The equation of plane which passes through the point of intersection of lines \[\vec{r}=\hat{i}+2\hat{j}+3\hat{k}+\lambda (3\hat{i}+\hat{j}+2\hat{k})\] and \[\vec{r}=3\hat{i}+\hat{j}+2\hat{k}+\mu (\hat{i}+2\hat{j}+3\hat{k})\] where \[\lambda ,\mu \in R\] and has the greatest distance from the origin is
Given three points P, Q, R with P (5, 3) and R lies on the x-axis. If equation of RQ is \[x-2y=2\] and PQ is parallel to the x-axis, then the centroid of \[\Delta PQR\] lies on the line
If M is a \[3\times 3\] matrix such that \[{{M}^{2}}=0\], then det. \[\left( {{(I+M)}^{50}}-50M \right)\] where I is an identity matrix of order 3, is equal to
If Rolle's theorem holds for the function \[f(x)=2{{x}^{3}}+a{{x}^{2}}+bx\] in \[x\in \] [-1,1] for the point \[c=\frac{1}{2}\], then the value of (2a + b) is equal to
Let a and b be any two non-zero real numbers satisfying \[\frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}=\frac{1}{4}\]. Then, the foot of perpendicular drawn from origin on the variable line \[\frac{x}{a}+\frac{y}{b}=1\], lies on
A)
circle of radius 2.
doneclear
B)
parabola of length of latus-rectum 4.
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C)
ellipse with length of semi-major axis 2.
doneclear
D)
hyperbola with length of semi-transverse axis \[\sqrt{2}\].
Let G be the geometric mean of two positive numbers a and b, and M be the arithmetic mean of \[\frac{1}{a}\] and \[\frac{1}{b}\]. If \[\frac{1}{M}:G\] is \[4:5\], then \[a:b\] can be
A number x is chosen at random from the set {1,2, 3,4,.....,100}. Define the event A = the chosen number \[x\] satisfies, \[\frac{(x-10)(x-50)}{(x-30)}\ge 0\]. Then P is equal to
If \[{{z}_{1}},{{z}_{2}}\] and \[{{z}_{3}},{{z}_{4}}\] are 2 pairs of complex conjugate numbers, then arg \[\left( \frac{{{z}_{i}}}{{{z}_{4}}} \right)\] arg \[\left( \frac{{{z}_{2}}}{{{z}_{3}}} \right)\]equals
AB is a vertical pole with B at the ground level and A at the top. A man finds that the angle of elevation of the point A from a certain point C on the ground is 60. He moves away from the role along the line BC to a point D such that CD = 7m from D, the angle of elevation from point A is \[{{45}^{o}}\], then the height of the pole is
Let A (2,3,5), B(-1,3,2) and \[C(\lambda ,5,\mu )\]be the vertices of \[\Delta ABC\]. If the median through A is equally inclined to the coordinate axes, then
The base of an equilateral triangle is along the line given by \[3x+4y=9\]. If a vertex of the triangle is (1, 2), then the length of a side of the triangle is
For the curve defined parametrically as \[y=3\sin \theta \,\cos \theta ,x={{e}^{\theta }}\sin \theta \] where \[\theta \in [0,\pi ]\], the tangent is parallel to x-axis when \[\theta \] is
Let \[{{x}_{1}},{{x}_{2}},{{x}_{3}},{{x}_{4}}\], ........ \[{{x}_{n}}\] be n observations and let \[\overline{x}\] be their arithmetic mean and \[{{\sigma }^{2}}\] be their variance.
STATEMENT-1: Variance of observations\[2{{x}_{1}},2{{x}_{2}},2{{x}_{3}}\], ....., \[2{{x}_{n}}\] is \[4{{\sigma }^{2}}\].
STATEMENT-2: Arithmetic mean of \[2{{x}_{1}},2{{x}_{2}},2{{x}_{3}}\],......, \[2{{x}_{n}}\] is \[4\overline{x}\].
A)
Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1.
doneclear
B)
Statement-1 is true, Statement-2 is false.
doneclear
C)
Statement-1 is false, Statement-2 is true.
doneclear
D)
Statement-1 is true, Statement-2 is true and Statement-2 is correct explanation for Statement-1.
If non-zero real numbers b and c are such that min. \[f(x)>\] max. \[g(x)\], where \[g(x)=-{{x}^{2}}-2cx+{{b}^{2}}(x\in R)\] and\[g(x)=-{{x}^{2}}-2cx+{{b}^{2}}(x\in R)\] then \[\left| \frac{c}{b} \right|\] lies in W interval
A chord is drawn through the focus of parabola \[{{y}^{2}}=6x\] such that its distance from the vertex of this parabola is \[\frac{\sqrt{5}}{2}\], then its slope can be
If the volume of a spherical ball is increasing at the rate of \[4\pi \,cc/\sec \]., then the rate of increase of its radius (in cm. / sec.), when the volume is 288\[\pi \]cc, is
If m is a non-zero real number then f \[\int{\frac{{{x}^{5m-1}}+2{{x}^{4m-1}}}{{{({{x}^{2m}}+{{x}^{m}}+1)}^{2}}}dx}\], is equal to [Note: C denotes constant of integration]
If \[(\alpha ,\beta ,\gamma )\] is a point on the line \[\frac{x-1}{2}=\frac{y+1}{-3}=z\]at distance \[2\sqrt{7}\] from the point M(1, -1, 0), then the coefficient of \[{{x}^{\alpha +\beta +\gamma }}\] in the expansion of \[{{\left( 2{{x}^{2}}-\frac{1}{x} \right)}^{9}}\]is equal to
Two vibrating strings of the same material but lengths L and 2L have radii 2r and r respectively. They are stretched under the same tension. Both the strings vibrate in their fundamental modes, the one of length L with frequency \[{{v}_{1}}\]. The ratio \[{{v}_{1}}/{{v}_{2}}\]
A particle moves along a parabolic path \[y=9{{x}^{2}}\] in such a way that the x-components of 1 velocity remains constant and has a value \[\frac{1}{3}m{{s}^{-1}}\]. The acceleration of the projectile is :
The ratio of radii of the earth to another planet is \[\frac{2}{3}\] and the ratio of their mean densities is \[\frac{4}{5}\]. If an astronaut can jump to a maximum height of \[1.5\] m on the earth, with the same effort, the maximum height he can jump on the planet is :
A steel wire of length \[4.7\] m and cross-section \[3.0\times {{10}^{-5}}{{m}^{2}}\] stretches by the same amount as a copper wire of length \[3.5\] m and cross-section \[4.0\times {{10}^{-5}}{{m}^{2}}\] under a given load. What is the ratio of the Young's modulus of steel to that of copper?
Two soap bubbles combine to form a single bubble. In this process, the change in volume and surface area are respectively V and A. If p is the atmospheric pressure and T is the surface tension of the soap solution, then which of the following relation is true?
A Carnot engine operating between temperatures \[{{T}_{1}}\] and \[{{T}_{2}}\] has efficiency \[0.2\]. When \[{{T}_{2}}\] is reduced by 50 K, its efficiency increases to \[0.4\]. Then, \[{{T}_{1}}\] and \[{{T}_{2}}\] are respectively.
A particle of mass m is executing oscillations about the origin on the X-axis with amplitude A. Its potential energy \[U(x)=a{{x}^{4}}\] where a is positive constant. The x-coordinate of mass where potential energy is one third of the kinetic energy of particle is:
The minimum force required to move a body up and inclined plane is three times the minimum force required to prevent it from sliding down the plane. If the coefficient of friction between the body and the inclined plane is \[\frac{1}{2\sqrt{3}}\] the angle of the inclined plane is:
A solid uniform sphere resting on a rough horizontal plane is given a horizontal impulse directed through its centre sot that it starts sliding with an initial velocity \[{{v}_{0}}\]. When it finally starts rolling without slipping the speed of its centre is :
A bullet of mass 20 g and moving with \[600\,m{{s}^{-1}}\] collides with a block of mass 4 kg hanging with the string. What is velocity of bullet when it comes out of block if block rises to height \[0.2\] m after collision?
A bob of mass m, suspended by a string of length \[{{l}_{1}}\], is given a minimum velocity required to complete a full circle in the vertical plane. At the highest point, it collides elastically with another bob of mass m suspended by a string of length \[{{l}_{2}}\], which is initially at rest. Both the strings are massless and inextensible. If the second bob, after collision acquires the maximum speed so that it will oscillate in the vertical plane, the ratio \[\frac{{{l}_{1}}}{{{l}_{2}}}\] is
A vernier callipers has 1 mm marks on the main scale. It has 20 equal divisions on the vernier scale which match with 16 main scale divisions. For this vernier callipers, the least count is :
The proper life of pion \[({{\pi }^{+}})\] is \[2.5\times {{10}^{-8}}s\] . In a beam of pions travelling with a speed of \[0.9\,c\], the pion in the libratory frame can travel a maximum distance of
When a monochromatic point source of light is at a distance \[0.2\] m from a photoelectric cell, the saturation current and cut-off voltage are \[12.0\] mA and \[0.5\] V. If the same source is placed \[0.4\] m away from the photoelectric cell, then the saturation current and the stopping potential respectively are:
Two polaroids are placed in the path of unpolarised beam of intensity \[{{I}_{0}}\] such that no light is emitted from the second polaroid. If a third polaroid whose polarisation axis makes an angle \[\theta \] with the polaration axis of first polaroid, is placed between these polaroids, then the intensity of light emerging from the last polaroid will be :
A ray of light strikes a material's slab at an angle of incidence \[{{60}^{o}}\]. If the reflected and refracted rays are perpendicular to each other, the refractive index of the materials is :
Light wave is travelling along y-direction. If the corresponding E vector at any time is along the x-axis, the direction of B vector at the time is along
An electric bulb has a rated power of 50 W at 100V. If it is used on an AC source 200 V, 50 Hz, a choke has to be used in series with it. They should have an inductance of:
An uniformly wound solenoid coil of self-inductance \[1.8\times {{10}^{-4}}H\] and resistance \[6\Omega \] is broken up into two identical coils. These identical coils are then connected in parallel across a \[12V\]battery of negligible resistance. The time constant for the current in the circuit is :
A bar magnet when placed at an angle of \[30{}^\circ \] to direction of magnetic field induction of \[5\times {{10}^{-2}}T,\]experience of moment of couple \[25\times {{10}^{-6}}Nm.\] Nm. If the length of the magnet is 5 cm, then its pole strength is:
A heating coil is used to heat water in a container from \[15{}^\circ C\] to \[50{}^\circ C\] in\[20\text{ }mm\]. Two such coils are then joined in series to heat the same amount of water for the same temperature difference from the same constant voltage source. The time taken now is:
A hollow cylinder has a charge q coulomb at the centre within it. If ([) is the electric flux in unit of voltmeter associated with the curved surface B, the flux linked with the plane surface A in unit of voltmeter will be:
Ab all A moving with kinetic energy E, makes a head on elastic collision with a stationary ball of mass n times that of A .The maximum potential energy due to deformation stored in the system during the collision is:
Consider the following equilibrium in a closed container \[{{N}_{2}}{{O}_{4}}(g)2N{{O}_{2}}(g)\] At a fixed temperature, the volume of the reaction container is halved. For this change, which of the following statements holds true regarding the equilibrium constant \[\mathbf{(}{{\mathbf{K}}_{\mathbf{p}}}\mathbf{)}\] and degree of dissociation \[\mathbf{(}\alpha \mathbf{)}\]?
The value of \[{{\log }_{10}}K\] for a reaction \[AB\] is: (Given: \[{{\Delta }_{r}}H_{298K}^{{}^\circ }=-54.07\,kJ\,mo{{l}^{-1}},\] \[{{\Delta }_{r}}S_{298K}^{{}^\circ }=10J{{K}^{-1}}\,mo{{l}^{-1}}\]and \[R=8,314\text{ }J{{K}^{-1}}\text{ }mo{{l}^{-1}};\]\[2.303\times 8.314\times 298=5705\])
\[0.1\text{ }M\] acetic acid solution is titrated against \[0.1\text{ }M\] \[NaOH\] solution What would be the difference in \[\mathbf{pH}\] between \[1/4\] and \[3/4\] stages of neutralisation of acid?
An element X has atomic mass of 200 amu. If it has two isotopes as \[{{\mathbf{X}}^{\mathbf{201}}}\] & \[{{\mathbf{X}}^{\mathbf{198}}},\] the percentage of \[{{\mathbf{X}}^{\mathbf{201}}}\] is:
The equilibrium constants for the reaction\[S{{O}_{3(g)}}S{{O}_{2(g)}}+\frac{1}{2}{{O}_{2}}(g)\] &\[2S{{O}_{2(g)}}+{{O}_{2(g)}}2S{{O}_{3(g)}}\] are \[{{K}_{1}}\] & \[{{K}_{2}}\] respectively. The relation between \[{{K}_{1}}\] & \[{{K}_{2}}\] is
The observed dipole moment is \[1.03\text{ }D\] and their intern clear distance is \[1.275\overset{{}^\circ }{\mathop{A}}\,\] (\[ID={{10}^{-18}}\]esu.cm). The % covalent character in \[H-Cl\]bond is present