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A resistance of 2 Ω is connected across one gap of a metre-bridge (the length of the wire is 100 cm) and an unknown resistance, greater than 2 Ω, is connected across the other gap. When these resistances are interchanged, the balance point shifts by 20 cm. Neglecting any corrections, the unknown resistance is
In an experiment to determine the focal length (f) of a concave mirror by the u-v method, a student places the object pin A on the principal axis at a distance x from the pole P. The student looks at the pin and its inverted image from a distance keeping his/her eye in line with PA. When the student shifts his/her eye towards left, the image appears to the right of the object pin. Then,
Two particles of mass m each are tied at the ends of a light string of length 2a. The whole system is kept on a frictionless horizontal surface with the string held tight so that each mass is at a distance ‘a’ from the center P (as shown in the figure). Now, the mid-point of the string is pulled vertically upwards with a small but constant force F. As a result, the particles move towards each other on the surface. The magnitude of acceleration, when the separation between them becomes 2x, is
A long, hollow conducting cylinder is kept coaxially inside another long, hollow conducting cylinder of larger radius. Both the cylinders are initially electrically neutral.
Consider a neutral conducting sphere. A positive point charge is placed outside the sphere. The net charge on the sphere is then,
A circuit is connected as shown in the figure with the switch S open. When the switch is closed, the total amount of charge that flows from Y to X is
A ray of light traveling in wateris incident on its surface open to air. The angle of incidence is $$\theta$$, which is less than the critical angle. Then there will be
In the options given below, let E denote the rest mass energy of a nucleus and n aneutron. The correct option is
The largest wavelength in the ultraviolet region of the hydrogen spectrum is 122 nm. The smallest wavelength in the infrared region of the hydrogen spectrum (to the nearest integer) is
STATEMENT-1
A block of mass m starts moving on a rough horizontal surface with a velocity v. It stops due to friction between the block and the surface after moving through a certain distance. The surface is now tilted to an angle of $$30^\circ$$ with the horizontal and the same block is made to go up on the surface with the same initial velocity v. The decrease in the mechanical energy in the second situation is smaller than thatin thefirst situation.
because
STATEMENT-2
The coefficient of friction between the block and the surface decreases with the increase in the angle of inclination.
STATEMENT-1
In an elastic collision between two bodies, the relative speed of the bodies after collision is equal to the relative speed before the collision.
because
STATEMENT-2
In anelastic collision, the linear momentum of the system is conserved.
STATEMENT-1
The formula connecting u, v and f for a spherical mirror is valid only for mirrors whosesizes are very small compared to their radii of curvature.
because
STATEMENT-2
Lawsof reflection are strictly valid for plane surfaces, but not for large spherical surfaces.
STATEMENT-1
If the accelerating potential in an X-ray tube is increased, the wavelengths of the characteristic X-rays do not change.
because
STATEMENT-2
Whenanelectron beam strikes the target in an X-ray tube, part of the kinetic energy is converted into X-ray energy.
The ratio $$\frac{x_1}{x_2}$$ is
When disc B is brought in contact with disc A, they acquire a common angular velocity in time t. The averagefrictional torque on one disc by the other during this period is
The loss of kinetic energy during the above process is
The piston is now pulled out slowly and held at a distance 2L from the top. The pressure in the cylinder between its top and the piston will then be
While the piston is at a distance 2L from the top, the hole at the top is sealed. The piston is then released, to a position where it can stay in equilibrium. In this condition, the distance of the piston from the top is
The piston is taken completely out of the cylinder. The hole at the top is sealed. A water tank is brought below the cylinder and put in a position so that the water surface in the tank is at the same level as the top of the cylinder as shown in the figure. The density of the water is $$\rho$$. In equilibrium, the height H of the water column in the cylinder satisfies
Some physical quantities are given in Column I and some possible SI units in which these quantities may be expressed are given in Column II. Match the physical quantities in Column I with the units in Column II and indicate your answer by darkening appropriate bubbles in the $$4 \times 4$$ matrix given in the ORS.
Column I gives certain situations in which a straight metallic wire of resistance R is used and Column II gives some resulting effects. Match the statements in Column I with the statements in Column II and indicate your answer by darkening appropriate bubbles in the $$4 \times 4$$ matrix given in the ORS.
Some laws / processes are given in Column I. Match these with the physical phenomena given in Column II and indicate your answer by darkening appropriate bubbles in the $$4 \times 4$$ matrix given in the ORS.
The species having bond order different from that in CO is
Among the following, the paramagnetic compound is
Extraction of zinc from zinc blende is achieved by
In the following reaction,
the structure of the major product 'X' is
The reagent(s) for the following conversion,
is/are
The numberof structural isomers for $$C_6H_{14}$$ is
The percentage of p-character in the orbitals forming P—P bondsin $$P_4$$ is
When 20 g of naphthoic acid $$(C_{11}H_8O_2)$$ is dissolved in 50 g of benzene $$(K_f$$ = 1.72 K kg mol$$^{-1}$$), a freezing point depression of 2 K is observed. The van't Hoff factor (i) is
The value of $$\log_{10} K$$ for a reaction $$A \rightleftharpoons B$$ is
(Given : $$\triangle_rH_{298K}^{\circ}$$ = -54.07 kJ mol$$^{-1}$$, $$\triangle_rS_{298K}^{\circ}$$ = 10 JK$$^{-1}$$ mol$$^{-1}$$ and R = 8.314 JK$$^{-1}$$ mol$$^{-1}$$; $$2.303 \times 8.314 \times 298 = 5705$$)
STATEMENT-1 : Boron always forms covalent bond.
because
STATEMENT-2: The small size of $$B^{3+}$$ favours formation of covalent bond.
STATEMENT-1: In water, orthoboric acid behaves as a weak monobasic acid.
because
STATEMENT-2: In water, orthoboric acid acts as a proton donor.
STATEMENT-1: p-Hydroxybenzoic acid has a lower boiling point than o-hydroxybenzoic acid.
because
STATEMENT-2 : o-Hydroxybenzoic acid has intramolecular hydrogen bonding.
STATEMENT-1 : Micelles are formed by surfactant molecules above the critical micellar concentration (CMC).
because
STATEMENT-2 : The conductivity of a solution having surfactant molecules decreases sharply at the CMC.
Argon is used in are welding because of its
The structure of $$XeO_3$$ is
$$XeF_4$$ and $$XeF_6$$ are expected to be
Chemical reactions involve interaction of atoms and molecules. A large number of atoms/molecules (approximately $$6.023 \times 10^{23}$$) are present in a few grams of any chemical compound varying with their atomic/molecular masses. To handle such large numbers conveniently, the mole concept was introduced. This concept has implications in diverse areas such as analytical chemistry, biochemistry, electrochemistry and radiochemistry. The following exampleillustrates a typical case, involving chemical/electrochemical reaction, which requires a clear understanding of the mole concept.
A 4.0 molar aqueous solution of NaCl is prepared and 500 mLofthis solution is electrolysed. This leads to the evolution of chlorine gas at one of the electrodes
(atomic mass : Na = 23, Hg = 200; 1 Faraday = 96500 coulombs).
The total number of moles of chlorine gas evolved is
If the cathode is a Hg electrode, the maximum weight (g) of amalgam formed from this solution is
The total charge (coulombs) required for complete electrolysis is
Match the complexes in Column I with their properties listed in Column II. Indicate your answer by darkening the appropriate bubbles of the $$4 \times 4$$ matrix given in the ORS.
Match the chemical substances in Column I with type of polymers/type of bonds in Column II. Indicate your answer by darkening the appropriate bubbles of the $$4 \times 4$$ matrix given in the ORS.
Match gases under specified conditions listed in Column I with their properties/laws in Column II. Indicate your answer by darkening the appropriate bubbles of the $$4 \times 4$$ matrix given in the ORS.
Let $$\alpha, \beta$$ be the roots of the equation $$x^2 - px + r = 0$$ and $$\frac{\alpha}{2}, 2 \beta$$ be the roots of the equation $$x^2 - qx + r = 0$$. Then the value of r is
Let f(x) be differentiable on the interval $$(0, \infty)$$ suh that f(1) = 1, and $$\lim_{t \rightarrow x}\frac{t^2 f(x) - x^2f(t)}{t-x} = 1$$ for each x > 0. Then f(x) is
One Indian and four American men and their wives are to be seated randomly around a circular table. Then the conditional probability that the Indian man is seated adjacent to his wife given that each American manis seated adjacent to his wife is
The tangent to the curve $$y = e^x$$ drawn at the point $$\left(c, e^c\right)$$ intersects the line joining the points $$\left(c-1, e^{c-1}\right)$$ and $$\left(c+1, e^{c+1}\right)$$
$$\lim_{x \rightarrow \frac{\pi}{4}}\frac{\int_{2}^{\sec^2 x}f(t)dt }{x^2 - \frac{\pi^2}{16}}$$ equals
A hyperbola, having the transverse axis of length $$2 \sin \theta$$, is confocal with the ellipse $$3x^2 + 4y^2 = 12$$. Then its equation is
The number of distinct real values of $$\lambda$$, for which the vectors $$-\lambda^2 \hat{i} + \hat{j} + \hat{k}, \hat{i} - \lambda^2 \hat{j} + \hat{k}$$ and $$\hat{i} + \hat{j} - \lambda^2 \hat{k}$$ are coplanar, is
A man walks a distance of 3 units from the origin towards the north-east $$(N 45^\circ E)$$ direction. From there, he walks a distance of 4 units towards the north-west $$(N 45^\circ W)$$ direction to reach a point P. Then the position of P in the Argand plane is
The number of solutions of the pair of equations
$$2 \sin^2 \theta - \cos 2 \theta = 0$$
$$2 \cos^2 \theta - 3 \sin \theta = 0$$
in the interval $$[0, 2 \pi]$$ is
Let $$H_1, H_2, ...., H_n$$ be mutually exclusive and exhaustive events with $$P(H_i) > 0, i = 1, 2, 3, ...., n$$. Let E be any other event with $$0 < P(E) < 1$$.
STATEMENT-1: $$P(H_i \mid E) > P(E \mid H_i).P(H_i)$$ for i = 1, 2, ..., n.
because
STATEMENT-2: $$\sum_{i=1}^n P(H_i) = 1$$
Tangents are drawn from the point (17, 7) to the circle $$x^2 + y^2 = 169$$.
STATEMENT-1: The tangents are mutually perpendicular.
because
STATEMENT-2: The locus of the points from which mutually perpendicular tangents can be drawn to the givencircle is $$x^2 + y^2 = 338$$.
Let the vectors $$\overrightarrow{PQ}, \overrightarrow{QR}, \overrightarrow{RS}, \overrightarrow{ST}, \overrightarrow{TU}$$ and $$\overrightarrow{UP}$$ represent the sides of a regular
hexagon.
STATEMENT-1: $$\overrightarrow{PQ}\times \left(\overrightarrow{RS} + \overrightarrow{ST}\right) \neq \overrightarrow{0}.$$
because
STATEMENT-2: $$\overrightarrow{PQ}\times \overrightarrow{RS} = \overrightarrow{0}$$ and $$\overrightarrow{PQ}\times \overrightarrow{ST} \neq \overrightarrow{0}$$
Let $$F(x)$$ be an indefinite integral of $$\sin^2 x$$.
STATEMENT-1 : The function $$F(x)$$ satisfies $$F(x + \pi) = F(x)$$ for all real x.
because
STATEMENT-2: $$\sin^2 (x + \pi) = \sin^2 x$$ for all real x.
Let $$V_r$$ denote the sum of the first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is $$(2r - 1)$$. Let
$$T_r = V_{r+1}-V_r-2$$ and $$Q_r = T_{r+1}-T_r$$ for r = 1, 2, ....
The sum $$V_1 + V_2 + ... + V_n$$ is
$$T_r$$ is always
Which one of the following is a correct statement?
Consider the circle $$x^2 + y^2 = 9$$ and the parabola $$y^2 = 8x$$. They intersect at P and Q in the first and the fourth quadrants, respectively. Tangents to the circle at P and Q intersect the x-axis at R and tangents to the parabola at P and Q intersect the x-axis at S.
The ratio of the areas of the triangles PQS and PQR is
The radius of the circumcircle of the triangle PRS is
The radius of the incircle of the triangle PQF is
Consider the following linear equations
$$ax + by + cz = 0$$
$$bx + cy + az = 0$$
$$cx + ay + bz = 0$$
Match the conditions/expressions in Column I with statements in Column II and indicate your answer by darkening the appropriate bubbles in the $$4 \times 4$$ matrix given in the ORS.
In the following [x] denotes the greatest integer less than or equal to x.
Match the functions in Column I with the properties in Column II and indicate your answer by darkening the appropriate bubbles in the 4 x 4 matrix given in the ORS.
Match the integrals in Column I with the values in Column II and indicate your answer by darkening the appropriate bubbles in the $$4 \times 4$$ matrix given in the ORS.
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