JEE Advanced 2021 — Paper 2

One end of a horizontal uniform beam of weight W and length L is hinged on a vertical wall at point O and its other end is supported by a light inextensible rope. The other end of the rope is fixed at point Q , at a height L above the hinge at point O . A block of weight $\alpha \mathrm{W}$ is attached at the point P of the beam, as shown in the figure (not to scale). The rope can sustain a maximum tension of $(2 \sqrt{2}) \mathrm{W}$. Which of the following statement(s) is(are) correct?

A source, approaching with speed $u$ towards the open end of a stationary pipe of length $L$, is emitting a sound of frequency $f_{s}$.

The farther end of the pipe is closed. The speed of sound in air is $v$ and $f_{0}$ is the fundamental frequency of the pipe.

For which of the following combination(s) of $u$ and $f_{s}$, will the sound reaching the pipe lead to a resonance?

For a prism of prism angle $\theta=60^{\circ}$, the refractive indices of the left half and the right half are, respectively, $\mathrm{n}_{1}$ and $\mathrm{n}_{2}\left(\mathrm{n}_{2} \geq \mathrm{n}_{1}\right)$ as shown in the figure. The angle of incidence $i$ is chosen such that the incident light rays will have minimum deviation if $n_{1}=n_{2}=n=1.5$. For the case of unequal refractive indices, $n_{1}=n$ and $n_{2}=$ $\mathrm{n}+\Delta \mathrm{n}$ (where $\Delta \mathrm{n} \ll \mathrm{n}$ ), the angle of emergence $e=i+\Delta e$. Which of the following statement(s) is(are) correct?

A physical quantity $\vec{S}$ is defined as $\vec{S}=(\vec{E} \times \vec{B}) / \mu_{0}$, where $\vec{E}$ is electric field, $\vec{B}$ is magnetic field and

$\mu_{0}$ is the permeability of free space. The dimensions of $\vec{S}$ are the same as the dimensions of which of the following quantity(ies)?

A heavy nucleus $N$, at rest, undergoes fission $N \rightarrow P+Q$, where $P$ and Q are two lighter nuclei.

Let $\delta=\mathrm{M}_{\mathrm{N}}$ $-M_{P}-M_{Q}$, where $M_{P}, M_{Q}$ and $M_{N}$ are the masses of $P, Q$ and $N$, respectively.

$E_{P}$ and $E_{Q}$ are the kinetic energies of $P$ and $Q$, respectively. The speeds of $P$ and $Q$ are $v_{P}$ and $v_{Q}$, respectively.

If $c$ is the speed of light, which of the following statement(s) is(are) correct ?

Two concentric circular loops, one of radius R and the other of radius 2 R , lie in the xy-plane with the origin as their common centre, as shown in the figure.

The smaller loop carries current $\mathrm{I}_{1}$ in the anti-clockwise direction and the larger loop carries current $I_{2}$ in the clock wise direction, with

$I_{2}>2 I_{1}$. $\vec{B}(x, y)$ denotes the magnetic field at a point ( $\mathrm{x}, \mathrm{y}$ ) in the xy-plane.

Which of the following statement(s) is(are) correct?

In order to measure the internal resistance $r_{1}$ of a cell of emf $E$, a meter bridge of wire resistance $R_{0}=50 \Omega$, a resistance

$R_{0} / 2$, another cell of emf $E / 2$ (internal resistance $r$ ) and a galvanometer $G$ are used in a circuit, as shown in the figure.

If the null point is found at $\ell=72 \mathrm{~cm}$, then the value of $\mathrm{r}_{1}=$ $\qquad$ $\Omega$.

The distance between two stars of masses $3 \mathrm{M}_{\mathrm{S}}$ and $6 \mathrm{M}_{\mathrm{S}}$ is $9 R$. Here $R$ is the mean distance between the centers of the Earth and the Sun, and $\mathrm{M}_{\mathrm{S}}$ is the mass of the Sun. The two stars orbit around their common centre of mass in circular orbits with period nT , where T is the period of Earth's revolution around the Sun. The value of $n$ is $\qquad$

In a photoemission experiment, the maximum kinetic energies of photoelectrons from metals $\mathrm{P}, \mathrm{Q}$ and R are

$E_{P}, E_{Q}$ and $E_{R}$, respectively, and they are related by $E_{P}=2 E_{Q}=2 E_{R}$. In this experiment, the same source of monochromatic light is used for metal P and Q while a different source of monochromatic light is used for the metal R.The work functions for metals $\mathrm{P}, \mathrm{Q}$ and R are $4.0 \mathrm{eV}, 4.5 \mathrm{eV}$ and 5.5 eV ,

respectively. The energy of the incident photon used for metal $R$, in eV , is $\qquad$ _.

Correct option(s) for the following sequence of reactions is (are)

For the following reaction $2 \mathrm{X}+\mathrm{Y} \xrightarrow{\mathrm{k}} \mathrm{P}$ the rate of reaction is $\frac{\mathrm{d}[\mathrm{P}]}{\mathrm{dt}}=\mathrm{k}[\mathrm{X}]$. Two moles of $\mathbf{X}$ are mixed with one mole of $\mathbf{Y}$ to make 1.0 L of solution. At $50 \mathrm{~s}, 0.5$ mole of $\mathbf{Y}$ is left in the reaction mixture. The correct statement(s) about the reaction is(are) (Use: $\ln 2=0.693$ )

Some standard electrode potentials at 298 K are given below:

To a solution containing 0.001 M of $\mathbf{X}^{2+}$ and 0.1 M of $\mathbf{Y}^{2+}$, the metal rods $\mathbf{X}$ and $\mathbf{Y}$ are inserted (at 298 K ) and connected by a conducting wire. This resulted in dissolution of $\mathbf{X}$. The correct combination(s) of $\mathbf{X}$ and $\mathbf{Y}$, respectively, is(are) (Given: Gas constant, $\mathrm{R}=8.314 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$, Faraday constant, $\mathrm{F}=96500 \mathrm{C} \mathrm{mol}^{-1}$ )

The pair(s) of complexes wherein both exhibit tetrahedral geometry is(are) (Note: py = pyridine Given: Atomic numbers of $\mathrm{Fe}, \mathrm{Co}, \mathrm{Ni}$ and Cu are 26, 27, 28 and 29, respectively)

The correct statement(s) related to oxoacids of phosphorous is(are)

One mole of an ideal gas at 900 K , undergoes two reversible processes, I followed by II, as shown below. If the work done by the gas in the two process are same, the value of $\ln \frac{\mathrm{V}_{3}}{\mathrm{~V}_{2}}$

Consider a helium (He) atom that absorbs a photon of wavelength 330 nm . The change in the velocity (in $\mathrm{cm} \mathrm{s}^{-1}$ ) of He atom after the photon absorption is $\qquad$ . (Assume: Momentum is conserved when photon is absorbed. Use: Planck constant $=6.6 \times 10^{-34} \mathrm{~J} \mathrm{~s}$, Avogadro number $=6 \times 10^{23} \mathrm{~mol}^{-1}$, Molar mass of $\mathrm{He}=4 \mathrm{~g} \mathrm{~mol}^{-1}$ )

Ozonolysis of $\mathrm{ClO}_{2}$ produces an oxide of chlorine. The average oxidation state of chlorine in this oxide is

Let

and

If the total number of elements in the set $S_{r}$ is $n_{r}$, $r = 1,2,3,4$, then which of the following statements is (are) \textbf{TRUE}?

Let $f:\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \rightarrow R$ be a continuous function such that $f(0)=1$ and $\int_{0}^{\frac{\pi}{3}} f(t) d t=0$. Then which of the following statements is(are) TRUE?

For any real numbers $\alpha$ and $\beta$, let $y_{\alpha, \beta}(x), x \in R$, be the solution of the differential equation

Let $S=\left\{y_{\alpha, \beta}(x): \alpha, \beta \in R\right\}$. Then which of the following functions belong(s) the set $S$ ?

Let O be the origin and $\overrightarrow{\mathrm{OA}}=2 \hat{i}+2 \hat{j}+\hat{k}, \overrightarrow{\mathrm{OB}}=\hat{\mathrm{i}}-2 \hat{j}+2 \hat{k}$ and $\overrightarrow{\mathrm{OC}}=\frac{1}{2}(\overrightarrow{\mathrm{OB}}-\lambda \overrightarrow{\mathrm{OA}})$ for some $\lambda>0$. If $|\overrightarrow{\mathrm{OB}} \times \overrightarrow{\mathrm{OC}}|=\frac{9}{2}$, then which of the following statements is(are) TRUE?

Let E denote the parabola $\mathrm{y}^{2}=8 \mathrm{x}$. Let $\mathrm{P}=(-2,4)$ and let Q and $\mathrm{Q}^{\prime}$ be two distinct points on E such that the lines PQ and $\mathrm{PQ}^{\prime}$ are tangents to E . Let F be the focus of E . Then which of the following statements is(are) TRUE?

A number is chosen at random from the set $\{1,2,3, \ldots . ., 2000\}$. Let $p$ be the probability that the chosen number is a multiple of 3 or a multiple of 7 . Then the value of 500 p is $\qquad$

Let $E$ be the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$. For any three distinct points $P, Q$ and $Q^{\prime}$ on $E$, let $M(P, Q)$ be the midpoint of the line segment joining $P$ and $Q$, and $M\left(P, Q^{\prime}\right)$ be the mid-point of the line segment joining $P$ and $\mathrm{Q}^{\prime}$. Then the maximum possible value of the distance between $\mathrm{M}(\mathrm{P}, \mathrm{Q})$ and $\mathrm{M}\left(\mathrm{P}, \mathrm{Q}^{\prime}\right)$, as $\mathrm{P}, \mathrm{Q}$ and $\mathrm{Q}^{\prime}$ vary on E , is $\qquad$

For any real number x , let $[\mathrm{x}$ ] denote the largest integer less than or equal to x . If

then the value of 9I is $\qquad$

A Soft plastic bottle, filled with water of density $1 \mathrm{gm} / \mathrm{cc}$, carries an inverted glass test-tube with some air (ideal gas)

trapped as shown in the figure. The test-tube has a mass of 5 gm , and it is made of a thick glass of density $2.5 \mathrm{gm} / \mathrm{cc}$.

Initially the bottle is sealed at atmospheric pressure $p_{0}=10^{5} \mathrm{~Pa}$ so that the volume of the trapped air is

$\mathrm{v}_{0}=3.3 \mathrm{cc}$. When the bottle is squeezed from outside at constant temperature, the pressure inside rises and the volume of the trapped air reduces. It is found that the test tube begins to sink at pressure $p_{0}+\Delta p$ without changing its orientation. At this pressure,

the volume of the trapped air is $\mathrm{v}_{0}-\Delta \mathrm{v}$. Let $\Delta \mathrm{v}=\mathrm{X}$ cc and

$\Delta \mathrm{p}=\mathrm{Y} \times 10^{3} \mathrm{~Pa}$.

The value of X is $\text{ }\!\!~\!\!\text{ }$ .

A Soft plastic bottle, filled with water of density $1 \mathrm{gm} / \mathrm{cc}$, carries an inverted glass test-tube with some air (ideal gas)

trapped as shown in the figure. The test-tube has a mass of 5 gm , and it is made of a thick glass of density $2.5 \mathrm{gm} / \mathrm{cc}$.

Initially the bottle is sealed at atmospheric pressure $p_{0}=10^{5} \mathrm{~Pa}$ so that the volume of the trapped air is

$\mathrm{v}_{0}=3.3 \mathrm{cc}$. When the bottle is squeezed from outside at constant temperature, the pressure inside rises and the volume of the trapped air reduces. It is found that the test tube begins to sink at pressure $p_{0}+\Delta p$ without changing its orientation. At this pressure,

the volume of the trapped air is $\mathrm{v}_{0}-\Delta \mathrm{v}$. Let $\Delta \mathrm{v}=\mathrm{X}$ cc and

$\Delta \mathrm{p}=\mathrm{Y} \times 10^{3} \mathrm{~Pa}$.

The value of Y is $\text{ }\!\!~\!\!\text{ }$ .

A pendulum consists of a bob of mass $m=0.1 \mathrm{~kg}$ and a massless inextensible string of length $\mathrm{L}=1.0 \mathrm{~m}$.

It is suspended from a fixed point at height $\mathrm{H}=0.9 \mathrm{~m}$ above a frictionless horizontal floor. Initially,

the bob of the pendulum is lying on the floor at rest vertically below the point of suspension. A horizontal impulse $\mathrm{P}=0.2 \mathrm{~kg}-\mathrm{m} / \mathrm{s}$ is imparted to the bob at some instant. After the bob slides for some distance, the string becomes taut and the bob lifts off the floor.

The magnitude of the angular momentum of the pendulum about the point of suspension just before the bob lifts off is

$\mathrm{J} \mathrm{kg}-\mathrm{m}^{2} / \mathrm{s}$. The kinetic energy of the pendulum just after the lift-off is K Joules.

The value of J is $\text{ }\!\!~\!\!\text{ }$ .

In the circuit, a metal filament lamp is connected in series with a capacitor of capacitance $\mathrm{C} \mu \mathrm{F}$ across a $200 \mathrm{~V}, 50$ Hz supply. The power consumed by the lamp is 500 W while the voltage drop across it is 100 V . Assume that there is no inductive load in the circuit. Take rms values of the voltages. The magnitude of the phase-angle (in degrees) between the current and supply voltage is $\phi$. Assume, $\pi \sqrt{3} \approx 5$.

The value of C is $\text{ }\!\!~\!\!\text{ }$ .

In the circuit, a metal filament lamp is connected in series with a capacitor of capacitance $\mathrm{C} \mu \mathrm{F}$ across a $200 \mathrm{~V}, 50$ Hz supply. The power consumed by the lamp is 500 W while the voltage drop across it is 100 V . Assume that there is no inductive load in the circuit. Take rms values of the voltages. The magnitude of the phase-angle (in degrees) between the current and supply voltage is $\phi$. Assume, $\pi \sqrt{3} \approx 5$.

The value of $\phi$ is $\text{ }\!\!~\!\!\text{ }$ .

A special metal S conducts electricity without any resistance. A closed wire loop, made of S, does not allow any change in flux through itself by inducing a suitable current to generate a compensating flux. The induced current in the loop cannot decay due to its zero resistance. This current gives rise to a magnetic moment which in turn repels the source of magnetic field or flux. Consider such a loop, of radius a, with its centre at the origin. A magnetic dipole of moment $m$ is brought along the axis of this loop from infinity to a point at distance $r(\gg$ a) from the centre of the loop with its north pole always facing the loop, as shown in the figure below.

The magnitude of magnetic field of a dipole $m$, at a point on its axis at distance $r$, is $\frac{\mu_{0}}{2 \pi} \frac{m}{r^{3}}$,

where $\mu_{0}$ is the permeability of free space. The magnitude of the force between two magnetic dipoles with moments, $m_{1}$ and $m_{2}$, separated by a distance $r$ on the common axis, with their north poles facing each other, is $\frac{k m_{1} m_{2}}{r^{4}}$, where $k$ is a constant of appropriate dimensions. The direction of this force is along the line joining the two dipoles.

When the dipole

is placed at a distance

from the center of the loop (as shown in the figure), the current induced in the loop will be proportional to

A special metal S conducts electricity without any resistance. A closed wire loop, made of S, does not allow any change in flux through itself by inducing a suitable current to generate a compensating flux. The induced current in the loop cannot decay due to its zero resistance. This current gives rise to a magnetic moment which in turn repels the source of magnetic field or flux. Consider such a loop, of radius a, with its centre at the origin. A magnetic dipole of moment $m$ is brought along the axis of this loop from infinity to a point at distance $r(\gg$ a) from the centre of the loop with its north pole always facing the loop, as shown in the figure below.

The magnitude of magnetic field of a dipole $m$, at a point on its axis at distance $r$, is $\frac{\mu_{0}}{2 \pi} \frac{m}{r^{3}}$,

where $\mu_{0}$ is the permeability of free space. The magnitude of the force between two magnetic dipoles with moments, $m_{1}$ and $m_{2}$, separated by a distance $r$ on the common axis, with their north poles facing each other, is $\frac{k m_{1} m_{2}}{r^{4}}$, where $k$ is a constant of appropriate dimensions. The direction of this force is along the line joining the two dipoles.

The work done in bringing the dipole from infinity to a distance $r$ from the center of the loop by the given process is proportional to

A thermally insulating cylinder has a thermally insulating and frictionless movable partition in the middle, as shown in the figure below. On each side of the partition, there is one mole of an ideal gas, with specific heat at constant volume, $\mathrm{C}_{\mathrm{V}}=2 \mathrm{R}$. Here, R is the gas constant. Initially, each side has a volume $\mathrm{V}_{0}$ and temperature $T_{0}$. The left side has an electric heater, which is turned on at very low power to transfer heat Q to the gas on the left side. As a result the partition moves slowly towards the right reducing the right side volume to $V_{0} / 2$. Consequently, the gas temperatures on the left and the right sides become $T_{L}$ and $T_{R}$, respectively. Ignore the changes in the temperatures of the cylinder, heater and the partition.

The value of $\frac{\text{Q}}{\text{R}{{\text{T}}_{0}}}$ is

At 298 K, the limiting molar conductivity of a weak monobasic acid is

At 298 K, for an aqueous solution of the acid the degree of dissociation is $\alpha$ and the molar conductivity is

At 298 K, upon 20 times dilution with water, the molar conductivity of the solution becomes

The value of $\boldsymbol{\alpha}$ is $\qquad$ .

At 298 K, the limiting molar conductivity of a weak monobasic acid is

At 298 K, for an aqueous solution of the acid the degree of dissociation is $\alpha$ and the molar conductivity is

At 298 K, upon 20 times dilution with water, the molar conductivity of the solution becomes

The value of $\mathbf{y}$ is $\qquad$ .

Reaction of $x\,\mathrm{g}$ of $\mathrm{Sn}$ with $\mathrm{HCl}$ quantitatively produced a salt. Entire amount of the salt reacted with $y\,\mathrm{g}$ of nitrobenzene in the presence of required amount of $\mathrm{HCl}$ to produce $1.29\,\mathrm{g}$ of an organic salt (quantitatively).

(Use molar masses in $\mathrm{g\,mol^{-1}}$ of $\mathrm{H,C,N,O,Cl,Sn}$ as $1,12,14,16,35,119$ respectively.)

The value of $x$ is $\_\_\_$ and the value of $y$ is $\_\_\_$.

The value of $\mathbf{x}$ is $\_\_\_\_\_\_$.

Reaction of $x\,\mathrm{g}$ of $\mathrm{Sn}$ with $\mathrm{HCl}$ quantitatively produced a salt. Entire amount of the salt reacted with $y\,\mathrm{g}$ of nitrobenzene in the presence of required amount of $\mathrm{HCl}$ to produce $1.29\,\mathrm{g}$ of an organic salt (quantitatively).

(Use molar masses in $\mathrm{g\,mol^{-1}}$ of $\mathrm{H,C,N,O,Cl,Sn}$ as $1,12,14,16,35,119$ respectively.)

The value of $x$ is $\_\_\_$ and the value of $y$ is $\_\_\_$.

The value of $\mathbf{y}$ is $\_\_\_\_\_$.

A sample ( 5.6 g ) containing iron is completely dissolved in cold dilute HCl to prepare a 250 mL of solution. Titration of 25.0 mL of this solution requires 12.5 mL of $0.03 \mathrm{M} \mathrm{KMnO}_{4}$ solution to reach the end point. Number of moles of $\mathrm{Fe}^{2+}$ present in 250 mL solution is $\mathbf{x} \times 10^{-2}$ (consider complete dissolution of $\mathrm{FeCl}_{2}$ ). The amount of iron present in the sample is $\mathbf{y} \%$ by weight. (Assume: $\mathrm{KMnO}_{4}$ reacts only with $\mathrm{Fe}^{2+}$ in the solution Use: Molar mass of iron as $56 \mathrm{~g} \mathrm{~mol}^{-1}$ )

The value of $\mathbf{x}$ is $\qquad$ .

A sample ( 5.6 g ) containing iron is completely dissolved in cold dilute HCl to prepare a 250 mL of solution. Titration of 25.0 mL of this solution requires 12.5 mL of $0.03 \mathrm{M} \mathrm{KMnO}_{4}$ solution to reach the end point. Number of moles of $\mathrm{Fe}^{2+}$ present in 250 mL solution is $\mathbf{x} \times 10^{-2}$ (consider complete dissolution of $\mathrm{FeCl}_{2}$ ). The amount of iron present in the sample is $\mathbf{y} \%$ by weight. (Assume: $\mathrm{KMnO}_{4}$ reacts only with $\mathrm{Fe}^{2+}$ in the solution Use: Molar mass of iron as $56 \mathrm{~g} \mathrm{~mol}^{-1}$ )

The value of $\mathbf{y}$ is $\qquad$ .

The amount of energy required to break a bond is same as the amount of energy released when the same bond is formed. In gaseous state, the energy required for homolytic cleavage of a bond is called Bond Dissociation Energy (BDE) or Bond Strength. BDE is affected by s-character of the bond and the stability of the radicals formed. Shorter bonds are typically stronger bonds. BDEs for some bonds are given below:

For the following reaction $\mathrm{CH}_{4}(\mathrm{~g})+\mathrm{Cl}_{2}(\mathrm{~g}) \xrightarrow{\text { light }} \mathrm{CH}_{3} \mathrm{Cl}(\mathrm{g})+\mathrm{HCl}(\mathrm{g})$ The correct statement is

The reaction of $\mathrm{K}_{3}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]$ with freshly prepared $\mathrm{FeSO}_{4}$ solution produces a dark blue precipitate called Turnbull's blue. Reaction of $\mathrm{K}_{4}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]$ with the $\mathrm{FeSO}_{4}$ solution in complete absence of air produces a white precipitate $\mathbf{X}$, which turns blue in air. Mixing the $\mathrm{FeSO}_{4}$ solution with $\mathrm{NaNO}_{3}$, followed by a slow addition of concentrated $\mathrm{H}_{2} \mathrm{SO}_{4}$ through the side of the test tube produces a brown ring.

Precipitate $\mathbf{X}$ is

The reaction of $\mathrm{K}_{3}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]$ with freshly prepared $\mathrm{FeSO}_{4}$ solution produces a dark blue precipitate called Turnbull's blue. Reaction of $\mathrm{K}_{4}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]$ with the $\mathrm{FeSO}_{4}$ solution in complete absence of air produces a white precipitate $\mathbf{X}$, which turns blue in air. Mixing the $\mathrm{FeSO}_{4}$ solution with $\mathrm{NaNO}_{3}$, followed by a slow addition of concentrated $\mathrm{H}_{2} \mathrm{SO}_{4}$ through the side of the test tube produces a brown ring.

Among the following, the brown ring is due to the formation of

The radius of the circle C is $\qquad$

The value of $\alpha$ is $\qquad$

Let $f_{1}:(0, \infty) \rightarrow R$ and $f_{2}:(0, \infty) \rightarrow R$ be defined by $f_{1}(x)=\int_{0}^{x} \prod_{j=1}^{21}(t-j)^{j} d t, x>0$ and $f_{2}(x)=98(x-1)^{50}-600(x-1)^{49}+2450, x>0$, where, for any positive integer $n$ and real numbers $a_{1}, a_{2}$, $\ldots \ldots, a_{n}, \prod_{i=1}^{n} a_{i}$ denotes the product of $a_{1}, a_{2}, \ldots . ., a_{n}$. Let $m_{i}$ and $n_{i}$, respectively, denote the number of points of local minima and the number of points of local maxima of function $f_{i}, i=1,2$, in the interval ( 0 , $\infty)$.

The value of $2 m_{1}+3 n_{1}+m_{1} n_{1}$ is $\qquad$

Let $f_{1}:(0, \infty) \rightarrow R$ and $f_{2}:(0, \infty) \rightarrow R$ be defined by $f_{1}(x)=\int_{0}^{x} \prod_{j=1}^{21}(t-j)^{j} d t, x>0$ and $f_{2}(x)=98(x-1)^{50}-600(x-1)^{49}+2450, x>0$, where, for any positive integer $n$ and real numbers $a_{1}, a_{2}$, $\ldots \ldots, a_{n}, \prod_{i=1}^{n} a_{i}$ denotes the product of $a_{1}, a_{2}, \ldots . ., a_{n}$. Let $m_{i}$ and $n_{i}$, respectively, denote the number of points of local minima and the number of points of local maxima of function $f_{i}, i=1,2$, in the interval ( 0 , $\infty)$.

The value of $6 m_{2}+4 n_{2}+8 m_{2} n_{2}$ is $\qquad$

Let $g_{i}:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow R, i=1,2$ and $f:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow R$ be functions such that $g_{1}(x)=1, g_{2}(x)=|4 x-\pi|$ and $f(x)=\sin ^{2} x$, for all $x \in\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right]$. Define $S_{i}=\int_{\frac{\pi}{8}}^{\frac{3 \pi}{8}} f(x) \cdot g_{i}(x) d x, i=1,2$

The value of $\frac{16 S_{1}}{\pi}$ is $\qquad$

Let $g_{i}:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow R, i=1,2$ and $f:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow R$ be functions such that $g_{1}(x)=1, g_{2}(x)=|4 x-\pi|$ and $f(x)=\sin ^{2} x$, for all $x \in\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right]$. Define $S_{i}=\int_{\frac{\pi}{8}}^{\frac{3 \pi}{8}} f(x) \cdot g_{i}(x) d x, i=1,2$

The value of $\frac{48 \mathrm{~S}_{2}}{\pi^{2}}$ is $\qquad$

Which of the following statements is TRUE?