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If the velocity of light $$c$$, universal gravitational constant $$G$$ and planck's constant $$h$$ are chosen as fundamental quantities. The dimensions of mass in the new system is:
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For a train engine moving with speed of $$20 \text{ ms}^{-1}$$, the driver must apply brakes at a distance of 500 m before the station for the train to come to rest at the station. If the brakes were applied at half of this distance, the train engine would cross the station with speed $$\sqrt{x} \text{ ms}^{-1}$$. The value of $$x$$ is ______. (Assuming same retardation is produced by brakes)
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As shown in the figure a block of mass 10 kg lying on a horizontal surface is pulled by a force F acting at an angle $$30°$$, with horizontal. For $$\mu_s = 0.25$$, the block will just start to move for the value of $$F$$: [Given $$g = 10 \text{ m} \cdot \text{s}^{-2}$$]
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A block is fastened to a horizontal spring. The block is pulled to a distance $$x = 10$$ cm from its equilibrium position (at $$x = 0$$) on a frictionless surface from rest. The energy of the block at $$x = 5$$ cm is 0.25 J. The spring constant of the spring is ______ $$\text{N m}^{-1}$$.
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A force $$F = (5 + 3y^2)$$ acts on a particle in the $$y$$-direction, where $$F$$ is newton and $$y$$ is in meter. The work done by the force during a displacement from $$y = 2$$ m to $$y = 5$$ m is ______ J.
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Figures (a), (b), (c) and (d) show variation of force with time
The impulse is highest in figure.
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Moment of inertia of a disc of mass $$M$$ and radius '$$R$$' about any of its diameter is $$\frac{MR^2}{4}$$. The moment of inertia of this disc about an axis normal to the disc and passing through a point on its edge will be, $$\frac{x}{2}MR^2$$. The value of $$x$$ is ______.
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The escape velocities of two planets $$A$$ and $$B$$ are in the ratio $$1 : 2$$. If the ratio of their radii respectively is $$1 : 3$$, then the ratio of acceleration due to gravity of planet $$A$$ to the acceleration of gravity of planet $$B$$ will be:
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For a body projected at an angle with the horizontal from the ground, choose the correct statement
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The Young's modulus of a steel wire of length 6 m and cross-sectional area $$3 \text{ mm}^2$$, is $$2 \times 10^{11} \text{ N/m}^2$$. The wire is suspended from its support on a given planet. A block of mass 4 kg is attached to the free end of the wire. The acceleration due to gravity on the planet is $$\frac{1}{4}$$ of its value on the earth. The elongation of wire is (Take $$g$$ on the earth $$= 10 \text{ m/s}^2$$):
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The surface of water in a water tank of cross section area $$750 \text{ cm}^2$$ on the top of a house is $$h$$ m. above the tap level. The speed of water coming out through the tap of cross section area $$500 \text{ mm}^2$$ is $$30 \text{ cm s}^{-1}$$. At that instant, $$\frac{dh}{dt}$$ is $$x \times 10^{-3} \text{ m s}^{-1}$$. The value of $$x$$ will be ______.
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For three low density gases $$A, B, C$$ pressure versus temperature graphs are plotted while keeping them at constant volume, as shown in the figure. The temperature corresponding to the point '$$K$$' is:
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A Carnot engine operating between two reservoirs has efficiency $$\frac{1}{3}$$. When the temperature of cold reservoir raised by $$x$$, its efficiency decreases to $$\frac{1}{6}$$. The value of $$x$$, if the temperature of hot reservoir is $$99°C$$, will be
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Choose the correct length $$(L)$$ versus square of time period $$(T_2)$$ graph for a simple pendulum executing simple harmonic motion
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A cubical volume is bounded by the surfaces $$x = 0, x = a, y = 0, y = a, z = 0, z = a$$. The electric field in the region is given by $$\vec{E} = E_0 x\hat{i}$$. Where $$E_0 = 4 \times 10^4 \text{ NC}^{-1} \text{ m}^{-1}$$. If $$a = 2$$ cm, the charge contained in the cubical volume is $$Q \times 10^{-14}$$ C. The value of $$Q$$ is ______.
(Take $$\epsilon_0 = 9 \times 10^{-12} \text{ C}^2 \text{ N}^{-1}\text{m}^{-2}$$)
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Given below are two statements : One is labelled as Assertion A and the other is labelled as Reason R.
Assertion A : Two metallic spheres are charged to the same potential. One of them is hollow and another is solid, and both have the same radii. Solid sphere will have lower charge than the hollow one.
Reason R : Capacitance of metallic spheres depend on the radii of spheres.
In the light of the above statements, choose the correct answer from the options given below.
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Given below are two statements : One is labelled as Assertion A and the other is labelled as Reason R.
Assertion A : For measuring the potential difference across a resistance of $$600 \Omega$$, the voltmeter with resistance $$1000 \Omega$$ will be preferred over voltmeter with resistance $$4000 \Omega$$.
Reason R : Voltmeter with higher resistance will draw smaller current than voltmeter with lower resistance.
In the light of the above statements, choose the most appropriate answer from the options given below
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Equivalent resistance between the adjacent corners of a regular $$n$$-sided polygon of uniform wire of resistance $$R$$ would be :
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In the given circuit, the value of $$\left|\frac{I_1+I_3}{I_2}\right|$$ is:
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A coil or radius R is placed in a constant magnetic field B .The magnetic field is parallel to the plane of the coil as shown in figure. Find magnetic flux passing through the coil.
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As shown in the figure, a long straight conductor with a semicircular arc of radius $$\frac{\pi}{10}$$ m is carrying a current $$I = 3A$$. The magnitude of the magnetic field at the center $$O$$ of the arc is: (The permeability of the vacuum $$= 4\pi \times 10^{-7} \text{ NA}^{-2}$$)
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A square shaped coil of area $$70 \text{ cm}^2$$ having 600 turns rotates in a magnetic field of $$0.4 \text{ Wb m}^{-2}$$, about an axis which is parallel to one of the side of the coil and perpendicular to the direction of field. If the coil completes 500 revolution in a minute, the instantaneous emf when the plane of the coil is inclined at $$60°$$ with the field, will be ______ V.
(Take $$\pi = \frac{22}{7}$$)
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The ratio of average electric energy density and total average energy density of electromagnetic wave is:
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Two objects $$A$$ and $$B$$ are placed at 15 cm and 25 cm from the pole in front of a concave mirror having radius of curvature 40 cm. The distance between images formed by the mirror is:
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As shown in the figure, in Young's double slit experiment, a thin plate of thickness $$t = 10 \mu m$$ and refractive index $$\mu = 1.2$$ is inserted infront of slit $$S_1$$. The experiment is conducted in air ($$\mu = 1$$) and uses a monochromatic light of wavelength $$\lambda = 500$$ nm. Due to the insertion of the plate, central maxima is shifted by a distance of $$x\beta_0$$. $$\beta_0$$ is the fringe-width before the insertion of the plate. The value of the $$x$$ is ______
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The threshold frequency of metal is $$f_0$$. When the light of frequency $$2f_0$$ is incident on the metal plate, the maximum velocity of photoelectron is $$v_1$$. When the frequency of incident radiation is increased to $$5f_0$$, the maximum velocity of photoelectrons emitted is $$v_2$$. The ratio of $$v_1$$ to $$v_2$$ is:
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An electron of a hydrogen like atom, having $$Z = 4$$, jumps from $$4^{th}$$ energy state to $$2^{nd}$$ energy state. The energy released in this process, will be: (Given $$Rch = 13.6$$ eV)
Where $$R =$$ Rydberg constant, $$c =$$ Speed of light in vacuum, $$h =$$ Planck's constant
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Nucleus $$A$$ having $$Z = 17$$ and equal number of protons and neutrons has 1.2 MeV binding energy per nucleon. Another nucleus $$B$$ of $$Z = 12$$ has total 26 nucleons and 1.8 MeV binding energy per nucleons. The difference of binding energy of $$B$$ and $$A$$ will be ______ MeV.
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Choose the correct statement about Zener diode:
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In an amplitude modulation, a modulating signal having amplitude of $$X$$ V is superimposed with a carrier signal of amplitude $$Y$$ V in first case. Then, in second case, the same modulating signal is superimposed with different carrier signal of amplitude $$2Y$$ V. The ratio of modulation index in the two case respectively will be :
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The molality of a $$10\%(v/V)$$ solution of di-bromine solution in $$CCl_4$$ (carbon tetrachloride) is '$$x$$'. $$x =$$ ______ $$\times 10^{-2}$$ M. (Nearest integer)
[Given : molar mass of $$Br_2 = 160 \text{ g mol}^{-1}$$, atomic mass of $$C = 12 \text{ g mol}^{-1}$$, atomic mass of $$Cl = 35.5 \text{ g mol}^{-1}$$, density of dibromine $$= 3.2 \text{ g cm}^{-3}$$, density of $$CCl_4 = 1.6 \text{ g cm}^{-3}$$]
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Which one of the following sets of ions represents a collection of isoelectronic species?
(Given : Atomic Number : F : 9, Cl : 17, Na = 11, Mg = 12, Al = 13, K = 19, Ca = 20, Sc = 21)
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For electron gain enthalpies of the elements denoted as $$\Delta_{eg}H$$, the incorrect option is :
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0.3 g of ethane undergoes combustion at $$27°C$$ in a bomb calorimeter. The temperature of calorimeter system (including the water) is found to rise by $$0.5°C$$. The heat evolved during combustion of ethane at constant pressure is ______ $$\text{kJmol}^{-1}$$.
(Nearest integer)
[Given : The heat capacity of the calorimeter system is $$20 \text{ kJ K}^{-1}$$, $$R = 8.3 \text{ JK}^{-1} \text{ mol}^{-1}$$. Atomic mass of C and H are 12 and 1 $$\text{g mol}^{-1}$$ respectively]
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The effect of addition of helium gas to the following reaction in equilibrium state at constant volume, is :
$$PCl_5(g) \rightleftharpoons PCl_3(g) + Cl_2(g)$$
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Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : An aqueous solution of KOH when for volumetric analysis, its concentration should be checked before the use.
Reason (R) : On aging, KOH solution absorbs atmospheric $$CO_2$$.
In the light of the above statements, choose the correct answer from the options given below.
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O − O bond length in $$H_2O_2$$ is $$\underline{X}$$ than the O − O bond length in $$F_2O_2$$. The O − H bond length in $$H_2O_2$$ is $$\underline{Y}$$ than that of the O − F bond in $$F_2O_2$$. Choose the correct option for $$\underline{X}$$ and $$\underline{Y}$$ from the given below.
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The starting material for convenient preparation of deuterated hydrogen peroxide $$(D_2O_2)$$ in laboratory is:
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Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): Gypsum is used for making fireproof wall boards.
Reason (R): Gypsum is unstable at high temperatures.
In the light of the above statements, choose the correct answer from the options given below:
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The correct order of bond enthalpy ($$\text{kJ mol}^{-1}$$) is:
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Given below are two statements:
Statement I: Sulphanilic acid gives esterification test for carboxyl group.
Statement II: Sulphanilic acid gives red colour in Lassigne's test for extra element detection.
In the light of the above statements, choose the most appropriate answer from the options given below:
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Testosterone, which is a steroidal hormone, has the following structure. The total number of asymmetric carbon atom/s in testosterone is ______.
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'X' is:
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The industrial activity held least responsible for global warming is :
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A metal M crystallizes into two lattices :- face centred cubic (fcc) and body centred cubic (bcc) with unit cell edge length of $$2.0$$ and $$2.5$$ Å respectively. The ratio of densities of lattices fcc to bcc for the metal M is ______ (Nearest integer)
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20% of acetic acid is dissociated when its 5 g is added to 500 mL of water. The depression in freezing point of such water is ______ $$\times 10^{-3}°C$$. Atomic mass of C, H and O are 12, 1 and 16 a.m.u. respectively.
[Given : Molal depression constant and density of water are $$1.86 \text{ K kg mol}^{-1}$$ and $$1 \text{ g cm}^{-3}$$ respectively.
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$$1 \times 10^{-5}$$ M $$AgNO_3$$ is added to 1 L of saturated solution of AgBr. The conductivity of this solution at 298 K is ______ $$\times 10^{-8} \text{ S m}^{-1}$$.
[Given : $$K_{sp}(AgBr) = 4.9 \times 10^{-13}$$ at 298 K, $$\lambda^0_{Ag^+} = 6 \times 10^{-3} \text{ Sm}^2 \text{ mol}^{-1}$$, $$\lambda^0_{Br^-} = 8 \times 10^{-3} \text{ Sm}^2 \text{ mol}^{-1}$$, $$\lambda^0_{NO_3^-} = 7 \times 10^{-3} \text{ Sm}^2 \text{ mol}^{-1}$$]
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The graph which represents the following reaction is:
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$$A \rightarrow B$$
The above reaction is of zero order. Half life of this reaction is 50 min. The time taken for the concentration of A to reduce to one-fourth of its initial value is ______ min. (Nearest integer)
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In figure, a straight line is given for Freundrich Adsorption $$(y = 3x + 2.505)$$. The value of $$\frac{1}{n}$$ and $$\log K$$ are respectively.
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Among following compounds, the number of those present in copper matte is
A. $$CuCO_3$$
B. $$Cu_2S$$
C. $$Cu_2O$$
D. FeO
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Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : $$Cu^{2+}$$ in water is more stable than $$Cu^+$$.
Reason (R) : Enthalpy of hydration for $$Cu^{2+}$$ is much less than that of $$Cu^+$$.
In the light of the above statements, choose the correct answer from the options given below :
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Which element is not present in Nessler's reagent?
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The complex cation which has two isomers is:
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The spin only magnetic moment of $$[Mn(H_2O)_6]^{2+}$$ complexes is ______ B.M. (Nearest integer)
(Given atomic number of Mn = 25)
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The structures of major products A, B and C in the following reaction are sequence
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In a reaction,
Reagents 'X' and 'Y' respectively are:
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Among the following, the number of tranquilizer/s is/are
A. Chlorodiazepoxide
B. Veronal
C. Valium
D. Salvarsan
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All structures given below are of vitamin C. Most stable of them is
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Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): $$\alpha$$-halocarboxylic acid on reaction with dil. $$NH_3$$ gives good yield of $$\alpha$$-amino carboxylic acid whereas the yield of amines is very low when prepared from alkyl halides.
Reason (R): Amino acids exist in zwitter ion form in aqueous medium.
In the light of the above statements, choose the correct answer from the options given below :
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The number of integral values of $$k$$, for which one root of the equation $$2x^2 - 8x + k = 0$$ lies in the interval $$(1, 2)$$ and its other root lies in the interval $$(2, 3)$$, is :
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Let $$a, b$$ be two real numbers such that $$ab < 0$$. If the complex number $$\frac{1+ai}{b+i}$$ is of unit modulus and $$a + ib$$ lies on the circle $$|z - 1| = |2z|$$, then a possible value of $$\frac{1+[a]}{4b}$$, where $$[t]$$ is greatest integer function, is :
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Number of integral solutions to the equation $$x + y + z = 21$$, where $$x \geq 1, y \geq 3, z \geq 4$$, is equal to _____.
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The total number of six digit numbers, formed using the digits 4, 5, 9 only and divisible by 6, is _____.
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The sum $$\sum_{n=1}^{\infty} \frac{2n^2+3n+4}{(2n)!}$$ is equal to :
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The sum of the common terms of the following three arithmetic progressions.
$$3, 7, 11, 15, \ldots, 399$$
$$2, 5, 8, 11, \ldots, 359$$ and
$$2, 7, 12, 17, \ldots, 197$$, is equal to _____.
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If the term without $$x$$ in the expansion of $$\left(x^{2/3} + \frac{\alpha}{x^3}\right)^{22}$$ is 7315, then $$|\alpha|$$ is equal to _____.
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Let the sixth term in the binomial expansion of $$\left(\sqrt{2^{\log_2(10-3^x)}} + \sqrt[5]{2^{(x-2)\log_2 3}}\right)^m$$ powers of $$2^{(x-2)\log_2 3}$$ be 21. If the binomial coefficients of the second, third and fourth terms in the expansion are respectively the first, third and fifth terms of an A.P., then the sum of the squares of all possible values of $$x$$ is _____.
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If the $$x$$-intercept of a focal chord of the parabola $$y^2 = 8x + 4y + 4$$ is 3, then the length of this chord is equal to _____.
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The line $$x = 8$$ is the directrix of the ellipse $$E : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ with the corresponding focus $$(2, 0)$$. If the tangent to $$E$$ at the point $$P$$ in the first quadrant passes through the point $$(0, 4\sqrt{3})$$ and intersects the $$x$$-axis at $$Q$$, then $$(3PQ)^2$$ is equal to _____.
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Let $$P(x_0, y_0)$$ be the point on the hyperbola $$3x^2 - 4y^2 = 36$$, which is nearest to the line $$3x + 2y = 1$$. Then $$\sqrt{2}(y_0 - x_0)$$ is equal to :
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Which of the following statements is a tautology?
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Let $$9 = x_1 < x_2 < \ldots < x_7$$ be in an A.P. with common difference $$d$$. If the standard deviation of $$x_1, x_2, \ldots, x_7$$ is 4 and the mean is $$\bar{x}$$, then $$\bar{x} + x_6$$ is equal to :
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Let $$P(S)$$ denote the power set of $$S = \{1, 2, 3, \ldots, 10\}$$. Define the relations $$R_1$$ and $$R_2$$ on $$P(S)$$ as $$AR_1B$$ if $$(A \cap B^c) \cup (B \cap A^c) = \phi$$ and $$AR_2 B$$ if $$A \cup B^c = B \cup A^c, \forall A, B \in P(S)$$. Then :
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If $$A = \frac{1}{2}\begin{bmatrix} 1 & \sqrt{3} \\ -\sqrt{3} & 1 \end{bmatrix}$$ then,
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For the system of linear equations $$ax + y + z = 1$$, $$x + ay + z = 1$$, $$x + y + az = \beta$$, which one of the following statements is NOT correct?
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Let $$S = \left\{x \in R : 0 \lt x \lt 1 \text{ and } 2\tan^{-1}\left(\frac{1-x}{1+x}\right) = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right\}$$. If $$n(S)$$ denotes the number of elements in $$S$$ then :
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Let $$f : R - \{0, 1\} \to R$$ be a function such that $$f(x) + f\left(\frac{1}{1-x}\right) = 1 + x$$. Then $$f(2)$$ is equal to :
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If $$y(x) = x^x, x > 0$$, then $$y''(2) - 2y'(2)$$ is equal to :
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The sum of the absolute maximum and minimum values of the function $$f(x) = |x^2 - 5x + 6| - 3x + 2$$ in the interval $$[-1, 3]$$ is equal to :
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The value of the integral $$\int_{-\pi/4}^{\pi/4} \frac{x + \frac{\pi}{4}}{2 - \cos 2x} dx$$ is :
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If $$\int_0^{\pi} \frac{5^{\cos x}(1+\cos x \cos 3x+\cos^2 x+\cos^3 x \cos 3x)dx}{1+5^{\cos x}} = \frac{k\pi}{16}$$, then $$k$$ is equal to _____.
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The area of the region given by $$\{(x, y) : xy \leq 8, 1 \leq y \leq x^2\}$$ is :
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Let $$\alpha x = \exp(x^\beta y^\gamma)$$ be the solution of the differential equation $$2x^2 y dy - (1 - xy^2)dx = 0$$, $$x > 0, y(2) = \sqrt{\log_e 2}$$. Then $$\alpha + \beta - \gamma$$ equals :
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Let $$\vec{a} = 5\hat{i} - \hat{j} - 3\hat{k}$$ and $$\vec{b} = \hat{i} + 3\hat{j} + 5\hat{k}$$ be two vectors. Then which one of the following statements is TRUE?
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Let $$\vec{a} = 2\hat{i} - 7\hat{j} + 5\hat{k}$$, $$\vec{b} = \hat{i} + \hat{k}$$ and $$\vec{c} = \hat{i} + 2\hat{j} - 3\hat{k}$$ be three given vectors. If $$\vec{r}$$ is a vector such that $$\vec{r} \times \vec{a} = \vec{c} \times \vec{a}$$ and $$\vec{r} \cdot \vec{b} = 0$$, then $$|\vec{r}|$$ is equal to:
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Let the plane $$P$$ pass through the intersection of the planes $$2x + 3y - z = 2$$ and $$x + 2y + 3z = 6$$, and be perpendicular to the plane $$2x + y - z + 1 = 0$$. If $$d$$ is the distance of $$P$$ from the point $$(-7, 1, 1)$$, then $$d^2$$ is equal to :
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Let $$\alpha x + \beta y + \gamma z = 1$$ be the equation of a plane passing through the point $$(3, -2, 5)$$ and perpendicular to the line joining the points $$(1, 2, 3)$$ and $$(-2, 3, 5)$$. Then the value of $$\alpha \beta y$$ is equal to _____.
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The point of intersection $$C$$ of the plane $$8x + y + 2z = 0$$ and the line joining the points $$A(-3, -6, 1)$$ and $$B(2, 4, -3)$$ divides the line segment $$AB$$ internally in the ratio $$k : 1$$. If $$a, b, c$$ ($$|a|, |b|, |c|$$ are coprime) are the direction ratios of the perpendicular from the point $$C$$ on the line $$\frac{1-x}{1} = \frac{y+4}{2} = \frac{z+2}{3}$$, then $$|a + b + c|$$ is equal to _____.
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Two dice are thrown independently. Let $$A$$ be the event that the number appeared on the $$1^{st}$$ die is less than the number appeared on the $$2^{nd}$$ die, $$B$$ be the event that the number appeared on the $$1^{st}$$ die is even and that on the second die is odd, and $$C$$ be the event that the number appeared on the $$1^{st}$$ die is odd and that on the $$2^{nd}$$ is even. Then
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