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If the slope of the normal to the curve x3 = 8a2y, a > 0, at a point in the first quadrant is - 2/3, then the point is
If ω is a cube of unity and equation has one root . Then
⇒ x = 0
∵ x2 + 5 = 0. Not possible ∀ x ∈ R
If | z1 − 1 | < 1 , | z2 − 2 | < 2 , | z3 − 3 | < 3 then | z1 + z2 + z3 |
The order of the differential equation y(dy/dx)=x/(dy/dx)+(dy/dx)3 is
The solution of differential equation (xdy/dx) = y+x2 is
The order of differential equation (d2y/dx2)2 = (1+dy/dx)1/2 is
If f x = x − x2 + x3 − x4 + … ∞ , | x | < 1, then f′(x) =
If f(x) = log (1 + x/1 - x), then f(2x/1 + x2) is equal to
In how many ways can 4 boys and 3 girls be seated in a row of 7 chairs if the boys and girls alternate?
If A and B are two independent events in a given sample space and the probability that both A and B occur is 0.16 while the probability that neither occurs is 0.36. The probabilities P(A) and P(B) are respectively
If the mean of a Poisson's Distribution is λ, then its standard deviation is
If in a Δ A B C , 3 a = b + c then tanB/2 .tan C/2 is equal to
If A and B are two mutually exclusive events such that P(B) = 2P(A) and A ∪ B = S. Then P(A) is equal to
If (a, b), (c, d), (e, f) are the verices of a triangle such that a, c, e are in G.P. with common ratio r and b, d, f are in G.P. with common ratio's then the area of the triangle is
For a frequency distribution mean deviation from mean is computed by
The mode of the following items is 0, 1, 6, 7, 2, 3, 7, 6, 6, 2, 6, 0, 5, 6, 0
p and q are the roots of x2 + ax + b = 0, then equation whose roots are p2q, q2p is
p + q = - a, p.q = b
Now, p2 q + q2 p = pq(p + q)
= b(-a) = - ab
p2q . q2p = (pq)3 = b3
The required equation is
x2 - (p2q + q2p) x+ (p2q . q2p) = 0
x2 + abx + b3 = 0
If the two roots of the equation x2-3x+2=0 are equal, roots are
The smallest radius of the sphere passing thro'(1,0,0),(0,1,0) and (0,0,1) is
Let the sphere be
x2 + y2 + z2 + 2ux + 2vy + 2wz + d' = 0 ...(i)
It passes through the points (1, 0, 0), (0, 1, 0), (0, 0, 1)
∴ 1 + 2y + d' = 0...(ii)
1 + 2v + d' = 0 ...(iii)
1 + 2w + d' = 0 ...(iv)
From (ii), (iii), (iv), we get u = v = w = - d' + 1/2
Let r denote the radius of (i)
∴ r2 = u2 + v2 + w2 - d'
= 3 ( d' + 1/2 )2 - d'
= 3/4 (d' + 1)2 - d'
r will be minimum if d is minimum.
The equation of the sphere with centre (3, 6, -4) and touching the plane
2x - 2y - z - 10 = 0 is
As the required sphere touches the plane
2x - 2y - z - 10 = 0 .....(1)
∴ Radius of the required sphere = ⊥ from the center (3, 6, -4) of the required sphere to the plane (1)
∴ The required equation of the sphere is given by
(x - 3)2 + (y - 6)2 + (z + 4)2 = 16 (Central form)
⇒ x2 + 9 - 6x + y2 + 36 - 12y + z2 + 16 + 8z = 16
⇒ x2 + y2 + z2 - 6x - 12y + 8z + 45 = 0
Two common tangents to the circle x2 + y2 = 2a2 and parabola y2 = 8ax are
Consider A, B, C and D with position vectors 7î - 4ĵ + 7k̂, î - 6ĵ + 10k̂, -î -3ĵ + 4k̂ and 5î - ĵ + 5k̂ respectively. Then ABCD is a
A step-up transformer is used on a 120V line to provide a potential difference of 2400 V. If the primary coil has 75 turns, the number of turns in the secondary coil is
A positron and a proton are accelerated by the same accelerating potential. Then the ratio of the associated wavelengths of positron and proton will be [M = mass of proton; m = mass of positron]
Since both proton and positron have the same charge
A particle of mass 0.6 gm and having charge of 25 nC is moving horizontally with a uniform velocity 1.2 x 104 m s-1 in a uniform magnetic field, then the value of the magnetic induction is (g = 10 m s-2)
A condenser is connected across another charged condenser. The energy in the two condensers will be
A laser beam is used for carrying out surgery because it
As the laser beam is highly monochromatic, directional and coherent, it can be sharply focussed.
Therefore it is used for carrying out surgery
The temperature coefficient of resistance for a wire is 0.00125/ oC. At 300 K its resistance is 1 ohm. The temperature at which the resistance becomes 2 ohm is
In a Wheatstone bridge all the four arms have equal resistance R. If the resistance of the galvanometer arm is also R, The equivalent resistance of the combination as seen by the battery is
The resistance of an ammeter is 13 Ω and its scale is graduated for a current upto 100A. After an additional shunt has been connected to this ammeter it becomes possible to measure currents upto 750A by this meter. The value of shunt resistance is
The effective resistance of two resistors in parallel is 12/7 Ω. If one of the resistors is disconnected the resistance becomes 4 Ω. The resistance of the other resistor is
A caesium photocell, with a steady potential difference of 60V across, is illuminated by a bright point source of light 50cm away. When the same light is placed 1m away the photoelectrons emitted from the cell
When current cahnge from + 2 A to -2 A in 0.05 s, an e.m.f. of 8 V is induced in a coil. The coefficient of self inductance of the coil is
A coil of wire of radius r has 600 turns and a self inductance of 108 mH. The self inductance of a similar coil of 500 turns will be
As L ∝ N2
∴ L = 108 x (500/600)2 = 75 mH
A charge q is placed at the corner of a cube of side l. The electric flux passing through the cube is
Eight cubes, each side l, are required to form a Gaussian surface
So that the charge q at the corner of a small cube appears at the centre of the bigger cube
According to Gauss's law, the electric flux through the bigger cubes = q/∈0
Hence the electric flux through the given small cube =q/8∈0
Consider the situation shown in figure. Work done in carrying a point charge from k to A, B, C are WA , WB , Wc respectively. Which one is true?
Two equal charges q of opposite sign separated by a distance 2a constitute an electric dipole of dipole moment p. If P is a point at a distance r from the centre of the dipole and the line joining centre to this point makes an angle θ with the axis of the dipole, then potential at the point P is given by
A conducting sphere of radius 10 cm is charged with 10 μC. Another uncharged sphere of radius 20 cm in allowed to touch it for some time. If both the spheres are separated, then surface density of charges on the spheres will be in the ratio of
When a body is connected to the earth, then electron from the earth, flow into the body. It means that the body is
Inside a hollow charged spherical conductor, the potential at every point
Two wires of equal length are placed perpendicular to the plane of paper at a distance of 5 cm from each other. If the currents flowing in first and second wires are 2 A and 5 A respectively in opposite direction, then the resultant magnetic field at mid-point of both the wires is
What is the particle x in the following nuclear reaction?
Find the kinetic energy of the photoelectrons emitted when light of wavelength 4000 Å is incident on a metal of work function 2 eV (find approximately)
In photoelectric effect experiment, the stopping potential for incident light of wave-length 4000 Å is 2 volt. If the wavelength is changed to 3000 Å, the stopping potential is
When a glass prism of refracting angle 600 is immersed in a liquid its angle of minimum deviation is 300 . The critical angle of glass with respect to the liquid medium is
An n - type and p - silicon can be obtained by doping pure silicon with:
For obtaining ‘n’ type semi conductor V group element is doped with silicon.
For obtaining ‘p’ type semiconductor III group element is doped with silicon.
On increasing the reverse bias to a large value, in a p-n junction diode, the current
The dominant contribution to current comes from holes in case of
In intrinsic semiconductor at room temperature, number of electrons and holes are
The sun is rotating about is own axis. The spectral lines emitted from the two ends of its equator, for an observer on the earth, will show
If the phase differenc between light waves from two slits of Young's double slit experiment is π radian, then the central fringe will be
In the propagation of light waves, the angle between the direction of vibration and plane of polarisation is
Both light and sound waves produce diffraction. It is more difficult to observe the diffraction with light waves because
Consider the following compounds :
I. HCHO II.CH₃CHO III. CH₃COCH₃
The reactivities of these compounds are such that
Which of the following reagent react differently with HCHO and CH₃CHO and CH₃COCH₃
12 g of Mg (at. mass 24) on reacting completely with acid gives hydrogen gas, the volume of which at S.T.P would be
How many monochlorobutanes will be obtained on chlorination of n-butane?
Which of the following element is responsible for oxidation of water to O₂ in biological processes?
When three different amino acid molecules condense, then how many different tripeptide molecules are formed?
When two amino acids condense they form a dipeptide and water. A tripeptide is made up from three amino acids. If the three amino acids are all different then there are six possible ways in which they can combine
Which of the following compounds on boiling with KMnO₄ and the subsequent acidification will not give benzoic acid ?
Hydrolysis of an ester gives a carboxylic acid which on Kolbe's electrolysis yields ethane. The ester is
Which one of the following informations can be obtained on the basis of Le Chatelier principle ?
Two moles of HI was heated in a sealed tube at 440 oC till the equilibrium was reached. HI was found to be 22% decomposed. The equilibrium constant for dissociation is
In which of the following reactions, heat liberated is known as standard heat of formation of CO2?
Which of the following is an example of intensive property?
The number of possible structural isomers of C₄H₁₀O is
Which one of the following cyano complexes would exhibit the lowest value of paramagnetic behaviour?
(At. No. : Cr = 24, Mn = 25, Fe = 26, Co = 27)
In the following reaction C₆H₆NH₂ + CHCl₃ + 3NaOH → A + 3B + 3C the product A is
When electricity is passed through aqueous solution of aluminium chloride, 13.5 g of Al are deposited. The number of faradays of electricity passed must be
Which one of the following is not the correct reaction of aryl diazonium salts?
A certain current liberate 0.5 g of hydrogen in 2 h. How many grams of copper can be liberated by the same current flowing for the same time in a copper sulphate solution?
If the standard electrode potential of Cu2+/Cu electrode is 0.34 V, what is the electrode potential at 0.01 M concentration of Cu2+? (T = 298K)
White P when boiled with strong solution of caustic soda produces
Oxygen and cyclopropane at partial pressure of 570 torr and 170 torr respectively are mixed in a gas cylinder. What is the ratio of the number of moles of cyclopropane to the number of moles of oxygen ?
The edge length of face centred unit cubic cell is 508 pm. If the radius of the cation is 110 pm, the radius of the anion is
Which of the following can't be explained by adsorption?
Among the following ions in aqueous solution, the ion is likely to be colourless
Among the following which is used as an electrolyte in electroplating a baser metal with gold?
Fill in the blank with appropriate word.
Waking at three o'clock, I heard the........of thunder
Choose the alternative which best expresses the meaning of the idiom/phrase.
To take with a grain of salt
Improve the sentence by choosing best alternative for capitalised part of the sentence.
He should move on to the next point, and not HARP ONE STRING ONLY.