JEE Main 2025 Jan 24 Shift 2 — Question Paper

The function $f:(-\infty, \infty) \to (-\infty, 1)$, defined by $f(x) = \dfrac{2^x - 2^{-x}}{2^x + 2^{-x}}$ is :

Let $A = \left\{x \in (0, \pi) - \left\{\dfrac{\pi}{2}\right\} : \log_{(2/\pi)}|\sin x| + \log_{(2/\pi)}|\cos x| = 2\right\}$ and

$B = \left\{x \geq 0 : \sqrt{x}\left(\sqrt{x} - 4\right) - 3\left|\sqrt{x} - 2\right| + 6 = 0\right\}.$ Then $\mathrm{n(A \cup B)}$ is equal to :

The number of real solution(s) of the equation $x^2 + 3x + 2 = \min\{|x - 3|, |x + 2|\}$ is :

For some a, b, let $f(x) = \begin{vmatrix} a + \dfrac{\sin x}{x} & 1 & b \\ a & 1 + \dfrac{\sin x}{x} & b \\ a & 1 & b + \dfrac{\sin x}{x} \end{vmatrix}$, $x \neq 0$, $\lim\limits_{x \to 0} f(x) = \lambda + \mu a + \nu b$. Then $(\lambda + \mu + \nu)^2$ is equal to :

If the system of equations

$x + 2y - 3z = 2$

$2x + \lambda y + 5z = 5$

$14x + 3y + \mu z = 33$

has infinitely many solutions, then $\lambda + \mu$ is equal to :

If $7 = 5 + \dfrac{1}{7}(5+\alpha) + \dfrac{1}{7^2}(5 + 2\alpha) + \dfrac{1}{7^3}(5 + 3\alpha) + \ldots\ldots \infty$, then the value of $\alpha$ is :

In an arithmetic progression, if $\mathrm{S_{40}} = 1030$ and $\mathrm{S_{12}} = 57$, then $\mathrm{S_{30}} - \mathrm{S_{10}}$ is equal to :

Let $\mathrm{A} = [a_{ij}]$ be a square matrix of order 2 with entries either 0 or 1. Let E be the event that A is an invertible matrix. Then the probability P(E) is :

Suppose A and B are the coefficients of $30^{\text{th}}$ and $12^{\text{th}}$ terms respectively in the binomial expansion of $(1+x)^{2n-1}$. If $2\mathrm{A} = 5\mathrm{B}$, then $n$ is equal to :

Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group A and the remaining 3 from group B, is equal to :

Let the points $\left(\dfrac{11}{2}, \alpha\right)$ lie on or inside the triangle with sides $x + y = 11$, $x + 2y = 16$ and $2x + 3y = 29$. Then the product of the smallest and the largest values of $\alpha$ is equal to :

If the equation of the parabola with vertex $\mathrm{V}\left(\dfrac{3}{2}, 3\right)$ and the directrix $x + 2y = 0$ is $\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0$, then $\alpha + \beta + \gamma$ is equal to :

The equation of the chord, of the ellipse $\dfrac{x^2}{25} + \dfrac{y^2}{16} = 1$, whose mid-point is $(3, 1)$ is :

If $\alpha > \beta > \gamma > 0$, then the expression

$\cot^{-1}\left\{\beta + \dfrac{(1 + \beta^2)}{(\alpha - \beta)}\right\} + \cot^{-1}\left\{\gamma + \dfrac{(1 + \gamma^2)}{(\beta - \gamma)}\right\} + \cot^{-1}\left\{\alpha + \dfrac{(1 + \alpha^2)}{(\gamma - \alpha)}\right\}$ is equal to :

Let $\vec{a} = 3\hat{i} - \hat{j} + 2\hat{k}$, $\vec{b} = \vec{a} \times \left(\hat{i} - 2\hat{k}\right)$ and $\vec{c} = \vec{b} \times \hat{k}$. Then the projection of $\vec{c} - 2\hat{j}$ on $\vec{a}$ is :

Let the position vectors of three vertices of a triangle be $4\vec{p} + \vec{q} - 3\vec{r}$, $-5\vec{p} + \vec{q} + 2\vec{r}$ and $2\vec{p} - \vec{q} + 2\vec{r}$. If the position vectors of the orthocenter and the circumcenter of the triangle are $\dfrac{\vec{p} + \vec{q} + \vec{r}}{4}$ and $\alpha\vec{p} + \beta\vec{q} + \gamma\vec{r}$ respectively, then $\alpha + 2\beta + 5\gamma$ is equal to :

Let $[x]$ denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function $f(x) = [x] + |x - 2|$, $-2 < x < 3$, is not continuous and not differentiable. Then $\mathrm{m + n}$ is equal to :

Let $(2, 3)$ be the largest open interval in which the function $f(x) = 2\log_e(x - 2) - x^2 + \mathrm{a}x + 1$ is strictly increasing and $(\mathrm{b}, \mathrm{c})$ be the largest open interval, in which the function $\mathrm{g}(x) = (x-1)^3(x + 2 - \mathrm{a})^2$ is strictly decreasing. Then $100(\mathrm{a} + \mathrm{b} - \mathrm{c})$ is equal to :

The area of the region enclosed by the curves $y = e^x$, $y = |e^x - 1|$ and y-axis is :

Let $f : (0, \infty) \to \mathbf{R}$ be a function which is differentiable at all points of its domain and satisfies the condition $x^2 f'(x) = 2xf(x) + 3$, with $f(1) = 4$. Then $2f(2)$ is equal to :

Number of functions $f : \{1, 2, \ldots, 100\} \to \{0, 1\}$, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to __________.

Let P be the image of the point $\mathrm{Q}(7, -2, 5)$ in the line $\mathrm{L} : \dfrac{x-1}{2} = \dfrac{y+1}{3} = \dfrac{z}{4}$ and $\mathrm{R}(5, \mathrm{p}, \mathrm{q})$ be a point on L. Then the square of the area of $\Delta\mathrm{PQR}$ is __________.

If $\displaystyle\int \dfrac{2x^2 + 5x + 9}{\sqrt{x^2 + x + 1}}\, \mathrm{d}x = x\sqrt{x^2 + x + 1} + \alpha\sqrt{x^2 + x + 1} + \beta \log_e\left|x + \dfrac{1}{2} + \sqrt{x^2 + x + 1}\right| + \mathrm{C}$,

where C is the constant of integration, then $\alpha + 2\beta$ is equal to __________.

Let $y = y(x)$ be the solution of the differential equation $2\cos x \dfrac{\mathrm{d}y}{\mathrm{d}x} = \sin 2x - 4y\sin x$, $x \in \left(0, \dfrac{\pi}{2}\right)$. If $y\left(\dfrac{\pi}{3}\right) = 0$, then $y'\left(\dfrac{\pi}{4}\right) + y\left(\dfrac{\pi}{4}\right)$ is equal to __________.

Let $\mathrm{H_1} : \dfrac{x^2}{\mathrm{a}^2} - \dfrac{y^2}{\mathrm{b}^2} = 1$ and $\mathrm{H_2} : -\dfrac{x^2}{\mathrm{A}^2} + \dfrac{y^2}{\mathrm{B}^2} = 1$ be two hyperbolas having length of latus rectums $15\sqrt{2}$ and $12\sqrt{5}$ respectively. Let their ecentricities be $\mathrm{e_1} = \sqrt{\dfrac{5}{2}}$ and $\mathrm{e_2}$ respectively. If the product of the lengths of their transverse axes is $100\sqrt{10}$, then $25\mathrm{e_2^2}$ is equal to __________.

The energy E and momentum p of a moving body of mass m are related by some equation. Given that c represents the speed of light, identify the correct equation

Which of the following figure represents the relation between Celsius and Fahrenheit temperatures ?

A solid sphere is rolling without slipping on a horizontal plane. The ratio of the linear kinetic energy of the centre of mass of the sphere and rotational kinetic energy is :

The position vector of a moving body at any instant of time is given as $\vec{r} = \left(5t^2\,\hat{i} - 5t\,\hat{j}\right)\,\mathrm{m}$. The magnitude and direction of velocity at $t = 2\,\mathrm{s}$ is,

A solid sphere and a hollow sphere of the same mass and of same radius are rolled on an inclined plane. Let the time taken to reach the bottom by the solid sphere and the hollow sphere be $t_1$ and $t_2$, respectively, then

The temperature of a body in air falls from $40°\mathrm{C}$ to $24°\mathrm{C}$ in 4 minutes. The temperature of the air is $16°\mathrm{C}$. The temperature of the body in the next 4 minutes will be :

The magnitude of heat exchanged by a system for the given cyclic process ABCA (as shown in figure) is (in SI unit) :

[Figure: P–V diagram with P (in kPa) on the vertical axis and V (in cc) on the horizontal axis; a circle spans 200–400 on both axes with A at the right (400 cc), B at the top (400 kPa), C at the left (200 cc) and D at the bottom (200 kPa); the cycle follows the upper semicircle through B and the horizontal line C→A through the centre, the lower semicircle through D being dotted]

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A) : In an insulated container, a gas is adiabatically shrunk to half of its initial volume. The temperature of the gas decreases.

Reason (R) : Free expansion of an ideal gas is an irreversible and adiabatic process.

In the light of the above statements, choose the correct answer from the options given below :

A particle oscillates along the $x$-axis according to the law, $x(t) = x_0 \sin^2\left(\frac{t}{2}\right)$ where $x_0 = 1$ m. The kinetic energy (K) of the particle as a function of $x$ is correctly represented by the graph

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A) : A electron in a certain region of uniform magnetic field is moving with constant velocity in a straight line path.

Reason (R) : The magnetic field in that region is along the direction of velocity of the electron.

In the light of the above statements, choose the correct answer from the options given below :

A long straight wire of circular cross-section of radius 'a' carries a steady current I. The current I is uniformly distributed across this cross-section. The variation of magnitude of magnetic field B with distance r from the centre of the wire is represented by the graph :

A small uncharged conducting sphere is kept in contact with an identical sphere having charge $4 \times 10^{-8}\,\mathrm{C}$ and then removed to a distance such that the force of repulsion between them is $9 \times 10^{-3}\,\mathrm{N}$. The distance between them is (Take $\frac{1}{4\pi\epsilon_0}$ as $9 \times 10^{9}$ in SI units)

[Figure: two squares of side 'a'. Configuration (1): identical charges $+q_0$ at the four corners A, B, C, D; G, E, H, F are the midpoints of sides AB, BC, DC and AD respectively, with centre O. Configuration (2): the same charges $+q_0$ placed at the midpoints G, E, H and F]

In the first configuration (1) as shown in the figure, four identical charges $(q_0)$ are kept at the corners A, B, C and D of square of side length 'a'. In the second configuration (2), the same charges are shifted to mid points G, E, H and F, of the square. If $K = \frac{1}{4\pi\epsilon_0}$, the difference between the potential energies of configuration (2) and (1) is given by :

N charges, each of value q, are placed at equal intervals on a circle of radius R. The circle rotates about its axis with an angular velocity $\omega$ as shown in the figure. A relatively large Amperian loop B encircles the whole circle whereas a relatively small Amperian loop A encircles a small segment. The difference between the enclosed currents, $I_A - I_B$, for the given Amperian loops is

[Figure: a circle of radius R carrying N equally spaced charges, rotating with angular velocity ω about its axis; a large Amperian loop B threads the entire circle while a small Amperian loop A encircles a small segment of it]

Arrange the following in the ascending order of wavelength ($\lambda$) :

(A) Microwaves ($\lambda_1$)

(B) Ultraviolet rays ($\lambda_2$)

(C) Infrared rays ($\lambda_3$)

(D) X-rays ($\lambda_4$)

Choose the most appropriate answer from the options given below :

A photograph of a landscape is captured by a drone camera at a height of 18 km. The size of the camera film is $2\ \mathrm{cm} \times 2\ \mathrm{cm}$ and the area of the landscape photographed is $400\ \mathrm{km}^2$. The focal length of the lens in the drone camera is :

Young's double slit inteference apparatus is immersed in a liquid of refractive index 1.44. It has slit separation of 1.5 mm. The slits are illuminated by a parallel beam of light whose wavelength in air is 690 nm. The fringe-width on a screen placed behind the plane of slits at a distance of 0.72 m, will be :

In a Young's double slit experiment, three polarizers are kept as shown in the figure. The transmission axes of $\mathrm{P}_1$ and $\mathrm{P}_2$ are orthogonal to each other. The polarizer $\mathrm{P}_3$, whose transmission axis is at $45°$ to the transmission axes of $\mathrm{P}_1$ and $\mathrm{P}_2$, covers both the slits. An unpolarized light of wavelength $\lambda$ and intensity $\mathrm{I}_0$ is incident on $\mathrm{P}_1$ and $\mathrm{P}_2$. At a point after $\mathrm{P}_3$, where the path difference between the light waves from $s_1$ and $s_2$ is $\frac{\lambda}{3}$, the intensity is :

[Figure: unpolarized light of wavelength λ and intensity I₀ is incident on polarizers P₁ and P₂ placed over slits s₁ and s₂ respectively; polarizer P₃ placed after them covers both slits]

In photoelectric effect, the stopping potential $(\mathrm{V}_0)$ v/s frequency $(\nu)$ curve is plotted.

(h is the Planck's constant and $\phi_0$ is work function of metal)

(A) $\mathrm{V}_0$ v/s $\nu$ is linear.

(B) The slope of $\mathrm{V}_0$ v/s $\nu$ curve $= \frac{\phi_0}{h}$

(C) h constant is related to the slope of $\mathrm{V}_0$ v/s $\nu$ line.

(D) The value of electric charge of electron is not required to determine h using the $\mathrm{V}_0$ v/s $\nu$ curve.

(E) The work function can be estimated without knowing the value of h.

Choose the correct answer from the options given below :

The output of the circuit is low (zero) for :

[Figure: logic circuit — inputs X and Y feed an OR gate; the OR gate's output passes through a NOT (inverting) gate to give the final output]

(A) $X = 0,\ Y = 0$

(B) $X = 0,\ Y = 1$

(C) $X = 1,\ Y = 0$

(D) $X = 1,\ Y = 1$

Choose the correct answer from the options given below :

A string of length L is fixed at one end and carries a mass M at the other end. The mass makes $\left(\frac{3}{\pi}\right)$ rotations per second about the vertical axis passing through the end of the string as shown in the figure. The tension in the string is ______ ML.

[Figure: a string of length L fixed to the ceiling makes angle θ with the vertical; tension T acts along the string and the mass at its end moves in a horizontal circle of radius R]

Acceleration due to gravity on the surface of earth is 'g'. If the diameter of earth is reduced to one third of its original value and mass remains unchanged, then the acceleration due to gravity on the surface of the earth is ______ g.

The increase in pressure required to decrease the volume of a water sample by 0.2% is $\mathrm{P} \times 10^5\ \mathrm{Nm}^{-2}$. Bulk modulus of water is $2.15 \times 10^9\ \mathrm{Nm}^{-2}$. The value of P is ______.

A tightly wound long solenoid carries a current of 1.5 A. An electron is executing uniform circular motion inside the solenoid with a time period of 75 ns. The number of turns per metre in the solenoid is ______.

$\left[\text{Take mass of electron } m_e = 9 \times 10^{-31}\ \mathrm{kg}, \text{ charge of electron } |q_e| = 1.6 \times 10^{-19}\ \mathrm{C}, \ \mu_0 = 4\pi \times 10^{-7}\ \mathrm{\frac{N}{A^2}}, \ 1\ \mathrm{ns} = 10^{-9}\ \mathrm{s}\right]$

The ratio of the power of a light source $\mathrm{S}_1$ to that of the light source $\mathrm{S}_2$ is 2. $\mathrm{S}_1$ is emitting $2 \times 10^{15}$ photons per second at 600 nm. If the wavelength of the source $\mathrm{S}_2$ is 300 nm, then the number of photons per second emitted by $\mathrm{S}_2$ is ______ $\times 10^{14}$.

The elemental composition of a compound is 54.2% C, 9.2% H and 36.6% O. If the molar mass of the compound is $132\ \mathrm{g\ mol^{-1}}$, the molecular formula of the compound is :

[Given : The relative atomic mass of $\mathrm{C:H:O = 12:1:16}$]

For hydrogen atom, the orbital/s with lowest energy is/are :

(A) $\mathrm{4s}$ (B) $\mathrm{3p_x}$ (C) $\mathrm{3d_{x^2-y^2}}$ (D) $\mathrm{3d_{z^2}}$ (E) $\mathrm{4p_z}$

Choose the correct answer from the options given below :

Which of the following mixing of 1M base and 1M acid leads to the largest increase in temperature ?

$$\mathrm{S(g)} + \frac{3}{2}\mathrm{O_2(g)} \rightarrow \mathrm{SO_3(g)} + 2x\ \mathrm{kcal}$$

$$\mathrm{SO_2(g)} + \frac{1}{2}\mathrm{O_2(g)} \rightarrow \mathrm{SO_3(g)} + y\ \mathrm{kcal}$$

The heat of formation of $\mathrm{SO_2(g)}$ is given by :

For the reaction, $$\mathrm{H_2(g) + I_2(g) \rightleftharpoons 2HI(g)}$$ Attainment of equilibrium is predicted correctly by :

[Figure: each option is a plot of molar concentration (y-axis) versus time (x-axis) for the three species $\mathrm{H_2}$, $\mathrm{I_2}$ and $\mathrm{HI}$ approaching equilibrium plateaus]

Based on the data given below :

$$\mathrm{E^\circ_{Cr_2O_7^{2-}/Cr^{3+}}} = 1.33\ \mathrm{V} \qquad \mathrm{E^\circ_{Cl_2/Cl^-}} = 1.36\ \mathrm{V}$$

$$\mathrm{E^\circ_{MnO_4^-/Mn^{2+}}} = 1.51\ \mathrm{V} \qquad \mathrm{E^\circ_{Cr^{3+}/Cr}} = -0.74\ \mathrm{V}$$

the strongest reducing agent is :

Given below are two statements :

Statement (I): [Figure: a plot of $t_{1/2}$ (y-axis) versus $[R]_0$ (x-axis) shown as a horizontal straight line] is valid for a first order reaction.

Statement (II): [Figure: a plot of $\log\frac{[R]}{[R]_0}$ (y-axis) versus time (x-axis) shown as a straight line through the origin with slope $= \frac{k}{2.303}$] is valid for a first order reaction.

In the light of the above statements, choose the correct answer from the options given below :

Given below are two statements :

Statement (I): Experimentally determined oxygen-oxygen bond lengths in the $\mathrm{O_3}$ are found to be same and the bond length is greater than that of a $\mathrm{O=O}$ (double bond) but less than that of a single $\mathrm{(O-O)}$ bond.

Statement (II): The strong lone pair-lone pair repulsion between oxygen atoms is solely responsible for the fact that the bond length in ozone is smaller than that of a double bond $\mathrm{(O=O)}$ but more than that of a single bond $\mathrm{(O-O)}$.

In the light of the above statements, choose the correct answer from the options given below :

Find the compound 'A' from the following reaction sequences.

\mathrm{A} \xrightarrow{\text{aqua-regia}} \mathrm{B} \xrightarrow[\text{(2) AcOH}]{\text{(1) KNO_2 | NH_4OH}} \text{yellow ppt}

When ethane-1,2-diamine is progressively added to an aqueous solution of nickel(II) chloride, the observed order of colour change will be :

The conditions and consequence that favours the $\mathrm{t_{2g}^3\ e_g^1}$ configuration in a metal complex are :

Match List - I with List - II.

List - I (Transition metal ion)	List - II (Spin only magnetic moment (B.M.))
(A) $\mathrm{Ti^{3+}}$	(I) 3.87
(B) $\mathrm{V^{2+}}$	(II) 0.00
(C) $\mathrm{Ni^{2+}}$	(III) 1.73
(D) $\mathrm{Sc^{3+}}$	(IV) 2.84

Choose the correct answer from the options given below :

Given below are two statements :

Statement (I): The first ionization energy of Pb is greater than that of Sn.

Statement (II): The first ionization energy of Ge is greater than that of Si.

In the light of the above statements, choose the correct answer from the options given below :

The five successive ionization energies of an element are respectively 800, 2427, 3658, 25024 and $32824\ \mathrm{kJ/mol}$. Using the above values, predict the group in which it is present :

In the given structure, number of sp and $\mathrm{sp^2}$ hybridized carbon atoms present respectively are :

[Figure: an open-chain structure $\mathrm{CH_3-CO-CH=CH-CH_2-C{\equiv}C-CH=CH-C{\equiv}N}$ — a methyl ketone bearing a C=C, then a $\mathrm{CH_2}$, then a $\mathrm{C{\equiv}C}$ triple bond, another C=C, terminating in a nitrile $\mathrm{C{\equiv}N}$]

Identify correct statement/s :

(A) $\mathrm{-OCH_3}$ and $\mathrm{-NHCOCH_3}$ are activating group. (B) $\mathrm{-CN}$ and $\mathrm{-OH}$ are meta directing group. (C) $\mathrm{-CN}$ and $\mathrm{-SO_3H}$ are meta directing group. (D) Activating groups act as ortho- and para directing groups. (E) Halides are activating groups.

Choose the correct answer from the options given below :

The structure of the major product formed in the following reaction is :

[Figure: 1-(chloromethyl)-3-bromo-5-iodobenzene (a benzene ring bearing $\mathrm{-CH_2Cl}$, and at the two meta positions an $\mathrm{I}$ and a $\mathrm{Br}$)] $\xrightarrow{\text{AgCN}}$ major product

Match List - I with List - II.

List - I	List - II
(A) \mathrm{RCN} \xrightarrow[\text{(ii) H_3O^+}]{\text{(i) SnCl_2, HCl}} \mathrm{RCHO}	(I) Etard reaction
(B) \mathrm{C_6H_5COCl} \xrightarrow{\text{H_2 / Pd-BaSO_4}} \mathrm{C_6H_5CHO}	(II) Gatterman-Koch reaction
(C) \mathrm{C_6H_5CH_3} \xrightarrow[\text{(ii) H_3O^+}]{\text{(i) CrO_2Cl_2, CS_2}} \mathrm{C_6H_5CHO}	(III) Rosenmund reduction
(D) \mathrm{C_6H_6} \xrightarrow[\text{(ii) anhydrous AlCl_3 / CuCl}]{\text{(i) CO, HCl}} \mathrm{C_6H_5CHO}	(IV) Stephen reaction

Choose the correct answer from the options given below :

For reaction

[Figure: aniline (benzene ring with $\mathrm{-NH_2}$) $\rightarrow$ 2-bromoaniline (benzene ring with $\mathrm{-NH_2}$ and an ortho $\mathrm{-Br}$), labelled (Major)]

The correct order of set of reagents for the above conversion is :

Match List - I with List - II.

List - I (nitrogenous base) : (A) Adenine (B) Cytosine (C) Thymine (D) Uracil

List - II (structures) : (I) [Figure: pyrimidine ring with two C=O groups and an N-H — structure of uracil] (II) [Figure: pyrimidine ring with a $\mathrm{CH_3}$ (H_3C) substituent, two C=O groups and an N-H — structure of thymine] (III) [Figure: purine ring bearing an $\mathrm{NH_2}$ group — structure of adenine] (IV) [Figure: pyrimidine ring bearing an $\mathrm{NH_2}$ group and one C=O — structure of cytosine]

Choose the correct answer from the options given below :

The observed and normal molar masses of compound $\mathrm{MX_2}$ are respectively 65.6 and 164. The percentage degree of ionization of $\mathrm{MX_2}$ is _______ % (nearest integer).

Consider a complex reaction taking place in three steps with rate constants $k_1$, $k_2$ and $k_3$ respectively. The overall rate constant $k$ is given by the expression $k = \sqrt{\dfrac{k_1 k_3}{k_2}}$. If the activation energies of the three steps are 60, 30 and $10\ \mathrm{kJ\ mol^{-1}}$ respectively, then the overall energy of activation in $\mathrm{kJ\ mol^{-1}}$ is _______. (Nearest integer)

In Carius method of estimation of halogen, 0.25 g of an organic compound gave 0.15 g of silver bromide (AgBr). The percentage of Bromine in the organic compound is _______ $\times 10^{-1}\%$ (Nearest integer).

(Given : Molar mass of Ag is 108 and Br is $80\ \mathrm{g\ mol^{-1}}$)

The hydrocarbon (X) with molar mass $80\ \mathrm{g\ mol^{-1}}$ and 90% carbon has _______ degree of unsaturation.

The possible number of stereoisomers for 5-phenylpent-4-en-2-ol is _______.