A tangential force of \[20\,N\] is applied on a cylinder of mass \[4\text{ }kg\] and moment of inertia \[0.02\text{ }kg\text{ }{{m}^{2}}\] about its own axis. If the cylinder rolls without slipping, then linear acceleration of its centre of mass will be
Two soap bubbles A and B are kept in a dased chamber where the air is maintained at pressure\[8\text{ }N/{{m}^{2}}\]. The radii of bubbles A and B are 2cm and 4cm, respectively. Surface tension of the soap-water used to make bubbles is\[0.04\text{ }N/m\]. Find the ratio \[{{\eta }_{B}}/{{\eta }_{A}},\] where \[{{\eta }_{A}}\] and \[{{\eta }_{B}}\] are the number of moles of air in bubbles A and B, respectively. [Neglect the effect of gravity]
A bullet of mass 20g and moving with 600 m/s 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 \[0.2\text{ }m\] after collision?
Two straight long conductors AOB and COD are perpendicular to each other and carry currents \[{{i}_{1}}\] and \[{{i}_{2}}\]. The magnitude of the magnetic induction at a point P at a distance a from the point O in a direction perpendicular to the plane ABCD is
Two block of mass 8kg and 4kg are connected by a string as shown in the figure below. Calculate their acceleration, if they are initially at rest on the floor, when a force of 100N is applied on the massless pulley in upward direction \[(g=10\,m{{s}^{-2}})\]
A large number of bullets are fired in all directions with the same speed v. What is the maximum area on the ground on which these bullets will spread?
Photoelectric effect experiments are performed using three different metal plates p, q and r having work functions \[{{\phi }_{p}}=2.0eV,\] \[{{\phi }_{q}}=2.5eV\] and \[{{\phi }_{r}}=3.0eV,\] respectively. Alight beam containing wavelengths of \[550\text{ }nm,\] \[450\text{ }nm\] and \[350\text{ }nm\] with equal intensities illuminates each of the plates. The correct \[I-V\] graph for the experiment is [Take\[hc=1240\text{ }eV\,nm\]]
The wavelength \[{{K}_{\alpha }}\]of X-rays for two metals 'A and 'B' are \[\frac{4}{1875\,R}\] and \[\frac{1}{675\,R}\]respectively, where 'R' is Rydberg constant. Find the number of elements lying between A and B according to their atomic numbers
The half-life of a radioactive nucleus is 50 days. The time interval \[({{t}_{2}}-{{t}_{1}})\] between the time \[{{t}_{2}}\]when \[\frac{2}{3}\] of it has decayed and the time \[{{t}_{1}}\] when \[\frac{1}{3}\] of it had decayed is
In a vertical U-tube containing a liquid, the two arms are maintained at different temperatures \[{{t}_{1}}\]and \[{{t}_{2}}\]. The liquid columns in the two arms have heights \[{{l}_{1}}\] and \[{{l}_{2}}\] respectively. The coefficient of volume expansion of the liquid is equal to
A speeding motorcyclist sees traffic jam ahead of him. He slows down to\[36\text{ }km/hour\]. He finds that traffic has eased and a car moving ahead of him at \[18\text{ }km/hour\] is honking at a frequency of \[1392\text{ }Hz\]. If the speeds of sound is \[343\text{ }m/s,\] the frequency of the honk as heard by him will be:
The P-V diagram of a system undergoint thermodynamic transformation is shown in figure. The work done on the system in going from \[A\to B\to C\] is \[50\text{ }J\]and 20 cat heat is given to the system. The change in internal energy between A and C is
A conducting square loop of side L and resistance R moves in its plane with a uniform velocity v perpendicular to one of its sides. A magnetic induction B constant in time and space, pointing perpendicular and into the plane at the loop exists everywhere with half the loop outside the field, as shown in figure. The induced emf is
In a series LCR circuit \[R=200\Omega \] and the voltage and the frequency of the main supply is \[~220V\] and \[50\text{ }Hz\] respectively. On taking out the capacitance from the circuit the current lags behind the voltage by \[30{}^\circ \]. On taking out the inductor from the circuit the current leads the voltage by\[30{}^\circ \]. The power dissipated in the LCR circuit is
A circular coil of 16 turns and radius 10cm carries a current of \[0.75\text{ }A\] and rest with its plane normal to an external magnetic field of \[5.0\times {{10}^{-2}}T.\] The coil is free to rotate about its stable equilibrium position with a frequency of \[2.0\text{ }{{s}^{-1}}\]. Compute the moment of inertia (in \[kg{{m}^{2}}\]) of the coil about its axis of rotation.
A wire is stretched by \[0.01\text{ }m\]by a certain force F. Another wire of same material whose diameter and length are double to the original wire is stretched by the same force. Then, find its elongation (in metre).
Interference fringes were produced in Young's double slit experiment using light of wave length\[5000\text{ }A{}^\circ \]. When a film of material \[2.5\times {{10}^{-3}}\text{ }cm\]thick was placed over one of the slits, the fringe pattern shifted by a distance equal to 20 fringe width. What is the refractive index of the material of the film?
The concentration of hole - electron pairs in pure silicon at \[T=300K\]is \[7\times {{10}^{15}}\] per cubic meter. Antimony is doped into silicon in a proportion of 1 atom in \[{{10}^{7}}\text{ }Si\] atoms. Assuming that half of the impurity atoms contribute electron in the conduction band, calculate the factor by which the number of charge carriers increases due to doping the number of silicon atoms per cubic meter is \[5\times {{10}^{28}}\].
The heat of formation of \[C{{H}_{3}}OC{{H}_{3(g)}}\]is [Given: \[B.E{{.}_{H-H}}=103\]kcal, \[B.E{{.}_{C-H}}=87\]kcal \[B.E{{.}_{C-O}}=70\]kcal, \[B.E{{.}_{O=O}}=177\]kcal; Heat of vaporisation of 1 gram atom of carbon = 125 kcal.]
A solid 'X' on heating gives \[C{{O}_{2}}\]and a residue. The residue with \[{{H}_{2}}O\] form 'Y'. On passing an excess of \[{{H}_{2}}O\]through 'Y in \[{{H}_{2}}O\], a clear solution of 'Z' is obtained. On boiling 'Z; 'X' is reformed. 'X' is
A solution of a metal ion when treated with KI gives a red precipitate which dissolves in excess of KI to give a colourless solution.
Moreover, the solution of metal ion on treatment with a solution of cobalt (II) thiocyanate gives rise to deep blue crystalline precipitate. The metal ion is
An organic compound 'A' having molecular formula \[{{C}_{2}}{{H}_{3}}N\] on reduction gave another compound 'B'. Upon treatment with nitrous acid, 'B' gave ethyl alcohol. On warming with chloroform and alcoholic KOH, it formed an offensive smelling compound 'C'. The compound 'C' is
At temperature of 298 K the emf of the following electrochemical cell \[A{{g}_{(s)}}|A{{g}^{+}}(0.1M)||Z{{n}^{2+}}(0.1M)|Z{{n}_{(s)}}\] will be (given \[E_{cell}^{{}^\circ }=-\,1.562\,V\])
The \[p{{K}_{a}}\]of acetylsalicylic acid (aspirin) is 3.5. The pH of gastric juice in human stomach is about 2-3 and the pH in the small intestine is about 8. Aspirin will be
A)
unionised in the small intestine and in the stomach
doneclear
B)
completely ionised in the small intestine and in the stomach
doneclear
C)
ionised in the stomach and almost unionised in the small intestine
doneclear
D)
ionised in the small intestine and almost unionised in the stomach.
p-Cresol reacts with chloroform in alkaline medium to give a compound 'A' which adds hydrogen cyanide to form the compound 'B'. Compound 'B' on acidic hydrolysis gives chiral carboxylic acid. The structure of the carboxylic acid is
0.008 g of starch is required to prevent coagulation of 10 mL of gold sol when 1 mL of 10% NaCI solution is present. The gold number of starch sol is ______.
A hydrocarbon (X) having molecular weight 70 gives a single mono chloride but three di chlorides on chlorination in the presence of ultraviolet light. The number of C-atoms in hydrocarbon (X) is ______.
In the disproportionation reaction, \[3HCl{{O}_{3}}\xrightarrow[{}]{{}}HCl{{O}_{4}}+C{{l}_{2}}+2{{O}_{2}}+{{H}_{2}}O\] the equivalent mass of the oxidising agent is ______. (Molar mass of \[HCl{{O}_{3}}=84.45\])
A diatomic molecule has a dipole moment of 1.2 D. If the bond distance is \[1.0\overset{\text{o}}{\mathop{\text{A}}}\,,1/x\] of an electronic charge exists on each atom. The value of x is __.
Let \[f(x)=\sqrt{\frac{{{x}^{2}}+px+1}{{{x}^{2}}-p}}\]. If \[f(x)\] is discontinuous at exactly two values of x, then number of integers in the range of p is
If \[{{z}_{1}},{{z}_{2}}\] and \[{{z}_{3}}\] are any three roots of the equation \[{{z}^{6}}={{(z+1)}^{6}},\] then \[\arg \left( \frac{{{z}_{1}}-{{z}_{3}}}{{{z}_{2}}-{{z}_{3}}} \right)\] can be equal to
If \[\alpha \] and \[\beta \] are the roots of the equation \[{{x}^{2}}-5x-1=0,\] then the value of \[\frac{{{\alpha }^{15}}+{{\alpha }^{11}}+{{\beta }^{15}}+{{\beta }^{11}}}{{{\alpha }^{13}}+{{\beta }^{13}}}\] is
Let \[<{{a}_{n}}>\] be an arithmetic sequence such that arithmetic mean of \[{{a}_{1}},{{a}_{3}},{{a}_{5}},.....{{a}_{97}},{{a}_{99}}\]is 1. Then the value of \[\left| \sum\limits_{r=1}^{50}{{{(-1)}^{\frac{r(r+1)}{2}}}.{{a}_{2r-1}}} \right|\] is
If A and B are events such that \[P(\bar{A}\cup \bar{B})=\frac{3}{4},\] \[P(\bar{A}\cap \bar{B})=\frac{1}{4}\] and \[P(A)=\frac{1}{3},\] then \[P(\bar{A}\cap B)\] is
If \[\tan (2A-3B).\] \[\tan (4B-A)=1,\] where A, \[B\in \left( 0,\frac{\pi }{2} \right),\] then the value of \[\tan \left( \frac{A+B}{6} \right)\] is equal to
The equations of perpendicular bisectors of two sides AB and AC of a triangle ABC are \[x+y+1=0\] and \[x-y+1=0,\] respectively. If circumradius of \[\Delta ABC\] is 2 units, then locus of vertex A is
From a point, perpendicular tangents are drawn to the ellipse \[{{x}^{2}}+2{{y}^{2}}=2\]. The chord of contact touches a circle which is concentric with the given ellipse. Find the ratio of maximum and minimum areas of circle.
The solution of differential equation \[\frac{dy}{dx}=-\left( \frac{y+\sin x}{x} \right)\] satisfying condition \[y(0)=1\] does not pass through the point
Consider the area bounded by \[f(x)={{e}^{-x}}+3,\] \[g(x)=\log (x+2),\] \[x=-1\] and y-axis. The areas divided by the line passing through the points of intersection of these curves with the coordinate axes are in the ratio
Let \[f(x)\] be a non-negative continuous function defined on R such that \[f(x)+f\left( x+\frac{1}{2} \right)=3,\]\[\forall \,x\,\in .R\]?. Then the value of \[\int\limits_{0}^{100}{f(x)\,dx}\] is ____.
Let vectors \[\vec{a}=-\hat{i}+\hat{j}+\hat{k},\] \[\vec{b}=2\hat{i}+\hat{k}\] and \[\vec{c}\] be such that \[[\vec{a}\,\vec{b}\,\vec{c}]=0,\] \[\vec{b}\,.\,\vec{c}=0\] and \[\vec{a}\,.\,\vec{c}=7\]. Then the value of \[\frac{2}{7}|\vec{c}{{|}^{2}}\] is
If the distance between the planes \[Ax-2y+z=d\] and the plane containing the lines \[\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}\] and \[\frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}\] is \[\sqrt{6},\] then \[|d|\] is