JEE (Advanced) 2018 Paper-1

Instructions

For each question, enter the correct numerical value (in decimal notation, truncated/rounded-off to the second decimal place; e.g. 6.25, 7.00, -0.33, -.30, 30.27, -127.30) using the mouse and the on-screen virtual numeric keypad in the place designated to enter the answer.

Question 11

Three identical capacitors $$๐ถ_{1}, ๐ถ_{2}$$ and $$๐ถ_{3}$$ have a capacitance of 1.0 $$\mu$$๐น each and they are uncharged initially. They are connected in a circuit as shown in the figure and $$๐ถ_{1}$$ is then filled completely with a dielectric material of relative permittivity $$\epsilon_{r}$$.The cell electromotive force (emf) $$๐‘‰_{0}$$ = 8 ๐‘‰. First the switch $$๐‘†_{1}$$ is closed while the switch $$๐‘†_{2}$$ is kept open. When the capacitor $$๐ถ_{3}$$ is fully charged, $$๐‘†_{1}$$ is opened and $$๐‘†_{2}$$ is closed simultaneously. When all theย capacitors reach equilibrium, the charge on $$๐ถ_{3}$$ is found to be 5 $$\mu$$๐ถ. The value of $$\epsilon_{r}$$ =____________.

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Question 12

In the xy-plane, the region y > 0 has a uniform magnetic field $$B_{1} \hat{k}$$ and the region y < 0 has another uniform magnetic field $$B_{2} \hat{k}$$. A positively charged particle is projected from the origin along the positive y-axis with speed $$v_{0} = \pi m s^{-1}$$ and t = 0 as shown in the figure.Neglect gravity in this problem. Let $$t = T$$ be the time when the particle crosses the x-axis from below for the first time. If $$B_2 = 4B_1$$, the average speed of the particle, in $$ m s^{-1}$$, along the x-axis in the time interval T is __________.

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Question 13

Sunlight of intensity 1.3 $$kW ๐‘š^{โˆ’2}$$ is incident normally on a thin convex lens of focal length 20 ๐‘๐‘š. Ignore the energy loss of light due to the lens and assume that the lens aperture size is much smaller than its focal length. The average intensity of light, in $$kW ๐‘š^{โˆ’2}$$, at a distance 22 ๐‘๐‘š from the lens on the other side is __________

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Question 14

Two conducting cylinders of equal length but different radii are connected in series between two heat baths kept at temperatures $$๐‘‡_{1} = 300 ๐พ and ๐‘‡_{2}$$ = 100 ๐พ, as shown in the figure. The radius of the bigger cylinder is twice that of the smaller one and the thermal conductivities of the materials of the smaller and the larger cylinders are $$๐พ_{1} and ๐พ_{2}$$ respectively. If the temperature at the junction of the two cylinders in the steady state is 200 ๐พ, then $$\frac{๐พ_{1}}{๐พ_{2}}$$ =__________.

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Instructions

In electromagnetic theory, the electric and magnetic phenomena are related to each other. Therefore, the dimensions of electric and magnetic quantities must also be related to each other. In the questions below, [E] and [B] stand for dimensions of electric and magnetic fields respectively, while $$[\epsilon_{0}]$$ and $$[\mu_{0}]$$ stand for dimensions of the permittivity and permeability of free space respectively. [L] and [T] are dimensions of length and time respectively. All the quantities are given in SI units.

Question 15

The relation between $$[E]$$ and $$[B]$$ is

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Question 16

The relation between $$[\epsilon_{0}]$$ and $$[\mu_{0}]$$ is

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Instructions

If the measurement errors in all the independent quantities are known, then it is possible to determine the error in any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first power of the error. For example, consider the relation $$Z = \frac{x}{y}$$. If the errors in x, y and z are $$\triangle x, \triangle y$$ and $$\triangle z$$ respectively, then $$Z \pm \triangle Z = \frac{x \pm \triangle x}{y \pm \triangle y}=\frac{x}{y} \left(1 \pm \frac{\triangle x}{x} \right) \left(1 \pm \frac{\triangle y}{y} \right)^{-1}$$.ย The series expansion for $$\left(1 \pm \frac{\triangle y}{y}\right)^{-1}$$, to to first power in $$\frac{\triangle y}{y}$$, is $$1 \mp \left(\frac{\triangle y}{y}\right)$$ The relative errors in independent variables are always added. So the error in z will beย $$\triangle z = z(\frac{\triangle x}{x} + \frac{\triangle y}{y})$$. The above derivation makes the assumption that $$\frac{\triangle x}{x} \ll 1, \frac{\triangle y}{y} \ll 1$$.Therefore, the higher powers of these quantities are neglected.

Question 17

Consider the ratio $$\gamma=\frac{(1-a)}{(1+a)}$$ to be determined by measuring a dimensionless quantity ๐‘Ž. If the error in the measurement of ๐‘Ž is $$\triangle a(\frac{\triangle a}{a} \ll 1)$$,then what is the error $$\triangle ๐‘Ÿ $$ in determining r?

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Question 18

In an experiment the initial number of radioactive nuclei is 3000. It is found that $$1000 \pm 40$$ nuclei decayed in the first 1.0 s. For $$\mid x \mid \ll 1$$, $$\ln(1 + x)$$ = x up to first power in x. The error $$\triangle \lambda$$, in the determination of the decay constant $$\lambda$$, in $$s^{-1}$$, is , is

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Instructions

For the following questions answer them individually

Question 19

The compound(s) which generate(s) N2 gas upon thermal decomposition below $$300^\circ$$ is (are)

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Question 20

The correct statement(s) regarding the binary transition metal carbonyl compounds is (are) (Atomic numbers: Fe = 26, Ni = 28)

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