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Let \(x_{0}\) be the real number such that \(e^{x_{0}}+x_{0}=0\). For a given real number \(\alpha\), define \[ g(x)=\frac{3 x e^{x}+3 x-\alpha e^{x}-\alpha x}{3\left(e^{x}+1\right)} \] for all real numbers \(x\). Then which one of the following statements is TRUE?
(A) For \(\alpha=2, \lim _{x \rightarrow x_{0}}\left|\frac{g(x)+e^{x_{0}}}{x-x_{0}}\right|=0\)
(B) For \(\alpha=2, \lim _{x \rightarrow x_{0}}\left|\frac{g(x)+e^{x_{0}}}{x-x_{0}}\right|=1\)
(C) For \(\alpha=3, \lim _{x \rightarrow x_{0}}\left|\frac{g(x)+e^{x_{0}}}{x-x_{0}}\right|=0\)
(D) For \(\alpha=3, \lim _{x \rightarrow x_{0}}\left|\frac{g(x)+e^{x_{0}}}{x-x_{0}}\right|=\frac{2}{3}\)
" "Let \(\mathbb{R}\) denote the set of all real numbers. Then the area of the region \[ \left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x>0, y>\frac{1}{x}, 5 x-4 y-1>0,4 x+4 y-17<0\right\} \] is
(A) \(\frac{17}{16}-\log _{e} 4\) (B) \(\frac{33}{8}-\log _{e} 4\)
(C) \(\frac{57}{8}-\log _{e} 4\) (D) \(\frac{17}{2}-\log _{e} 4\)
" "The total number of real solutions of the equation \[ \theta=\tan ^{-1}(2 \tan \theta)-\frac{1}{2} \sin ^{-1}\left(\frac{6 \tan \theta}{9+\tan ^{2} \theta}\right) \] is
(Here, the inverse trigonometric functions \(\sin ^{-1} x\) and \(\tan ^{-1} x\) assume values in \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) and ( \(-\frac{\pi}{2}, \frac{\pi}{2}\) ), respectively.)
(A) 1 (B) 2 (C) 3 (D) 5
" "Let \(S\) denote the locus of the point of intersection of the pair of lines \[ \begin{gathered} 4 x-3 y=12 \alpha \\ 4 \alpha x+3 \alpha y=12 \end{gathered} \] where \(\alpha\) varies over the set of non-zero real numbers. Let \(T\) be the tangent to \(S\) passing through the points ( \(p, 0\) ) and ( \(0, q\) ), \(q>0\), and parallel to the line \(4 x-\frac{3}{\sqrt{2}} y=0\).
Then the value of \(p q\) is (A) \(-6 \sqrt{2}\) (B) \(-3 \sqrt{2}\) (C) \(-9 \sqrt{2}\) (D) \(-12 \sqrt{2}\)
" "Let \(I=\left(\begin{array}{ll}1 0 \\ 0 1\end{array}\right)\) and \(P=\left(\begin{array}{ll}2 0 \\ 0 3\end{array}\right)\). Let \(Q=\left(\begin{array}{ll}x y \\ z 4\end{array}\right)\) for some non-zero real numbers \(x, y\), and \(z\), for which there is a \(2 \times 2\) matrix \(R\) with all entries being non-zero real numbers, such that \(Q R=R P\). \end{itemize} Then which of the following statements is (are) TRUE? (A) The determinant of \(Q-2 I\) is zero
(B) The determinant of \(Q-6 I\) is 12
(C) The determinant of \(Q-3 I\) is 15
(D) \(y z=2\)
" " Let \(S\) denote the locus of the mid-points of those chords of the parabola \(y^{2}=x\), such that the area of the region enclosed between the parabola and the chord is \(\frac{4}{3}\). Let \(\mathcal{R}\) denote the region lying in the first quadrant, enclosed by the parabola \(y^{2}=x\), the curve \(S\), and the lines \(x=1\) and \(x=4\). Then which of the following statements is (are) TRUE? (A) \((4, \sqrt{3}) \in S\)
(B) \((5, \sqrt{2}) \in S\)
(C) Area of \(\mathcal{R}\) is \(\frac{14}{3}-2 \sqrt{3}\)
(D) Area of \(\mathcal{R}\) is \(\frac{14}{3}-\sqrt{3}\)
" " Let \(P\left(x_{1}, y_{1}\right)\) and \(Q\left(x_{2}, y_{2}\right)\) be two distinct points on the ellipse \[ \frac{x^{2}}{9}+\frac{y^{2}}{4}=1 \] such that \(y_{1}>0\), and \(y_{2}>0\). Let \(C\) denote the circle \(x^{2}+y^{2}=9\), and \(M\) be the point \((3,0)\). Suppose the line \(x=x_{1}\) intersects \(C\) at \(R\), and the line \(x=x_{2}\) intersects C at \(S\), such that the \(y\)-coordinates of \(R\) and \(S\) are positive. Let \(\angle R O M=\frac{\pi}{6}\) and \(\angle S O M=\frac{\pi}{3}\), where \(O\) denotes the origin \((0,0)\). Let \(|X Y|\) denote the length of the line segment \(X Y\). Then which of the following statements is (are) TRUE? (A) The equation of the line joining \(P\) and \(Q\) is \(2 x+3 y=3(1+\sqrt{3})\)
(B) The equation of the line joining \(P\) and \(Q\) is \(2 x+y=3(1+\sqrt{3})\)
(C) If \(N_{2}=\left(x_{2}, 0\right)\), then \(3\left|N_{2} Q\right|=2\left|N_{2} S\right|\)
(D) If \(N_{1}=\left(x_{1}, 0\right)\), then \(9\left|N_{1} P\right|=4\left|N_{1} R\right|\)
" "\(8 \quad\) Let \(\mathbb{R}\) denote the set of all real numbers. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by \[ f(x)= \begin{cases}\frac{6 x+\sin x}{2 x+\sin x} \text { if } x \neq 0 \\ \frac{7}{3} \text { if } x=0\end{cases} \] Then which of the following statements is (are) TRUE? (A) The point \(x=0\) is a point of local maxima of \(f\)
(B) The point \(x=0\) is a point of local minima of \(f\)
(C) Number of points of local maxima of \(f\) in the interval \([\pi, 6 \pi]\) is 3
(D) Number of points of local minima of \(f\) in the interval [ \(2 \pi, 4 \pi\) ] is 1
" "Let \(y(x)\) be the solution of the differential equation \[ x^{2} \frac{d y}{d x}+x y=x^{2}+y^{2}, \quad x>\frac{1}{e} \] satisfying \(y(1)=0\). Then the value of \(2 \frac{(y(e))^{2}}{y\left(e^{2}\right)}\) is \(\qquad\) .
" "Let \(a_{0}, a_{1}, \ldots, a_{23}\) be real numbers such that \[ \left(1+\frac{2}{5} x\right)^{23}=\sum_{i=0}^{23} a_{i} x^{i} \] for every real number \(x\). Let \(a_{r}\) be the largest among the numbers \(a_{j}\) for \(0 \leq j \leq 23\).
Then the value of \(r\) is \(\qquad\) .
" "A factory has a total of three manufacturing units, \(M_{1}, M_{2}\), and \(M_{3}\), which produce bulbs independent of each other. The units \(M_{1}, M_{2}\), and \(M_{3}\) produce bulbs in the proportions of \(2: 2: 1\), respectively. It is known that \(20 \%\) of the bulbs produced in the factory are defective. It is also known that, of all the bulbs produced by \(M_{1}, 15 \%\) are defective. Suppose that, if a randomly chosen bulb produced in the factory is found to be defective, the probability that it was produced by \(M_{2}\) is \(\frac{2}{5}\). If a bulb is chosen randomly from the bulbs produced by \(M_{3}\), then the probability that it is defective is \(\qquad\) .
" " Consider the vectors \[ \vec{x}=\hat{\imath}+2 \hat{\jmath}+3 \hat{k}, \quad \vec{y}=2 \hat{\imath}+3 \hat{\jmath}+\hat{k}, \quad \text { and } \quad \vec{z}=3 \hat{\imath}+\hat{\jmath}+2 \hat{k} \] For two distinct positive real numbers \(\alpha\) and \(\beta\), define \[ \vec{X}=\alpha \vec{x}+\beta \vec{y}-\vec{z}, \quad \vec{Y}=\alpha \vec{y}+\beta \vec{z}-\vec{x}, \quad \text { and } \quad \vec{Z}=\alpha \vec{z}+\beta \vec{x}-\vec{y} \] If the vectors \(\vec{X}, \vec{Y}\), and \(\vec{Z}\) lie in a plane, then the value of \(\alpha+\beta-3\) is \(\qquad\) .
" " For a non-zero complex number \(z\), let \(\arg (z)\) denote the principal argument of \(z\), with \(-\pi<\arg (z) \leq \pi\). Let \(\omega\) be the cube root of unity for which \(0<\arg (\omega)<\pi\). Let \[ \alpha=\arg \left(\sum_{n=1}^{2025}(-\omega)^{n}\right) . \] Then the value of \(\frac{3 \alpha}{\pi}\) is \(\qquad\) .
" "Let \(\mathbb{R}\) denote the set of all real numbers. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) and \(g: \mathbb{R} \rightarrow(0,4)\) be functions defined by \[ f(x)=\log _{e}\left(x^{2}+2 x+4\right), \text { and } g(x)=\frac{4}{1+e^{-2 x}} \] Define the composite function \(f \circ g^{-1}\) by \(\left(f \circ g^{-1}\right)(x)=f\left(g^{-1}(x)\right)\), where \(g^{-1}\) is the inverse of the function \(g\). Then the value of the derivative of the composite function \(f \circ g^{-1}\) at \(x=2\) is \(\qquad\) .
" "Let \[ \alpha=\frac{1}{\sin 60^{\circ} \sin 61^{\circ}}+\frac{1}{\sin 62^{\circ} \sin 63^{\circ}}+\cdots+\frac{1}{\sin 118^{\circ} \sin 119^{\circ}} . \] Then the value of \[ \left(\frac{\operatorname{cosec} 1^{\circ}}{\alpha}\right)^{2} \] is \(\qquad\) .
" "If \[ \alpha=\int_{\frac{1}{2}}^{2} \frac{\tan ^{-1} x}{2 x^{2}-3 x+2} d x \] then the value of \(\sqrt{7} \tan \left(\frac{2 \alpha \sqrt{7}}{\pi}\right)\) is \(\qquad\) .
(Here, the inverse trigonometric function \(\tan ^{-1} x\) assumes values in \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\).) " "A temperature difference can generate e.m.f. in some materials. Let \(S\) be the e.m.f. produced per unit temperature difference between the ends of a wire, \(\sigma\) the electrical conductivity and \(\kappa\) the thermal conductivity of the material of the wire. Taking \(M, L, T, I\) and \(K\) as dimensions of mass, length, time, current and temperature, respectively, the dimensional formula of the quantity \(Z=\frac{S^{2} \sigma}{\kappa}\) is: (A) \(\left[M^{0} L^{0} T^{0} I^{0} K^{0}\right]\) (B) \(\left[M^{0} L^{0} T^{0} I^{0} K^{-1}\right]\)
(C) \(\left[M^{1} L^{2} T^{-2} I^{-1} K^{-1}\right]\) (D) \(\left[M^{1} L^{2} T^{-4} I^{-1} K^{-1}\right]\)
" "Two co-axial conducting cylinders of same length \(\ell\) with radii \(\sqrt{2} R\) and \(2 R\) are kept, as shown in Fig. 1. The charge on the inner cylinder is \(Q\) and the outer cylinder is grounded. The annular region between the cylinders is filled with a material of dielectric constant \(\kappa=5\). Consider an imaginary plane of the same length \(\ell\) at a distance \(R\) from the common axis of the cylinders. This plane is parallel to the axis of the cylinders. The cross-sectional view of this arrangement is shown in Fig. 2. Ignoring edge effects, the flux of the electric field through the plane is ( \(\epsilon_{0}\) is the permittivity of free space): \begin{figure}[h] \includegraphics[width=\textwidth]{2025_08_07_b4fa520a65d5b2bc96aeg-08(1)} \captionsetup{labelformat=empty} \caption{Fig. 1} \end{figure} \begin{figure}[h] \includegraphics[width=\textwidth]{2025_08_07_b4fa520a65d5b2bc96aeg-08} \captionsetup{labelformat=empty} \caption{Fig. 2} \end{figure} (A) \(\frac{Q}{30 \epsilon_{0}}\) (B) \(\frac{Q}{15 \epsilon_{0}}\) (C) \(\frac{Q}{60 \epsilon_{0}}\) (D) \(\frac{Q}{120 \epsilon_{0}}\)
" " As shown in the figures, a uniform rod \(O O^{\prime}\) of length \(l\) is hinged at the point \(O\) and held in place vertically between two walls using two massless springs of same spring constant. The springs are connected at the midpoint and at the top-end ( \(O^{\prime}\) ) of the rod, as shown in Fig. 1 and the rod is made to oscillate by a small angular displacement. The frequency of oscillation of the rod is \(f_{1}\). On the other hand, if both the springs are connected at the midpoint of the rod, as shown in Fig. 2 and the rod is made to oscillate by a small angular displacement, then the frequency of oscillation is \(f_{2}\). Ignoring gravity and assuming motion only in the plane of the diagram, the value of \(\frac{f_{1}}{f_{2}}\) is: \begin{figure}[h] \includegraphics[width=\textwidth]{2025_08_07_b4fa520a65d5b2bc96aeg-09(1)} \captionsetup{labelformat=empty} \caption{Fig. 1} \end{figure} \begin{figure}[h] \includegraphics[width=\textwidth]{2025_08_07_b4fa520a65d5b2bc96aeg-09} \captionsetup{labelformat=empty} \caption{Fig. 2} \end{figure} (A) 2 (B) \(\sqrt{2}\) (C) \(\sqrt{\frac{5}{2}}\) (D) \(\sqrt{\frac{2}{5}}\)
" " Consider a star of mass \(m_{2} \mathrm{~kg}\) revolving in a circular orbit around another star of mass \(m_{1} \mathrm{~kg}\) with \(m_{1} \gg m_{2}\). The heavier star slowly acquires mass from the lighter star at a constant rate of \(\gamma \mathrm{kg} / \mathrm{s}\). In this transfer process, there is no other loss of mass. If the separation between the centers of the stars is \(r\), then its relative rate of change \(\frac{1}{r} \frac{d r}{d t}\left(\right.\) in \(\left.^{-1}\right)\) is given by: (A) \(-\frac{3 \gamma}{2 m_{2}}\) (B) \(-\frac{2 \gamma}{m_{2}}\) (C) \(-\frac{2 \gamma}{m_{1}}\) (D) \(-\frac{3 \gamma}{2 m_{1}}\)
" "A positive point charge of \(10^{-8} \mathrm{C}\) is kept at a distance of 20 cm from the center of a neutral conducting sphere of radius 10 cm . The sphere is then grounded and the charge on the sphere is measured. The grounding is then removed and subsequently the point charge is moved by a distance of 10 cm further away from the center of the sphere along the radial direction. Taking \(\frac{1}{4 \pi \epsilon_{0}}=9 \times 10^{9} \mathrm{Nm}^{2} / \mathrm{C}^{2}\) (where \(\epsilon_{0}\) is the permittivity of free space), which of the following statements is/are correct: \end{itemize} (A) Before the grounding, the electrostatic potential of the sphere is 450 V .
(B) Charge flowing from the sphere to the ground because of grounding is \(5 \times 10^{-9} \mathrm{C}\).
(C) After the grounding is removed, the charge on the sphere is \(-5 \times 10^{-9} \mathrm{C}\).
(D) The final electrostatic potential of the sphere is 300 V .
" " Two identical concave mirrors each of focal length \(f\) are facing each other as shown in the schematic diagram. The focal length \(f\) is much larger than the size of the mirrors. A glass slab of thickness \(t\) and refractive index \(n_{0}\) is kept equidistant from the mirrors and perpendicular to their common principal axis. A monochromatic point light source \(S\) is embedded at the center of the slab on the principal axis, as shown in the schematic diagram. For the image to be formed on \(S\) itself, which of the following distances between the two mirrors is/are correct:
\includegraphics[max width=\textwidth, center]{2025_08_07_b4fa520a65d5b2bc96aeg-11} (A) \(4 f+\left(1-\frac{1}{n_{0}}\right) t\) (B) \(2 f+\left(1-\frac{1}{n_{0}}\right) t\)
(C) \(4 f+\left(n_{0}-1\right) t\) (D) \(2 f+\left(n_{0}-1\right) t\)
" "Six infinitely large and thin non-conducting sheets are fixed in configurations I and II. As shown in the figure, the sheets carry uniform surface charge densities which are indicated in terms of \(\sigma_{0}\). The separation between any two consecutive sheets is \(1 \mu \mathrm{~m}\). The various regions between the sheets are denoted as \(1,2,3,4\) and 5 . If \(\sigma_{0}=9 \mu \mathrm{C} / \mathrm{m}^{2}\), then which of the following statements is/are correct: (Take permittivity of free space \(\epsilon_{0}=9 \times 10^{-12} \mathrm{~F} / \mathrm{m}\) ) \[ \begin{array}{ccccccccccc} +\sigma_{0} -\sigma_{0} +\sigma_{0} -\sigma_{0} +\sigma_{0} -\sigma_{0} +\frac{\sigma_{0}}{2} -\sigma_{0} +\sigma_{0} -\sigma_{0} \end{array}+\sigma_{0} \frac{-\sigma_{0}}{2} \] (A) In region 4 of the configuration I, the magnitude of the electric field is zero.
(B) In region 3 of the configuration II, the magnitude of the electric field is \(\frac{\sigma_{0}}{\epsilon_{0}}\).
(C) Potential difference between the first and the last sheets of the configuration I is 5 V .
(D) Potential difference between the first and the last sheets of the configuration II is zero.
" "The efficiency of a Carnot engine operating with a hot reservoir kept at a temperature of 1000 K is 0.4. It extracts 150 J of heat per cycle from the hot reservoir. The work extracted from this engine is being fully used to run a heat pump which has a coefficient of performance 10. The hot reservoir of the heat pump is at a temperature of 300 K . Which of the following statements is/are correct: (A) Work extracted from the Carnot engine in one cycle is 60 J .
(B) Temperature of the cold reservoir of the Carnot engine is 600 K .
(C) Temperature of the cold reservoir of the heat pump is 270 K .
(D) Heat supplied to the hot reservoir of the heat pump in one cycle is 540 J .
" A conducting solid sphere of radius \(R\) and mass \(M\) carries a charge \(Q\). The sphere is rotating about an axis passing through its center with a uniform angular speed \(\omega\). The ratio of the magnitudes of the magnetic dipole moment to the angular momentum about the same axis is given as \(\alpha \frac{Q}{2 M}\). The value of \(\alpha\) is
A hydrogen atom, initially at rest in its ground state, absorbs a photon of frequency \(v_{1}\) and ejects the electron with a kinetic energy of 10 eV . The electron then combines with a positron at rest to form a positronium atom in its ground state and simultaneously emits a photon of frequency \(v_{2}\). The center of mass of the resulting positronium atom moves with a kinetic energy of 5 eV . It is given that positron has the same mass as that of electron and the positronium atom can be considered as a Bohr atom, in which the electron and the positron orbit around their center of mass. Considering no other energy loss during the whole process, the difference between the two photon energies (in eV) is
"An ideal monatomic gas of \(n\) moles is taken through a cycle \(W X Y Z W\) consisting of consecutive adiabatic and isobaric quasi-static processes, as shown in the schematic \(V-T\) diagram. The volume of the gas at \(W, X\) and \(Y\) points are, \(64 \mathrm{~cm}^{3}, 125 \mathrm{~cm}^{3}\) and \(250 \mathrm{~cm}^{3}\), respectively. If the absolute temperature of the gas \(T_{W}\) at the point \(W\) is such that \(n R T_{W}=1 \mathrm{~J}\) ( \(R\) is the universal gas constant), then the amount of heat absorbed (in J ) by the gas along the path \(X Y\) is \(\qquad\)
\includegraphics[max width=\textwidth, center]{2025_08_07_b4fa520a65d5b2bc96aeg-14(1)}
" A geostationary satellite above the equator is orbiting around the earth at a fixed distance \(r_{1}\) from the center of the earth. A second satellite is orbiting in the equatorial plane in the opposite direction to the earth's rotation, at a distance \(r_{2}\) from the center of the earth, such that \(r_{1}=1.21 r_{2}\). The time period of the second satellite as measured from the geostationary satellite is \(\frac{24}{p}\) hours. The value of \(p\) is \(\qquad\)
"The left and right compartments of a thermally isolated container of length \(L\) are separated by a thermally conducting, movable piston of area \(A\). The left and right compartments are filled with \(\frac{3}{2}\) and 1 moles of an ideal gas, respectively. In the left compartment the piston is attached by a spring with spring constant \(k\) and natural length \(\frac{2 L}{5}\). In thermodynamic equilibrium, the piston is at a distance \(\frac{L}{2}\) from the left and right edges of the container as shown in the figure. Under the above conditions, if the pressure in the right compartment is \(P=\frac{k L}{A} \alpha\), then the value of \(\alpha\) is \(\qquad\)
\includegraphics[max width=\textwidth, center]{2025_08_07_b4fa520a65d5b2bc96aeg-14}" "In a Young's double slit experiment, a combination of two glass wedges \(A\) and \(B\), having refractive indices 1.7 and 1.5, respectively, are placed in front of the slits, as shown in the figure. The separation between the slits is \(d=2 \mathrm{~mm}\) and the shortest distance between the slits and the screen is \(D=2 \mathrm{~m}\). Thickness of the combination of the wedges is \(t=12 \mu \mathrm{~m}\). The value of \(l\) as shown in the figure is 1 mm . Neglect any refraction effect at the slanted interface of the wedges. Due to the combination of the wedges, the central maximum shifts (in mm ) with respect to O by \(\qquad\)
\includegraphics[max width=\textwidth, center]{2025_08_07_b4fa520a65d5b2bc96aeg-15(1)}
" "A projectile of mass 200 g is launched in a viscous medium at an angle \(60^{\circ}\) with the horizontal, with an initial velocity of \(270 \mathrm{~m} / \mathrm{s}\). It experiences a viscous drag force \(\vec{F}=-c \vec{v}\) where the drag coefficient \(c=0.1 \mathrm{~kg} / \mathrm{s}\) and \(\vec{v}\) is the instantaneous velocity of the projectile. The projectile hits a vertical wall after 2 s . Taking \(e=2.7\), the horizontal distance of the wall from the point of projection (in m ) is
" "An audio transmitter (T) and a receiver (R) are hung vertically from two identical massless strings of length 8 m with their pivots well separated along the \(X\) axis. They are pulled from the equilibrium position in opposite directions along the \(X\) axis by a small angular amplitude \(\theta_{0}=\cos ^{-1}(0.9)\) and released simultaneously. If the natural frequency of the transmitter is 660 Hz and the speed of sound in air is \(330 \mathrm{~m} / \mathrm{s}\), the maximum variation in the frequency (in Hz ) as measured by the receiver (Take the acceleration due to gravity \(g=10 \mathrm{~m} / \mathrm{s}^{2}\) ) is \(\qquad\)
\includegraphics[max width=\textwidth, center]{2025_08_07_b4fa520a65d5b2bc96aeg-15}" "During sodium nitroprusside test of sulphide ion in an aqueous solution, one of the ligands coordinated to the metal ion is converted to (A) \(\mathrm{NOS}^{-}\) (B) \(\mathrm{SCN}^{-}\) (C) \(\mathrm{SNO}^{-}\) (D) \(\mathrm{NCS}^{-}\)
" "The complete hydrolysis of \(\mathrm{ICl}, \mathrm{ClF}_{3}\) and \(\mathrm{BrF}_{5}\), respectively, gives (A) \(\mathrm{IO}^{-}, \mathrm{ClO}_{2}{ }^{-}\)and \(\mathrm{BrO}_{3}{ }^{-}\)
(B) \(\mathrm{IO}_{3}{ }^{-}, \mathrm{ClO}_{2}{ }^{-}\)and \(\mathrm{BrO}_{3}{ }^{-}\)
(C) \(\mathrm{IO}^{-}, \mathrm{ClO}^{-}\)and \(\mathrm{BrO}_{2}{ }^{-}\)
(D) \(\mathrm{IO}_{3}{ }^{-}, \mathrm{ClO}_{4}{ }^{-}\)and \(\mathrm{BrO}_{2}{ }^{-}\)
" "Monocyclic compounds \(\mathbf{P}, \mathbf{Q}, \mathbf{R}\) and \(\mathbf{S}\) are the major products formed in the reaction sequences given below.
\includegraphics[max width=\textwidth, center]{2025_08_07_b4fa520a65d5b2bc96aeg-17(5)}
\includegraphics[max width=\textwidth, center]{2025_08_07_b4fa520a65d5b2bc96aeg-17(3)}
\includegraphics[max width=\textwidth, center]{2025_08_07_b4fa520a65d5b2bc96aeg-17(4)}
(i) \(\mathrm{O}_{3}, \mathrm{Zn}-\mathrm{H}_{2} \mathrm{O}\)
(ii) \(\mathrm{CH}_{3} \mathrm{MgBr}\) (2 equiv.)
\includegraphics{smile-f6cdc79583d4f0219ea374222ad450035bb907ed}
\(\xrightarrow{\text { (iii) } \mathrm{H}^{+}, \Delta} \mathrm{S}\) The product having the highest number of unsaturated carbon atom(s) is (A) \(\mathbf{P}\) (B) \(\mathbf{Q}\)
(C) \(\mathbf{R}\) (D) \(\mathbf{S}\)
" "The correct reaction/reaction sequence that would produce a dicarboxylic acid as the major product is (A) \includegraphics[max width=\textwidth]{2025_08_07_b4fa520a65d5b2bc96aeg-17(2)}
(B) \includegraphics[max width=\textwidth]{2025_08_07_b4fa520a65d5b2bc96aeg-17}
(C) \includegraphics[max width=\textwidth]{2025_08_07_b4fa520a65d5b2bc96aeg-17(6)}
(D) \includegraphics[max width=\textwidth]{2025_08_07_b4fa520a65d5b2bc96aeg-17(1)}
" "The correct statement(s) about intermolecular forces is(are) \end{itemize} (A) The potential energy between two point charges approaches zero more rapidly than the potential energy between a point dipole and a point charge as the distance between them approaches infinity.
(B) The average potential energy of two rotating polar molecules that are separated by a distance \(r\) has \(1 / r^{3}\) dependence.
(C) The dipole-induced dipole average interaction energy is independent of temperature.
(D) Nonpolar molecules attract one another even though neither has a permanent dipole moment.
" "The compound(s) with \(\mathrm{P}-\mathrm{H}\) bond(s) is(are) (A) \(\mathrm{H}_{3} \mathrm{PO}_{4}\)
(B) \(\mathrm{H}_{3} \mathrm{PO}_{3}\)
(C) \(\mathrm{H}_{4} \mathrm{P}_{2} \mathrm{O}_{7}\)
(D) \(\mathrm{H}_{3} \mathrm{PO}_{2}\)
" "For the reaction sequence given below, the correct statement(s) is(are)
\includegraphics[max width=\textwidth, center]{2025_08_07_b4fa520a65d5b2bc96aeg-19(1)} (A) Both \(\mathbf{X}\) and \(\mathbf{Y}\) are oxygen containing compounds.
(B) \(\mathbf{Y}\) on heating with \(\mathrm{CHCl}_{3} / \mathrm{KOH}\) forms isocyanide.
(C) \(\mathbf{Z}\) reacts with Hinsberg's reagent.
(D) \(\mathbf{Z}\) is an aromatic primary amine.
" " For the reaction sequence given below, the correct statement(s) is(are)
\includegraphics[max width=\textwidth, center]{2025_08_07_b4fa520a65d5b2bc96aeg-19} (A) \(\mathbf{P}\) is optically active.
(B) S gives Bayer's test.
(C) Q gives effervescence with aq. \(\mathrm{NaHCO}_{3}\).
(D) \(\mathbf{R}\) is an alkyne.
" "The density (in \(\mathrm{g} \mathrm{cm}^{-3}\) ) of the metal which forms a cubic close packed (ccp) lattice with an axial distance (edge length) equal to 400 pm is \(\qquad\) . Use: Atomic mass of metal \(=105.6\) amu and Avogadro's constant \(=6 \times 10^{23} \mathrm{~mol}^{-1}\)
" "The solubility of barium iodate in an aqueous solution prepared by mixing 200 mL of 0.010 M barium nitrate with 100 mL of 0.10 M sodium iodate is \(\boldsymbol{X} \times 10^{-6} \mathrm{~mol} \mathrm{dm}^{-3}\). The value of \(\boldsymbol{X}\) is
\(\qquad\) . Use: Solubility product constant ( \(K_{\text {sp }}\) ) of barium iodate \(=1.58 \times 10^{-9}\)
" "Q. 11 Adsorption of phenol from its aqueous solution on to fly ash obeys Freundlich isotherm. At a given temperature, from \(10 \mathrm{mg} \mathrm{g}^{-1}\) and \(16 \mathrm{mg} \mathrm{g}^{-1}\) aqueous phenol solutions, the concentrations of adsorbed phenol are measured to be \(4 \mathrm{mg} \mathrm{g}^{-1}\) and \(10 \mathrm{mg} \mathrm{g}^{-1}\), respectively. At this temperature, the concentration (in \(\mathrm{mg} \mathrm{g}^{-1}\) ) of adsorbed phenol from \(20 \mathrm{mg} \mathrm{g}^{-1}\) aqueous solution of phenol will be
\(\qquad\) . Use: \(\log _{10} 2=0.3\)
" " Q. 13 At 300 K , an ideal dilute solution of a macromolecule exerts osmotic pressure that is expressed in terms of the height ( h ) of the solution (density \(=1.00 \mathrm{~g} \mathrm{~cm}^{-3}\) ) where h is equal to 2.00 cm . If the concentration of the dilute solution of the macromolecule is \(2.00 \mathrm{~g} \mathrm{dm}^{-3}\), the molar mass of the macromolecule is calculated to be \(\boldsymbol{X} \times 10^{4} \mathrm{~g} \mathrm{~mol}^{-1}\). The value of \(\boldsymbol{X}\) is \(\qquad\) . Use: Universal gas constant \((\mathrm{R})=8.3 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\) and acceleration due to gravity \((g)=10 \mathrm{~m} \mathrm{~s}^{-2}\)
" " An electrochemical cell is fueled by the combustion of butane at 1 bar and 298 K . Its cell potential is \(\frac{\boldsymbol{X}}{F} \times 10^{3}\) volts, where \(F\) is the Faraday constant. The value of \(\boldsymbol{X}\) is \(\qquad\) . Use: Standard Gibbs energies of formation at 298 K are: \(\Delta_{f} G_{\mathrm{CO}_{2}}^{o}=-394 \mathrm{~kJ} \mathrm{~mol}^{-1} ; \Delta_{f} G_{\text {water }}^{o}=\) \(-237 \mathrm{~kJ} \mathrm{~mol}^{-1} ; \Delta_{f} G_{\text {butane }}^{o}=-18 \mathrm{~kJ} \mathrm{~mol}^{-1}\)
" "The sum of the spin only magnetic moment values (in B.M.) of \(\left[\mathrm{Mn}(\mathrm{Br})_{6}\right]^{3-}\) and \(\left[\mathrm{Mn}(\mathrm{CN})_{6}\right]^{3-}\) is
\(\qquad\) .
" "A linear octasaccharide (molar mass \(=1024 \mathrm{~g} \mathrm{~mol}^{-1}\) ) on complete hydrolysis produces three monosaccharides: ribose, 2-deoxyribose and glucose. The amount of 2-deoxyribose formed is \(58.26 \%(\mathrm{w} / \mathrm{w})\) of the total amount of the monosaccharides produced in the hydrolyzed products. The number of ribose unit(s) present in one molecule of octasaccharide is \(\qquad\) . Use: Molar mass (in \(\mathrm{g} \mathrm{mol}^{-1}\) ): ribose = 150, 2-deoxyribose = 134, glucose = 180;
Atomic mass (in amu): \(\mathrm{H}=1, \mathrm{O}=16\) " " Consider a reaction \(A+R \rightarrow\) Product. The rate of this reaction is measured to be \(k[A][R]\). At the start of the reaction, the concentration of \(R,[R]_{0}\), is 10 -times the concentration of \(A,[A]_{0}\). The reaction can be considered to be a pseudo first order reaction with assumption that \(k[R]=k^{\prime}\) is constant. Due to this assumption, the relative error (in \%) in the rate when this reaction is \(40 \%\) complete, is \(\qquad\) .
[0pt] [ \(k\) and \(k^{\prime}\) represent corresponding rate constants]
" "\(1 \quad\) Let \(\mathbb{R}\) denote the set of all real numbers. Let \(a_{i}, b_{i} \in \mathbb{R}\) for \(i \in\{1,2,3\}\). Define the functions \(f: \mathbb{R} \rightarrow \mathbb{R}, g: \mathbb{R} \rightarrow \mathbb{R}\), and \(h: \mathbb{R} \rightarrow \mathbb{R}\) by \[ \begin{aligned} f(x)=a_{1}+10 x+a_{2} x^{2}+a_{3} x^{3}+x^{4} \\ g(x)=b_{1}+3 x+b_{2} x^{2}+b_{3} x^{3}+x^{4} \\ h(x)=f(x+1)-g(x+2) \end{aligned} \] If \(f(x) \neq g(x)\) for every \(x \in \mathbb{R}\), then the coefficient of \(x^{3}\) in \(h(x)\) is (A) 8
(B) 2
(C) -4
(D) -6
" "Three students \(S_{1}, S_{2}\), and \(S_{3}\) are given a problem to solve. Consider the following events:
\(U:\) At least one of \(S_{1}, S_{2}\), and \(S_{3}\) can solve the problem,
\(V: S_{1}\) can solve the problem, given that neither \(S_{2}\) nor \(S_{3}\) can solve the problem,
\(W\) : \(S_{2}\) can solve the problem and \(S_{3}\) cannot solve the problem,
\(T\) : \(S_{3}\) can solve the problem. For any event \(E\), let \(P(E)\) denote the probability of \(E\). If \[ P(U)=\frac{1}{2}, \quad P(V)=\frac{1}{10}, \quad \text { and } \quad P(W)=\frac{1}{12} \] then \(P(T)\) is equal to (A) \(\frac{13}{36}\) (B) \(\frac{1}{3}\) (C) \(\frac{19}{60}\) (D) \(\frac{1}{4}\)
" "Let \(\mathbb{R}\) denote the set of all real numbers. Define the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) by \[ f(x)=\left\{\begin{array}{cc} 2-2 x^{2}-x^{2} \sin \frac{1}{x} \text { if } x \neq 0 \\ 2 \text { if } x=0 \end{array}\right. \] Then which one of the following statements is TRUE? (A) The function \(f\) is NOT differentiable at \(x=0\)
(B) There is a positive real number \(\delta\), such that \(f\) is a decreasing function on the interval ( \(0, \delta\) )
(C) For any positive real number \(\delta\), the function \(f\) is NOT an increasing function on the interval ( \(-\delta, 0\) )
(D) \(x=0\) is a point of local minima of \(f\)
" "Consider the matrix \[ P=\left(\begin{array}{lll} 2 0 0 \\ 0 2 0 \\ 0 0 3 \end{array}\right) \] Let the transpose of a matrix \(X\) be denoted by \(X^{T}\). Then the number of \(3 \times 3\) invertible matrices \(Q\) with integer entries, such that \[ Q^{-1}=Q^{T} \text { and } P Q=Q P \] is (A) 32 (B) 8 (C) 16 (D) 24
\end{center " " Let \(L_{1}\) be the line of intersection of the planes given by the equations \end{itemize} \[ 2 x+3 y+z=4 \text { and } x+2 y+z=5 . \] Let \(L_{2}\) be the line passing through the point \(P(2,-1,3)\) and parallel to \(L_{1}\). Let \(M\) denote the plane given by the equation \[ 2 x+y-2 z=6 \] Suppose that the line \(L_{2}\) meets the plane \(M\) at the point \(Q\). Let \(R\) be the foot of the perpendicular drawn from \(P\) to the plane \(M\). Then which of the following statements is (are) TRUE? (A) The length of the line segment \(P Q\) is \(9 \sqrt{3}\)
(B) The length of the line segment \(Q R\) is 15
(C) The area of \(\triangle P Q R\) is \(\frac{3}{2} \sqrt{234}\)
(D) The acute angle between the line segments \(P Q\) and \(P R\) is \(\cos ^{-1}\left(\frac{1}{2 \sqrt{3}}\right)\)
" "Let \(\mathbb{N}\) denote the set of all natural numbers, and \(\mathbb{Z}\) denote the set of all integers. Consider the functions \(f: \mathbb{N} \rightarrow \mathbb{Z}\) and \(g: \mathbb{Z} \rightarrow \mathbb{N}\) defined by \[ f(n)= \begin{cases}(n+1) / 2 \text { if } n \text { is odd } \\ (4-n) / 2 \text { if } n \text { is even }\end{cases} \] and \[ g(n)= \begin{cases}3+2 n \text { if } n \geq 0 \\ -2 n \text { if } n<0\end{cases} \] Define \((g \circ f)(n)=g(f(n))\) for all \(n \in \mathbb{N}\), and \((f \circ g)(n)=f(g(n))\) for all \(n \in \mathbb{Z}\). Then which of the following statements is (are) TRUE? (A) \(g \circ f\) is NOT one-one and \(g \circ f\) is NOT onto
(B) \(f \circ g\) is NOT one-one but \(f \circ g\) is onto
(C) \(g\) is one-one and \(g\) is onto
(D) \(f\) is NOT one-one but \(f\) is onto
" "Let \(\mathbb{R}\) denote the set of all real numbers. Let \(z_{1}=1+2 i\) and \(z_{2}=3 i\) be two complex numbers, where \(i=\sqrt{-1}\). Let \[ S=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}:\left|x+i y-z_{1}\right|=2\left|x+i y-z_{2}\right|\right\} \] Then which of the following statements is (are) TRUE? (A) \(S\) is a circle with centre ( \(-\frac{1}{3}, \frac{10}{3}\) )
(B) \(S\) is a circle with centre \(\left(\frac{1}{3}, \frac{8}{3}\right)\)
(C) \(S\) is a circle with radius \(\frac{\sqrt{2}}{3}\)
(D) \(S\) is a circle with radius \(\frac{2 \sqrt{2}}{3}\)
" "Let the set of all relations \(R\) on the set \(\{a, b, c, d, e, f\}\), such that \(R\) is reflexive and symmetric, and \(R\) contains exactly 10 elements, be denoted by \(\mathcal{S}\). Then the number of elements in \(\mathcal{S}\) is \(\qquad\) .
" "For any two points \(M\) and \(N\) in the \(X Y\)-plane, let \(\overrightarrow{M N}\) denote the vector from \(M\) to \(N\), and \(\overrightarrow{0}\) denote the zero vector. Let \(P, Q\) and \(R\) be three distinct points in the \(X Y\)-plane. Let \(S\) be a point inside the triangle \(\triangle P Q R\) such that \[ \overrightarrow{S P}+5 \overrightarrow{S Q}+6 \overrightarrow{S R}=\overrightarrow{0} \] Let \(E\) and \(F\) be the mid-points of the sides \(P R\) and \(Q R\), respectively. Then the value of \[ \frac{\text { length of the line segment } E F}{\text { length of the line segment } E S} \] is \(\qquad\) .
" " Let \(S\) be the set of all seven-digit numbers that can be formed using the digits 0,1 and 2 . For example, 2210222 is in \(S\), but 0210222 is NOT in \(S\).
Then the number of elements \(x\) in \(S\) such that at least one of the digits 0 and 1 appears exactly twice in \(x\), is equal to \(\qquad\) .
" "Let \(\alpha\) and \(\beta\) be the real numbers such that \[ \lim _{x \rightarrow 0} \frac{1}{x^{3}}\left(\frac{\alpha}{2} \int_{0}^{x} \frac{1}{1-t^{2}} d t+\beta x \cos x\right)=2 \] Then the value of \(\alpha+\beta\) is \(\qquad\) .
" "Let \(\mathbb{R}\) denote the set of all real numbers. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a function such that \(f(x)>0\) for all \(x \in \mathbb{R}\), and \(f(x+y)=f(x) f(y)\) for all \(x, y \in \mathbb{R}\). Let the real numbers \(a_{1}, a_{2}, \ldots, a_{50}\) be in an arithmetic progression. If \(f\left(a_{31}\right)=64 f\left(a_{25}\right)\), and \[ \sum_{i=1}^{50} f\left(a_{i}\right)=3\left(2^{25}+1\right), \] then the value of \[ \sum_{i=6}^{30} f\left(a_{i}\right) \] is \(\qquad\) .
" "For all \(x>0\), let \(y_{1}(x), y_{2}(x)\), and \(y_{3}(x)\) be the functions satisfying \[ \begin{aligned} \frac{d y_{1}}{d x}-(\sin x)^{2} y_{1}=0, \quad y_{1}(1)=5 \\ \frac{d y_{2}}{d x}-(\cos x)^{2} y_{2}=0, \quad y_{2}(1)=\frac{1}{3} \\ \frac{d y_{3}}{d x}-\left(\frac{2-x^{3}}{x^{3}}\right) y_{3}=0, \quad y_{3}(1)=\frac{3}{5 e} \end{aligned} \] respectively. Then \[ \lim _{x \rightarrow 0^{+}} \frac{y_{1}(x) y_{2}(x) y_{3}(x)+2 x}{e^{3 x} \sin x} \] is equal to \(\qquad\)" "Suppose that the sum of the frequencies is 19 and the median of this frequency distribution is 6.
For the given frequency distribution, let \(\alpha\) denote the mean deviation about the mean, \(\beta\) denote the mean deviation about the median, and \(\sigma^{2}\) denote the variance. Match each entry in List-I to the correct entry in List-II and choose the correct option. \section*{List-I} (P) \(7 f_{1}+9 f_{2}\) is equal to
(Q) \(19 \alpha\) is equal to
(R) \(19 \beta\) is equal to
(S) \(19 \sigma^{2}\) is equal to \section*{List-II} (1) 146
(2) 47
(3) 48
(4) 145
(5) 55 (A) \((\mathrm{P}) \rightarrow(5)\) \((\mathrm{Q}) \rightarrow(3)\) \((\mathrm{R}) \rightarrow(2)\) \((\mathrm{S}) \rightarrow(4)\)
(B) \((\mathrm{P}) \rightarrow(5)\) \((\mathrm{Q}) \rightarrow(2)\) \((\mathrm{R}) \rightarrow(3)\) (S) \(\rightarrow\) (1)
(C) \((\mathrm{P}) \rightarrow(5)\) \((\mathrm{Q}) \rightarrow(3)\) \((\mathrm{R}) \rightarrow(2)\) (S) \(\rightarrow\) (1)
(D) \((\mathrm{P}) \rightarrow(3)\) \((\mathrm{Q}) \rightarrow(2)\) \((\mathrm{R}) \rightarrow(5)\) (S) \(\rightarrow\) (4)
" "Let \(\mathbb{R}\) denote the set of all real numbers. For a real number \(x\), let \([x]\) denote the greatest integer less than or equal to \(x\). Let \(n\) denote a natural number. Match each entry in List-I to the correct entry in List-II and choose the correct option. \section*{List-I} (P) The minimum value of \(n\) for which the function \[ f(x)=\left[\frac{10 x^{3}-45 x^{2}+60 x+35}{n}\right] \] is continuous on the interval \([1,2]\), is
(Q) The minimum value of \(n\) for which \[ g(x)=\left(2 n^{2}-13 n-15\right)\left(x^{3}+3 x\right), \] \(x \in \mathbb{R}\), is an increasing function on \(\mathbb{R}\), is
(R) The smallest natural number \(n\) which is greater than 5 , such that \(x=3\) is a point of local minima of \[ h(x)=\left(x^{2}-9\right)^{n}\left(x^{2}+2 x+3\right), \] is
(S) Number of \(x_{0} \in \mathbb{R}\) such that \[ l(x)=\sum_{k=0}^{4}\left(\sin |x-k|+\cos \left|x-k+\frac{1}{2}\right|\right), \] \(x \in \mathbb{R}\), is NOT differentiable at \(x_{0}\), is \section*{List-II} (1) 8
(2) 9
(3) 5
(4) 6
(5) 10 (A) \((\mathrm{P}) \rightarrow(1)\) \((\mathrm{Q}) \rightarrow(3)\) \((\mathrm{R}) \rightarrow(2)\) \((\mathrm{S}) \rightarrow(5)\)
(B) \((\mathrm{P}) \rightarrow(2)\) \((\mathrm{Q}) \rightarrow(1)\) \((\mathrm{R}) \rightarrow(4)\) (S) \(\rightarrow\) (3)
(C) \((\mathrm{P}) \rightarrow(5)\) \((\mathrm{Q}) \rightarrow(1)\) \((\mathrm{R}) \rightarrow(4)\) \((\mathrm{S}) \rightarrow(3)\)
(D) \((\mathrm{P}) \rightarrow(2)\) \((\mathrm{Q}) \rightarrow(3)\) \((\mathrm{R}) \rightarrow(1)\) \((\mathrm{S}) \rightarrow(5)\)
" "Let \(\vec{w}=\hat{\imath}+\hat{\jmath}-2 \hat{k}\), and \(\vec{u}\) and \(\vec{v}\) be two vectors, such that \(\vec{u} \times \vec{v}=\vec{w}\) and \(\vec{v} \times \vec{w}=\vec{u}\). Let \(\alpha, \beta, \gamma\), and \(t\) be real numbers such that \(\vec{u}=\alpha \hat{\imath}+\beta \hat{\jmath}+\gamma \hat{k}, \quad-t \alpha+\beta+\gamma=0, \quad \alpha-t \beta+\gamma=0, \quad\) and \(\alpha+\beta-t \gamma=0\). Match each entry in List-I to the correct entry in List-II and choose the correct option. \section*{List-I} (P) \(|\vec{v}|^{2}\) is equal to
(Q) If \(\alpha=\sqrt{3}\), then \(\gamma^{2}\) is equal to
(R) If \(\alpha=\sqrt{3}\), then \((\beta+\gamma)^{2}\) is equal to
(S) If \(\alpha=\sqrt{2}\), then \(t+3\) is equal to \section*{List-II} (1) 0
(2) 1
(3) 2
(4) 3
(5) 5 (A) \((\mathrm{P}) \rightarrow(2)\) \((\mathrm{Q}) \rightarrow(1)\) \((\mathrm{R}) \rightarrow(4)\) \((\mathrm{S}) \rightarrow(5)\)
(B) \((\mathrm{P}) \rightarrow(2)\) \((\mathrm{Q}) \rightarrow(4)\) \((\mathrm{R}) \rightarrow(3)\) \((\mathrm{S}) \rightarrow(5)\)
(C) \((\mathrm{P}) \rightarrow(2)\) \((\mathrm{Q}) \rightarrow(1)\) \((\mathrm{R}) \rightarrow(4)\) (S) \(\rightarrow\) (3)
(D) \((\mathrm{P}) \rightarrow(5)\) \((\mathrm{Q}) \rightarrow(4)\) \((\mathrm{R}) \rightarrow(1)\) (S) \(\rightarrow\) (3)
" "The center of a disk of radius \(r\) and mass \(m\) is attached to a spring of spring constant \(k\), inside a ring of radius \(R>r\) as shown in the figure. The other end of the spring is attached on the periphery of the ring. Both the ring and the disk are in the same vertical plane. The disk can only roll along the inside periphery of the ring, without slipping. The spring can only be stretched or compressed along the periphery of the ring, following the Hooke's law. In equilibrium, the disk is at the bottom of the ring. Assuming small displacement of the disc, the time period of oscillation of center of mass of the disk is written as \(T=\frac{2 \pi}{\omega}\). The correct expression for \(\omega\) is ( \(g\) is the acceleration due to gravity):
\includegraphics[max width=\textwidth, center]{2025_08_07_00c198eec9d2d28d09ffg-11} (A) \(\sqrt{\frac{2}{3}\left(\frac{g}{R-r}+\frac{k}{m}\right)}\) (B) \(\sqrt{\frac{2 g}{3(R-r)}+\frac{k}{m}}\)
(C) \(\sqrt{\frac{1}{6}\left(\frac{g}{R-r}+\frac{k}{m}\right)}\) (D) \(\sqrt{\frac{1}{4}\left(\frac{g}{R-r}+\frac{k}{m}\right)}\)
" "In a scattering experiment, a particle of mass \(2 m\) collides with another particle of mass \(m\), which is initially at rest. Assuming the collision to be perfectly elastic, the maximum angular deviation \(\theta\) of the heavier particle, as shown in the figure, in radians is:
\includegraphics[max width=\textwidth, center]{2025_08_07_00c198eec9d2d28d09ffg-11(1)} (A) \(\pi\) (B) \(\tan ^{-1}\left(\frac{1}{2}\right)\) (C) \(\frac{\pi}{3}\) (D) \(\frac{\pi}{6}\)
" "A conducting square loop initially lies in the \(X Z\) plane with its lower edge hinged along the \(X\)-axis. Only in the region \(y \geq 0\), there is a time dependent magnetic field pointing along the \(Z\)-direction, \(\vec{B}(t)=B_{0}(\cos \omega t) \hat{k}\), where \(B_{0}\) is a constant. The magnetic field is zero everywhere else. At time \(t=0\), the loop starts rotating with constant angular speed \(\omega\) about the \(X\) axis in the clockwise direction as viewed from the \(+X\) axis (as shown in the figure). Ignoring self-inductance of the loop and gravity, which of the following plots correctly represents the induced e.m.f. ( \(V\) ) in the loop as a function of time:
\includegraphics[max width=\textwidth, center]{2025_08_07_00c198eec9d2d28d09ffg-12(5)} (A) \includegraphics[max width=\textwidth]{2025_08_07_00c198eec9d2d28d09ffg-12(1)} (B) \includegraphics[max width=\textwidth]{2025_08_07_00c198eec9d2d28d09ffg-12}
(C) \includegraphics[max width=\textwidth]{2025_08_07_00c198eec9d2d28d09ffg-12(4)} (D) \includegraphics[max width=\textwidth]{2025_08_07_00c198eec9d2d28d09ffg-12(3)}
" "Figure 1 shows the configuration of main scale and Vernier scale before measurement. Fig. 2 shows the configuration corresponding to the measurement of diameter \(D\) of a tube. The measured value of \(D\) is: \begin{figure}[h] \includegraphics[width=\textwidth]{2025_08_07_00c198eec9d2d28d09ffg-12(2)} \captionsetup{labelformat=empty} \caption{Fig. 2} \end{figure} (A) 0.12 cm
(B) 0.11 cm
(C) 0.13 cm
(D) 0.14 cm
" "A conducting square loop of side \(L\), mass \(M\) and resistance \(R\) is moving in the \(X Y\) plane with its edges parallel to the \(X\) and \(Y\) axes. The region \(y \geq 0\) has a uniform magnetic field, \(\vec{B}=B_{0} \widehat{k}\). The magnetic field is zero everywhere else. At time \(t=0\), the loop starts to enter the magnetic field with an initial velocity \(v_{0} \hat{J} \mathrm{~m} / \mathrm{s}\), as shown in the figure. Considering the quantity \(K=\frac{B_{0}^{2} L^{2}}{R M}\) in appropriate units, ignoring self-inductance of the loop and gravity, which of the following statements is/are correct:
\includegraphics[max width=\textwidth, center]{2025_08_07_00c198eec9d2d28d09ffg-14} \end{itemize} (A) If \(v_{0}=1.5 K L\), the loop will stop before it enters completely inside the region of magnetic field.
(B) When the complete loop is inside the region of magnetic field, the net force acting on the loop is zero.
(C) If \(v_{0}=\frac{K L}{10}\), the loop comes to rest at \(t=\left(\frac{1}{K}\right) \ln \left(\frac{5}{2}\right)\).
(D) If \(v_{0}=3 K L\), the complete loop enters inside the region of magnetic field at time \(t=\left(\frac{1}{K}\right) \ln \left(\frac{3}{2}\right)\).
" "Length, breadth and thickness of a strip having a uniform cross section are measured to be 10.5 cm , 0.05 mm , and \(6.0 \mu \mathrm{~m}\), respectively. Which of the following option(s) give(s) the volume of the strip in \(\mathrm{cm}^{3}\) with correct significant figures: (A) \(3.2 \times 10^{-5}\) (B) \(32.0 \times 10^{-6}\) (C) \(3.0 \times 10^{-5}\) (D) \(3 \times 10^{-5}\)
" "Consider a system of three connected strings, \(S_{1}, S_{2}\) and \(S_{3}\) with uniform linear mass densities \(\mu\) \(\mathrm{kg} / \mathrm{m}, 4 \mu \mathrm{~kg} / \mathrm{m}\) and \(16 \mu \mathrm{~kg} / \mathrm{m}\), respectively, as shown in the figure. \(S_{1}\) and \(S_{2}\) are connected at the point \(P\), whereas \(S_{2}\) and \(S_{3}\) are connected at the point \(Q\), and the other end of \(S_{3}\) is connected to a wall. A wave generator 0 is connected to the free end of \(S_{1}\). The wave from the generator is represented by \(y=y_{0} \cos (\omega t-k x) \mathrm{cm}\), where \(y_{0}, \omega\) and \(k\) are constants of appropriate dimensions. Which of the following statements is/are correct:
\includegraphics[max width=\textwidth, center]{2025_08_07_00c198eec9d2d28d09ffg-15}
\includegraphics[max width=\textwidth, center]{2025_08_07_00c198eec9d2d28d09ffg-15(1)} (A) When the wave reflects from \(P\) for the first time, the reflected wave is represented by \(y=\alpha_{1} \mathrm{y}_{0} \cos (\omega t+k x+\pi) \mathrm{cm}\), where \(\alpha_{1}\) is a positive constant.
(B) When the wave transmits through \(P\) for the first time, the transmitted wave is represented by \(y=\alpha_{2} \mathrm{y}_{0} \cos (\omega t-k x) \mathrm{cm}\), where \(\alpha_{2}\) is a positive constant.
(C) When the wave reflects from \(Q\) for the first time, the reflected wave is represented by \(y=\alpha_{3} \mathrm{y}_{0} \cos (\omega t-k x+\pi) \mathrm{cm}\), where \(\alpha_{3}\) is a positive constant.
(D) When the wave transmits through \(Q\) for the first time, the transmitted wave is represented by \(y=\alpha_{4} \mathrm{y}_{0} \cos (\omega t-4 k x) \mathrm{cm}\), where \(\alpha_{4}\) is a positive constant.
" A person sitting inside an elevator performs a weighing experiment with an object of mass 50 kg . Suppose that the variation of the height \(y\) (in m ) of the elevator, from the ground, with time \(t\) (in s ) is given by \(y=8\left[1+\sin \left(\frac{2 \pi t}{T}\right)\right]\), where \(T=40 \pi \mathrm{~s}\). Taking acceleration due to gravity, \(g=10\) \(\mathrm{m} / \mathrm{s}^{2}\), the maximum variation of the object's weight (in N ) as observed in the experiment is
A cube of unit volume contains \(35 \times 10^{7}\) photons of frequency \(10^{15} \mathrm{~Hz}\). If the energy of all the photons is viewed as the average energy being contained in the electromagnetic waves within the same volume, then the amplitude of the magnetic field is \(\alpha \times 10^{-9} \mathrm{~T}\). Taking permeability of free space \(\mu_{0}=4 \pi \times 10^{-7} \mathrm{Tm} / \mathrm{A}\), Planck's constant \(h=6 \times 10^{-34} \mathrm{Js}\) and \(\pi=\frac{22}{7}\), the value of \(\alpha\) is
Two identical plates P and Q , radiating as perfect black bodies, are kept in vacuum at constant absolute temperatures \(\mathrm{T}_{\mathrm{P}}\) and \(\mathrm{T}_{\mathrm{Q}}\), respectively, with \(\mathrm{T}_{\mathrm{Q}}<\mathrm{T}_{\mathrm{P}}\), as shown in Fig. 1. The radiated power transferred per unit area from P to Q is \(W_{0}\). Subsequently, two more plates, identical to P and Q , are introduced between P and Q , as shown in Fig. 2. Assume that heat transfer takes place only between adjacent plates. If the power transferred per unit area in the direction from P to Q (Fig. 2) in the steady state is \(W_{S}\), then the ratio \(\frac{W_{0}}{W_{S}}\) is \(\qquad\) "A solid glass sphere of refractive index \(n=\sqrt{3}\) and radius \(R\) contains a spherical air cavity of radius \(\frac{R}{2}\), as shown in the figure. A very thin glass layer is present at the point O so that the air cavity (refractive index \(n=1\) ) remains inside the glass sphere. An unpolarized, unidirectional and monochromatic light source \(S\) emits a light ray from a point inside the glass sphere towards the periphery of the glass sphere. If the light is reflected from the point 0 and is fully polarized, then the angle of incidence at the inner surface of the glass sphere is \(\theta\). The value of \(\sin \theta\) is \(\qquad\)
\includegraphics[max width=\textwidth, center]{2025_08_07_00c198eec9d2d28d09ffg-17}
" A single slit diffraction experiment is performed to determine the slit width using the equation, \(\frac{b d}{D}=\) \(m \lambda\), where \(b\) is the slit width, \(D\) the shortest distance between the slit and the screen, \(d\) the distance between the \(m^{\text {th }}\) diffraction maximum and the central maximum, and \(\lambda\) is the wavelength. \(D\) and \(d\) are measured with scales of least count of 1 cm and 1 mm , respectively. The values of \(\lambda\) and \(m\) are known precisely to be 600 nm and 3, respectively. The absolute error (in \(\mu \mathrm{m}\) ) in the value of \(b\) estimated using the diffraction maximum that occurs for \(m=3\) with \(d=5 \mathrm{~mm}\) and \(D=1 \mathrm{~m}\) is \(\qquad\)
Consider an electron in the \(n=3\) orbit of a hydrogen-like atom with atomic number \(Z\). At absolute temperature \(T\), a neutron having thermal energy \(k_{\mathrm{B}} T\) has the same de Broglie wavelength as that of this electron. If this temperature is given by \(T=\frac{Z^{2} h^{2}}{\alpha \pi^{2} a_{0}^{2} m_{\mathrm{N}} k_{\mathrm{B}}}\), (where \(h\) is the Planck's constant, \(k_{B}\) is the Boltzmann constant, \(m_{\mathrm{N}}\) is the mass of the neutron and \(a_{0}\) is the first Bohr radius of hydrogen atom) then the value of \(\alpha\) is \(\qquad\) "Choose the option that describes the correct match between the entries in List-I to those in List-II. \section*{List-I} (P)
\includegraphics[max width=\textwidth, center]{2025_08_07_00c198eec9d2d28d09ffg-20(1)} \section*{(Q)
\includegraphics[max width=\textwidth]{2025_08_07_00c198eec9d2d28d09ffg-20}} (R)
\includegraphics[max width=\textwidth, center]{2025_08_07_00c198eec9d2d28d09ffg-20(3)}
(S)
\includegraphics[max width=\textwidth, center]{2025_08_07_00c198eec9d2d28d09ffg-20(2)} (A) \(\mathrm{P} \rightarrow 3, \mathrm{Q} \rightarrow 1, \mathrm{R} \rightarrow 2, \mathrm{~S} \rightarrow 4\)
(B) \(\mathrm{P} \rightarrow 4, \mathrm{Q} \rightarrow 5, \mathrm{R} \rightarrow 3, \mathrm{~S} \rightarrow 1\)
(C) \(\mathrm{P} \rightarrow 2, \mathrm{Q} \rightarrow 1, \mathrm{R} \rightarrow 4, \mathrm{~S} \rightarrow 5\)
(D) \(\mathrm{P} \rightarrow 2, \mathrm{Q} \rightarrow 1, \mathrm{R} \rightarrow 3, \mathrm{~S} \rightarrow 5\)
\section*{List-II} (1) \(\vec{E}=0\)
(2) \(\vec{E}=-\frac{p}{2 \pi \epsilon_{0} r^{3}} \hat{\jmath}\)
(3) \(\vec{E}=-\frac{p}{4 \pi \epsilon_{0} r^{3}}(\hat{\imath}-\hat{\jmath})\)
(4) \(\vec{E}=\frac{p}{4 \pi \epsilon_{0} r^{3}}(2 \hat{\imath}-\hat{\jmath})\)
(5) \(\vec{E}=\frac{p}{\pi \epsilon_{0} r^{3}} \hat{\imath}\)
" "A circuit with an electrical load having impedance \(Z\) is connected with an AC source as shown in the diagram. The source voltage varies in time as \(V(t)=300 \sin (400 t) \mathrm{V}\), where \(t\) is time in s. List-I shows various options for the load. The possible currents \(i(t)\) in the circuit as a function of time are given in List-II.
\includegraphics[max width=\textwidth, center]{2025_08_07_00c198eec9d2d28d09ffg-21} Choose the option that describes the correct match between the entries in List-I to those in ListII. \section*{List-I} (P) \(\quad \begin{gathered}30 \Omega \\ \bigvee\end{gathered}\)
(P)
\includegraphics[max width=\textwidth, center]{2025_08_07_00c198eec9d2d28d09ffg-21(1)}
(Q) \(\oiiint^{30 \Omega} \begin{array}{r}100 \mathrm{mH} \\ 000000\end{array}\)
(Q)
\includegraphics[max width=\textwidth, center]{2025_08_07_00c198eec9d2d28d09ffg-21(7)}
(R) \(\quad 50 \mu \mathrm{~F} \quad 30 \Omega \quad 25 \mathrm{mH}\)
\(-16 M M \sim 000000\)
(S) \[ \begin{array}{lrr} 50 \mu \mathrm{~F} 60 \Omega 125 \mathrm{mH} \\ - \mapsto \end{array} \] \section*{List-II} (1)
\includegraphics[max width=\textwidth, center]{2025_08_07_00c198eec9d2d28d09ffg-21(2)}
(2)
\includegraphics[max width=\textwidth, center]{2025_08_07_00c198eec9d2d28d09ffg-21(4)}
(3)
\includegraphics[max width=\textwidth, center]{2025_08_07_00c198eec9d2d28d09ffg-21(5)}
(4)
\includegraphics[max width=\textwidth, center]{2025_08_07_00c198eec9d2d28d09ffg-21(6)}
(5)
\includegraphics[max width=\textwidth, center]{2025_08_07_00c198eec9d2d28d09ffg-21(3)} (A) \(\mathrm{P} \rightarrow 3, \mathrm{Q} \rightarrow 5, \mathrm{R} \rightarrow 2, \mathrm{~S} \rightarrow 1\)
(B) \(\mathrm{P} \rightarrow 1, \mathrm{Q} \rightarrow 5, \mathrm{R} \rightarrow 2, \mathrm{~S} \rightarrow 3\)
(C) \(\mathrm{P} \rightarrow 3, \mathrm{Q} \rightarrow 4, \mathrm{R} \rightarrow 2, \mathrm{~S} \rightarrow 1\)
(D) \(\mathrm{P} \rightarrow 1, \mathrm{Q} \rightarrow 4, \mathrm{R} \rightarrow 2, \mathrm{~S} \rightarrow 5\)
" "List-I shows various functional dependencies of energy \((E)\) on the atomic number \((Z)\). Energies associated with certain phenomena are given in List-II. Choose the option that describes the correct match between the entries in List-I to those in ListII. \section*{List-I} (P) \(E \propto Z^{2}\)
(Q) \(E \propto(Z-1)^{2}\)
(R) \(E \propto Z(Z-1)\)
(S) \(E\) is practically independent of \(Z\) \section*{List-II} (1) energy of characteristic x-rays
(2) electrostatic part of the nuclear binding energy for stable nuclei with mass numbers in the range 30 to 170
(3) energy of continuous x-rays
(4) average nuclear binding energy per nucleon for stable nuclei with mass number in the range 30 to 170
(5) energy of radiation due to electronic transitions from hydrogen-like atoms (A) \(\mathrm{P} \rightarrow 4, \mathrm{Q} \rightarrow 3, \mathrm{R} \rightarrow 1, \mathrm{~S} \rightarrow 2\)
(B) \(\mathrm{P} \rightarrow 5, \mathrm{Q} \rightarrow 2, \mathrm{R} \rightarrow 1, \mathrm{~S} \rightarrow 4\)
(C) \(\mathrm{P} \rightarrow 5, \mathrm{Q} \rightarrow 1, \mathrm{R} \rightarrow 2, \mathrm{~S} \rightarrow 4\)
(D) \(\mathrm{P} \rightarrow 3, \mathrm{Q} \rightarrow 2, \mathrm{R} \rightarrow 1, \mathrm{~S} \rightarrow 5\)
" "The heating of \(\mathrm{NH}_{4} \mathrm{NO}_{2}\) at \(60-70^{\circ} \mathrm{C}\) and \(\mathrm{NH}_{4} \mathrm{NO}_{3}\) at \(200-250^{\circ} \mathrm{C}\) is associated with the formation of nitrogen containing compounds \(\mathbf{X}\) and \(\mathbf{Y}\), respectively. \(\mathbf{X}\) and \(\mathbf{Y}\), respectively, are (A) \(\mathrm{N}_{2}\) and \(\mathrm{N}_{2} \mathrm{O}\)
(B) \(\mathrm{NH}_{3}\) and \(\mathrm{NO}_{2}\)
(C) NO and \(\mathrm{N}_{2} \mathrm{O}\)
(D) \(\mathrm{N}_{2}\) and \(\mathrm{NH}_{3}\)
" "The correct order of the wavelength maxima of the absorption band in the ultraviolet-visible region for the given complexes is (A) \(\left[\mathrm{Co}(\mathrm{CN})_{6}\right]^{3-}<\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}\right]^{3+}<\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{5}\left(\mathrm{H}_{2} \mathrm{O}\right)\right]^{3+}<\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{5}(\mathrm{Cl})\right]^{2+}\)
(B) \(\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{5}(\mathrm{Cl})\right]^{2+}<\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{5}\left(\mathrm{H}_{2} \mathrm{O}\right)\right]^{3+}<\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}\right]^{3+}<\left[\mathrm{Co}(\mathrm{CN})_{6}\right]^{3-}\)
(C) \(\left[\mathrm{Co}(\mathrm{CN})_{6}\right]^{3-}<\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{5}(\mathrm{Cl})\right]^{2+}<\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{5}\left(\mathrm{H}_{2} \mathrm{O}\right)\right]^{3+}<\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}\right]^{3+}\)
(D) \(\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}\right]^{3+}<\left[\mathrm{Co}(\mathrm{CN})_{6}\right]^{3-}<\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{5}(\mathrm{Cl})\right]^{2+}<\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{5}\left(\mathrm{H}_{2} \mathrm{O}\right)\right]^{3+}\)
" "One of the products formed from the reaction of permanganate ion with iodide ion in neutral aqueous medium is (A) \(\mathrm{I}_{2}\) (B) \(\mathrm{IO}_{3}^{-}\) (C) \(\mathrm{IO}_{4}^{-}\) (D) \(\mathrm{IO}_{2}^{-}\)
" " Consider the depicted hydrogen ( \(\mathbf{H}\) ) in the hydrocarbons given below. The most acidic hydrogen \((\mathbf{H})\) is (A) \includegraphics{smile-e7ec6d16b1bf3758a536ff0c85f41b45a13fb25e} (B) \includegraphics{smile-4be8e160ca1641eaab30e4cc6b17947f1c54d424}
(C) \includegraphics{smile-719277614b8d77206307cff6d616f967c174d464} (D) \includegraphics{smile-2355779a6299ee8460332dfdafbf00072d64dd24}
" "Regarding the molecular orbital (MO) energy levels for homonuclear diatomic molecules, the INCORRECT statement(s) is(are) \end{itemize} (A) Bond order of \(\mathrm{Ne}_{2}\) is zero.
(B) The highest occupied molecular orbital (HOMO) of \(\mathrm{F}_{2}\) is \(\sigma\)-type.
(C) Bond energy of \(\mathrm{O}_{2}^{+}\)is smaller than the bond energy of \(\mathrm{O}_{2}\).
(D) Bond length of \(\mathrm{Li}_{2}\) is larger than the bond length of \(\mathrm{B}_{2}\).
" "The pair(s) of diamagnetic ions is(are) (A) \(\mathrm{La}^{3+}, \mathrm{Ce}^{4+}\)
(B) \(\mathrm{Yb}^{2+}, \mathrm{Lu}^{3+}\)
(C) \(\mathrm{La}^{2+}, \mathrm{Ce}^{3+}\)
(D) \(\mathrm{Yb}^{3+}, \mathrm{Lu}^{2+}\)
" "For the reaction sequence given below, the correct statement(s) is(are)
\includegraphics[max width=\textwidth, center]{2025_08_07_00c198eec9d2d28d09ffg-26}
(In the options, \(X\) is any atom other than carbon and hydrogen, and it is different in \(\mathbf{P}, \mathbf{Q}\) and \(\mathbf{R}\) ) (A) \(\mathrm{C}-\mathrm{X}\) bond length in \(\mathbf{P}, \mathbf{Q}\) and \(\mathbf{R}\) follows the order \(\mathbf{Q}>\mathbf{R}>\mathbf{P}\).
(B) C-X bond enthalpy in \(\mathbf{P}, \mathbf{Q}\) and \(\mathbf{R}\) follows the order \(\mathbf{R}>\mathbf{P}>\mathbf{Q}\).
(C) Relative reactivity toward \(\mathrm{S}_{\mathrm{N}} 2\) reaction in \(\mathbf{P}, \mathbf{Q}\) and \(\mathbf{R}\) follows the order \(\mathbf{P}>\mathbf{R}>\mathbf{Q}\).
(D) \(\mathrm{p} K_{\mathrm{a}}\) value of the conjugate acids of the leaving groups in \(\mathbf{P}, \mathbf{Q}\) and \(\mathbf{R}\) follows the order \(\mathbf{R}>\mathbf{Q}>\mathbf{P}\).
" "In an electrochemical cell, dichromate ions in aqueous acidic medium are reduced to \(\mathrm{Cr}^{3+}\). The current (in amperes) that flows through the cell for 48.25 minutes to produce 1 mole of \(\mathrm{Cr}^{3+}\) is \(\qquad\) . Use: 1 Faraday \(=96500 \mathrm{C} \mathrm{mol}^{-1}\)
" At \(25^{\circ} \mathrm{C}\), the concentration of \(\mathrm{H}^{+}\)ions in \(1.00 \times 10^{-3} \mathrm{M}\) aqueous solution of a weak monobasic acid having acid dissociation constant ( \(K_{\mathrm{a}}\) ) of \(4.00 \times 10^{-11}\) is \(\boldsymbol{X} \times 10^{-7} \mathrm{M}\). The value of \(\boldsymbol{X}\) is \(\qquad\) "10 Molar volume ( \(V_{\mathrm{m}}\) ) of a van der Waals gas can be calculated by expressing the van der Waals equation as a cubic equation with \(V_{\mathrm{m}}\) as the variable. The ratio (in \(\mathrm{mol} \mathrm{dm}^{-3}\) ) of the coefficient of \(V_{\mathrm{m}}^{2}\) to the coefficient of \(V_{\mathrm{m}}\) for a gas having van der Waals constants \(a=6.0 \mathrm{dm}^{6} \mathrm{~atm} \mathrm{~mol}{ }^{-2}\) and \(b=0.060\) \(\mathrm{dm}^{3} \mathrm{~mol}^{-1}\) at 300 K and 300 atm is \(\qquad\) . Use: Universal gas constant \((\mathrm{R})=0.082 \mathrm{dm}^{3} \mathrm{~atm} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\)
" "Considering ideal gas behavior, the expansion work done (in kJ ) when 144 g of water is electrolyzed completely under constant pressure at 300 K is \(\qquad\) . Use: Universal gas constant \((\mathrm{R})=8.3 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\); Atomic mass (in amu): \(\mathrm{H}=1, \mathrm{O}=16\)
" "The monomer ( \(\mathbf{X}\) ) involved in the synthesis of Nylon 6,6 gives positive carbylamine test. If 10 moles of \(\mathbf{X}\) are analyzed using Dumas method, the amount (in grams) of nitrogen gas evolved is
\(\qquad\) . Use: Atomic mass of N (in amu) = 14
" "The reaction sequence given below is carried out with 16 moles of \(\mathbf{X}\). The yield of the major product in each step is given below the product in parentheses. The amount (in grams) of \(\mathbf{S}\) produced is \(\qquad\) .
\includegraphics[max width=\textwidth, center]{2025_08_07_00c198eec9d2d28d09ffg-28} Use: Atomic mass (in amu): \(\mathrm{H}=1, \mathrm{C}=12, \mathrm{O}=16, \mathrm{Br}=80\)" "The correct match of the group reagents in List-I for precipitating the metal ion given in List-II from solutions, is \section*{List-I} (P) Passing \(\mathrm{H}_{2} \mathrm{~S}\) in the presence of \(\mathrm{NH}_{4} \mathrm{OH}\)
(Q) \(\left(\mathrm{NH}_{4}\right)_{2} \mathrm{CO}_{3}\) in the presence of \(\mathrm{NH}_{4} \mathrm{OH}\)
(R) \(\mathrm{NH}_{4} \mathrm{OH}\) in the presence of \(\mathrm{NH}_{4} \mathrm{Cl}\)
(S) Passing \(\mathrm{H}_{2} \mathrm{~S}\) in the presence of dilute HCl \section*{List-II} (1) \(\mathrm{Cu}^{2+}\)
(2) \(\mathrm{Al}^{3+}\)
(3) \(\mathrm{Mn}^{2+}\)
(4) \(\mathrm{Ba}^{2+}\)
(5) \(\mathrm{Mg}^{2+}\) (A) \(\mathrm{P} \rightarrow 3 ; \mathrm{Q} \rightarrow 4 ; \mathrm{R} \rightarrow 2 ; \mathrm{S} \rightarrow 1\)
(B) \(\mathrm{P} \rightarrow 4 ; \mathrm{Q} \rightarrow 2 ; \mathrm{R} \rightarrow 3 ; \mathrm{S} \rightarrow 1\)
(C) \(\mathrm{P} \rightarrow 3 ; \mathrm{Q} \rightarrow 4 ; \mathrm{R} \rightarrow 1 ; \mathrm{S} \rightarrow 5\)
(D) \(\mathrm{P} \rightarrow 5 ; \mathrm{Q} \rightarrow 3 ; \mathrm{R} \rightarrow 2 ; \mathrm{S} \rightarrow 4\)
" "The major products obtained from the reactions in List-II are the reactants for the named reactions mentioned in List-I. Match each entry in List-I with the appropriate entry in List-II and choose the correct option. \section*{List-I} (P) Stephen reaction
(Q) Sandmeyer reaction
(R) Hoffmann bromamide degradation reaction
(S) Cannizzaro reaction \section*{List-II} (1) Toluene \(\xrightarrow{\begin{array}{l}\text { (i) } \mathrm{CrO}_{2} \mathrm{Cl}_{2} / \mathrm{CS}_{2} \\ \text { (ii) } \mathrm{H}_{3} \mathrm{O}^{+}\end{array}}\)
(2)
(i) \(\mathrm{PCl}_{5}\)
(ii) \(\mathrm{NH}_{3}\)
(iii) \(\mathrm{P}_{4} \mathrm{O}_{10}, \Delta\) Benzoic acid
(3)
\includegraphics[max width=\textwidth, center]{2025_08_07_00c198eec9d2d28d09ffg-30}
(4)
(i) \(\mathrm{Cl}_{2} / \mathrm{h} v, \mathrm{H}_{2} \mathrm{O}\)
(ii) Tollen's reagent
(iii) \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) Toluene (iv) \(\mathrm{NH}_{3}\)
(5)
(i) \(\left(\mathrm{CH}_{3} \mathrm{CO}\right)_{2} \mathrm{O}\), Pyridine
(ii) \(\mathrm{HNO}_{3}, \mathrm{H}_{2} \mathrm{SO}_{4}, 288 \mathrm{~K}\) Aniline (iii) aq. NaOH (A) \(\mathrm{P} \rightarrow 2 ; \mathrm{Q} \rightarrow 4 ; \mathrm{R} \rightarrow 1 ; \mathrm{S} \rightarrow 3\)
(B) \(\mathrm{P} \rightarrow 2 ; \mathrm{Q} \rightarrow 3 ; \mathrm{R} \rightarrow 4 ; \mathrm{S} \rightarrow 1\)
(C) \(\mathrm{P} \rightarrow 5 ; \mathrm{Q} \rightarrow 3 ; \mathrm{R} \rightarrow 4 ; \mathrm{S} \rightarrow 2\)
(D) \(\mathrm{P} \rightarrow 5 ; \mathrm{Q} \rightarrow 4 ; \mathrm{R} \rightarrow 2 ; \mathrm{S} \rightarrow 1\)
" "Match the compounds in List-I with the appropriate observations in List-II and choose the correct option. \section*{List-I} (P)
\includegraphics{smile-085b9ea8f57e8518c0debb257e25a8779dc728dc}
(Q)
\includegraphics{smile-6a2404cab8e0f27c25cfc7234380cc149e7ec8aa}
(R)
\includegraphics{smile-f2145d0460624d6bf6277cc4af46e4d514df3366}
(S)
\includegraphics{smile-53197f5d307c3be7a2a259c2d687d6b88a0a5d79} \section*{List-II} (1) Reaction with phenyl diazonium salt gives yellow dye.
(2) Reaction with ninhydrin gives purple color and it also reacts with \(\mathrm{FeCl}_{3}\) to give violet color.
(3) Reaction with glucose will give corresponding hydrazone.
(4) Lassiagne extract of the compound treated with dilute HCl followed by addition of aqueous \(\mathrm{FeCl}_{3}\) gives blood red color.
(5) After complete hydrolysis, it will give ninhydrin test and it DOES NOT give positive phthalein dye test. (A) \(\mathrm{P} \rightarrow 1 ; \mathrm{Q} \rightarrow 5 ; \mathrm{R} \rightarrow 4 ; \mathrm{S} \rightarrow 2\)
(B) \(\mathrm{P} \rightarrow 2 ; \mathrm{Q} \rightarrow 5 ; \mathrm{R} \rightarrow 1 ; \mathrm{S} \rightarrow 3\)
(C) \(\mathrm{P} \rightarrow 5 ; \mathrm{Q} \rightarrow 2 ; \mathrm{R} \rightarrow 1 ; \mathrm{S} \rightarrow 4\)
(D) \(\mathrm{P} \rightarrow 2 ; \mathrm{Q} \rightarrow 1 ; \mathrm{R} \rightarrow 5 ; \mathrm{S} \rightarrow 3\)
"