{[A]}_{0}},\] has a half-life of 0.2 sec. If one starts with the concentration 2[A]y then the half- life is:
(I) When copper ore is mixed with silica, in a reverberatory furnace copper matte is produced. The copper matte contains sulphides of \[C{{u}^{+2}}\] and \[F{{e}^{+2}}|\]ion.
(II) Zone refining is based on the principle that impurities are more soluble in molten metal than in solid metal.
(III) In the metallurgy of aluminum, graphite anode is oxidised to carbon monoxide and carbon dioxide.
A crystal of formula \[\mathbf{A}{{\mathbf{B}}_{\mathbf{3}}}\] has A ions at the cube corners and B ions at the edge centers. The coordination numbers of A and B are respectively
Graph between log k and \[\frac{1}{T}\] (k is rate constant in \[{{s}^{-1}}\] and T is the temperature in K) is a straight line. As shown in figure if \[OX=5\]and slope of the line = \[-\frac{1}{2.303}\]then \[{{E}_{a}}\] is:
Equivalent conductance of \[1\,M\,\,C{{H}_{3}}COOH\] is \[10\text{ }oh{{m}^{-1}}\text{ }c{{m}^{2}}\text{ }equi{{v}^{-1}}.\]and that at infinite dilution is \[200\text{ }oh{{m}^{-1}}\text{ }c{{m}^{2}}\text{ }equi{{v}^{-1}}.\]Hence % ionisation of \[C{{H}_{3}}COOH\] is:
A solution containing one mol per liter each of \[Cu{{(N{{O}_{3}})}_{2}},\] \[AgN{{O}_{3}},\] \[Hg{{(N{{O}_{3}})}_{2}}\] and \[Mg{{(N{{O}_{3}})}_{2}}\] is being electrolysed by using inert electrodes. The values of the standard oxidation potentials in volts are \[Ag/A{{g}^{+}}=-0.8V;\] \[Hg/H{{g}^{2+}}=-0.79V;\]. The order in which metals will be formed at cathode will be -