Let \[f(x)\] is differentiable function in \[[2,5]\] such that \[f(2)=\frac{1}{5}\] and \[f(5)=\frac{1}{2}\], then there exists a number \[c,2<c<5\] for which \[f(c)\] equals
If \[{{z}_{1}},{{z}_{2}},{{z}_{3}}\in C,\,i{{z}^{3}}+{{z}^{2}}-z+i=0\], where \[i=\sqrt{-1}\] then the area of triangle formed by \[{{z}_{1}},{{z}_{2}}\], and \[{{z}_{3}}\] is equal to
If for positive integers \[r>1,\,\,n>2\], the coefficients of the \[{{(3r)}^{th}}\] and \[{{(r+2)}^{th}}\] power of \[x\] in the expansion of \[{{(1+x)}^{2n}}\] are equal, then n is equal to
A tangent to the hyperbola \[{{x}^{2}}-2{{y}^{2}}=4\] meets x-axis at P and y-axis at Q. Line PR and QR are drawn such that OPRQ is a rectangle (where 0 is origin). The locus of R is
If \[\hat{a},\hat{b}\] and \[\hat{c}\] are unit vectors satisfying \[\hat{a}-\sqrt{3}\hat{b}+\hat{c}=\vec{0}\], then the angle between the vectors \[\hat{a}\] and \[\hat{c}\] is
If the area bounded by the parabolas \[{{y}^{2}}=4\alpha (x+\alpha )\] and \[{{y}^{2}}=-4\alpha (x-\alpha )\], where \[\alpha >0\] is 48 square units then \[\alpha \] is equal to
Let\[a=\operatorname{Im}.\left( \frac{1+{{z}^{2}}}{2iz} \right)\], where z is any non-zero complex number and \[i=\sqrt{-1}\]. The set \[A=\{a:\left| z \right|=1\,and\,\ne \pm 1\}\] is equal to
Let\[A=\{1,2,3,4\}\]. The number of different ordered pairs (B, C) that can be formed such that \[B\subseteq AC\subseteq A\] and \[B\cap C\] is empty, is
Let the equations of two ellipses be \[{{E}_{1}}:\frac{{{x}^{2}}}{3}+\frac{{{y}^{2}}}{2}=1\] and \[{{E}_{2}}:\frac{{{x}^{2}}}{16}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\], If the product of their eccentricities is \[\frac{1}{2}\], then the length of the minor axis of ellipse \[{{E}_{2}}\] is
If \[\left\{ x\left| x\in N\,and \right.{{x}^{2}}\le 6\frac{1}{4} \right\}\] and\[B=\left\{ x\left| x\in N\,and\,{{x}^{2}}\le 5 \right. \right\}\]. Then the number of subsets of set \[A\times (A\cap B)\] which contains 3 elements is
If two lines \[{{L}_{1}}\] and \[{{L}_{2}}\] in space are defined by\[{{L}_{1}}=\left\{ x=\sqrt{\lambda }y+(\sqrt{\lambda }-1),\,z=(\sqrt{\lambda }-1)y+\sqrt{\lambda } \right\}\] and \[{{L}_{2}}=\left\{ x=\sqrt{\mu }y+(1-\sqrt{\mu }),\,z=(1-\sqrt{\mu })y+\sqrt{\mu } \right\}\]then \[{{L}_{1}}\] is perpendicular to \[{{L}_{2}}\] for all non-negative reals \[\lambda \] and \[\mu \], such that
If \[{{z}_{1}}\ne 0\] and \[{{z}_{2}}\] be two complex numbers such that \[\frac{{{z}_{2}}}{{{z}_{1}}}\] is a purely imaginary number, then \[\left| \frac{2{{z}_{1}}+3{{z}_{2}}}{2{{z}_{1}}-3{{z}_{2}}} \right|\] is equal to
On the sides AB, BC, CA of a \[\Delta ABC\], 3,4,5 district points (excluding vertices A,B,C) are respectally chosen. The number of triangles that can be constructed using these chosen points as vertices are
The time period of oscillation of a simple pendulum is given by \[T=2\pi \sqrt{\frac{\ell }{g}}\]. Find percentage error in calculation g while \[\ell =10\pm -0.1\,cm\] and \[T=0.5\pm 0.02\,cm\]
In a new system of units and 1 unit mass is 9 kg, 1 unit length is 2 m, 1 unit time is 3 sec and 1 unit work is \[x\] Joule then value of \['x'\] will be :
A particle is moving on straight path. If initial velocity is \[7\,m{{s}^{-1}}\] and acceleration is \[-2\,m{{s}^{-2}}\] then distance travelled by the particle in fourth second will be:
A 2kg particle is dropped from height\[20\,\,m\]. Just before hitting the ground its velocity is \[19\,m{{s}^{-1}}\] . Find work done by resistance on the particle during its flight \[(g=10\,m{{s}^{-2}})\]
A uniform rod of length \['l'\] which is free to rotate about horizontal axis passing through one of its end is released. Find acceleration of free end of the rod just after release.
If three particle of equal mass \['m'\] is equidistance from each other and doing motion on circular path having radius r due to their gravitational interaction. then their speed v will be :
A hollow sphere of radius \[2R\] is charged to V volts and another smaller sphere of radius R is charged to \[V/2\] volts. Then the smaller sphere is placed inside the bigger sphere without changing the net charge on each sphere. The potential difference between the two spheres would be :
Null point in the galvanometer is obtained when a cell of emf E and internal resistance r is connected across the length of 22 cm wire of the potentiometer. Now a resistance of \[10\,\,\Omega \] is connected across the terminals of the cell (by closing the key K) and null point is obtained against the length of 20 cm. Then the internal resistance r of the cell is:
Switch S of the circuit shown in figure is closed at\[t=0\]. If \[\varepsilon \] denotes the induced emf in L and I the current flowing through the circuit at time t, then which of the following graphs is correct:
A very long uniformly charged nonconducting rod (linear charge density\[=\lambda \]) is moving with a constant velocity v through the centre of a circular conducting loop as shown. If radius of the loop is r, then induced emf in loop is:
A light ray is incident perpendicular to one face of a \[{{90}^{o}}\] prism and is totally internally reflected at the glass - air interface. If the angle of reflection is\[{{45}^{o}}\]. We conclude that the refractive index is :
In the Young's slit experiment, when a glass plate \[(\mu =1.5)\] of thickness t is introduced in the path of one of the interfering beams (wavelength\[=\lambda \]), the intensity at the position where central maxima occurred previous remains unchanged. The minimum thickness of the glass plate is:
A sinusoidal voltage of peak value 200 volt is I connected to a diode and resistor R in the circuit shown so that half wave rectification occurs. If the forward resistance of the diode is negligible compared to R, the rms value of voltage across R in volts is approximately.
The intensity of X-rays from a coolidge tube is plotted against wavelength \[\lambda \] as shown in figure. The minimum wavelength found is \[{{\lambda }_{C}}\] and the wavelength of the \[{{K}_{a}}\] lines is \[{{\lambda }_{K}}\] the accelerating voltage is increased.
Which one of the following rods will conduct the highest quantity of heat assuming that their ends are maintained at the same steady temperatures \[{{T}_{1}}\] and\[{{T}_{2}}\]?
The work done requires to increase the area of soap film of \[10\,cm\times 6\,cm\] to \[10\,cm\times 11\,cm\] is \[3.0\times {{10}^{-4}}J\]. The surface tension of film is :
\[8\,gm\,{{O}_{2}},\,14\,gm\,{{N}_{2}}\] and \[22\,gm\,C{{O}_{2}}\] is mixed in a container of 10 litre capacity at \[{{27}^{o}}C\]. The pressure exerted by the mixture in terms of atmospheric pressure will be:
\[1.44\,g\] of titanium (Ti) reacts with excess of \[{{O}_{2}}\] and produced \[x\] g of a non-stoichiometric compound. \[T{{i}_{1.44}}O\]. The value x is (Atomic mass of \[Ti=48\,u\])
The two particles A and B have de-Broglie wavelength 1 nm and 5 nm respectively. If mass of A is four times the mass of B, then the ratio of kinetic energies of A and B would be
Ionisation energy and electron affinity of fluorine are respectively \[17.42\,eV\] and \[3.45\,\,eV\], then electronegativity of F atom on pauling scale will be
A mixture of 32 g of \[{{O}_{2}}\] gas and \[8g\,{{D}_{2}}\] gas is allowed to effuse through an orifice, the relative rate of effusion \[\left( \frac{{{r}_{{{D}_{2}}}}}{{{r}_{{{O}_{2}}}}} \right)\] at the start of effusion is
\[A\to 2B+C\]. The above first order reaction has total pressure \[{{P}_{t}}\] after time t and after long time \[(t\to \infty )\] was\[{{P}_{\infty }}\], then k in terms of \[{{P}_{t}}\], \[{{P}_{\infty }}\] and t is
In face-centered lattice with all positions occupied by A atoms, the body centered octahedral hole is occupied by an atom B of an appropriate size. The formula of the compound is
For the reaction,\[{{N}_{2}}(g)+3{{H}_{2}}(g)\underset{{{K}_{2}}}{\overset{{{K}_{1}}}{\longleftrightarrow}}2N{{H}_{3}}(g)\],the rate law for the disappearance of \[N{{H}_{3}}\] is
A decimolar solution of potassium ferrocyanide is \[50%\] dissociated at \[300\,K\]. The osmotic pressure of the solution is \[(R=0.0821\,L\,atm\,{{K}^{-1}}mo{{l}^{-1}})\]
In Ellingham diagram two line representing variation of \[\Delta {{G}^{o}}\] with temperature for metal to metal oxide formation intersect at point P a shown in diagram
A)
Metal A can reduce BO only
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B)
Metal A can reduce AO only
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C)
Metal A and B can reduce each other simultaneously
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D)
Neither A nor B can reduce the oxide of each other's oxide
, histidine has\[p{{K}_{{{a}_{1}}}}=1.8\], \[p{{K}_{{{a}_{2}}}}=9.2\] and \[p{{K}_{{{a}_{3}}}}=6.0\]. The isoelectric point, \[(PI)\] of histidine is likely to be