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The rate of change of the volume of a sphere w.r.t. its surface area, when the radius is 2cm, is
A real value of x will satisfy the equation
[((3  4 i x)/( 3 + 4i x))] = α  iβ ( α ,β real),if
The area bounded by the curve y = (x + 1)^{2}, y = (x  1)^{2} and the line y = 1/4 is
Given curves are
(x + 1)^{2} = y .....(1)
and (x  1)^{2} = y .....(2)
Axis of parabola (1) is x =  1 and vertex is ( 1, 0) and it meets yaxis at (0,1)
Axis of parabola (2) is x = 1 and vertex is (1, 0) and it cuts y  axis at (0, 1)
When y = 1/4
from (2), x  1 = 1/2
∴ x = 1/2 , 3/2
∴ C ≡ ( 1/2 , 1/4 )
Required area = 2 area BACB
where [x] denotes the greatest integer less than or equal to x, is continuous at x = 2, then the value of a is equal to
All the roots of the equation z^{n} cos θ_{o} + z^{n − 1} cos θ_{1} + … + cos θ_{n} = 2 , where θ_{o} , θ_{1} , θ_{2} , … , θ_{n} ∈ R will satisfy
If f(x) = tanx/x , x ≠ 0 ; f x = 1, x = 0 , then the function at x = 0 is
Multiply R_{1}, R_{2}, R_{3} by a, b, c respectively and divide by abc
Take abc common from C_{3} so that C_{1}, C_{3} are identical
A solution of the differential equation x (dy/dx)^{2} + (yx) dy/dx  y = 0 is
y = A e^{x} + B e^{2x} + C e^{3x }satisfies the diferential equation
The solution of differential equation (dy/dx)=[(1+y^{2})/(1+x^{2})] is
If tan^{1} x + tan^{1} y + tan^{1} z = π/2 , then 1  xy  yz  zx =
Let tan^{1} x = A, tan^{1} y = B, tan^{1} z = C,
Then x = tan A, y = tan B, z = tan C, A + B + C = π/2
A + B + C = π/2
⇒ tan (A + B + C) is not defined
⇒ 1  S_{2} = 0
⇒ 1  tan A tan B  tan B tan C  tan C tan A = 0
⇒ 1  xy  yz  zx = 0
A and B are square matrices of the same order. Which is correct?
"It has two pairs of opposite sides parallel." Which of the following make this open sentence true?
Parallelograms and rectangles have two pairs (sets) of parallel sides. A trapezoid has one pair (set) of parallel sides.
A parallelogram is cut by two sets of m lines parallel to the sides, the number of parallelogram thus formed is
The chance of throwing a total of 7 or 12 with 2 dice, is
From a box containing 10 cards, numbered 1, 2,3,......,10. Four cards are drawn together, what is the probability of their sum is even ?
The probabilities that three men hit a target are 1/6 , 1/4 , 1/3 respectively. Each shoots once the target what is the probability that exactly one of the them hits the target?
In the expansion of (1 + x)^{n},coefficients of 2nd ,3rd and 4th terms are in A.P.Then n is equal to
The mean age of a combined group of men and women is 30 years. If the means of the age of men and women are respectively 32 and 27, then the percentage of women in the group is
If equations x = cy + bz, y = az + cx and z = bx + ay have any solution except x = 0, y = 0, z = 0, then the relation between a, b and c is
The system of homogeneous equations
x − cy − bz = 0......(i)
cx − y  bz = 0
bx  ay − z = 0
has nontrivial solution (since x, y, z are not all zero)
The given expression be
x^{2} + 2bx + c
Comparing it with Ax^{2} + Bx + c, we get
A = 1, B = 2b, C = c
Also A = 1 > 0
x^{2} + 2bx + c will be positive i.e.
will be of the same sign as that of A,
if B^{2}  4AC < 0
4b^{2}  4(1) (c) < 0
or b^{2} < c
The equation of the plane through intersection of planes x + 2y + 3z = 4 and 2x + y  z = 5 and perpendicular to the plane 5x + 3y + 6z + 8 = 0 is
The radius of the sphere which passes thro' the points (0,0,0),(1,0,0),(0,1,0) and (0,0,1) is
If 2sec2α = tanβ + cotβ, then one of the value of (α+β) is
The number of tangents to the circle x^{2} + y^{2}  2x + 4y = 0 through the point (1, 2) is
If a circle passes through the point (a, b) and cuts the cirlce x^{2}+ y^{2} = p^{2} orthogonally, then the equation of the locus of its centre is
If vectors i3j+2k, i+2j represent the diagonals of a parallelogram, then its area is
are three non coplanar vectors, then the value of the expression
Light of wave length λ and photon energy 2eV falling on a metal surface produces photoelectrons of maximum velocity V. If the λ is decreased by 25% and the maximum velocity of electron is doubled, the work function of the metal is (in eV)
If the electric flux entering and leaving an enclosed surface respectively is φ_{1} and φ_{2} , the electric charge inside the surface will be
Hint: Here, we have to apply gauss's law to find electric charge.
Solution:
We know that, electric flux ϕ1 (or electric field lines) entering in a closed surface is ve and electric flux ϕ2 (or electric field lines) leaving a closed surface is +ve.
Hence, net electric flux through the closed surface,
ϕ=ϕ2−ϕ1
Now, according to Gauss' theorem, the net electric flux ϕ passing through a closed surface is equal to the 1/ε0 times of the total charge q, inside the surface.
Step1: Apply gauss's law
Given, Net electric flux, ϕ=(ϕ2−ϕ1)
ϕ=ε0qin
⇒qin=ε0ϕ
∴qin=(ϕ2−ϕ1)ε0
n small drops of same size are charged to V volt each. They coalesce to form a bigger drop. The potential of the bigger drop is
A coil of n number of turns is wound tightly in the form of a spiral with inner and outer radii a and b respectively. When a current of strength I is passed through the coil, the magnetic field at its centre is
If 1000 droplets each of charge q and radius r are mixed to form a big drop, then the potential of big drop, as compared to small droplet will be
If the value of potential on each small drop be V and potential on each small drop be V ^{'}
V ^{'} = n ^{2/3} V
⇒ V ^{'} = 1000 ^{2/3} V
The masses of neutron, proton and deuteron in a.m.u are 1.00893 1.00813 and 2.01473 respectively. The packing fraction of the deuteron in a.m.u is
Packing fraction = (M  A)/A , where M is the atomic mass and A is the mass number.
P = M  A/A = (2.01473  2) /2 = 73.6 x 10^{4}
Capacitance of a capacitor becomes 7/6 times its original value if a dielectric slab of thickness, t = 2/3 d is introduced in between the plates. 'd' is the separation between the plates. The dielectric constant of the dielectric slab is
A ray of light is incident on a plane mirror at an angle of 60^{0} . The angle of deviation produced by the mirror is
For a transistor the parameter β = 99. The value of the parameter α is
In Young's double slit experiment, the width of the fringes obtained with light of wavelength 6000 Angstrom is 2.0mm. The fringe width, if the entire apparatus is immersed in a liquid of refractive index 1.33, will be
Fringe width in Young's double slit experiment β = Dλ/d
When apparatus is immersed in liquid, only wavelength of light (λ) changes
Wavelength of light is liquid,
λ' = λ/n , where n = refractive index of medium
Initial fringe width, β = Dλ/d .....(i)
Fringe width in liquid, β' = Dλ'/d .....(ii)
Dividing Equation (ii) by Equation (i), we get
β/ β = λ'/λ
or β'/β = λ n/λ = 1/n
or β' = β/n
The new fringe width, β' = 2.0/1.33 mm = 1.5 mm
In the propagation of light waves, the angle between the direction of vibration and plane of polarisation is
Which of the following method is most appropriate for the manufacture of methane?
Which of the following does not turn Schiff's reagent to pink?
Ozonolysis of C₇H₁₄ gave 2methyl3pentanone toluene. The alkene is
The energy stored in the cells of a living body is in the form of
Which of the following substance when boiled with caustic soda evolve ammonia ?
To become a carbohydrate a compound must contain at least
Which salt can be produced by the reaction of carbon monoxide and caustic soda (NaOH)?
The acid showing salt like character in aqueous solutions is
A chemical reaction is catalyzed by a catalyst X. Hence X
A glass bulb is filled with NO₂ gas and immersed in an ice bath at 0^{o}C which becomes colourless after some time . This colourless gas will be
In a closed insulated container a liquid is stirred with a paddle to increase the temperature. Which of the following is true
Identify the reaction in which the heat liberated corresponds to the heat of formation ( Δ H )
[Cr(H₂O)₆]Cl₃(at no. of Cr=24) has a magnetic moment of 3.83 BM. The correct distribution of 3d electrons in the Chromium of the complex is
Which of the following is produced by reaction of RCN in sodium and alcohol ?
The indicator that is obtained by coupling the diazonium salt of sulphanilic acid with N,Ndimethylaniline is
The cell potential of the following cell at 25 ^{0}C (in volts) is
Pt , H_{2} (1 atm)  H^{+} 0.01 M ∥ Cu^{2+} (0.1 M)  Cu (E^{o} Cu^{2+} ∕ Cu = + 0.337 V)
A smuggler could not carry gold by chemically depositing iron on the gold surface since
The formal oxidation state of the nitrogen atom in a nitrite is +3. This means that it is can be either oxidised to oxidation states +4 and +5 or reduced to oxidation states as low as 3.
A container contains 1 mole of a gas at 1 atm pressure and 27^{o}C, and its volume is 24.6 ltres. If pressure is 10 atm and temperature 327^{o}C, then new volume is approximately
P_{4}O_{10} has short and long PO bonds. The number of short PO bonds in this compound is
P_{4}O_{10} has four P = O bonds which are shorter than PO single bonds.
Each P atom has three PO single bonds and one P = O bond, i.e.,
a total of four PO linkages
The catalytic activity of the transition metals and their compounds is ascribed to
The separation of colloidal particles from those of molecular dimensions is called
The brown ring complex compounds is formulated as [Fe(H_{2}O)_{5}NO] SO_{4}. The oxidation state of iron is
Fill in the blank with appropriate word.
The Hubble Space Telescope will search for planets around other stars, a key to the..........for extra terrestrial life, and examine interstellar dust gases out of which stars are born.
Out of the given alternatives, choose the one which can be substituted for the given capitalised word.
The staff have felt ON EDGE since they heard the rumour about retrenchment.
Improve the sentence by choosing best alternative for capitalised part of the sentence.
It was indeed a shock for her, but she has LATER recovered from it.
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