The position vectors of three points A, B, C are \[\hat{i}+2\hat{j}+3\hat{k},\] \[2\hat{i}+3\hat{j}+\hat{k}\] & \[3\hat{i}+\hat{j}+2\hat{k}.\] A unit vector perpendicular to the plane of the triangle ABC is -
Let \[{{(1+{{x}^{2}})}^{2}}{{(1+x)}^{n}}={{A}_{0}}+{{A}_{1}}x+{{A}_{2}}{{x}^{2}}+.....\] If \[{{A}_{0}},\] \[{{A}_{1}},\] \[{{A}_{2}}\] are in A. P. then the value of n is-
If \[{{z}_{1}}\] and \[{{z}_{2}}\] are two complex numbers such that \[\operatorname{Re}\,\,({{z}_{2}})\ne 0,\] \[Re\,\,({{z}_{1}}+{{z}_{2}})=0\] and \[\operatorname{Im}\,({{z}_{1}}{{z}_{2}})=0\]then -
y = f(x) is a continuous function such that its graph passes through (a, 0). Then \[\underset{x\to a}{\mathop{\text{Limit}}}\,\,\,\frac{{{\log }_{e}}(1+3f(x))}{2f(x)}\] is-
If \[f(x)\] is a differentiable function satisfying \[f(x)<2\] for all \[x\in R\] and \[f(1)=2,\] then greatest possible integral value of \[f(3)\] is -
Tangents are drawn from a point on the directrix to the parabola \[{{y}^{2}}=4ax.\] The locus of foot of perpendicular drawn from this point to its chord of contact is a -
A missile is fired at a plane on which there are two targets I and II. The probability of hitting target I is \[{{P}_{1}}\] & that of hitting the II is \[{{P}_{2}}\] If it is known that target I is not hit, then the probability that the target II is hit is -
If A. M. between \[{{p}^{th}}\] and \[{{q}^{th}}\] terms of an A.P. be equal to the A.M. between \[{{r}^{th}}\] and \[{{s}^{th}}\] term of the A.P. then p + q is equal to -
The lines \[y=-\frac{3}{2}x\] and \[y=-\frac{2}{5}\,x\] intersect the curve \[3{{x}^{2}}+4xy+5{{y}^{2}}-4=0\] at the points P and Q respectively. The tangents drawn to the curve at P and Q.
The lengths of the diagonals of a parallelogram constructed on the vectors \[\vec{p}=2\,\,\vec{a}+\vec{b}\] and \[\vec{q}=\text{\vec{a}}-2\vec{b}\] where \[\vec{a}\] & \[\vec{b}\] are unit vectors forming an angle of \[60{}^\circ \] are -
Let \[{{z}_{1}},{{z}_{2}},{{z}_{3}},{{z}_{4}}\] be the vertices A, B, C, D respectively of a square on the argand plane taken in anticlockwise direction, then -
In an experiment to determine the acceleration due to gravity g, the formula used for the time Period of a periodic motion is \[T=2\pi \sqrt{\frac{7\left( R-r \right)}{5g}}\]. The values of R and r are Measured to be \[\left( 60\text{ }\pm \text{ }1 \right)\]mm and \[\left( 10\text{ }\pm \text{ }1 \right)\]mm, respectively. In five successive measurements, the time period is found to be\[0.52s,\text{ }0.56s,\text{ }0.57s,\text{ }0.54s\text{ }and\text{ }0.59s\].The least count of the watch used for the measurement of time period is 0.01 s. Which of the following statement is incorrect?
A)
The error in the measurement of r is \[~10%\]
doneclear
B)
The error in the determined value of g is \[11\text{ }%\]
A thin uniform rod, pivoted at \[O\], is rotating in the horizontal plane with constant angular speed \[\omega \], as shown in the figure. At time \[t\text{ }=\text{ 0},\] a small insect starts from \[O\] and moves with constant speed v, with respect to the rod towards the other end. It reaches the end of the rod at \[t=T\] and stops. The angular speed of the system remains \[\omega \] throughout. The magnitude of the torque about 0, as a function of time is best represented by which plot?
Two mirrors, one concave and the other convex, are placed 60 cm apart with their reflecting surfaces facing each other. An object is placed 30 cm from the pole of either of them on their axis. If the focal lengths of both the mirrors are 15 cm, the position of the image formed by reflection, first at the convex and then at the Concave mirror, is:
Five identical plates each of area A are joined as shown in the figure. The distance between the plates is d. The plates are connected to a potential difference of volts. The charge on plates 1 and 4 will be
Ionization potential of hydrogen atom is \[13.6V\], Hydrogen atoms in the ground state are excited by monochromatic radiation of photon energy\[12.1\text{ }\operatorname{eV}\]. The spectral lines emitted by hydrogen atoms according to Bohr's theory will be
A bar magnet of dipole moment \[1 A-{{m}^{2}}\]and moment of inertia \[{{10}^{-5}}kg-{{m}^{2}}\]when suspended freely has a time period of \[\sqrt{10}s\] at a certain place. Then the earth's magnetic field at that place if angle of dip is \[60{}^\circ \] will be (Take \[{{\pi }^{2}}= 10\])
A cylindrical vessel of cross-section A contains water to a height h. There is a hole in the bottom of radius 'a'. The time in which it will be emptied is:
An asteroid of mass m is approaching earth initially at a distance of \[10{{R}_{e}}\] with speed \[{{v}_{i}}.\] It hits the earth with a speed \[{{v}_{f}}\](\[{{R}_{e}}\] and \[{{M}_{e}}\]are radius and mass of earth), then
Two Polaroid?s are placed in the path of unpolarized beam of intensity \[{{I}_{0}}\] such that no light is emitted from the second Polaroid. If a third polaroid whose polarization axis makes an angle \[\theta \] with the polarization axis of first polaroid, is placed between these polaroid then the intensity of light emerging from the last polaroid will be
An infinite cylinder of radius r with surface charge density \[\sigma \] is rotated about its central axis with angular speed co. Then the magnetic field at any point inside the cylinder is
A 2 kg cylinder rests on a cart as shown in the figure. The coefficient of static friction between the cylinder and the cart is \[0.5.\] The cylinder is 4 cm in diameter and 10 cm in height. The minimum acceleration of the cart needed to make the cylinder tip over is about:
A conducting rod of mass in and \[\ell \] is placed on a smooth horizontal surface in a region where transverse uniform magnetic field B exists in the region.\[\operatorname{At}\,t= 0\], constant force F starts acting on the rod at its mid-point as shown. Potential difference between ends of the rod, \[{{V}_{P}}-\text{ }{{\text{V}}_{Q}}\]at any time \[t\] is
A glass sinker has a mass \[M\]in air. When weighed in a liquid at temperature \[{{t}_{1}},\] the apparent mass is \[{{M}_{1}}\] and when weighed in the same liquid at temperature \[{{t}_{2}}\], the apparent mass is \[{{M}_{2}}\]. If the coefficient of cubical expansion of the glass is \[{{\gamma }_{g}},\] then the real coefficient of expansion of the liquid is:
A radioactive material decays by simultaneous emission of two particles with respective half-lives 1620 and 810 years. The time (in years) after which one-fourth of the material remains is
Calculate the rate of loss of heat through a glass window of area \[1000{{\operatorname{cm}}^{2}}\]and thickness 0.4 cm when temperature inside is \[37{}^\circ C\] and outside is \[-5{}^\circ C.\]Coefficient of thermal conductivity of glass is \[2.2\times {{10}^{-3}}cal\,{{s}^{-1}}c{{m}^{-1}}{{K}^{-1}}.\]
From the following combinations of physical constants (expressed through their usual symbols) the only combination, that would have the same value in different systems of units, is:
On a hypotenuse of a right prism \[(30{}^\circ -60{}^\circ -90{}^\circ )\] of refractive index \[1.50,\] a drop of liquid is placed as shown in figure. Light is allowed to fall normally on the short face of the prism. In order that the ray of light may get totally reflected, the maximum value of refractive index is:
The voltage drop across a forward biased diode is 0.7 volt. In the following circuit, the voltages across the 10 ohm resistance in series with the Diode and 20 ohm resistance are:
A gas is a mixture of two parts by volume of hydrogen and one part by volume of nitrogen. If the velocity of sound in hydrogen at \[0{}^\circ C\]is \[1300m/s\] then the velocity of sound in the gaseous mixture at \[27{}^\circ \operatorname{C}\]will be
For a linear plot of log \[(x/m)\] versus log p in a Freundlich adsorption isotherm, which of the following statements is correct? (k and n are constants)
A)
1/n appears as the intercept
doneclear
B)
Only 1/n appears as the slope
doneclear
C)
\[\log \left( \frac{1}{n} \right)\]appears as the intercept
A gaseous substance dissolves in water giving a pale blue solution which decolorises \[KMn{{O}_{4}}\]and oxidises \[KI\]to \[{{I}_{2}}.\]Gaseous substance is
Given, \[E_{C{{r}^{3+}}/Cr}^{{}^\circ }=-0.74V;\] \[E_{MnO_{4}^{-}/M{{n}^{2+}}}^{{}^\circ }=1.51V\] \[E_{CrO_{7}^{2-}/C{{r}^{3+}}}^{{}^\circ }=1.33V;\] \[E_{Cl/C{{l}^{-}}}^{{}^\circ }=1.36V\] Based on the data given above strongest oxidizing agent will be
Element A burns in nitrogen to give an ionic compound B. Compound B reacts with water to give C and D. A solution of C becomes 'milky' on bubbling carbon dioxide. The element A is
Iron oxide crystallises in a hexagonal close-packed array of oxide ions with two out of every three octahedral holes occupied by iron ions. Derive the formula of the iron oxide.
If at 298 K the bond energies of \[C-H,C-C,C=C\] and \[H-H\] bonds are respectively 414, 347, 615 and \[435\,kJ\,mo{{l}^{-1}},\] the value of enthalpy change for the reaction, at 298 K will be
If \[x=\frac{4\lambda }{1+{{\lambda }^{2}}}\] and \[y=\frac{2-2{{\lambda }^{2}}}{1+{{\lambda }^{2}}}\] where a is a real parameter and \[{{x}^{2}}-xy+{{y}^{2}}\] lies between [a, b] then (a + b) is-
If the shaded portion represents the set of complex numbers then which of the following set of complex numbers satisfy the inequality \[{{\tan }^{-1}}({{\log }_{3}}\left| 2z-1 \right|>{{\tan }^{-1}}(\log 3\left| 2z+1 \right|)\]
The tangent at a point whose eccentric angle \[60{}^\circ \] on the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\,\,(a>b)\] meet the auxiliary circle at L and M. If LM subtends a right angle at the centre, then eccentricity of the ellipse is-
If \[\alpha =\underset{n\to \infty }{\mathop{\lim }}\,\left( \frac{1}{{{n}^{3}}+1}+\frac{4}{{{n}^{3}}+1}+......+\frac{{{n}^{2}}}{{{n}^{3}}+1} \right)\] and \[\beta =\underset{n\to \infty }{\mathop{\lim }}\,\,\frac{\sin 2x}{\sin 8x}\] then the quadratic equation whose roots are \[\alpha ,\]\[\beta \] is
A function \[y\text{ }=\text{ }f(x)\] satisfies the condition \[f'(x)\,\,sin\,x+f\,(x)\,\,\cos \,x=1,\,\,f(x)\] being bounded when \[x\to 0.\] If \[I=\int\limits_{0}^{\pi /2}{f(x)dx,}\] then
If the sum \[\sum\limits_{k=1}^{\infty }{\,\frac{1}{(k+2)\sqrt{k}+\sqrt{k+2}}=\frac{\sqrt{a}+\sqrt{b}}{\sqrt{c}}}\] \[a,b,c\text{ }\in \text{ }N\] and lie in [1, 15], then \[a\text{ }+\text{ }b\text{ }+\text{ }c\] equals to-
A musician using an open flute of length 50 cm produces second harmonic sound waves. A person runs towards the musician from another end of a hall at a speed of 10 km/h. If the wave speed is \[330 m/s,\] the frequency heard by the running person shall be close to:
You may have seen in a circus a motorcyclist driving in vertical loops inside a 'death well' (a hollow spherical chamber with holes, so the spectators can watch from outside). What is the minimum speed required at the uppermost position of motorcyclist to perform a vertical loop, if the radius of the chamber is 25 m?
A capacitor is initially connected to a battery of \[emf3\operatorname{V}.\text{ At}\,t\text{ }=\text{ }0\], switch is thrown to 5 state. Now charge on capacitor at any instant is given by
A diatomic molecule is made of two masses \[{{m}_{1}}\] and \[{{m}_{2}}\] which are separated by a distance r. If we calculate its rotational energy by applying Bohr's rule of angular momentum quantization, Its energy will be given by: (\[n\] is an integer)
In the experiment of calibration of voltmeter, a standard cell of e. m .f. 1.1 volt is balanced against \[440 cm\] of potential wire. The potential difference across the ends of resistance is found to balance against 220 cm of the wire. The corresponding reading of voltmeter is 0.5 volt. The error in the reading of voltmeter will be:
AB is a cylinder of length 1.0 m filled with a thin flexible diaphragm C (see figure) at the middle and two other thin flexible diaphrams A and B at the ends. The portions AC and BC contain hydrogen and oxygen gases respectively. The diaphrams A and B are set into vibrations of same frequency. What is the minimum frequency of these vibrations for which the diaphrams C is a node? Under the conditions of the experiment, the velocity of sound in hydrogen is \[1100 m/s\]and in oxygen is \[300 m/s.\]
Graph shows a hypothetical speed distribution for a sample of \[N\]gas particle (for \[v>{{v}_{0;}}\]) \[\frac{dN}{dv}=0\] Then the Correct statement is/are
A)
The value of \[a{{v}_{0}}\]is\[2N\]
doneclear
B)
The ratio \[v{{~}_{avg}}/{{v}_{0}}\]is equal to \[2/3~\]
Figure shows a method for measuring the acceleration due to gravity. The ball is projected upward by a "gun". The ball passes the electronic "gates" 1 and 2 as it rises and again as it falls. Each gate is connected to a separate timer. The first passage of the ball through each gate starts the corresponding timer, and the second passage through the same gate stops the timer. The time intervals \[\Delta {{t}_{1}}\] and \[\Delta {{t}_{2}}\] are thus measured. The vertical distance between the two gates is d. If \[\operatorname{d}=5m.\Delta {{t}_{1}}=3s,\Delta {{t}_{2}}=2s,\] then find the measured value of acceleration due to gravity (in\[\operatorname{m}/{{s}^{2}}\]).
A disc revolves with a speed of \[33\frac{1}{3}\] rev/min and has a radius of 15 cm. Two coins \[A\] and \[B\] are placed at 4 cm and 14 cm away from the Centre of the record respectively. If the coefficient of friction between the coins and the record is 0.15, which of the coins will revolve with the record?
Calculate the mass of a non-volatile solute (molar mass \[40\,g\,mo{{l}^{-\,1}}),\] which should be dissolved in 114 g octane to reduce its vapour pressure to 80%.
The bond dissociation energy of B-F in \[B{{F}_{3}}\]is \[646\,kJ\,mo{{l}^{-\,1}}\]. The correct reason For higher B-F bond dissociation energy as compared to that of C-F is
A)
smaller size of B-atom as compared to that of C-atom.
doneclear
B)
stronger or-bond between B and F in \[B{{F}_{3}}\] to that between C and F in \[C{{F}_{4}}.\]
doneclear
C)
significant \[p\pi -p\pi \] interaction between B and F in \[B{{F}_{3}}\]whereas there is no possibility of such interaction between C and F in \[C{{F}_{4}}.\]
doneclear
D)
lower degree of \[p\pi -p\pi \] interaction between B and F in \[B{{F}_{4}}\] than that between C and F in \[C{{F}_{4}}.\]
A metal complex having composition \[Cr{{(N{{H}_{3}})}_{4}}C{{l}_{2}}Br\] has been isolated in two forms A and B. A reacts with\[AgN{{O}_{3}}\]producing a white precipitate which was soluble in dilute ammonia solution. B reacts with \[AgN{{O}_{3}}\] producing a pale yellow precipitate soluble in concentrated ammonia solution. The formulae of A and B are respectively
On reaction with Cig, phosphorus forms two types of halides A and B. Halide A is yellowish white powder, but halide B is colourless oily liquid. Which of the following is/are their hydrolysis products?
Basic nature of above compounds is in the order as: 
can be achieved by using 
A and B are 
Identify the final product X. 
