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The density of a material in the shape of a cube is determines by measuring three sides of the cube and its mass. If the relative errors in measuring the mass and length are respectively 1.5% and 1%, the maximum error in determining the density is :
All the graphs below are intended to represent the same motion. One of them does it incorrectly. Pick it up.
In graph ‘2’ initial slope is zero which is not possible, since initial velocity is non zero in all other three graphs.
Two masses m_{1 }= 5 kg and m_{2} = 10 kg, connected by an inextensible string over a frictionless pulley, are moving as shown in the figure. The coefficient of friction of horizontal surface is 0.15. The minimum weight m that should be put on top of m_{2} to stop the motion is :
A particle is moving in a circular path of radius a under the action of an attractive potential Its total energy is :
‘ – ‘ sign of force implies attractive force So,
In a collinear collisio n, a particle with an init ial speed ν_{0} strikes a stationary particle of the same mass. If the final total kinetic energy is 50% greater than the original kinetic energy, the magnitude of the relative velocity between the two particles, after collision, is:
By cons of linear momentum,
So relative velocity :
Hence the Solution is Option (2)
Seven identical circular planar disks, each of mass M and radius R are welded symmetrically as shown. The moment of inertia of the arrangement about the axis normal to the plane and passing through the point P is :
Moment of inertia of each disc about the given axis is,
From a uniform circular disc of radius R and mass 9 M, a small disc of radius R/3 is removed as shown in the figure. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through centre of disc is:
Mass of disc = Volume x Density
9M = A x T x ρ (Area x thickness x density)
(i) Divided by (ii)
Moment of inertia of complete disc about an axis passes through O.
Moment of inertia of cut off disc about an axis passes through O
So, moment of inertia of remaining disc = I_{1} I_{2}
A particle is moving with a uniform speed in a circular orbit of radius R in a central force inversely proportional to the n^{th} power of R. If the period of rotation of the particle is T, then :
A solid sphere of radius r made of a soft material of bulk modulus K is surrounded by a liquid in a cylindrical container. A massless piston of area a floats on the surface of the liquid, covering entire cross section of cylindrical container. When a mass m is placed on the surface of the piston to compress the liquid, the fractional decrement in the radius of the sphere, (dr/r) is:
Bulk Modulus,
Two moles of an ideal monoatomic gas occupies a volume V at 27^{0}C. The gas expands adiabatically to a volume 2 V. Calculate (a) the final temperature of the gas and (b) change in its internal energy.
Initially n = 2, v, T= 300k
Finally V_{d }= 2v
Gas is monoatomic, So, r = 5/3
So,
Since gas undergoes expensed.
The mass of a hydrogen molecule is 3.32 x 10^{27} kg. If 10^{23} hydrogen molecules strike, per second, a fixed wall of area 2 cm^{2} at an angle of 45^{0} to the normal, and rebound elastically with a speed of 10^{3} m/s, then the pressure on the wall is nearly :
Change in momentum normal to the wall
A silver atom in a solid oscillates in simple harmonic motion in some direction with a frequency of 10^{12}/sec. What is the force constant of the bonds connecting one atom with the other ? (Mole wt. of silver = 108 and Avagadro number = 6.02 x 10^{23} gm mole^{1})
A granite rod of 60 cm length is clamped at its middle point and is set into longitudinal vibrations. The density of granite is 2.7 x 10^{3} kg/m^{3} and its young's modulus is 9.27 x 10^{10} Pa. What will be the fundamental frequency of the longitudinal vibrations ?
Since rod is clamped at centre. So centre it behaves as node & end it behave as antinode.
So,
Three concentric metal shells A, B and C of respective radii a, b and c (a < b < c) have surface charge densit ies +σ ,σ and +σ respectively. The potent ial of shell B is :
Charge in sphere
A parallel plate capacitor of capacitance 90 pF is connected to a battery of emf 20 V. If a dielectric material constant K = 5/3 is inserted between the plates, the magnitude of the induced charge will be :
q_{i} = C_{0}V
= 90 x 10^{12} x 20
= 1800 x 10^{12}
= 1.8nC
In an a.c. circuit, the instantaneous e.m.f. and current are given by:
e = 100 sin 30 t
.
In one cycle of a.c., the average power consumed by the circuit and the wattless current are, respectively :
Two batteries with e.m.f. 12 V and 13 V are connected in parallel across a load resistor of 10Ω. The internal resistances of the two batteries are 1Ω and 2Ω respectively. The voltage across the load lies between :
The circuit may be drawn as shown in the figure.
An electron, a proton and an alpha particle having the same kinetic energy are moving in circular orbits of radii r_{e}, r_{p}, r_{α} respectively in a uniform magnetic field B. The relation between r_{e}, r_{p}, r_{α} is :
The dipole moment of a circular loop carrying a current I, is m and the magnetic field at the centre of the loop is B_{1}. When the dipole moment is doubled by keeping the current constant, the magnetic field at the centre of the loop is B2. The ratio B_{1}/B_{2 }is:
Magnetic moment, and magnetic field at the centre of circle
For an RLC circuit driven with voltage of amplitude and frequency the current exhibits resonance. The quality factor, Q is given by
Alternate solution
is the only dimensionless quantity, hence must be the quality factor.
An EM wave from air enters a medium. The electric fields are in air and in medium, where the wave number k and frequency refer to their values in air. The medium is nonmagnetic. ∈_{r1 }and ∈_{r2 }refer to relative permitivit ies of air and medium respectively, which of the following options is correct ?
Where
So volume in medium 1 = C
Volume in medium 2 = C/2
Unpolarized light of intensity I passes through an ideal polarizer A. Another identical polarizer B is placed behind A. The intensity of light beyond B is found to be 1/2. Now another identical polarizer C is placed between A and B. The intensity beyond B is now found to be 1/8 The angle between polarizer A and C is :
Unpolarized light of intensity I, when passed through a polarizer A, its intensity becomes I/2
Since intensity of light emerging from polarizer B = I/2
So, A & B are parallel placed.
Let, C makes angle θ with A.
So,
The angular width of the central maximum in a single slit diffraction pattern is 60^{0}. The width of the slit is 1μm. The slit is illuminated by monochromatic plane waves. If another slit of same width is made near it, Young’s fringes can be observed on a screen placed at a distance of 50 cm from the slits. If the observed fringe width is 1 cm, what is slit separation distance ?
(i.e. distance between the centres of each slit.)
An electron from various excited states of hydrogen atom emit radiation to come to the ground state. Let λ_{n},λ_{g} be the de Broglie wavelength of the electron in the n^{th} state and the ground state respectively. Let ∧_{n} be the wavelength of the emitted photon in the transit ion fro m the n^{th} state to the ground state. For large n, (A, B are constants)
2πr = nλ_{n}
_{}
_{}
_{}
From equation (1) and (2)
If the series limit frequency of the Lyman series is then the series limit frequency of the Pfund series is :
Series limit frequency of the Lyman series is given by
Series limit frequency of the Pfund series
It is found that if a neutron suffers an elastic collinear collision with deuterium at rest, fractional loss of its energy is p_{d}; while for its similar collision with carbon nucleus at rest, fractional loss of energy is pc. The values of p_{d} and p_{c} are respectively :
(from conservation of momentum)
and K_{0} = K_{1} + K_{2} (for elastic collision)
So after solving
The reading of the ammeter for a silicon diode in the given circuit is :
A telephonic communication service is working at carrier frequency of 10 GHz. Only 10% of it is utilized for transmission. How many telephonic channels can be transmitted simultaneously if each channel requires a bandwidth of 5 kHz ?
No. of telephonic channels that can be transmitted simultaneously
In a potentiometer experiment, it is found that no current passes through the galvanometer when the terminals of the cell are connected across 52 cm of the potentiometer wire. If the cell is shunted by a resistance of 5Ω, a balance is found when the cell is connected across 40 cm of the wire. Find the internal resistance of the cell.
When the cell is shunted by a resistance of 5 Ω
On interchanging the resistances, the balane point of a meter bridge shifts to the left by 10cm. The resistance of their series combinations is 1 kΩ. How much was the resistance on the left slot before interchanging the resistances?
(i)
(ii)
From (i) and (ii)
⇒ l = 55 cm
(1)
and R1 + R 2 = 1000 Ω (iii)
from (i) and (iii)
R = 550 Ω
The ratio of mass present of C and H of an organic compound (C_{X}H_{Y}O_{Z}) is 6 : 1. If one molecule of the above compound (C_{X}H_{Y}O_{Z}) contains half as much oxygen as required to burn one molecule of compound C_{X}H_{Y} completely to CO_{2} and H_{2}O. The empirical formula of compound C_{X}H_{Y}O_{Z} is :
Which type of ‘defect’ has the presence of cations in the interstitial sites?
In Frenkel defect, smaller ion displaces from its actual lattice site into the interstitial sites.
According to molecular orbital theory, which of the following will not be a viable molecule?
electronic configuration
Number of electrons =
Molecule does not exist.
Which of the following lines correctly show the temperature dependence of equilibrium constant K, for an exothermic reaction?
For exothermic reaction ΔH = ve.
The combustion of benzene (1) gives CO_{2}(g) and H_{2}O(I). Given that heat of combustion of benzene at constant volume is 3263.9 kJ mol^{1} at 25^{0} C; heat of combust ion (in kJ mol^{1}) of benzene at constant pressure will be :
For 1 molal aqueous solution of the following compounds, which one will show the highest freezing point?
An aqueous solution contains 0.10 M H_{2}S and 0.20 M HCl. If the equilibrium constants for the format ion of HS^{} from H_{2}S is 1.0 x 10^{7} and that of S^{2} from HS^{} ions is 1.2 x 10^{13} then the concentration of S^{2} ions in aqueous solution is :
An aqueous solution contains an unknown concentration of Ba^{2+}. When 50 mL of a 1M solution of Na_{2}SO_{4} is added, BaSO_{4} just begins to precipitate. The final volume is 500 mL. The solubility product of BaSO_{4} is 1 x 10^{10}. What is the original concentration of Ba^{2+} ?
At 518^{0}C, the rate of decomposition of a sample of gaseous acetaldehyde, initially ar a pressure of 363 Torr, was 1.00 Torr s^{1} when 5% had reacted and 0.5 Torr s^{1} when 33% had reacted. The order of the reaction is:
How long (approximate) should water be electrolysed by passing through 100 amperes current so that the oxygen released can completely burn 27.66 g of diborane?
(Atomic weight of B = 10.8 u)
Moles of O_{2} required = 3
Gram equivalent of O_{2} = 3 x 4= 12
Gram equivalent
The recommended concentration of fluoride ion in drinking water is up to 1 ppm as fluoride ion is required to make teeth enamel harder by converting [3Ca_{3} (PO_{4})_{2} . Ca(OH)_{2}]
Which of the following compounds contain(s) no covalent bond(s)?
KCl, PH_{3}, O_{2}, B_{2}H_{6}, H_{2}SO_{4}
KCl is an ionic compound. It cannot from covalent bond. Elements of s – block & p – block combine to form ionic compounds.
Which of the following are Lewis acids?
Both BCl3 and AlCl3 are Lewis acids as both ‘B’ and ‘Al’ has vacant porbitals. SiCl4 is also a Lewis acid as silicon atom has vacant 3dorbital.
Therefore,both b and d options are possible.
Total number of lone pair of electrons in I^{}_{3} ion is :
The total number of lone pair of electrons is I^{}_{3} is 9
Which of the following salts is the most basic in aqueous solution?
CH_{3}COOK is a salt of weak acid and strong base. Hydrolysis of potassium acetate gives strong base KOH
Hydrogen peroxide oxides [Fe(CN)_{6} ]^{4} to [Fe(CN)_{6}]^{3 } in acidic medium but reduces [Fe(CN)_{6} ]^{3} to [Fe(CN)_{6}]^{4 }in alkaline medium. The other products formed are, respectively:
The oxidation states of Cr in [Cr(H_{2}O)_{6}] Cl_{3 }, [Cr(C_{6}H_{6})_{2}] and K_{2}[Cr(CN)_{2} (O)_{2} (O_{2}) NH_{3 }] respectively are :
The Oxidation state of Cr in
[Cr(H_{2}O)_{6}]CI_{3}
x + 6(0)  3 = 0
x = +3
[Cr(C_{6}H_{6})_{2}]
x + 2(0) = 0
x = 0
K_{2} [Cr (CN )_{2} (O)_{2} (O_{2}) NH_{3}]
+2 + x + 2 (1) + 2 (2) +1 (2) + 0
= 6
∴ x =+6
The compound that does not produce nitrogen gas by the thermal decomposition is :
When metal ‘M’ is treated with NaOH, a white gelatinous precipitate ‘X’ is obtained, which is soluble in excess of NaOH. Compound ‘X’ when heated strongly gives an oxide which is used in chromatography as an adsorbent. The metal ‘M’ is :
The Gelatinous precipitate formed in Al(OH)_{3} ; Al(OH)_{3} on strong heating gives Al2O_{3 }which is used in chromatography as an adsorbent. So the metal is A1.
Consider the following reaction and statements:
[Co(NH_{3})_{4}Br_{2}]^{+} +Br^{ }→ [Co(NH_{3})_{3}Br_{3}] + NH_{3}
(I) Two isomers are produced if the reactant complex ion is a cisisomer.
(II) Two isomers are produced if the reactant complex ion is a transisomer.
(III) Only one isomer is produced if the reactant complex ion is a transisomer.
(IV) Only one isomer is produced if the reactant complex ion is a cisisomer.
The correct statements are :
As all the NH3 posit ions are identical only one product can be formed.
Glucose on prolonged heating with HI gives :
The transalkenes are formed by the reduction of alkynes with :
Trans alkenes are formed by the reaction of alkynes with Na/liq. NH_{3} (birch Reduction)
Which of the following compounds will be suitable for Kjeldahl’s method for nitrogen estimation?
Kjeldahl’s method:
Organic compounds nitrogen +
Nitro compounds, A_{2}O compounds & Nitrogen part of the aromatic ring will not give positive result for Kjeldahl’s method. Aniline is the best suitable to estimate nitrogen using Kjeldahl’s method.
Phenol on treatment with CO_{2} in the presence of NaOH followed by acidification produces compound X as the major product. X on treatment with (CH_{3}CO)_{2}O in the presence of catalytic amount of H_{2}SO_{4} produces :
An alkali is titrated against an acid with methyl orange as indicator, which of the following is a correct combination?
When a weak base is titrated with string acid with methyl orange as an indicator then at end point The colour change will be yellow to pinkish red.
The predominant form of histamine present in human blood is (pk_{a} , Histidine = 6.0)
Phenol reacts with methyl chloroformate in the presence of NaOH to form product A. A reacts with Br_{2} to form product B. A and B are respectively :
The increasing order of basicity of the following compounds is :
Order of base nature depends on electron donation tendency.
It compounds (b) nitrogen is sp2 hybridized so least basic among all given compound
Compound (c) is a very strong nitrogeneous organic base as a lone pair of one nitrogen delocalize in resonance and make another nitrogen negatively charged and conjugate acid have two equivalent resonating structure.
Thus it is most basic in given compound.
(d) is secondary amin more than (a) which is a primary amine.
Order of base nature depends on electron donation tendency.
It compounds (b) nitrogen is sp2 hybridized so least basic among all given compound
Compound (c) is a very strong nitrogeneous organic base as a lone pair of one nitrogen delocalize in resonance and make another nitrogen negatively charged and conjugate acid have two equivalent resonating structure.
Thus it is most basic in given compound.
(d) is secondary amin more than (a) which is a primary amine
The major product formed in the following reaction is :
The major product of the following reaction is :
Let S= {x ∈ R : x ≥ 0 and 2  √x  3 + √x (√x  6) + 6 = 0} . Then S :
There are exactly two elements in the given set.
If α ,β ∈ c are the distinct roots, of the equation x^{2} x +1 =0 , then α^{101} + β^{107} is equal to :
x^{2} – x+ 1 = 0
Let a_{1 }, a_{2} , a_{3} , ......, a_{49} be in A.P. such that = 416 a_{9} + a_{43} = 66.
If a_{1}^{2} + a_{2}^{2} + .... + a_{17}^{2 }= 140m then m is equal to:
a_{1} + a_{5} + a_{9} = 416 ⇒ a + 24d = 32 ......(i)
a_{9} + a_{43} = 66 ⇒ a + 25d = 33......(ii)
From (i) and (ii) d = 1 and a = 8
Let A be the sum of the first 20 terms and B be the sum of the first 40 terms of the series
1 + 2•2^{2} + 3^{2} + 2•4^{2} + 5^{2} + 2•6^{2} + .....
If B2A=100λ, then λ is equal to:
A = 1^{2} + 2.2^{2} + 3^{2} + 2.4^{2}+ ………+ A^{2}+2.20^{2}
= (1^{2} + 2.2^{2} + 3^{2} + 4^{2}+ ………+20^{2})+ (2^{2}+ 4^{2}+ ………+ 20^{2})
= 2870 + 1540 = 4410 = 2870 + 1540 = 4410
= 540 x 41 + 41 x 280 = 41 x 820 = 33620
33620  8820 = 100λ
100λ = 24800
λ = 248
If the curve y^{2} = 6x, 9x^{2} + by^{2} = 16 intersect each other at right angles, then the value of b is:
2yy’= 6
Let g(x) = cos x^{2} ,f(x) = √x, and α ,β (α < β) be the roots of the quadratic equation 18x^{2}  9πx +π^{2} = 0. Then the area (in sq. units) bounded by the curve y = (gof )(x) and the lines x =α , x = β and y = 0 , is :
g(x) = cos x^{2}
f(x) = √x
g(f (x)) = cos x
Let y=y(x) be the solution of the differential equation
Integrating both sides we get y sin x = 2x^{2}+ C
A straight line through a fixed point (2, 3) intersects the coordinate axes at distinct points P and Q. If O is the origin and the rectangle OPRQ is completed, then the locus of R is:
Let R = (h,k)
P = (0, k)
Q = (h,0)
Equation of line would be,
2k + 3h = hk
Locus of (h, k) is 2y + 3x = xy
Tangent and normal are drawn at P(16, 16) on the parabola y^{2} = 16x, which intersect the axis of the parabola at A and B, respectively. If C is the centre of the circle through the points P, A and B and ∠CPB=θ, then a value of tanq is :
The equation of tangent at P
The normal is y = y – 16 = 2(x – 16)
B = (24, 0)
AB is the diameter
Centre of the circle C = (4, 0)
lope of PB = 2 = m_{1}
Tangents are drawn to the hyperbola 4x^{2} y^{2}= 36 at the points P and Q. If these tangents intersects at the point T(0, 3) then the area (in sq. units) of ΔPTQ is :
Equation of PQ,
4x.(0) 3y = 36
Y = 12
Let be a vector coplanar with the vectors and perpendicular to then, is equal to :
PQR is a triangular park with PQ=PR=200 m. A T.V. tower stands at the midpoint of QR. If the angle of elevation of the top of the tower at P,Q and R are respectively 45^{0}, 30^{0} and 30^{0} then the height of tower (in m) is :
200 = 3h^{2} +h^{2}
4h^{2 }= (200)^{2}
4h^{2 }= 40000
h = 100
The Boolean expression ~(p ν q) ν (~p ∧ q) is equivalent to:
Two sets A and B are as under :
A={(a, b) ∈ R x R : a  5 <1 and b  5<1};
B={(a, b) ∈ R x R : 4(a  6)^{2} + 9(b  5)^{2} ≤ 36 }. Then :
If = ( A + Bx)( x A)^{2} , then the ordered pair (A, B) is equal to :
If the system of linear equations
x + ky + 3z = 0
3x + ky  2z = 0
2x + 4y  3z = 0
has a nonzero solution (x, y, z), then xz/y^{2} is equal to:
From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. The number of such arrangements is :
The sum of the coefficients of all odd degree terms in the expansion of
For each t ∈ R, let [t] be the greatest integer less than or equal to t. Than
Let S={t ∈ R:f(x)= x  π • (e^{x}  1) sin x is not different iable at t}. Then the set S is equal to:
Let and g(x) = x  1/x, x∈R  {1, 0,1} . If then the local minimum value of value of h(x) is :
Let the orthocenter and centroid of a triangle be A(3, 5) and B(3, 3) respectively. If C is the circumcentre of this triangle, than the radius of the circle having line segment AC as diameter, is :
If the tangent at (1,7) to the curve x^{2} =y 6 touches the circle x^{2} + y^{2} +16x +12 y + c = 0 than the value of c is :
If L_{1} is the line of intersect ion of the planes 2x  2y + 3z = 0, x  y + z = 0 and L_{2} is the line of intersection of the planes x + 2y  z  3 = 0, 3x  y + 2z = 0 then the distance of the origin fro m the plane, containing the lane L_{1} and L_{2} , is:
The length of the project ion of the line segment joining the points (5, 1, 4) and (4, 1, 3) on the plane, x + y + z = 7 is:
A bag contains 4 red and 6 black balls. A ball is drawn at random from the bag, its colour is observed and this ball along with two addit ional balls of the same colour is returned to the bag. If now a ball is drawn at random from the bag, then probability that this drawn ball is red, is:
If and then the standard deviation of the 9 items x_{1} x_{2}, …., x_{9} is:
If sum of all the solution of equation in [0,π ] is kπ ,then k is equal to :
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