A large open tank has two holes in the wall. One is a square hole of side L at a depth y from the top and the other is a circular hole of radius R at a depth 4y from the top. When the tank is completely filled with water the quantities of water flowing out per second from both the holes are the same. Then R is equal to
A wheel of radius 0.4 m can rotate freely about its axis as shown in the figure. A string is wrapped over its rim and a mass of 4 kg is hung. An angular acceleration of \[8 rad {{s}^{-}}^{2}\] is produced in it due to the torque. Then, moment of inertia of the wheel is \[\left( g = 10 m{{s}^{-}}^{2} \right)\]
If the ratio of lengths, radii and Young?s modulus of steel and brass wires shown in the figure are a, b, and c, respectively. The ratio between the increase in lengths of brass and steel wires would be
A magnetic needle suspended parallel to a magnetic field requires \[\sqrt{3}\,J\] of work to turn it through \[60{}^\circ \]. The torque needed to maintain the needle in this position will be:
The energy stored in the capacitor as shown in the figure is \[4.5 \times 1{{0}^{-}}^{6}J\]. If the battery is replaced by another capacitor of 900 pF as shown in the figure (b), then the total energy of system is
If the speed of light (c), acceleration due to gravity (g) and pressure (p) are taken as the fundamental quantities, then the dimension of gravitational constant is
A man standing on the roof of a house of height h throws one particle vertically downwards and another particle horizontally with the same velocity u. The ratio of their velocities when they reach the earth?s surface will be
A balloon is at a height of 81m and is ascending upwards with a velocity of 12 m/s. A body of 2 kg weight is dropped from it. If \[\operatorname{g}=10 m/{{s}^{2}}\], the body will reach the surface of the earth in
The graph between\[1/\lambda \], and stopping potential (V) of three metals having work functions \[\phi 1,\,\,\phi 2\,\,and\,\,\phi 3\] in an experiment of photo-electric effect is plotted as shown in the figure. Which of the following statement(s) is/are correct? [Here \[\lambda \] is the wavelength of the incident ray]
A)
Ratio of work functions \[{{\phi }_{1}}:{{\phi }_{2}}:{{\phi }_{3}}=\,\,1:2:5\]
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B)
Ratio of work functions \[{{\phi }_{1}}:{{\phi }_{2}}:{{\phi }_{3}}=\,\,4:2:1\]
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C)
tan 6 is directly proportional to hc/e, where h is Planck?s constant and c is the speed of light.
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D)
The violet colour light can eject photoelectrons from metals 2 and 3.
Half-lives of two radioactive elements A and B are 20 minutes and 40 minutes, respectively, initially, the samples have equal number of nuclei. After 80 minutes, the ratio of decayed number of A and B nuclei will be:
Three rods of same dimensions are arranged as shown in figure they have thermal conductivities\[{{\operatorname{K}}_{1}},\,\,{{K}_{2}}\,\,and\,\,{{K}_{3}}\]. The points P and Q are maintained at different temperatures for the heat to flow at the same rate along PRQ and PQ then which of the following option is correct?
A body executes simple harmonic motion under the action of a force \[{{F}_{1}}\] with a time period \[\frac{4}{5}s\]. If the force is changed to \[{{F}_{2}}\], it executes S.H.M. with time period \[\frac{3}{5}s\]. If both the forces \[{{\operatorname{F}}_{1}}\,\,and\,\,{{F}_{2}}\] act simultaneously in the same direction on the body, its time period in second is
If 2 mole of an ideal monatomic gas at temperature \[{{T}_{0}}\] is mixed with 4 moles of another ideal monatomic gas at temperature \[2{{T}_{0}}\], then the temperature of the mixture is
A long solenoid of diameter 0.1 m has \[2 \times 1{{0}^{4}}\] turns per meter. At the centre of the solenoid, a coil of 100 turns and radius 0.01 m is placed with its axis coinciding with the solenoid axis. The current in the solenoid reduces at a constant rate to OA from 4 A in 0.05 s. If the resistance of the coil is \[10{{\pi }^{2}}\Omega \] the total charge flowing through the coil during this time is:
A diode detector is used to detect an amplitude modulated wave of \[60%\] modulation by using a condenser of capacity 250 picofarad in parallel with a load resistance 100 kilo ohm. Find the maximum modulated frequency which could be detected by it.
The intensity of gamma radiation from a given source is I. On passing through 36 mm of lead, it is reduced to \[\frac{1}{8}\]. The thickness of lead which will reduce the intensity to \[\frac{1}{2}\] will be
In double slit experiment, the angular width of the fringes is \[0.20{}^\circ \] for the sodium light \[(\lambda =5890\,A)\]. In order to increase the angular width of the fringes by \[10%\], the necessary change in wavelength is
A)
zero
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B)
\[\operatorname{increased} by 6479 \overset{{}^\circ }{\mathop{A}}\,\]
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C)
\[\operatorname{decreased} by 589 \overset{{}^\circ }{\mathop{A}}\,\]
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D)
\[\operatorname{increased} by 589\,\,\overset{{}^\circ }{\mathop{A}}\,\]
The heart of man pumps 5 litres of blood through die arteries per minute at a pressure of 150 mm of mercury. If the density of mercury be \[13.6 \times \,1{{0}^{3}}kg/{{m}^{3}}\] and \[\operatorname{g} = 10m/{{s}^{2}}\] then the power of heart in watt is:
One end of a cylindrical rod is grounded to a hemispherical surface of radius \[\operatorname{R} = 20 mm\]. It is immersed in water \[\left( \mu = 4/3 \right)\]. If the refractive index of the rod is 1.5 and an object is placed in water on the axis at a distance of 10 cm from the pole, then determine the position of the image from the pole on the axis (in cm).
A source of sound of frequency 256 Hz is moving rapidly towards a wall with a velocity of 5m/s. How many beats per second will be heard if sound travels at a speed of 330 m/s by an observer behind the source?
In a CE transistor amplifier, the audio signal voltage across the collector resistance of \[2k\Omega \] is 2V. If the base resistance is \[1k\Omega \] and the current amplification of the transistor is 100, the input signal voltage (in mV) is
Anhydrous \[AlC{{l}_{3}}\]is covalent. From the data given below, predict whether it would remain covalent or become ionic in aqueous solution (ionisation energy of \[Al=5137\,\text{kJ}\,\text{mo}{{\text{l}}^{-1}}\],
The dispersed phase in colloidal iron (III) hydroxide and colloidal gold is positively and negatively charged respectively. Which of the following is not correct?
A)
Magnesium chloride solution coagulates the gold sol more readily than iron (III) hydroxide sol.
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B)
Sodium sulphate solution causes coagulation in both sols.
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C)
Mixing of the sols has no effect.
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D)
Coagulation in both sols can be brought about by electrophoresis.
For the reaction \[X+Y\xrightarrow[{}]{{}}P\], the rate law is expressed as rate \[=k[X]{{[Y]}^{2}}\]. Which of the following statements will be false for the reaction?
A)
If [Y] is held constant and [X] is doubled, reaction rate will be doubled.
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B)
If [X] is held constant and [Y] is reduced to one-fourth, the rate of reaction will be halved.
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C)
If [X] and [Y] both are doubled, the rate of reaction will be increased by 8 times.
Consider the following metallurgical process : Metal sulphide \[\xrightarrow[{}]{x}\]Metal oxide \[\xrightarrow[{}]{y}\] Impure metal \[\xrightarrow[{}]{z}\]Pure metal x, y and z are respectively
H. In a normal spinel type structure, the oxide ions are arranged in ccp, whereas, 1/8 tetrahedral holes are occupied by \[Z{{n}^{2+}}\]ions and 50% of octahedral holes are occupied by \[F{{e}^{3+}}\] ions. The formula of the compound is
Among the following the number of paramagnetic as well as coloured species is____. \[{{O}_{2}},N{{O}_{2}},C{{u}^{2+}},Hg_{2}^{2+},F{{e}^{2+}},F{{e}^{3+}},{{[Fe{{(CN)}_{6}}]}^{4-}}\], \[{{[Fe{{(CN)}_{6}}]}^{3-}},{{[Ni{{({{H}_{2}}O)}_{6}}]}^{2+}},{{[Ni{{(CN)}_{4}}]}^{2-}}\]
A decimolar solution of potassium Ferro cyanide, \[{{K}_{4}}[Fe{{(CN)}_{6}}]\]is 50% dissociated at 300 K. The osmotic pressure (in atm) of the solution is _____.
The electronegativity of carbon if \[{{E}_{H-H}}=104.2\text{kcal}\,\text{mo}{{\text{l}}^{-1}}\],\[{{E}_{C-C}}=83.1\,\text{kcal}\,\text{mo}{{\text{l}}^{-1}}\]\[{{E}_{C-H}}=98.8\,\text{kcal}\,\text{mo}{{\text{l}}^{-1}},{{\chi }_{H}}=2.1\]is__________.
0.1 mole of \[C{{H}_{3}}N{{H}_{2}}({{K}_{b}}=5\times {{10}^{-4}})\]is mixed with 0.08 mole of HCl and diluted to one litre. The 1-F concentration in the solution is \[x\times {{10}^{-11}}\]. The value of x is ____.
If the angle between the normal to the parabola \[{{x}^{2}}=4ay\] at point P and the focal chord passing through P is \[\frac{\pi }{3},\] then the slope of the tangent at point P is
Let f be a continuous and differentiable function on R satisfying \[f(-x)=f(x)\] and \[f(2+x)=f(2-x)\forall x\in R\] and \[f'(1)=-5\]. Then the value of \[\sum\limits_{r=0}^{100}{{{(-1)}^{r}}f'(r)}\] is equal to
A curve \[y=f(x)\] which passes through \[(4,0)\] satisfies the differential equation \[x\,dy+2y\,dx=x(x-3)\,dx.\]The area bounded by \[y=f(x)\] and line \[y=x\](in square unit) is
Let \[{{\cos }^{2}}\theta +b\] and \[si{{n}^{2}}\theta +b\] be roots of the equation \[{{x}^{2}}+4x+\frac{61}{16}=0.\]Then the equation whose roots are \[{{\tan }^{2}}\theta \] and \[co{{t}^{2}}\theta \] is
If the straight line \[4ax+3by=24\] is a normal to the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] \[(a>b)\] then \[({{a}^{2}}-{{b}^{2}})\] is equal to
Consider three vectors \[{{\vec{v}}_{1}},{{\vec{v}}_{2}}\] and \[{{\vec{v}}_{3}}\] such that \[{{\vec{v}}_{1}}={{\vec{v}}_{2}}-{{\vec{v}}_{3}}.\] If \[{{\vec{v}}_{1}}=(\vec{a}\times \hat{i})\times \hat{i},\] \[{{\vec{v}}_{2}}=(\vec{a}\times \hat{j})\times \hat{j}\] and \[{{\vec{v}}_{3}}=(\vec{a}\times \hat{k})\times \hat{k},\] where \[\vec{a}\] is non-zero vector, then
A fair coin is tossed 10 times and the outcomes are listed. Let \[{{H}_{i}}\] be the event that the \[{{i}^{th}}\] outcome is a head and \[{{A}_{m}}\] be the event that the list contains exactly m heads. Then
The value of \[^{n}{{C}_{1}}\left( \sum\limits_{r=0}^{1}{^{1}{{C}_{r}}} \right){{+}^{n}}{{C}_{2}}\left( \sum\limits_{r=0}^{2}{^{2}{{C}_{r}}} \right){{+}^{n}}{{C}_{3}}\left( \sum\limits_{r=0}^{3}{^{3}{{C}_{r}}} \right)+...{{+}^{n}}{{C}_{n}}\left( \sum\limits_{r=0}^{n}{^{n}{{C}_{r}}} \right)\]is equal to
If the lines \[\vec{r}=-\hat{i}+\hat{j}-\hat{k}+\lambda (2\hat{i}+\hat{j}+3\hat{k})\] and \[\vec{r}=-2\hat{i}+\alpha \hat{j}+\hat{k}+\mu (2\hat{i}+4\hat{k})\] \[(\lambda ,\,\mu \in R)\] are coplanar, then the value of \[\alpha \] is _____.
If \[\sin 3\theta =2\sin \theta ,\] then \[\frac{\tan 2\theta }{\tan \theta }\] is equal to \[\left( \theta \ne \frac{n\pi }{2},n\in I \right)\_\_\_\_\_\_\_\_\_\_\_.\]
The number of ordered pairs \[(x,y)\] satisfying \[x\left( {{\sin }^{2}}x+\frac{1}{{{x}^{2}}} \right)=2\sin x{{\sin }^{2}}y,\]where \[x\in (-\pi ,0)\cup (0,\pi )\] and \[y\in [0,2\pi ]\] is