JEE Advanced 2026 Paper-2

A metal wire of cross-sectional area $$0.5\,\mathrm{mm^2}$$ and length $$100\,\mathrm{m}$$ is connected across a battery of e.m.f. $$2\,\mathrm{V}$$ and internal resistance $$1\,\Omega$$. The density, atomic mass and electrical conductivity of the metal are $$6.35\times 10^{3}\,\mathrm{kg\,m^{-3}}$$, $$63.5\,\mathrm{gm/mole}$$ and $$2\times 10^{8}\,\mathrm{mho\,m^{-1}}$$, respectively. Assuming one conduction electron per atom of the metal, the drift velocity (in $$\mathrm{mm\,s^{-1}}$$) of the electrons in the wire is:

[Take Avogadro's number as $$6\times 10^{23}$$ and charge of the electron as $$1.6\times 10^{-19}\,\mathrm{C}$$.]

A nuclear reactor starts producing a radioactive nuclide $$X$$ from $$t=0$$, at a constant rate of $$\alpha$$ per second. Each decay of $$X$$ produces energy $$E_0$$, which is utilized to heat a liquid of mass $$m$$ and specific heat $$s$$. Assuming no heat loss from the liquid and taking $$\lambda$$ as the decay constant of $$X$$, the rate of increase in the temperature of the liquid is:

A beam of polychromatic light passes through a thin prism of prism angle $$6^\circ$$. The refractive index of the material of the prism varies with wavelength $$(\lambda)$$ as $$n(\lambda)=\alpha\lambda+\dfrac{\beta}{\lambda^2}$$, where $$\alpha=3\,\mu\mathrm{m^{-1}}$$ and $$\beta=0.096\,\mu\mathrm{m^2}$$. If $$\lambda_{\min}$$ is the wavelength at which the angle of minimum deviation $$D_m$$ is smallest, then the correct value of $$D_m$$ at $$\lambda_{\min}$$ is

A particle of mass $$m$$, and angular momentum $$\ell$$ is moving in a circular orbit of radius $$r_0$$ under the influence of an attractive force $$\vec{F}(r)=-\dfrac{k}{r^2}\hat{r}$$. Keeping its angular momentum unchanged, the particle is displaced radially by a small distance $$\delta r\ll r_0$$, due to which its radial distance varies periodically. The corresponding time period is:

Consider two isosceles prisms 1 and 2 with prism angles $$A_1$$ and $$A_2$$ and refractive indices $$n_1$$ and $$n_2$$, respectively, as shown in the figure. The faces $$a_1b_1$$ and $$a_2b_2$$ are parallel to each other and perpendicular to the mirror $$M$$. If a ray of light is incident on the face $$a_1c_1$$ and emerges from the face $$a_2c_2$$, then the correct statement(s) is/are:

In a vacuum chamber, a particle of charge $$1\,\mu\mathrm{C}$$ and mass $$1\,\mathrm{mg}$$ is projected with a velocity $$(\hat{i}+2\hat{j})\,\mathrm{ms^{-1}}$$ from the $$XZ$$ plane at time $$t=0$$ in an electric field of $$1\hat{i}\,\mathrm{Vm^{-1}}$$. At $$t=0.2\,\mathrm{s}$$, the electric field is switched off and a magnetic field of $$6\hat{j}\,\mathrm{T}$$ is switched on. The acceleration due to gravity is $$-10\hat{j}\,\mathrm{ms^{-2}}$$. Correct option(s) is/are:

Two charges $$Q_1=q$$ and $$Q_2=mq$$ are placed at the points $$P_1(a,b)$$ and $$P_2(ma,mb)$$, respectively, in the $$XY$$ plane, where $$a,b\neq 0$$ and $$m\neq 0,1$$. If $$V_1$$ is the potential at a point in the $$XY$$ plane due to charge $$Q_1$$ and $$V_2$$ is the potential at that point due to charge $$Q_2$$. Correct statement(s) for the points at which $$|V_1|=|V_2|$$ is/are:

Consider an electric dipole comprising two charges $$+q$$ and $$-q$$ each with mass $$m$$, separated by a fixed distance $$d$$ and initially at rest with its dipole moment pointing along $$\hat{i}$$. A uniform electric field $$E\hat{j}$$ is turned on at time $$t=0$$ and it is turned off at $$t=t_f$$, when the dipole moment makes an angle $$\theta_f$$ with $$\hat{i}$$. Neglecting any sources of energy loss, correct option(s) is/are:

Ten moles of an ideal monatomic gas, initially in state $$\boldsymbol{a}$$ at atmospheric pressure and temperature $$T_a=27^\circ\mathrm{C}$$, is enclosed in a metal cylinder of volume $$V_0$$ fitted with a frictionless piston. The gas is suddenly compressed to state $$\boldsymbol{b}$$ with volume $$V_0/3$$. Now, keeping the piston stationary, the cylinder is submerged in a water bath of temperature $$11^\circ\mathrm{C}$$ until the gas reaches the temperature of the water bath, which is denoted as state $$\boldsymbol{c}$$. Finally, while still in the water bath, the piston is brought slowly to its initial position, which is denoted as state $$\boldsymbol{f}$$. If $$R$$ is universal gas constant, then the correct option(s) is/are:

[Given: $$9^{1/3}=2.08$$]

Two thin wires, Wire-1 of diameter $$0.650\,\mathrm{mm}$$ and Wire-2 of unknown diameter $$d$$ are given. To obtain the value of $$d$$, the diameters of the two wires are measured with a screw gauge. The screw gauge has a pitch of $$0.5\,\mathrm{mm}$$ and there are $$100$$ divisions on the circular scale (CS). The smallest division on the linear scale (LS) is $$0.5\,\mathrm{mm}$$. The table shows the readings of LS and CS for the measurements. The value of $$d$$ (in $$\mu\mathrm{m}$$) is:

In a single slit diffraction experiment, a slit of width $$(0.016\pm 0.002)\,\mathrm{mm}$$ is used to measure the wavelength of a monochromatic light source. In the diffraction pattern, the angular distance between the central maximum and first minimum is measured to be $$(2^\circ\pm 40')$$. The value of the fractional error in the measurement of wavelength is:

[Given: $$\sin(2^\circ)=0.035$$]

As shown in the figure, a ray $$AB$$ of unpolarized light enters from water of refractive index $$n_w=4/3$$ into a medium of refractive index $$n_p=4/\sqrt{3}$$ after passing through a glass plate of refractive index $$n_g=1.5$$ and a layer of water. At a particular incident angle $$i$$ the reflected ray $$CD$$ is polarized in the direction as shown in the figure. The value of $$i$$ (in degrees) is:

As shown in the figure, the resistance of a galvanometer $$G$$ can be found by the half-deflection method. Here the resistance $$R_2$$ is adjusted such that when the key $$K$$ is closed the deflection in the galvanometer becomes half of the value as compared to when $$K$$ is open. Half-deflection is obtained at $$R_2=4\,\Omega$$ and thus the galvanometer resistance is found to be $$6\,\Omega$$. In this half-deflection condition the current (in mA) through the resistor $$R_1$$ is:

In a new system of units, the units of mass, length, time and current are $$5\,\mathrm{kg}$$, $$5\,\mathrm{m}$$, $$5\,\mathrm{s}$$ and $$5\,\mathrm{A}$$, respectively. If $$\mu_0$$ and $$\epsilon_0$$ are the permeability and permittivity of free space, respectively, then in this new system of units, the magnitude of one SI unit of $$\sqrt{\mu_0/\epsilon_0}$$, is:

Question Stem for Question Nos. 15 and 16

A container of height $$2\,\mathrm{m}$$, length $$2\,\mathrm{m}$$ and breadth $$1\,\mathrm{m}$$ is made of insulating vertical walls and two large area horizontal metal plates ($$M_1$$ and $$M_2$$) which extend far beyond the vertical walls in all directions. The container is partitioned into two equal chambers with a thin insulating vertical wall. The partition wall contains a small hole of cross-sectional area $$\sqrt{10}\,\mathrm{cm^2}$$ near its bottom edge. Initially the hole is closed and the left chamber of the container is completely filled with a liquid of dielectric constant $$\epsilon_r=15$$ and the right chamber is empty ($$\epsilon_r=1$$). At time $$t=0$$, the hole is opened and the liquid flows from the left chamber to the right chamber. In both the chambers, the space above the liquid has $$\epsilon_r=1$$ and is maintained at atmospheric pressure. The schematic of the container at a time $$t>0$$ is shown in the figure.

[Given: acceleration due to gravity is $$10\,\mathrm{ms^{-2}}$$.]

The height (in m) of the liquid in left chamber at $$t=500\,\mathrm{s}$$ is:

Question Stem for Question Nos. 15 and 16

A container of height $$2\,\mathrm{m}$$, length $$2\,\mathrm{m}$$ and breadth $$1\,\mathrm{m}$$ is made of insulating vertical walls and two large area horizontal metal plates ($$M_1$$ and $$M_2$$) which extend far beyond the vertical walls in all directions. The container is partitioned into two equal chambers with a thin insulating vertical wall. The partition wall contains a small hole of cross-sectional area $$\sqrt{10}\,\mathrm{cm^2}$$ near its bottom edge. Initially the hole is closed and the left chamber of the container is completely filled with a liquid of dielectric constant $$\epsilon_r=15$$ and the right chamber is empty ($$\epsilon_r=1$$). At time $$t=0$$, the hole is opened and the liquid flows from the left chamber to the right chamber. In both the chambers, the space above the liquid has $$\epsilon_r=1$$ and is maintained at atmospheric pressure. The schematic of the container at a time $$t>0$$ is shown in the figure.

[Given: acceleration due to gravity is $$10\,\mathrm{ms^{-2}}$$.]

The difference in the capacitance (in F) between the metal plates at $$t=0$$ and that at $$t=500\,\mathrm{s}$$ is $$(8-n)\epsilon_0$$, where $$\epsilon_0$$ is the permittivity of free space. The value of $$n$$ is:

Question Stem for Question Nos. 17 and 18

A uniform circular disk of radius $$0.2\,\mathrm{m}$$ and mass $$1\,\mathrm{kg}$$ is pivoted at its top point $$C$$ such that it can rotate freely around $$C$$ in the $$XY$$ plane, as shown in the figure. Initially, when the disk is at rest, a particle of mass $$20\,\mathrm{g}$$, travelling along negative $$x$$ direction in the $$XY$$ plane with speed $$100\,\mathrm{ms^{-1}}$$, hits the circumference of the disk at a point $$P$$. After collision the particle moves along negative $$y$$ direction at a speed of $$90\,\mathrm{ms^{-1}}$$.

[Given: the acceleration due to gravity $$(g)=-10\hat{j}\,\mathrm{ms^{-2}}$$]

After the collision the disk starts to rotate around point $$C$$ in the $$XY$$ plane. The maximum change in the height (in m) of its center $$O$$ is:

Question Stem for Question Nos. 17 and 18

A uniform circular disk of radius $$0.2\,\mathrm{m}$$ and mass $$1\,\mathrm{kg}$$ is pivoted at its top point $$C$$ such that it can rotate freely around $$C$$ in the $$XY$$ plane, as shown in the figure. Initially, when the disk is at rest, a particle of mass $$20\,\mathrm{g}$$, travelling along negative $$x$$ direction in the $$XY$$ plane with speed $$100\,\mathrm{ms^{-1}}$$, hits the circumference of the disk at a point $$P$$. After collision the particle moves along negative $$y$$ direction at a speed of $$90\,\mathrm{ms^{-1}}$$.

[Given: the acceleration due to gravity $$(g)=-10\hat{j}\,\mathrm{ms^{-2}}$$]

Amount of energy loss (in J) in the collision is:

At 300 K, the molar conductivities of the aqueous solutions of three salts at two different concentrations are given below:

The conductivity of a saturated aqueous solution of AgCl is $$1.40\times 10^{-6}$$ S cm$$^{-1}$$ at 300 K. If the solubility of AgCl in water at 300 K is $$X$$ mol L$$^{-1}$$, then $$\log_{10}(X^{-1})$$ is

(Assume that AgCl dissolved in water ionizes completely and that the molar conductivity of saturated AgCl solution is equal to its limiting molar conductivity.)

The correct order of ONO bond angle in the given species is

Natural rubber on complete ozonolysis ($$\mathrm{O_3}$$/Zn-$$\mathrm{H_2O}$$) gives compound X as the major product. X gives positive iodoform and Tollen's tests. X on heating with aqueous NaOH gives Y as the major product. Y is

A known artificial sweetener X is composed of 4-chloro-4-deoxy-$$\alpha$$-D-galactose and 1,6-dichloro-1,6-dideoxy-$$\beta$$-D-fructose joined by a glycosidic linkage. Structure of D-galactose is given below (Fischer projection):

The correct structure of X is

For a first-order reaction $$\mathrm{R}\rightarrow\mathrm{P}$$ at a given temperature, $$k$$ is the rate constant. For this reaction, at the given temperature, the concentrations of $$\mathrm{R}$$ and $$\mathrm{P}$$ at a time $$t$$ are $$[\mathrm{R}]$$ and $$[\mathrm{P}]$$, respectively. The correct graphical representation(s) for this reaction is(are)

Correct statement(s) about the compounds P, Q and R is(are)

The correct statement(s) regarding the periodic properties of elements is(are)

In the following reaction sequence, P, Q, S and T are the major products.

The correct statement(s) about P, Q, S and T is(are)

The correct statement(s) regarding sugars is(are)

Given: Specific rotations of L-($$-$$)-glucose and L-(+)-fructose are $$-52.5^\circ$$ and $$+92.5^\circ$$, respectively.

$$\mathbf{X}^{a+}$$ and $$\mathbf{Y}^{b+}$$ are hydrogen-like species. The wavelength of light absorbed during the transition between the states with principal quantum numbers $$n=1$$ and $$n=2$$ of $$\mathbf{X}^{a+}$$ is $$\lambda$$. The wavelength of light absorbed during the transition between the states with principal quantum numbers $$n=2$$ and $$n=4$$ of $$\mathbf{Y}^{b+}$$ is $$9\lambda$$. The lowest possible value of $$(a+b)$$ is ___.

At a given temperature, 0.45 g of acetic acid in 50 mL of water is shaken with 1.0 g of charcoal and the pH of the resulting solution is 3.0. Assume, the adsorption of acetic acid from the aqueous solution by charcoal follows Freundlich isotherm,

$$\dfrac{x}{m}=kC^{1/n}$$

If the plot of $$\log_{10}(x/m)$$ against $$\log_{10}C$$ gives a straight line with slope 1, the value of $$k$$ in L mol$$^{-1}$$ is ___.

Given: The molar mass of acetic acid is 60 g mol$$^{-1}$$.
The acid dissociation constant of acetic acid is $$1.0\times 10^{-5}$$ at the given temperature.
$$x$$ is the mass (in grams) of acetic acid adsorbed. $$m$$ is the mass (in grams) of charcoal.
$$C$$ is the equilibrium concentration of acetic acid in the solution after the adsorption is complete.
$$k$$ and $$n$$ are constants for acetic acid$$-$$charcoal system at the given temperature.

In a solvent S, a compound B is partially dissociated into C and D as given below:

$$\mathbf{B}\rightleftharpoons 2\mathbf{C}+2\mathbf{D}$$

B, C and D are non-volatile in nature. The molar mass of B is 10 times the molar mass of S. The standard boiling point and the standard enthalpy of vaporization of S are 400 K and $$10R$$ J mol$$^{-1}$$, respectively ($$R$$ is the gas constant in J K$$^{-1}$$ mol$$^{-1}$$). A solution of B in S with an initial concentration of B as 0.25% (mass/mass) has a boiling point of 408 K at 1 bar pressure. In this solution, the mole percent of B that has been dissociated is ___.

Consider that the coordinating atoms of the ligands in cis-[Co(NH$$_3$$)$$_4$$Cl$$_2$$]Cl and mer-[Co(NH$$_3$$)$$_3$$Cl$$_3$$] octahedral complexes are at the vertices of an octahedron. The sum of total number of the triangular faces in both the complexes having one N atom and two Cl atoms at their corners is ___.

In the following reaction sequence, major products X and Y are acyclic monomers.

500 mol of X completely reacts with 500 mol of Y to give 1 mol of a single biodegradable acyclic copolymer Z as the only product. The amount of Z formed in grams is ___.

Given: Atomic mass (in amu): H : 1, C : 12, N : 14, O : 16, Br : 80

Question Stem (Q15 & Q16): Two volatile liquids A and B form an ideal solution. Consider a 5 molal solution of B in A inside a closed container having a total vapour pressure of 100 mm Hg at 300 K. The vapour pressure of pure A at 300 K is 105 mm Hg. Assume that A and B behave as ideal gases in the vapour phase.

Given: The gas constant $$R=0.08$$ L atm K$$^{-1}$$ mol$$^{-1}$$;
Molar mass of A is 50 g mol$$^{-1}$$;
Molar mass of B is 57 g mol$$^{-1}$$;
Density of liquid B at 300 K is 0.5 g/mL; 1 atm $$=760$$ mm Hg.

At 300 K, the ratio of the molar volume of pure B in vapour phase to its molar volume in liquid phase is ___.

Question Stem (Q15 & Q16): Two volatile liquids A and B form an ideal solution. Consider a 5 molal solution of B in A inside a closed container having a total vapour pressure of 100 mm Hg at 300 K. The vapour pressure of pure A at 300 K is 105 mm Hg. Assume that A and B behave as ideal gases in the vapour phase.

Given: The gas constant $$R=0.08$$ L atm K$$^{-1}$$ mol$$^{-1}$$;
Molar mass of A is 50 g mol$$^{-1}$$;
Molar mass of B is 57 g mol$$^{-1}$$;
Density of liquid B at 300 K is 0.5 g/mL; 1 atm $$=760$$ mm Hg.

The mole fraction of B in vapour phase which is in equilibrium with this solution is ___.

Question Stem (Q17 & Q18): Consider the following reaction sequence in which J, K, L and M are the major products.

Given: Atomic mass (in amu): H : 1, C : 12, N : 14, O : 16, S : 32, Br : 80, Ba : 137

The volume of 1 M aqueous $$\mathrm{H_2SO_4}$$ required to completely neutralize the ammonia evolved from 5.72 g of L in Kjeldahl's method of nitrogen estimation is ___ mL.

Question Stem (Q17 & Q18): Consider the following reaction sequence in which J, K, L and M are the major products.

Given: Atomic mass (in amu): H : 1, C : 12, N : 14, O : 16, S : 32, Br : 80, Ba : 137

In sulphur estimation by Carius method, the amount of $$\mathrm{BaSO_4}$$ formed from 3.79 g of M is ___ g.

Let $$\vec{a},\vec{b}$$ be two vectors, and let $$P,Q$$ and $$R$$ be the points with position vectors $$\vec{a}$$, $$\vec{b}$$ and $$\vec{a}+\vec{b}$$, respectively, with respect to the origin $$O$$. If $$|\vec{a}+\vec{b}|=\sqrt{21}$$, $$|\vec{a}-\vec{b}|=3$$, and $$\vec{a}$$ and $$(\vec{a}-\vec{b})$$ are perpendicular to each other, then the area of the triangle $$OPR$$ is

Let $$T$$ be the tangent to the parabola $$y^2=16x$$ at the point $$(64,32)$$. Let $$L$$ be the tangent to the same parabola at another point $$(x_1,\,y_1)$$ on the parabola. If $$L$$ and $$T$$ are perpendicular to each other, then the distance between the point $$(x_1,\,y_1)$$ and the focus of the parabola, is

Let $$y:(-\infty,\infty)\to(0,\infty)$$ be the solution of the differential equation

$$\dfrac{dy}{dx}=\dfrac{e^{5x}y^3+y^3}{e^x+e^x y^4},$$

satisfying $$y(0)=\dfrac{1}{\sqrt{2}}$$. Then the value of $$y(\log_e 2)$$ is

The value of the definite integral

$$\int_{0}^{2}\dfrac{1}{3^x+3}\,dx$$

is

Let $$\mathbb{R}$$ denote the set of all real numbers. Consider the polynomial function $$f:\mathbb{R}\to\mathbb{R}$$ defined by

$$f(x)=\dfrac{d^{10}}{dx^{10}}\big((x^2-1)^{10}\big),\qquad\text{for all }x\in\mathbb{R}.$$

Here $$\dfrac{d^{10}}{dx^{10}}\big((x^2-1)^{10}\big)$$ is the $$10^{\text{th}}$$ order derivative of the function $$(x^2-1)^{10}$$.

Then which of the following statements is (are) TRUE?

Let $$a,\,b,\,c$$ be positive integers in arithmetic progression such that the equation

$$ax^2+bx+c=0$$

has only integer solutions.

Then which of the following statements is (are) TRUE?

Let $$L$$ be the straight line joining the points $$P(1,2,-1)$$ and $$Q(2,3,1)$$. Let $$S$$ be the foot of the perpendicular drawn from the point $$R(4,-1,5)$$ to the line $$L$$. Another line passing through $$R$$ intersects $$L$$ at a point $$T$$ such that the point $$S$$ divides the line segment $$PT$$ internally in the ratio $$|PS|:|ST|=1:2$$, where $$|PS|$$ and $$|ST|$$ are the lengths of the line segments $$PS$$ and $$ST$$, respectively.

Then which of the following statements is (are) TRUE?

Let $$y=f(x)$$ be the real valued function defined on the interval $$(0,\infty)$$, satisfying $$y(1)=0$$ and the differential equation

$$x\dfrac{dy}{dx}=y-x^3.$$

Then which of the following statements is (are) TRUE?

Let $$\mathbb{R}$$ denote the set of all real numbers and let $$i=\sqrt{-1}$$. Consider the matrices

$$S=\begin{bmatrix}0&-1\\1&0\end{bmatrix}\quad\text{and}\quad T=\begin{bmatrix}1&1\\0&1\end{bmatrix}.$$

Let $$a,b,c,d$$ be real numbers such that

$$ST=\begin{bmatrix}a&b\\c&d\end{bmatrix}.$$

Let

$$H=\{\,x+iy:\;x,y\in\mathbb{R}\;\text{and}\;y>0\,\}.$$

Then which of the following statements is (are) TRUE?

Let $$\mathbb{N}$$ denote the set of all positive integers. Consider the sets

$$A=\{1,2,3,4,5\}\quad\text{and}\quad B=\{1,2,3,4,5,6,7\}.$$

Let $$S$$ be the set of all functions $$f:A\to B$$ such that $$f(2)\neq 2$$ and $$f(4)\neq 4$$. Consider the set

$$T=\big\{f\in S:\text{there exists a function }g:B\to\mathbb{N}\text{ such that }g\big(f(x)\big)=2^x\text{ for all }x\in A\big\}.$$

Then the number of elements in the set $$T$$ is ___.

A bookshelf contains 6 distinct books of Mathematics and 5 distinct books of Physics. From these 11 books, 6 books are chosen at random. Let $$X$$ be the absolute value of the difference between the number of Mathematics books chosen and the number of Physics books chosen. If $$\alpha$$ is the mean of the random variable $$X$$, then the value of $$77\alpha$$ is ___.

Consider a data consisting of 10 observations $$x_1,x_2,\dots,x_{10}$$, whose mean is $$5$$ and variance is $$7$$. If the mean and the variance of the first 8 observations $$x_1,x_2,\dots,x_8$$ are $$4$$ and $$3.5$$, respectively, and $$x_9 < x_{10}$$, then the value of $$3x_9 + 2x_{10}$$ is ___________.

Consider the ellipse $$E$$ given by $$\dfrac{x^2}{18}+\dfrac{y^2}{12}=1$$. Let $$H$$ be the hyperbola whose eccentricity is the reciprocal of the eccentricity of $$E$$ and whose foci are the same as that of $$E$$. Let $$P$$ and $$Q$$ be the points of intersection of $$H$$ and the parabola $$\sqrt{5}\,y=x^2$$ in the first quadrant. Let $$d$$ be the distance between $$P$$ and $$Q$$.

If $$a$$ and $$b$$ are the integers such that $$d^2=a+b\sqrt{5}$$, then the value of $$a-b$$ is ___.

For a real number $$\alpha$$, let $$[\alpha]$$ denote the greatest integer less than or equal to $$\alpha$$. For a finite set $$S$$, let $$|S|$$ denote the number of elements in the set $$S$$.

Consider the functions $$f:(-3,3)\to(-\infty,\,\infty)$$ and $$g:(-3,3)\to(-\infty,\,\infty)$$ defined by

$$f(x)=[x^3]\log_e\big(1+\sin^2(\pi(x-[x])))\big)$$

and

$$g(x)=x^3\sin^2(\pi\log_e(1+x-[x])).$$

Let

$$A=\{x\in(-3,3):f\text{ is discontinuous at }x\}$$

and

$$B=\{x\in(-3,3):g\text{ is discontinuous at }x\}.$$

Then the value of $$|A|+2|B|-|A\cap B|$$ is ___.

Question Stem for Question Nos. 15 and 16

Consider the curve $$C_1$$ given by $$y=e^{-x}$$ for $$x\in[0,10\pi]$$, and the curve $$C_2$$ given by $$y=e^{-x}(\sin x+\cos x)$$ for $$x\in[0,10\pi]$$.

Let $$n$$ be the total number of points of intersection of the curves $$C_1$$ and $$C_2$$.

Suppose that $$\alpha_1,\alpha_2,\dots,\alpha_n\in[0,10\pi]$$ are the $$x$$-coordinates of the points of intersection of the curves $$C_1$$ and $$C_2$$ such that $$\alpha_1<\alpha_2<\cdots<\alpha_n$$.

The value of $$n$$ is ___.

Question Stem for Question Nos. 15 and 16

Consider the curve $$C_1$$ given by $$y=e^{-x}$$ for $$x\in[0,10\pi]$$, and the curve $$C_2$$ given by $$y=e^{-x}(\sin x+\cos x)$$ for $$x\in[0,10\pi]$$.

Let $$n$$ be the total number of points of intersection of the curves $$C_1$$ and $$C_2$$.

Suppose that $$\alpha_1,\alpha_2,\dots,\alpha_n\in[0,10\pi]$$ are the $$x$$-coordinates of the points of intersection of the curves $$C_1$$ and $$C_2$$ such that $$\alpha_1<\alpha_2<\cdots<\alpha_n$$.

Let $$\beta$$ be the area of the region enclosed between the curves $$C_1$$, $$C_2$$, and the lines $$x=\alpha_1$$ and $$x=\alpha_4$$. Then the value of

$$-\dfrac{1}{\pi}\log_e\!\left(\beta-2\,e^{-\frac{\pi}{2}}\right)$$

is ___.

Consider the ellipses given by $$x^2+4y^2=1$$ and $$4x^2+y^2=1$$.

Let $$P$$ be the point in the first quadrant where the given ellipses intersect. If $$\theta$$ is the acute angle between the tangents to the given ellipses at the point $$P$$, then the value of $$4\tan\theta$$ is ___.

Consider the ellipses given by $$x^2+4y^2=1$$ and $$4x^2+y^2=1$$.

If $$\alpha$$ is the area of the common region that lies inside both the given ellipses, then the value of $$\cot\alpha$$ is ___.