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For a first order reaction $$A \rightarrow P$$, the temperature (T) dependent rate constant (k) was found to follow the equation $$\log k = -(2000)\frac{1}{T} + 6.0$$. The pre-exponential factor A and the activation energy $$E_a$$, respectively, are
The spin onlyimoment value (in Bohr magneton units) of $$Cr(CO)_6$$ is
In the following carbocation, $$H/CH_3$$ that is most likely to migrate to the positively charged carbon is
The correct stability order of the following resonance structures is
For the reduction of $$NO_{3}^{-}$$ ion in an aqueous solution, $$E^0$$ is +0.96 V. Values of $$E^0$$ for some metal ions are given below
$$V^{2+}(aq) + 2e^{-} \rightarrow V$$ $$E^0 = -1.19$$ V
$$Fe^{3+}(aq) + 3e^{-} \rightarrow Fe$$ $$E^0 = -0.04$$ V
$$Au^{3+}(aq) + 3e^{-} \rightarrow Au$$ $$E^0 = +1.40$$ V
$$Hg^{2+}(aq) + 2e^{-} \rightarrow Hg$$ $$E^0 = +0.86$$ V
The pair(s) of metals that is(are) oxidized by $$NO_{3}^{-}$$ in aqueous solution is(are)
Among the following, the state function(s) is(are)
In the reaction
$$2X + B_2H_6 \rightarrow [BH_2(X)_2]^{+}[BH_4]^{-}$$
the amine(s) X is (are)
The nitrogen oxide(s) that contain(s) N-N bond(s) is(are)
The correct statement(s) about the following sugars X andY is(are)
Figure
Each question contains statements given in two columns, which have to be matched. The statements in ColumnI are labelled A, B, C and D, while the statements in Column II are labelled p, q, r, s and t. Any given statement in Column I can have correct matching with ONE OR MORE statement(s) in Column II. The appropriate bubbles corresponding to the answers to these questions have to be darkened as illustrated in the following example: If the correct matches are A p, s and t; B — q and r; C p and q; and D —s and t; then the correct darkening of bubbles will look like the following.
Match each of the reactions given in Column I with the corresponding product(s) given in Column II.
Match each of the compounds given in Column I with the reaction(s), that they can undergo, given in Column II.
In a constant volume calorimeter, 3.5 g of a gas with molecular weight 28 was burnt in excess oxygen at 298.0 K. The temperature of the calorimeter was found to increase from 298.0 K to 298.45 K due to the combustion process. Given that the heat capacity of the calorimeteris 2.5 kJ K$$^{-1}$$, the numerical value for the enthalpy of combustion of the gas in kJ mol$$^{-1}$$ is
At 400 K, the root mean square (rms) speed of a gas X (molecular weight = 40) is equal to the most probable speed of gas Y at 60K. The molecular weight of the gas Y is
The dissociation constant of a substituted benzoic acid at $$25^\circ C$$ is $$1.0 \times 10^{-4}$$. The pH of a 0.01 M solutionofits sodium salt is
The total number of $$\alpha$$ and $$\beta$$ particles emitted in the nuclear reaction $$_{92}^{238}U \rightarrow _{82}^{214}Pb$$ is
The oxidation number of Mnin the product of alkaline oxidative fusion of $$MnO_2$$ is
The number of water molecule(s) directly bonded to the metal centre in $$CuSO_4.5H_2O$$ is
The coordination numberof Al in the crystalline state of $$AlCl_3$$, is
The total number of cyclic structural as well as stereo isomers possible for a compound with the molecular formula $$C_5H_{10}$$ is
If the sum of first n terms of an A.P. is $$cn^2$$, then the sum of squares of these n terms is
A line with positive direction cosines passes through the point P(2,-1, 2) and makes equal angles with the coordinate axes. The line meets the plane $$2x + y + z = 9$$ at point Q. The length of the line segment PQ equals
The normal at a point P on the ellipse $$x^2 + 4y^2 = 16$$ meets the x-axis at Q. If M is the mid point of the line segment PQ, then the locus of M intersects the latus rectums of the given ellipse at the points
The locus of the orthocentre of the triangle formed by the lines
(1 + p)x - py + p(1 + p) = 0,
(1 + q)x - qy + q(1 + q) = 0,
and y = 0, where $$p \neq q$$, is
If $$I_n = \int_{-\pi}^{\pi}\frac{\sin nx}{(1 + \pi^x) \sin x}dx$$, n = 0, 1, 2, 3, ....,
then
An ellipse intersects the hyperbola $$2x^2 - 2y^2 = 1$$ orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the ellipse are along the
coordinate axes, then
For the function
$$f(x) = x \cos \frac{1}{x}, x \geq 1$$,
The tangent PT and the normal PN to the parabola $$y^2 = 4ax$$ at a point P on it meet its axis at points T and N, respectively. The locus of the centroid of the triangle PTN is a parabola whose
For $$0 < \theta < \frac{\pi}{2}$$, the solution(s) of
$$ \sum_{m=1}^{6}\cosec\left(\theta + \frac{(m - 1)\pi}{4}\right)\cosec\left(\theta + \frac{m\pi}{4}\right) = 4\sqrt{2}$$
is(are)
Each question contains statements given in two columns, which have to be matched. The statements in ColumnI are labelled A, B, C and D, while the statements in ColumnII are labelled p, q, r, s and t. Any given statement in Column I can have correct matching with ONE OR MORE statement(s) in Column II. The appropriate bubbles corresponding to the answers to these questions haveto be darkened as illustratedin thefollowing example: If the correct matches are A — p, s and t; B — q and r; C — p and q; and D — s and t; thenthe correct darkening of bubbles will look like the following.
Match the statements/expressions given in Column I with the values given in Column II.
Match the statements/expressions given in Column I with the values given in Column II.
The maximum value of the function $$f(x) = 2x^3 - 15x^2 + 36x - 48$$ on the set $$A = \left\{x \mid x^2 + 20 \leq 9x \right\}$$ is
Let (x, y, z) be points with integer coordinates satisfying the system of homogeneous equations:
$$3x - y - z = 0$$
$$-3x + z = 0$$
$$-3x + 2y + z = 0$$.
Then the number of such points for which $$x^2 + y^2 + z^2 \leq 100$$ is
Let ABC and ABC' be two non-congruent triangles with sides $$AB = 4, AC = AC' = 2\sqrt{2}$$ and angle $$B = 30^\circ$$. The absolute value ofthe difference between the areas of these triangles is
Let p(x) be a polynomial of degree 4 having extremumat x = 1, 2 and
$$\lim_{x \rightarrow 0}\left(1 + \frac{p(x)}{x^2}\right) = 2$$.
Then the value of p(2) is
Let $$f : R \rightarrow R$$ be a continuous function which satisfies
$$f(x) = \int_{0}^{x}f(t)dt$$.
Then the value of $$f(\ln 5)$$ is
The centres of two circles $$C_1$$ and $$C_2$$ each of unit radius are at a distance of 6 units from each other. Let P be the mid point of the line segment joining the centres of $$C_1$$ and $$C_2$$ and C be a circle touching circles $$C_1$$ and $$C_2$$ externally. If a common tangent to $$C_1$$ and C passing through Pis also a common tangent to $$C_2$$ and C, then the radius ofthe circle C is
The smallest value of k, for which both the roots of the equation
$$x^2 - 8kx + 16(k^2 - k + 1) = 0$$
are real, distinct and have values at least 4, is
If the function $$f(x) = x^3 + e^{\frac{x}{2}}$$ and $$g(x) = f^{-1}(x)$$, then the value of $$g'(1)$$ is
The mass M shown in the figure oscillates in simple harmonic motion with amplitude A. The amplitude of the point P is
A piece of wire is bent in the shape of a parabola $$y = kx^2$$ (y-axis vertical) with a bead of mass m onit. The bead canslide on the wire withoutfriction. It stays at the lowest point of the parabola when the wire is at rest. The wire is now accelerated parallel to the x-axis with a constant acceleration a. The distance of the new equilibrium position of the bead, where the bead can stay at rest with respect to the wire, from the y-axis is
Photoelectric effect experiments are performed using three different metal plates p, q and r having work functions $$\phi_p = 2.0$$ eV, $$\phi_q = 2.5$$ eV and $$\phi_r = 3.0$$ eV, respectively. A light beam containing wavelengths of 550 nm, 450 nm and 350 nm with equal intensities illuminates each of the plates. The correct I-V graph for the experiment is [Take hc = 1240 eV nm]
A uniform rod of length L and mass M is pivoted at the centre. Its two ends are attached to two springs of equal spring constants k. The springs are fixed to rigid supports as shown in the figure, and the rod is free to oscillate in the horizontal plane. The rod is gently pushed through a small angle $$\theta$$ in one direction and released. The frequency of oscillation is
Two metallic rings A and B, identical in shape and size but having different resistivities $$\rho_A$$ and $$\rho_B$$, are kept on top of two identical solenoids as shown in the figure. When current J is switched on in both the solenoids in identical manner, the rings A and B jump to heights $$h_A$$ and $$h_B$$, respectively, with $$h_A > h_B$$. The possible relation(s) between their resistivities and their masses $$m_A$$ and $$m_B$$ is(are)
A student performed the experiment to measure the speed of sound in air using resonance air-column method. Two resonances in the air-column were obtained by lowering the water level. The resonance with the shorter air-column is the first resonanceandthat with the longer air-column is the second resonance. Then,
The figure shows the P-V plot of an ideal gas taken through a cycle ABCDA. The part ABC is a semi-circle and CDA is half of an ellipse. Then,
Under the influence of the Coulomb field of charge +Q, a charge -q is moving around it in an elliptical orbit. Find out the correct statement(s).
A sphere is rolling without slipping on a fixed horizontal plane surface. In the figure, A is the point of contact, B is the centre of the sphere and C is its topmost point. Then,
This section contains 2 questions. Each question contains statements given in two columns, which have to be matched. The statements in Column I are labelled A, B, C and D, while the statements in ColumnII are labelled p, q, r, s and t. Any given statement in Column I can have correct matching with ONE OR MORE statement(s) in Column II. The appropriate bubbles corresponding to the answers to these questions have to be darkened as illustrated in the following example: If the correct matches are A — p, s and t; B — q and r; C — p and q; and D~s and t; then the correct darkening of bubbles will look like the following.
Column II gives certain systems undergoing a process. Column I suggests changes in some of the parameters related to the system. Match the statements in Column I to the appropriate process(es) from Column II.
Column I shows four situations of standard Young’s double slit arrangement with the screen placed far away from the slits $$S_1$$ and $$S_2$$. In each of these cases $$S_1P_0 = S_2P_0, S_1P_1 - S_2P_1 = \frac{\lambda}{4}$$ and $$S_1P_2 - S_2P_2 = \frac{\lambda}{3}$$, where $$\lambda$$ is the wavelength of the light used. In the cases B, C and D, a transparent sheet of refractive index $$\mu$$ and thickness t is pasted on slit $$S_2$$. The thicknesses of the sheets are different in different cases. The phase difference between thelight waves reaching a point P on the screen from the two slits is denoted by $$\delta(P)$$ and the intensity by $$I(P)$$. Match each situation given in Column I with the statement(s) in Column II valid for that situation.
A metal rod AB of length 10x has its one end A in ice at $$0^\circ C$$ and the other end B in water at $$100^\circ C$$. If a point P on the rod is maintained at $$400^\circ C$$, then it is found that equal amounts of water andice evaporate and melt per unit time. The latent heat of evaporation of water is 540 cal/g and latent heat of melting of ice is 80 cal/g. If the point P is at a distance of $$\lambda x$$ from the ice end A, find the value of $$\lambda$$. [Neglect any heat loss to the surrounding.]
A cylindrical vessel of height 500 mm has an orifice (small hole) at its bottom. The orifice is initially closed and wateris filled in it up to height H. Now the top is completely sealed with a cap and theorifice at the bottom is opened. Some water comes out from the orifice and the water level in the vessel becomes steady with height of water column being 200 mm. Find the fall in height (in mm) of water level due to opening ofthe orifice.
[Take atmospheric pressure $$= 1.0 \times 10^5 N/m^2$$, density of water $$= 1000 kg/m^3$$ and $$g = 10 m/s^2$$. Neglect any affect of surface tension.]
Two soap bubbles A and B are kept in a closed chamber where the air is maintained at pressure 8 N/m$$^2$$. The radii of bubbles A and B are 2 cm and 4 cm, respectively. Surface tension of the soap-water used to make bubbles is 0.04 N/m. Find the ratio $$\frac{n_B}{n_A}$$. Where $$n_A$$ and $$n_B$$ are the number of moles of air in bubbles A and B, respectively. [Neglect the effect of gravity.]
Three objects A. B and C are kept in a straight line on a frictionless horizontal surface. These have:masses m, 2m and m, respectively. The object A moves towards B with a speed 9 m/s and makes an elastic collision with it. Thereafter, B makes completely inelastic collision with C. All motions occur on the same straight line. Find the final speed (in m/s) of the object C.
A steady current I goes through a wire loop PQR having shape of a right angle triangle with PQ = 3x, PR = 4x and QR = 5x. If the magnitude of the magnetic field at P due to this loop is $$k\left(\frac{\mu_0 I}{48 \pi x}\right)$$, find the value of k.
A light inextensible string that goes over a smooth fixed pulley as shown in the figure connects two blocks of masses 0.36 kg and 0.72 kg. Taking g = 10 m/s$$^2$$, find the work
done (in joules) by the string on the block of mass 0.36 kg during the first second after the system is released from rest.
A solid sphere of radius R has a charge Q distributed in its volume with a charge densityy $$\rho = kr^a$$, where k and a are constants and r is the distance from its center. If the electric field at $$r = \frac{R}{2}$$ is $$\frac{1}{8}$$ times that at r = R, find the value of a,
A 20 cm long string, having a mass of 1.0 g, is fixed at both the ends. The tension in tne string is 0.5 N. The string is set into vibrations using an external vibrator of
frequency 100 Hz. Find the separation (in em) between the successive nodes on the string.
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