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A flat plate is moving normal to its plane through a gas under the action of a constant force F. The gas is kept at a very low pressure. The speed of the plate v is much less than the average speed u of the gas molecules. Which of the following options is/are true?
A block of mass M has a circular cut with a frictionless surface as shown. The block rests on the horizontal frictionless surface of a fixed table. Initially the right edge of the block is at x = 0, in a co-ordinate system fixed to the table. A point mass m is released from rest at the topmost point of the path as shown and it slides down. When the mass loses contact with the block, its position is x and the velocity is v. At that instant, which of the following options is/are correct?
A block M hangs vertically at the bottom end of a uniform rope of constant mass per unit length. The top end of the rope is attached to a fixed rigid support at O. A transverse wave pulse (Pulse 1) of wavelength $$\lambda_0$$ is produced at point O on the rope. The pulse takes time $$T_{OA}$$ to reach point A. If the wave pulse of wavelength $$\lambda_0$$ is produced at point A (Pulse 2) without disturbing the position of M it takes time $$T_{OA}$$ to reach point O. Which of the following options is/are correct?
A human body has a surface area of approximately 1 $$m^2$$. The normal body temperature is 10 K above the surrounding room temperature $$T_0$$. Take the room temperature to be $$T_0 = 300 K$$. For $$T_0 = 300 K$$, the value of $$\sigma T_0^4 = 460 Wm^{-2}$$ (where $$\sigma$$ is the Stefan-Boltzmann constant). Which of the following options is/are correct?
A circular insulated copper wire loop is twisted to form two loops of area 𝐴 and 2𝐴 as shown in the figure. At the point of crossing the wires remain electrically insulated from each other. The entire loop lies in the plane (of the paper). A uniform magnetic field 𝐵 points into the plane of the paper. At 𝑡 = 0, the loop starts rotating about the common diameter as axis with a constant angular velocity ω in the magnetic field. Which of the following options is/are correct?
In the circuit shown, $$L = 1 \mu H, C = 1 \mu F$$ and $$R = 1 kΩ.$$ They are connected in series with an a.c. source $$V = V_0 \sin \omega t$$ as shown. Which of the following options is/are correct?
For an isosceles prism of angle 𝐴 and refractive index $$\mu$$, it is found that the angle of minimum deviation $$\delta_m = A$$. Which of the following options is/are correct?
A drop of liquid of radius $$R = 10^{-2} m$$ having surface tension $$S = \frac{0.1}{4 \pi} Nm^{-1}$$ divides itself into 𝐾 identical drops. In this process the total change in the surface energy $$\triangle U = 10^{-3} J$$. If $$K = 10^\alpha$$ then the value of $$\alpha$$ is
An electron in a hydrogen atom undergoes a transition from an orbit with quantum number $$n_i$$ to another with quantum number $$n_f$$. $$V_i$$ and $$V_f$$ are respectively the initial and final potential energies of the electron. If $$\frac{V_i}{V_f} = 6.25$$, then the smallest possible $$n_f$$ is
A monochromatic light is travelling in a medium of refractive index n = 1.6. It enters a stack of glass layers from the bottom side at an angle $$\theta = 30^\circ$$. The interfaces of the glass layers are parallel to each other. The refractive indices of different glass layers are monotonically decreasing as $$n_m = n - m \triangle n$$, where $$n_m$$ is the refractive index of the $$m^{th}$$ slab and $$\triangle n = 0.1$$ (see the figure) The ray is refracted out parallel to the interface between the $$(m - 1)^{th}$$ and $$m^{th}$$ slabs from the right side of the stack. What is the value of m?
A stationary source emits sound of frequency $$f_0 = 492 Hz$$. The sound is reflected by a large car approaching the source with a speed of $$2 ms^{-1}$$. The reflected signal is received by the source and superposed with the original. What will be the beat frequency of the resulting signal in Hz? (Given that the speed of sound in air is $$330 ms^{-1}$$ and the car reflects the sound at the frequency it has received).
$$^{131}I$$ is an isotope of Iodine that $$\beta$$ decays to an isotope of Xenon with a half-life of 8 days. A small amount of a serum labelled with $$^{131}I$$ is injected into the blood of a person. The activity of the amount of $$^{131}I$$ injected was $$2.4 \times 10^5$$ Becquerel (Bq). It is known that the
injected serum will get distributed uniformly in the blood stream in less than half an hour. After 11.5 hours, 2.5 ml of blood is drawn from the person’s body, and gives an activity of 115 Bq. The total volume of blood in the person’s body, in liters is approximately (you may use $$e^x \approx 1 + x$$ for $$\mid x \mid \ll 1$$ and $$\ln 2 \approx 0.7$$).
Answer the Questions by appropriately matching the information given in the three columns of the following table.
A charged particle (electron or proton) is introduced at the origin $$(x = 0, y = 0, z = 0)$$ with a given initial velocity $$\overrightarrow{v}$$. A uniform electric field $$\overrightarrow{E}$$ and a uniform magnetic field $$\overrightarrow{B}$$ exist everywhere. The velocity $$\overrightarrow{v}$$, electric field $$\overrightarrow{E}$$ and magnetic field $$\overrightarrow{B}$$ are given in columns 1, 2 and 3 respectively. The quantities $$E_0, B_0$$ are positive in magnitude.
In which case will the particle move in a straight line with constant velocity?
In which case will the particle describe a helical path with axis along the positive z direction?
In which case would the particle move in a straight line along the negative direction of y-axis (i.e., move along $$-\hat{y}$$)?
Answer the Questions by appropriately matching the information given in the three columns of the following table.
An ideal gas is undergoing a cyclic thermodynamic process in different ways as shown in the corresponding P - V diagrams in column 3 of the table. Consider only the path from state 1 to state 2. W denotes the corresponding work done on the system. The equations and plots in the table have standard notations as used in thermodynamic processes. Here $$\gamma$$ is the ratio of heat capacities at constant pressure and constant volume. The number of moles in the gas is n.
Which of the following options is the only correct representation of a process in which $$\triangle U = \triangle Q - P \triangle V$$?
Which one of the following options is the correct combination?
Which one of the following options correctly represents a thermodynamic process that is used as a correction in the determination of the speed of sound in an ideal gas?
An ideal gas is expanded from $$(p_1, V_1, T_1)$$ to $$(p_2, V_2, T_2)$$ under different conditions. The correct statement(s) among the following is(are)
For a solution formed by mixing liquids L and M, the vapour pressure of L plotted against the mole fraction of M in solution is shown in the following figure. Here $$x_L$$ and $$x_M$$ represent mole fractions of L and M, respectively, in the solution. The correct statement(s) applicable to this system is(are)
The correct statement(s) about the oxoacids, $$HClO_4$$ and $$HClO$$, is(are)
The colour of the $$X_2$$ molecules of group 17 elements changes gradually from yellow to violet down the group. This is due to
Addition of excess aqueous ammonia to a pink coloured aqueous solution of $$MCl_2.6H_2O(X)$$ and $$NH_4Cl$$ gives an octahedral complex Y in the presence of air. In aqueous solution, complex Y behaves as 1:3 electrolyte. The reaction of X with excess HCl at room temperature results in the formation of a blue coloured complex Z. The calculated spin only magnetic moment of X and Z is 3.87 B.M., whereas it is zero for complex Y. Among the following options, which statement(s) is(are) correct?
The IUPAC name(s) of the following compound is(are)
The correct statement(s) for the following addition reactions is(are)
A crystalline solid of a pure substance has a face-centred cubic structure with a cell edge of 400 pm. If the density of the substance in the crystal is 8 g $$cm^{-3}$$, then the number of atoms present in 256 g of the crystal is $$N \times 10^{24}$$. The value of N is
The conductance of a 0.0015 M aqueous solution of a weak monobasic acid was determined by using a conductivity cell consisting of platinized Pt electrodes. The distance between the electrodes is 120 cm with an area of cross section of $$1 cm^2$$. The conductance of this solution was found to be $$5 \times 10^{-7} S$$. The pH of the solution is 4. The value of limiting molar conductivity $$\left(\bigwedge_m^0\right)$$ of this weak monobasic acid in aqueous solution is $$Z \times 10^2 S cm^{-1} mol^{-1}$$. The value of Z is
The sum of the number of lone pairs of electrons on each central atom in the following species is
$$\left[TeBr_6 \right]^{2−}, \left[BrF_2 \right]^+, SNF_3$$, and $$\left[XeF_3 \right]^−$$
(Atomic numbers: N = 7, F = 9, S = 16, Br = 35, Te = 52, Xe = 54)
Among $$H_2, He_2^+, Li_2, Be_2, B_2, C_2, N_2, O_2^−$$, and $$F_2$$, the number of diamagnetic species is
(Atomic numbers: H = 1, He = 2, Li = 3, Be = 4, B = 5, C = 6, N = 7, O = 8, F = 9)
Among the following, the number of aromatic compound(s) is
Answer the Questions by appropriately matching the information given in the three columns of the following table.
The wave function, $$\psi_{n, l, m_1}$$ is a mathematical function whose value depends upon spherical polar coordinates $$(r, \theta, \phi)$$ of the electron and characterized by the quantum numbers n, l and $$m_1$$. Here r is distance from nucleus, $$\theta$$ is colatitude and $$\phi$$ is azimuth. In the mathematical function given in the Table, Z is atomic number and $$a_0$$ is bohr radius.
For the given orbital in Column 1, the only CORRECT combination for any hydrogen-like species is
For hydrogen atom, the only CORRECT combination is
For $$He^+$$ ion, the only INCORRECT combination is
Answer the Questions by appropriately matching the information given in the three columns of the following table.
Columns 1, 2 and 3 contain starting materials, reaction conditions, and type of reactions, respectively.
For the synthesis of benzoic acid, the only CORRECT combination is
The only CORRECT combination that gives two different carboxylic acids is
The only CORRECT combination in which the reaction proceeds through radical mechanism is
If $$2x - y + 1 = 0$$ is a tangent to the hyperbola $$\frac{x^2}{a^2} - \frac{y^2}{16} = 1$$, then which of the following CANNOT be sides of a right angled triangle?
If a chord, which is not a tangent, of the parabola $$y^2 = 16 x$$ has the equation $$2x + y = p$$, and midpoint (h, k), then which of the following is(are) possible value(s) of p, h and k?
Let [x] be the greatest integer less than or equals to x. Then, at which of the following point(s) the function $$f(x) = x \cos (\pi(x + [x]))$$ is discontinuous?
Let $$f:R \rightarrow (0, 1)$$ be a continuous function. Then, which of the following function(s) has (have) the value zero at some point in the interval (0, 1)?
Which of the following is(are) NOT the square of a 3×3 matrix with real entries?
Let a, b, x and y be real numbers such that a - b = 1 and $$y \neq 0$$. If the complex number z = x + iy satisfies $$IM\left(\frac{az + b}{z+1}\right) = y$$, then which of the following is(are) possible value(s) of x?
Let X and Y be two events such that $$P(X) = \frac{1}{3}, P(X \mid Y) = \frac{1}{2}$$ and $$P(Y \mid X) = \frac{2}{5}$$. Then
For how many values of P, the circle $$x^2 + y^2 + 2x + 4y - p = 0$$ and the coordinate axes have exactly three common points?
Let $$R \rightarrow R$$ be a differentiable function such that $$f(0) = 0, f\left(\frac{\pi}{2}\right) = 3$$ and $$f'(0) = 1$$. If $$ g(x) = \int_{x}^{\frac{\pi}{2}}\left[f'(t) \cosec t - \cos t \cosec t f(t)\right] dt$$ for $$x \in \left(0, \frac{\pi}{2}\right]$$, then $$\lim_{x \rightarrow 0}g(x) =$$
For a real number $$\alpha$$, if the system $$\begin{bmatrix}1 & \alpha & \alpha^2 \\\alpha & 1 & \alpha\\\alpha^2 & \alpha & 1 \end{bmatrix}\begin{bmatrix}x\\y\\z \end{bmatrix} = \begin{bmatrix}1\\-1\\1 \end{bmatrix}$$ of linear equations, has infinitely many solutions, then $$1+ \alpha + \alpha^2 =$$
Words of length 10 are formed using the letters A, B, C, D, E, F, G,H, I, J. Let 𝑥 be the number of such words where no letter is repeated; and let 𝑦 be the number of such words where exactly one letter is repeated twice and no other letter is repeated. Then, $$\frac{y}{9x} =$$
The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side?
Answer Questions by appropriately matching the information given in the three columns of the following table.
Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.
For $$a = \sqrt{2}$$, if a tangent is drawn to a suitable conic (Column 1) at the point of contact (-1, 1), then which of the following options is the only CORRECT combination for obtaining equation?
If a tangent to a suitable conic (Column 1) is found to be y = x + 8 and its point of contact is (8, 16), then which of the following options is the only CORRECT combination?
The tangent to a suitable conic (Column 1) at $$\left(\sqrt{3}, \frac{1}{2}\right)$$ is found to be $$\sqrt{3}x + 2y = 4$$, then which of the following options is the only CORRECT combination?
Answer the Questions by appropriately matching the information given in the three columns of the following table.
Let $$f(x) = x + \log_e x - x \log_e x, x \in (0, \infty)$$.
Column 1 contains information about zeros of $$f(x), f'(x)$$ and $$f''(x)$$.
Column 2 contains information about the limiting behavior of $$f(x), f'(x)$$ and $$f''(x)$$ at infinity.
Column 3 contains information about increasing/decreasing nature of $$f(x)$$ and $$f'(x)$$.
Which of the following options is the only CORRECT combination?
Which of the following options is the only CORRECT combination?
Which of the following options is the only INCORRECT combination?
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