JEE Main 2024 Feb 1 Shift 1 — Question Paper

Let $S=\{z \in \mathbb{C} : |z-1|=1 \text{ and } (\sqrt{2}-1)(z+\bar{z}) - i(z-\bar{z}) = 2\sqrt{2}\}$. Let $z_1, z_2 \in S$ be such that $|z_1| = \max\limits_{z \in S} |z|$ and $|z_2| = \min\limits_{z \in S} |z|$. Then $\left|\sqrt{2}\, z_1 - z_2\right|^2$ equals :

Let $S=\{x \in \mathbb{R} : (\sqrt{3}+\sqrt{2})^x + (\sqrt{3}-\sqrt{2})^x = 10\}$. Then the number of elements in $S$ is :

If $A = \begin{bmatrix} \sqrt{2} & 1 \\ -1 & \sqrt{2} \end{bmatrix}$, $B = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$, $C = ABA^{T}$ and $X = A^{T}C^{2}A$, then $\det X$ is equal to :

If n is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then n is equal to :

If the system of equations

$$2x + 3y - z = 5$$ $$x + \alpha y + 3z = -4$$ $$3x - y + \beta z = 7$$

has infinitely many solutions, then $13\,\alpha\beta$ is equal to ________.

Let $3, a, b, c$ be in A.P. and $3, a-1, b+1, c+9$ be in G.P. Then, the arithmetic mean of $a, b$ and $c$ is :

If $5f(x) + 4f\left(\dfrac{1}{x}\right) = x^2 - 2, \forall\, x \neq 0$ and $y = 9x^2 f(x)$, then $y$ is strictly increasing in :

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as :

$$f(x) = \begin{cases} \dfrac{a - b\cos 2x}{x^2} & ;\ x < 0 \\ x^2 + cx + 2 & ;\ 0 \le x \le 1 \\ 2x + 1 & ;\ x > 1 \end{cases}$$

If $f$ is continuous everywhere in $\mathbb{R}$ and m is the number of points where $f$ is NOT differential then $m + a + b + c$ equals :

The area enclosed by the curves $xy + 4y = 16$ and $x + y = 6$ is equal to :

The value of the intergral $\displaystyle\int_0^{\pi/4} \dfrac{x\, dx}{\sin^4(2x) + \cos^4(2x)}$ equals :

Let $y = y(x)$ be the solution of the differential equation $\dfrac{dy}{dx} = 2x(x+y)^3 - x(x+y) - 1$, $y(0) = 1$.

Then, $\left(\dfrac{1}{\sqrt{2}} + y\left(\dfrac{1}{\sqrt{2}}\right)\right)^2$ equals :

For $0 < \theta < \pi/2$, if the eccentricity of the hyperbola $x^2 - y^2\operatorname{cosec}^2\theta = 5$ is $\sqrt{7}$ times eccentricity of the ellipse $x^2\operatorname{cosec}^2\theta + y^2 = 5$, then the value of $\theta$ is :

Let $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$, $a > b$ be an ellipse, whose eccentricity is $\dfrac{1}{\sqrt{2}}$ and the length of the latusrectum is $\sqrt{14}$. Then the square of the eccentricity of $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ is :

Let $f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = \begin{cases} \log_e x, & x > 0 \\ e^{-x}, & x \le 0 \end{cases}$ and $g(x) = \begin{cases} x, & x \ge 0 \\ e^{x}, & x < 0 \end{cases}$. Then, $gof: \mathbb{R} \to \mathbb{R}$ is :

Let $C: x^2 + y^2 = 4$ and $C': x^2 + y^2 - 4\lambda x + 9 = 0$ be two circles. If the set of all values of $\lambda$ so that the circles $C$ and $C'$ intersect at two distinct points, is $\mathbb{R} - [a, b]$, then the point $(8a + 12, 16b - 20)$ lies on the curve :

If the shortest distance between the lines $\dfrac{x - \lambda}{-2} = \dfrac{y - 2}{1} = \dfrac{z - 1}{1}$ and $\dfrac{x - \sqrt{3}}{1} = \dfrac{y - 1}{-2} = \dfrac{z - 2}{1}$ is 1, then the sum of all possible values of $\lambda$ is :

A bag contains 8 balls, whose colours are either white or black. 4 balls are drawn at random without replacement and it was found that 2 balls are white and other 2 balls are black. The probability that the bag contains equal number of white and black balls is :

Let the median and the mean deviation about the median of 7 observation 170, 125, 230, 190, 210, a, b be 170 and $\dfrac{205}{7}$ respectively. Then the mean deviation about the mean of these 7 observations is :

Let $\vec{a} = -5\hat{i} + \hat{j} - 3\hat{k}$, $\vec{b} = \hat{i} + 2\hat{j} - 4\hat{k}$ and $\vec{c} = \left(\left(\left(\vec{a} \times \vec{b}\right) \times \hat{i}\right) \times \hat{i}\right) \times \hat{i}$. Then $\vec{c} \cdot \left(-\hat{i} + \hat{j} + \hat{k}\right)$ is equal to :

If $\tan A = \dfrac{1}{\sqrt{x(x^2 + x + 1)}}$, $\tan B = \dfrac{\sqrt{x}}{\sqrt{x^2 + x + 1}}$ and

$\tan C = \left(x^{-3} + x^{-2} + x^{-1}\right)^{1/2}$, $0 < A, B, C < \dfrac{\pi}{2}$, then $A + B$ is equal to :

Let $A = \{1, 2, 3, \ldots, 20\}$. Let $R_1$ and $R_2$ two relation on $A$ such that $R_1 = \{(a, b) : b \text{ is divisible by } a\}$ $R_2 = \{(a, b) : a \text{ is an integral multiple of } b\}$. Then, number of elements in $R_1 - R_2$ is equal to ________.

Let $P = \{z \in \mathbb{C} : |z + 2 - 3i| \le 1\}$ and $Q = \{z \in \mathbb{C} : z(1 + i) + \bar{z}(1 - i) \le -8\}$. Let in $P \cap Q$, $|z - 3 + 2i|$ be maximum and minimum at $z_1$ and $z_2$ respectively. If $|z_1|^2 + 2|z_2|^2 = \alpha + \beta\sqrt{2}$, where $\alpha, \beta$ are integers, then $\alpha + \beta$ equals ________.

If the Coefficient of $x^{30}$ in the expansion of $\left(1 + \dfrac{1}{x}\right)^6 \left(1 + x^2\right)^7 \left(1 - x^3\right)^8$; $x \neq 0$ is $\alpha$, then $|\alpha|$ equals ________.

The number of elements in the set $S = \{(x, y, z) : x, y, z \in \mathbb{Z},\ x + 2y + 3z = 42,\ x, y, z \ge 0\}$ equals ________.

Let $3, 7, 11, 15, \ldots, 403$ and $2, 5, 8, 11, \ldots, 404$ be two arithmetic progressions. Then the sum, of the common terms in them, is equal to ________.

Let $\{x\}$ denote the fractional part of $x$ and $f(x) = \dfrac{\cos^{-1}\left(1 - \{x\}^2\right)\sin^{-1}(1 - \{x\})}{\{x\} - \{x\}^3}$, $x \neq 0$. If L and R respectively denotes the left hand limit and the right hand limit of $f(x)$ at $x = 0$, then $\dfrac{32}{\pi^2}\left(\mathrm{L}^2 + \mathrm{R}^2\right)$ is equal to ________.

If $\displaystyle\int_{-\pi/2}^{\pi/2} \dfrac{8\sqrt{2}\,\cos x\, dx}{\left(1 + e^{\sin x}\right)\left(1 + \sin^4 x\right)} = \alpha\pi + \beta \log_e\left(3 + 2\sqrt{2}\right)$, where $\alpha, \beta$ are integers, then $\alpha^2 + \beta^2$ equals ________.

If $x = x(t)$ is the solution of the differential equation $(t + 1)dx = \left(2x + (t + 1)^4\right)dt$, $x(0) = 2$, then, $x(1)$ equals ________.

Let the line $L: \sqrt{2}x + y = \alpha$ pass through the point of the intersection P (in the first quadrant) of the circle $x^2 + y^2 = 3$ and the parabola $x^2 = 2y$. Let the line L touch two circles $C_1$ and $C_2$ of equal radius $2\sqrt{3}$. If the centres $Q_1$ and $Q_2$ of the circles $C_1$ and $C_2$ lie on the $y$-axis, then the square of the area of the triangle $PQ_1Q_2$ is equal to ________.

Let the line of the shortest distance between the lines

$$L_1: \vec{r} = \left(\hat{i} + 2\hat{j} + 3\hat{k}\right) + \lambda\left(\hat{i} - \hat{j} + \hat{k}\right)$$ and

$$L_2: \vec{r} = \left(4\hat{i} + 5\hat{j} + 6\hat{k}\right) + \mu\left(\hat{i} + \hat{j} - \hat{k}\right)$$

intersect $L_1$ and $L_2$ at P and Q respectively. If $(\alpha, \beta, \gamma)$ is the mid point of the line segment PQ, then $2(\alpha + \beta + \gamma)$ is equal to ________.

The dimensional formula of angular impulse is :

A particle moving in a circle of radius R with uniform speed takes time T to complete one revolution. If this particle is projected with the same speed at an angle $\theta$ to the horizontal, the maximum height attained by it is equal to 4R. The angle of projection $\theta$ is then given by :

Consider a block and trolley system as shown in figure. If the coefficient of kinetic friction between the trolley and the surface is 0.04, the acceleration of the system in $\mathrm{ms^{-2}}$ is :

(Consider that the string is massless and unstretchable and the pulley is also massless and frictionless) :

[Figure: a 20 kg trolley on a horizontal surface (kinetic friction $f_k$ acting on it) connected by a string over a pulley at the table edge to a hanging 6 kg block with a 60 N downward force marked]

A ball of mass 0.5 kg is attached to a string of length 50 cm. The ball is rotated on a horizontal circular path about its vertical axis. The maximum tension that the string can bear is 400 N. The maximum possible value of angular velocity of the ball in rad/s is, :

A simple pendulum of length 1 m has a wooden bob of mass 1 kg. It is struck by a bullet of mass $10^{-2}$ kg moving with a speed of $2 \times 10^{2}\ \mathrm{ms^{-1}}$. The bullet gets embedded into the bob. The height to which the bob rises before swinging back is. (use $g = 10\ \mathrm{m/s^2}$)

If R is the radius of the earth and the acceleration due to gravity on the surface of earth is $g = \pi^2\ \mathrm{m/s^2}$, then the length of the second's pendulum at a height h = 2R from the surface of earth will be, :

With rise in temperature, the Young's modulus of elasticity :

The pressure and volume of an ideal gas are related as $PV^{\frac{3}{2}} = K$ (Constant). The work done when the gas is taken from state A $(P_1, V_1, T_1)$ to state B $(P_2, V_2, T_2)$ is :

Two moles a monoatomic gas is mixed with six moles of a diatomic gas. The molar specific heat of the mixture at constant volume is :

Two identical capacitors have same capacitance C. One of them is charged to the potential V and other to the potential 2V. The negative ends of both are connected together. When the positive ends are also joined together, the decrease in energy of the combined system is :

The reading in the ideal voltmeter (V) shown in the given circuit diagram is :

[Figure: a closed loop containing, in the top branch, three cells of 5V each with internal resistance 0.2 Ω each; in the left branch one 5V cell with 0.2 Ω; in the bottom branch three 5V cells with 0.2 Ω each; on the right, a 5V cell with 0.2 Ω connected across an ideal voltmeter (V)]

A galvanometer has a resistance of 50 $\Omega$ and it allows maximum current of 5 mA. It can be converted into voltmeter to measure upto 100 V by connecting in series a resistor of resistance :

In series LCR circuit, the capacitance is changed from C to 4C. To keep the resonance frequency unchanged, the new inductance should be :

A parallel plate capacitor has a capacitance C = 200 pF. It is connected to 230 V ac supply with an angular frequency 300 rad/s. The rms value of conduction current in the circuit and displacement current in the capacitor respectively are :

A monochromatic light of wavelength 6000$\text{Å}$ is incident on the single slit of width 0.01 mm. If the diffraction pattern is formed at the focus of the convex lens of focal length 20 cm, the linear width of the central maximum is :

The de Broglie wavelengths of a proton and an $\alpha$ particle are $\lambda$ and $2\lambda$ respectively. The ratio of the velocities of proton and $\alpha$ particle will be :

The minimum energy required by a hydrogen atom in ground state to emit radiation in Balmer series is nearly :

In the given circuit if the power rating of Zener diode is 10 mW, the value of series resistance $R_s$ to regulate the input unregulated supply is :

[Figure: an unregulated supply $V_s = 8\,\mathrm{V}$ in series with resistor $R_s$, feeding a Zener diode ($V_z = 5\,\mathrm{V}$) in parallel with a load resistor $R_L = 1\,\mathrm{k\Omega}$]

10 divisions on the main scale of a Vernier calliper coincide with 11 divisions on the Vernier scale. If each division on the main scale is of 5 units, the least count of the instrument is :

The radius (r), length ($l$) and resistance (R) of a metal wire was measured in the laboratory as

$r = (0.35 \pm 0.05)$ cm

$R = (100 \pm 10)$ ohm

$l = (15 \pm 0.2)$ cm

The percentage error in resistivity of the material of the wire is :

A particle is moving in one dimension (along $x$ axis) under the action of a variable force. It's initial position was 16 m right of origin. The variation of its position ($x$) with time (t) is given as $x = -3t^3 + 18t^2 + 16t$, where $x$ is in m and t is in s. The velocity of the particle when its acceleration becomes zero is ________ m/s.

The identical spheres each of mass 2M are placed at the corners of a right angled triangle with mutually perpendicular sides equal to 4 m each. Taking point of intersection of these two sides as origin, the magnitude of position vector of the centre of mass of the system is $\dfrac{4\sqrt{2}}{x}$, where the value of $x$ is ________

A plane is in level flight at constant speed and each of its two wings has an area of 40 $\mathrm{m^2}$. If the speed of the air is 180 km/h over the lower wing surface and 252 km/h over the upper wing surface, the mass of the plane is ________ kg. (Take air density to be 1 $\mathrm{kg\ m^{-3}}$ and $g = 10\ \mathrm{ms^{-2}}$)

A tuning fork resonates with a sonometer wire of length 1 m stretched with a tension of 6 N. When the tension in the wire is changed to 54 N, the same tuning fork produces 12 beats per second with it. The frequency of the tuning fork is ________ Hz.

Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle $\theta$ with each other. When suspended in water the angle remains the same. If density of the material of the sphere is 1.5 g/cc, the dielectric constant of water will be ________ (Take density of water = 1 g/cc)

The current in a conductor is expressed as $I = 3t^2 + 4t^3$, where I is in Ampere and t is in second. The amount of electric charge that flows through a section of the conductor during t = 1s to t = 2s is ________ C.

A regular polygon of 6 sides is formed by bending a wire of length $4\pi$ meter. If an electric current of $4\pi\sqrt{3}$ A is flowing through the sides of the polygon, the magnetic field at the centre of the polygon would be $x \times 10^{-7}$ T. The value of $x$ is ________.

A rectangular loop of sides 12 cm and 5 cm, with its sides parallel to the $x$-axis and $y$-axis respectively, moves with a velocity of 5 cm/s in the positive $x$ axis direction, in a space containing a variable magnetic field in the positive $z$ direction. The field has a gradient of $10^{-3}$ T/cm along the negative $x$ direction and it is decreasing with time at the rate of $10^{-3}$ T/s. If the resistance of the loop is 6 m$\Omega$, the power dissipated by the loop as heat is ________ $\times 10^{-9}$ W.

The distance between object and its 3 times magnified virtual image as produced by a convex lens is 20 cm. The focal length of the lens used is ________ cm.

The radius of a nucleus of mass number 64 is 4.8 fermi. Then the mass number of another nucleus having radius of 4 fermi is $\dfrac{1000}{x}$, where $x$ is ________.

According to the wave-particle duality of matter by de-Broglie, which of the following graph plot presents most appropriate relationship between wavelength of electron($\lambda$) and momentum of electron (p) ?

Arrange the bonds in order of increasing ionic character in the molecules. $\mathrm{LiF}$, $\mathrm{K_2O}$, $\mathrm{N_2}$, $\mathrm{SO_2}$ and $\mathrm{ClF_3}$ :

Choose the correct option for free expansion of an ideal gas under adiabatic condition from the following :

We have three aqueous solutions of $\mathrm{NaCl}$ labelled as 'A', 'B' and 'C' with concentration 0.1 M, 0.01 M and 0.001 M, respectively. The value of van 't Hoff factor(i) for these solutions will be in the order :

In case of isoelectronic species the size of $\mathrm{F^-}$, $\mathrm{Ne}$ and $\mathrm{Na^+}$ is affected by :

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

Assertion (A) : $\mathrm{PH_3}$ has lower boiling point than $\mathrm{NH_3}$.

Reason (R) : In liquid state $\mathrm{NH_3}$ molecules are associated through vander Waal's forces, but $\mathrm{PH_3}$ molecules are associated through hydrogen bonding.

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

In acidic medium, $\mathrm{K_2Cr_2O_7}$ shows oxidising action as represented in the half reaction :

$$\mathrm{Cr_2O_7^{2-} + XH^+ + Ye^{\ominus} \rightarrow 2A + ZH_2O}$$

X, Y, Z and A are respectively are :

Which of the following reactions are disproportionation reactions ?

(A) $\mathrm{Cu^+ \rightarrow Cu^{2+} + Cu}$

(B) $\mathrm{3MnO_4^{2-} + 4H^+ \longrightarrow 2MnO_4^- + MnO_2 + 2H_2O}$

(C) $\mathrm{2KMnO_4 \longrightarrow K_2MnO_4 + MnO_2 + O_2}$

(D) $\mathrm{2MnO_4^- + 3Mn^{2+} + 2H_2O \longrightarrow 5MnO_2 + 4H^+}$

Choose the correct answer from the options given below :

Given below are two statements :

Statement (I) : A solution of $\mathrm{[Ni(H_2O)_6]^{2+}}$ is green in colour.

Statement (II) : A solution of $\mathrm{[Ni(CN)_4]^{2-}}$ is colourless.

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

Which of the following complex is homoleptic ?

In Kjeldahl's method for estimation of nitrogen, $\mathrm{CuSO_4}$ acts as :

Given below are two statements :

Statement (I) : Aminobenzene and aniline are same organic compounds.

Statement (II) : Aminobenzene and aniline are different organic compounds.

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

Ionic reactions with organic compounds proceed through :

(A) homolytic bond cleavage

(B) heterolytic bond cleavage

(C) free radical formation

(D) primary free radical

(E) secondary free radical

Choose the correct answer from the options given below :

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

Assertion (A) : Haloalkanes react with KCN to form alkyl cyanides as a main product while with AgCN form isocyanide as the main product.

Reason (R) : KCN and AgCN both are highly ionic compounds.

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

Identify A and B in the following sequence of reaction

$$\mathrm{C_6H_5CH_3} \xrightarrow{\mathrm{Cl_2}/h\nu} \mathrm{A} \xrightarrow[\text{373 K}]{\mathrm{H_2O}} \mathrm{B}$$

[Figure: toluene (benzene ring with CH$_3$) treated with Cl$_2$/h$\nu$ to give A, then H$_2$O at 373 K to give B]

Match List - I with List - II.

List - I (Reactions)	List - II (Reagents)
(A) $\mathrm{CH_3(CH_2)_5\overset{\displaystyle O}{\underset{\phantom{O}}{\overset{\|}{C}}}\!-OC_2H_5 \longrightarrow CH_3(CH_2)_5CHO}$	(I) $\mathrm{CH_3MgBr,\ H_2O}$
(B) $\mathrm{C_6H_5COC_6H_5 \rightarrow C_6H_5CH_2C_6H_5}$	(II) $\mathrm{Zn(Hg)}$ and conc. $\mathrm{HCl}$
(C) $\mathrm{C_6H_5CHO \rightarrow C_6H_5CH(OH)CH_3}$	(III) $\mathrm{NaBH_4,\ H^+}$
(D) $\mathrm{CH_3COCH_2COOC_2H_5 \rightarrow CH_3\underset{\displaystyle H}{C}(OH)CH_2COOC_2H_5}$	(IV) DIBAL-H, $\mathrm{H_2O}$

Choose the correct answer from the options given below :

Given below are two statements :

Statement (I) : The $\mathrm{NH_2}$ group in Aniline is ortho and para directing and a powerful activating group.

Statement (II) : Aniline does not undergo Friedel-Craft's reaction (alkylation and acylation).

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

Which of the following compound will most easily be attacked by an electrophile ?

If one strand of a DNA has the sequence ATGCTTCA, sequence of the bases in complementary strand is :

Given below are two statements :

Statement (I) : Potassium hydrogen phthalate is a primary standard for standardisation of sodium hydroxide solution.

Statement (II) : In this titration phenolphthalein can be used as indicator.

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

Consider the following reaction :

$$\mathrm{3PbCl_2 + 2(NH_4)_3PO_4 \rightarrow Pb_3(PO_4)_2 + 6NH_4Cl}$$

If 72 mmol of $\mathrm{PbCl_2}$ is mixed with 50 mmol of $\mathrm{(NH_4)_3PO_4}$, then the amount of $\mathrm{Pb_3(PO_4)_2}$ formed is ________ mmol (nearest integer).

The number of molecules/ion/s having trigonal bipyramidal shape is ________.

$\mathrm{PF_5}$, $\mathrm{BrF_5}$, $\mathrm{PCl_5}$, $\mathrm{[PtCl_4]^{2-}}$, $\mathrm{BF_3}$, $\mathrm{Fe(CO)_5}$

$\mathrm{K_a}$ for $\mathrm{CH_3COOH}$ is $1.8 \times 10^{-5}$ and $\mathrm{K_b}$ for $\mathrm{NH_4OH}$ is $1.8 \times 10^{-5}$. The pH of ammonium acetate solution will be ________.