JEE Main 2024 — 1 February, Shift 1

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:

The value of the integral $\int_{0}^{\frac{\pi}{4}} \frac{x d x}{\sin ^{4}(2 x)+\cos ^{4}(2 x)}$ equals:

If $\mathrm{A}=\left[\begin{array}{cc}\sqrt{2} & 1\\ -1 & \sqrt{2}\end{array}\right], \mathrm{B}=\left[\begin{array}{ll}1 & 0\\ 1 & 1\end{array}\right], \mathrm{C}=\mathrm{ABA}^{\mathrm{T}}$ and X $=A^{T} C^{2} A$, then $\operatorname{det} \mathrm{X}$ is equal to :

If $\tan \mathrm{A}=\frac{1}{\sqrt{x\left(x^{2}+x+1\right)}}, \tan B=\frac{\sqrt{x}}{\sqrt{x^{2}+x+1}}$ and $\tan C=\left(x^{-3}+x^{-2}+x^{-1}\right)^{\frac{1}{2}},$ 0$<A,B,C<\frac{\pi }{2},\,then \,A+B$ 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:

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

Let the median and the mean deviation about the median of 7 observation$170,125,230,190,210, a, b$ be 170 and $\frac{205}{7}$ 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 $\vec{c}=(((\vec{a} \times \vec{b}) \times \hat{i}) \times \hat{i}) \times \hat{i}$. Then $\vec{c} \cdot(-\hat{i}+\hat{j}+\hat{k})$ is equal to

Let $S=\left\{x \in R:(\sqrt{3}+\sqrt{2})^{x}+(\sqrt{3}-\sqrt{2})^{x}=10\right\}$

Then the number of elements in S is :

The area enclosed by the curves $x y+4 y=16$ and $\mathrm{x}+\mathrm{y}=6$ is equal to :

Let $\mathrm{f}: \mathbf{R} \rightarrow \mathbf{R}$ and $\mathrm{g}: \mathbf{R} \rightarrow \mathbf{R}$ be defined

If the system of equations

$2 x+3 y-z=5$

$x+\alpha y+3 z=-4$

$3 x-y+\beta z=7$

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

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 $\mathrm{y}=\mathrm{y}(\mathrm{x})$ be the solution of the differential equation

$\frac{d y}{d x}=2 x(x+y)^{3}-x(x+y)-1, y(0)=1$. Then, $\left(\frac{1}{\sqrt{2}}+\mathrm{y}\left(\frac{1}{\sqrt{2}}\right)\right)^{2}$ equals :

Let $f: R \rightarrow R$ be defined as $f(x)=\left\{\begin{array}{ccc}\frac{a-b \cos 2 x}{x^{2}} & ; & x<0\\ x^{2}+c x+2 & ; & 0 \leq x \leq 1\\ 2 x+1 & ; & x>1\end{array}\right.$ If $f$ is continuous everywhere in $\mathbf{R}$ and m is the number of points where $f$ is NOT differential then $\mathrm{m}+\mathrm{a}+\mathrm{b}+\mathrm{c}$ equals :

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

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 :

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

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

If the shortest distance between the lines $\frac{x-\lambda}{-2}=\frac{y-2}{1}=\frac{z-1}{1}$ and

$\frac{x-\sqrt{3}}{1}=\frac{y-1}{-2}=\frac{z-2}{1}$ is 1 , then the sum of all possible values of $\lambda$ is :

If $x=x(t)$ is the solution of the differential equation $(t+1) d x=\left(2 x+(t+1)^{4}\right) d t, x(0)=2$, then, $x(1)$ equals $\qquad$

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

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

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 $\qquad$ .

Let $\{x\}$ denote the fractional part of $x$ and

$f(x)=\frac{\cos ^{-1}\left(1-\{x\}^{2}\right) \sin ^{-1}(1-\{x\})}{\{x\}-\{x\}^{3}}, x \neq 0$.

If $L$ and R respectively denotes the left hand limit and the right hand limit of $f(x)$ at $x=0$, then $\frac{32}{\pi^{2}}\left(L^{2}+R^{2}\right)$ is equal to $\qquad$ .

Let the line $L: \sqrt{2} \mathrm{x}+\mathrm{y}=\alpha$ pass through the point of the intersection P (in the first quadrant) of the circle $x^{2}+y^{2}=3$ and the parabola $x^{2}=2 y$. Let the line $L$ touch two circles $C_{1}$ and $C_{2}$ of equal radius $2 \sqrt{3}$. If the centres $\mathrm{Q}_{1}$ and $\mathrm{Q}_{2}$ of the circles $\mathrm{C}_{1}$ and $\mathrm{C}_{2}$ lie on the $y$-axis, then the square of the area of the triangle $\mathrm{PQ}_{1} \mathrm{Q}_{2}$ is equal to $\qquad$ .

Let $\mathrm{P}=\{\mathrm{z} \in \mathbb{C}:|\mathrm{z}+2-3 \mathrm{i}| \leq 1\}$ and $\mathrm{Q}=\{\mathrm{z} \in \mathbb{C}: \mathrm{z}(1+\mathrm{i})+\overline{\mathrm{z}}(1-\mathrm{i}) \leq-8\}$. Let in $\mathrm{P} \cap \mathrm{Q},|\mathrm{z}-3+2 \mathrm{i}|$ be maximum and minimum at $z_{1}$ and $z_{2}$ respectively. If $\left|z_{1}\right|^{2}+2|z|^{2}=\alpha+\beta \sqrt{2}$, where $\alpha, \beta$ are integers, then $\alpha+\beta$ equals $\qquad$ ・

If $\int_{-\pi / 2}^{\pi / 2} \frac{8 \sqrt{2} \cos x d x}{\left(1+e^{\sin x}\right)\left(1+\sin ^{4} x\right)}=\alpha \pi+\beta \log _{e}(3+2$ $\sqrt{2}$ ), where $\alpha, \beta$ are integers, then $\alpha^{2}+\beta^{2}$ equals $\qquad$ .

Let the line of the shortest distance between the lines

$\mathrm{L}_{1}: \overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})+\lambda(\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})$

and $\mathrm{L}_{2}: \overrightarrow{\mathrm{r}}=(4 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}+6 \hat{\mathrm{k}})+\mu(\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}})$

intersect $L_{1}$ and $L_{2}$ at $P$ and $Q$ respectively.

If $(\alpha, \beta, \gamma)$

is the midpoint of the line segment PQ , then $2(\alpha+\beta+\gamma)$ is equal to $\qquad$ .

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

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

he reading in the ideal voltmeter $(\mathrm{V})$ shown in the given circuit diagram is :

Two identical capacitors have same capacitance C. One of them is charged to the potential V and other to the potential 2 V . 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

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 :

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 $\mathrm{rad} / \mathrm{s}$ is,:

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

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

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 :

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 :

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

The radius (r), length $(l)$ and resistance (R) of a metal wire was measured in the laboratory as $\mathrm{r}=(0.35 \pm 0.05) \mathrm{cm}$ $\mathrm{R}=(100 \pm 10)$ ohm $l=(15 \pm 0.2) \mathrm{cm}$ The percentage error in resistivity of the material of the wire is :

The dimensional formula of angular impulse is :

A simple pendulum of length 1 m has a wooden bob of mass 1 kg . It is struck by a bullet of mass $10^{-2} \mathrm{~kg}$ 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 $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$ )

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 $4 R$. The angle of projection $\theta$ is then given by :

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

A monochromatic light of wavelength $6000\,\mathop{\mathrm{A}}\limits^{\circ}$ 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 :

A regular polygon of 6 sides is formed by bending a wire of length $4 \pi$ meter. If an electric current of $4 \pi \sqrt{3} \mathrm{~A}$ is flowing through the sides of the polygon, the magnetic field at the centre of the polygon would be $\mathrm{x} \times 10^{-7} \mathrm{~T}$. 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 $5 \mathrm{~cm} / \mathrm{s}$ 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 $10^{-3} \mathrm{~T} / \mathrm{cm}$ along the negative x direction and it is decreasing with time at the rate of $10^{-3} \mathrm{~T} / \mathrm{s}$. If the resistance of the loop is $6 \mathrm{~m} \Omega$, the power dissipated by the loop as heat is $\_\_\_\_\_\_\_$ $\times 10^{-9} \mathrm{~W}$.

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 $\qquad$ cm .

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 $1.5 \mathrm{~g} / \mathrm{cc}$, the dielectric constant of water will be $\_\_\_\_\_\_\_\_\_\_\_$ (Take density of water $=1 \mathrm{~g} / \mathrm{cc})$

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 $\frac{1000}{x}$, where x is $\qquad$ .

The identical spheres each of mass 2 M 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 $\frac{4 \sqrt{2}}{x}$, where the In air $\tan \frac{\theta}{2}=\frac{F}{m g}=\frac{q^{2}}{4 \pi \varepsilon_{0} r^{2} m g}$ value of $x$ is $\_\_\_\_\_\_\_$

A tuning fork resonates with a sonometer wire of length 1m stitched with a tension of 6N. 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.

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 $180 \mathrm{~km} / \mathrm{h}$ over the lower wing surface and $252 \mathrm{~km} / \mathrm{h}$ over the upper wing surface, the mass of the plane is $\_\_\_\_\_\_\_$ kg. (Take air density to be $1 \mathrm{~kg} \mathrm{~m}^{-3}$ and $\mathrm{g}=10 \mathrm{~ms}^{-2}$ )

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

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 $(\mathrm{t})$ is given as $x=-3 t^{3}+18 t^{2}+16 t$, where $x$ is in $m$ and $t$ is in $s$. The velocity of the particle when its acceleration becomes zero is $\_\_\_\_\_\_\_$ $\mathrm{m} / \mathrm{s}$.

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

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 statement, choose the most appropriate answer from the options given below:

In acidic medium, $\mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7}$ shows oxidising action as represented in the half reaction

$\mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}+\mathrm{XH}^{+}+\mathrm{Ye}^{-} \rightarrow 2 \mathrm{~A}+\mathrm{ZH}_{2} \mathrm{O}$ $\mathrm{X}, \mathrm{Y}, \mathrm{Z}$ and A are respectively are:

Which of the following reactions are disproportionation reactions?

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

(B) $3 \mathrm{MnO}_{4}^{2-}+4 \mathrm{H}^{+} \rightarrow 2 \mathrm{MnO}_{4}^{-}+\mathrm{MnO}_{2}+2 \mathrm{H}_{2} \mathrm{O}$

(C) $2 \mathrm{KMnO}_{4} \rightarrow \mathrm{~K}_{2} \mathrm{MnO}_{4}+\mathrm{MnO}_{2}+\mathrm{O}_{2}$

(D) $2 \mathrm{MnO}_{4}^{-}+3 \mathrm{Mn}^{2+}+2 \mathrm{H}_{2} \mathrm{O} \rightarrow 5 \mathrm{MnO}_{2}+4 \mathrm{H}^{+}$

Choose the correct answer from the options given below:

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

According to de-Broglie wave-particle duality, which graph correctly represents the relation between wavelength $\lambda$ and momentum $p$?

Given below are two statements, one is labelled as Statement I and the other is labelled as Statement II.

Statement I: A solution of $\left[\mathrm{Ni}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+}$ is green in colour.

Statement II: A solution of $\left[\mathrm{Ni}(\mathrm{CN})_{4}\right]^{2-}$ is colourless.

In the light of the above statements, choose the most appropriate 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) : $\mathrm{PH}_{3}$ 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:

Given below are two statements , one is labelled as Statement I and the other is labelled as Statement II.

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:

Which of the following complex is homoleptic?

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

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:

Arrange the bonds in order of increasing ionic character in the molecules. $\mathrm{LiF}, \mathrm{K}_{2} \mathrm{O}, \mathrm{N}_{2}, \mathrm{SO}_{2}$ and $\mathrm{CIF}_{3}$.

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

In Kjeldahl's method for estimation of nitrogen, $\mathrm{CuSO}_{4}$ acts as :

Given below are two statements, one is labelled as Statement I and the other is labelled as Statement II.

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:

Match List - I with List -II

Choose the correct answer from options given below:

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

Given below are two statements, one is labelled as Statement I and the other is labelled as Statement II.

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 :

The number of white coloured salts among the following is0

(A) $\mathrm{SrSO}_{4}$ (B) $\mathrm{Mg}\left(\mathrm{NH}_{4}\right) \mathrm{PO}_{4}$