NTA JEE Mains 24th Jan 2025 Shift 1

Let circle $$C$$ be the image of $$x^2 + y^2 - 2x + 4y - 4 = 0$$ in the line $$2x - 3y + 5 = 0$$ and $$A$$ be the point on $$C$$ such that $$OA$$ is parallel to the $$x$$-axis and $$A$$ lies on the right hand side of the centre $$O$$ of $$C$$. If $$B(\alpha, \beta)$$, with $$\beta < 4$$, lies on $$C$$ such that the length of the arc $$AB$$ is $$\left(1/6\right)^{\text{th}}$$ of the perimeter of $$C$$, then $$\beta + \sqrt{3}\alpha$$ is equal to

Let in a $$\triangle ABC$$, the length of the side $$AC$$ be $$6$$, the vertex $$B$$ be $$(1, 2, 3)$$ and the vertices $$A, C$$ lie on the line $$ \frac{x - 6}{3} = \frac{y - 7}{2} = \frac{z - 7}{-2}. $$ Then the area (in sq. units) of $$\triangle ABC$$ is:

Let the product of the focal distances of the point $$\left( \sqrt{3}, \frac{1}{2} \right)$$ on the ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad (a > b),$$ be $$\frac{7}{4}.$$ Then the absolute difference of the eccentricities of two such ellipses is

If the system of equations $$\begin{aligned} 2x - y + z &= 4, \\ 5x + \lambda y + 3z &= 12, \\ 100x - 47y + \mu z &= 212 \end{aligned}$$ has infinitely many solutions, then $$\mu - 2\lambda$$ is equal to:

For some $$n \ne 10,$$ let the coefficients of the 5th, 6th and 7th terms in the binomial expansion of $$(1+x)^{n+4}$$ be in A.P. Then the largest coefficient in the expansion of  $$(1+x)^{n+4}$$ is:

The product of all the rational roots of the equation $$(x^2 - 9x + 11)^2 - (x - 4)(x - 5) = 3$$ is equal to:

Let the line passing through the points $$(-1,2,1)$$ and parallel to the line $$\frac{x-1}{2}=\frac{y+1}{3}=\frac{z}{4}$$ intersect the line $$\frac{x+2}{3}=\frac{y-3}{2}=\frac{z-4}{1}$$ at the point $$P.$$ Then the distance of $$P$$ from the point $$Q(4,-5,1)$$ is:

$$ \text{Let the lines } 3x-4y-\alpha=0,\; 8x-11y-33=0,\; \text{ and } 2x-3y+\lambda=0$$ be concurrent. If the image of the point $$(1,2)$$ in the line $$2x-3y+\lambda=0$$ $$\text{ is } \left(\frac{57}{13},-\frac{40}{13}\right), \text{ then } |\alpha\lambda| \text{ is equal to:} $$

$$ \text{If } \alpha \text{ and } \beta \text{ are the roots of the equation } 2z^2-3z-2i=0 ,\; \text{ where } i=\sqrt{-1}, \text{ then }  16\cdot \operatorname{Re}\!\left(\frac{\alpha^{19}+\beta^{19}+\alpha^{11}+\beta^{11}}{\alpha^{15}+\beta^{15}}\right) \cdot \operatorname{Im}\!\left(\frac{\alpha^{19}+\beta^{19}+\alpha^{11}+\beta^{11}}{\alpha^{15}+\beta^{15}}\right) \text{ is equal to:} $$

For a statistical data $$x_1,x_2,\ldots,x_{10}$$ of 10 values, a student obtained the mean as  5.5 and  $$\sum_{i=1}^{10} x_i^2 = 371. $$  He later found that he had noted two values in the data incorrectly as 4 and 5, instead of the correct values  6 and 8  respectively. The variance of the corrected data is:

The area of the region $$\left\{(x,y) : x^2 + 4x + 2 \le y \le |x+2| \right\}$$ is equal to:

Let $$S_n=\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\cdots$$ upto  $$n$$ terms. If the sum of the first six terms of an A.P. with first term  $$-p$$ and common difference $$p$$  is  $$\sqrt{2026\, S_{2025}},$$  then the absolute difference between the 20th and 15th terms of the A.P. is:

$$ \text{Let } f:\mathbb{R}\setminus\{0\}\to\mathbb{R} \text{ be a function such that } f(x)-6f\!\left(\frac{1}{x}\right)=\frac{35}{3x}-\frac{5}{2}. \text{ If } \lim_{x\to 0}\left(\frac{1}{\alpha x}+f(x)\right)=\beta, \; \alpha,\beta\in\mathbb{R}, \text{ then } \alpha+2\beta \text{ is equal to:} $$

$$ \text{If } I(m,n)=\int_{0}^{1} x^{m-1}(1-x)^{\,n-1}\,dx,\quad m,n>0, \text{ then } I(9,14)+I(10,13) \text{ is:} $$

A and B alternately throw a pair of dice. A wins if he throws a sum of 5 before B throws a sum of 8,  and B wins if he throws a sum of 8 before A throws a sum of 5.  The probability that A wins if A makes the first throw, is:

Let $$f(x)=\dfrac{2^{x+2}+16}{2^{2x+1}+2^{x+4}+32}$$. Then the value of $$8\left(f\!\left(\dfrac{1}{15}\right)+f\!\left(\dfrac{2}{15}\right)+\cdots+f\!\left(\dfrac{59}{15}\right)\right)$$ is equal to:

Let  $$y=y(x)$$ be the solution of the differential equation  $$\left(xy-5x^2\sqrt{1+x^2}\right)dx+(1+x^2)dy=0, \quad y(0)=0.$$ Then  $$y(\sqrt{3})$$  is equal to:

$$\lim_{x\to 0}\cosec x \left( \sqrt{2\cos^2 x+3\cos x} - \sqrt{\cos^2 x+\sin x+4} \right)$$ is:

Consider the region $$R=\{(x,y): x \le y \le 9-\tfrac{11}{3}x^2,\; x\ge 0\}.$$  The area of the largest rectangle of sides parallel to the coordinate axes and inscribed in  $$R$$ is:

Let $$\vec a=\hat{i}+2\hat{j}+3\hat{k},  b=3\hat{i}+\hat{j}-\hat{k} $$ and  be three vectors such that  $$c$$ is coplanar with $$ \vec a$$  and  $$\vec b$$. If  $$\vec c $$ is perpendicular to  $$\vec b$$  and  $$\vec a\cdot \vec c=5,$$  then  $$|\vec c|$$  is equal to: 

Let $$ S=\{p_1,p_2,....,p_{10}\} $$ be the set of first ten prime numbers. Let $$ A=S\cup P, $$ where $$ P $$ is the set of all possible products of distinct elements of $$ S. $$ Then the number of all ordered pairs $$ (x,y),\; x\in S,\; y\in A, $$ such that $$ x $$ divides $$ y,$$ is: $$ \underline{ \hspace{2cm} } $$

$$ \text{If for some } \alpha,\beta;\; \alpha\le\beta,\; \alpha+\beta=8$$ and  $$\sec^2(\tan^{-1}\alpha)+\cosec^2(\cot^{-1}\beta)=36,$$ $$\alpha^2+\beta^2$$  is:_______

$$ \text{Let } A \text{ be a } 3\times 3 \text{ matrix such that } X^TAX=0 \text{ for all nonzero } 3\times1 \text{ matrices } X=\begin{bmatrix}x\\y\\z\end{bmatrix}. \text{ If } A\begin{bmatrix}1\\1\\1\end{bmatrix} = \begin{bmatrix}1\\4\\-5\end{bmatrix}, \; A\begin{bmatrix}1\\2\\1\end{bmatrix} = \begin{bmatrix}0\\4\\-8\end{bmatrix}, \text{ and } \det(\operatorname{adj}(2(A+I)))=2^\alpha 3^\beta 5^\gamma, \; \alpha,\beta,\gamma\in\mathbb{N}, \text{ then } \alpha^2+\beta^2+\gamma^2 \text{ is:} \underline{\hspace{2cm}}$$

Let $$f$$ be a differentiable function such that $$2(x+2)^2f(x)-3(x+2)^2 = 10\int_0^x (t+2)f(t)\,dt,\quad x\ge0.$$ Then $$f(2)$$ is equal to: $$\underline{\hspace{1cm}}$$

The number of 3-digit numbers that are divisible by 2 and 3, but not divisible by 4 and 9, is_______

During the transition of electron from state A to state C of a Bohr atom,  the wavelength of emitted radiation is  2000 A˚  and it becomes  6000 A˚  when the electron jumps from state B to state C.  Then the wavelength of the radiation emitted during the transition of electrons  from state A to state B is:

Consider the following statements: A. The junction area of solar cell is made very narrow compared to a photo diode. B. Solar cells are not connected with any external bias. C. LED is made of lightly doped p-n junction. D. Increase of forward current results in continuous increase of LED light intensity. E. LEDs have to be connected in forward bias for emission of light. Choose the correct answer from the options given below:

$$ \text{An alternating current is given by } I=I_A\sin\omega t + I_B\cos\omega t. \text{ The r.m.s. current will be:} $$

A car of mass  $$m$$ moves on a banked road having radius  $$' r '$$ and banking angle  $$\theta.$$ To avoid slipping from the banked road, the maximum permissible speed of the car is  $$v_0.$$  The coefficient of friction $$\mu$$ between the wheels of the car and the banked road is:

A satellite is launched into a circular orbit of radius $$R$$ around the earth. A second satellite is launched into an orbit of radius $$1.03R.$$ The time period of revolution of the second satellite is larger than the first one approximately by:

An ideal gas goes from an initial state to final state. During the process, the pressure of gas increases linearly with temperature.

A. The work done by gas during the process is zero. 

B. The heat added to gas is different from change in its internal energy. 

C. The volume of the gas is increased. 

D. The internal energy of the gas is increased. 

E. The process is isochoric (constant volume process) Choose the correct answer from the options given below:

An electron of mass $$m$$ with an initial velocity  $$\vec v=v_0\hat{i}\;(v_0>0)$$ enters an electric field $$\vec E=-E_0\hat{k}.$$ If the initial de Broglie wavelength is  $$\lambda_0,$$ the value after time $$t$$ would be:

What is the relative decrease in focal length of a lens for an increase in optical power  by $$0.1\,D$$ from  $$2.5\,D? \quad [\text{'D' stands for dioptre}] $$

A force $$F=\alpha+\beta x^2$$ acts on an object in the $$x$$-direction. The work done by the force is  $$5\,J$$ when the object is displaced by $$1\,m.$$ If the constant $$\alpha=1\,N$$  then $$\beta$$ will be:

A thin plano convex lens made of glass of refractive index 1.5 is immersed in a liquid of refractive index 1.2. When the plane side of the lens is silver coated for complete reflection, the lens immersed in the liquid behaves like a concave mirror of focal length 0.2 m. The radius of curvature of the curved surface of the lens is

A particle is executing simple harmonic motion with time period $$2\,s$$ and amplitude $$1\,cm.$$ If $$D$$ and $$d$$ are the total distance and displacement covered by the particle in  $$12.5\,s,$$ then  $$\frac{D}{d}$$ is:

The amount of work done to break a big water drop of radius  $$' R '$$  into  27  small drops of equal radius is  $$10\,J.$$ The work done required to break the same big drop into  64  small drops of equal radius will be:

A plano-convex lens having radius of curvature of first surface  $$2\,cm$$ exhibits focal length $$f_1$$ in air. Another plano-convex lens with first surface radius of curvature  $$3\,cm$$ has focal length $$f_2$$ when it is immersed in a liquid of refractive index  $$1.2.$$ If both the lenses are made of same glass of refractive index  $$1.5,$$ then the ratio $$f_1:f_2$$ will be:

An air bubble of radius $$0.1\,cm$$ lies at a depth of $$20\,cm$$ below the free surface of a liquid of density $$1000\,kg/m^3.$$ If the pressure inside the bubble is  $$2100\,N/m^2$$ greater than the atmospheric pressure, then the surface tension of the liquid in SI unit is  $$(g=10\,m/s^2):$$

A uniform solid cylinder of mass  $$m$$ and radius $$r$$ rolls along an inclined rough plane of inclination $$45^\circ.$$ If it starts to roll from rest from the top of the plane, then the linear acceleration of the cylinder's axis will be: 

The Young's double slit interference experiment is performed using light  consisting of  $$480\,nm$$ and  $$600\,nm$$ wavelengths. The least number of the bright fringes of  $$480\,nm$$  light that are  required for the first coincidence with the bright fringes formed by  $$600\,nm$$ light is:

A parallel plate capacitor was made with two rectangular plates, each with length $$l=3\,cm$$ and breadth $$b=1\,cm.$$ The distance between the plates is$$ 3\,\mu m.$$ Out of the following, which are the ways to increase the capacitance by a factor of $$10?$$  A. $$l=30cm,$$ $$b=1cm,$$ $$d=1\mu$$ $$m$$ $$B.$$ $$l=3cm,$$  $$b=1cm,$$ $$d=30\mu  m  C.$$ $$l=6cm,$$ $$b=5cm,$$ $$d=3\mu$$ $$m  D.$$ $$l=1cm,$$ $$b=1cm, d=10\mu\ m  E.$$ $$l=5cm,$$ $$b=2cm,$$  $$d=1\mu m$$ Choose the correct answer from the options given below:

Consider a parallel plate capacitor of area  $$A$$  (of each plate)  and separation  $$d$$ between the plates. If $$E$$ is the electric field and $$\varepsilon_0$$ is the permittivity of free space between the plates, then potential energy stored in the capacitor is: 

An object of mass $$m$$ is projected from origin in a vertical  $$xy$$ plane at an angle $$45^\circ$$ with the $$x$$ -axis with an initial velocity $$v_0.$$ The magnitude and direction of the angular momentum of the object  with respect to origin, when it reaches at the maximum height, will be  $$[g$$  is acceleration due to gravity]

For an experimental expression $$y=\frac{32.3\times1125}{27.4},$$ where all the digits are significant. Then to report the value of  $$y$$  we should write

A current of  $$5\,A$$  exists in a square loop of side  $$\frac{1}{\sqrt{2}}\,m.$$ Then the magnitude of the magnetic field  $$B$$ at the centre of the square loop will be $$p \times10^{-6}\,T,$$ where value of $$p$$  is  $$\underline{\hspace{2cm}}.$$ $$\left[\mu_0=4\pi\times10^{-7}\,TmA^{-1}\right] $$

A square loop of sides $$a=1\,m$$ is held normally in front of a point charge $$q=1\,C.$$ The flux of the electric field through the shaded region is $$\frac{5}{p}\times\frac{1}{\varepsilon_0}\,Nm^2C^{-1}$$, where the value of $$p$$ is $$\underline{\hspace{1cm}}.$$

The temperature of 1 mole of an ideal monoatomic gas is increased by $$50^\circ C$$ at constant pressure. The total heat added and change in internal energy are $$E_1$$ and $$E_2,$$  respectively. If  $$\frac{E_1}{E_2}=\frac{x}{9},$$  then the value of x  is  $$\underline{\hspace{2cm}}.$$

The least count of a screw gauge is $$0.01\,mm.$$ If the pitch is increased by $$75\%$$ and number of divisions on the circular scale is reduced by $$50\%,$$ the new least count will be $$\underline{\hspace{2cm}}\times10^{-3}\,mm.$$

A wire of resistance $$9\,\Omega$$ is bent to form an equilateral triangle. Then the equivalent resistance across any two vertices will be  $$\underline{\hspace{2cm}}\,\Omega.$$

The carbohydrate "Ribose" present in DNA, is A. A pentose sugar B. present in pyranose from C. in "D" configuration D. a reducing sugar, when free E. in $$\alpha$$-anomeric form Choose the correct answer from the options given below:

Given below are two statements: Statement I: The conversion proceeds well in the less polar medium.
$$\mathrm{CH_3-CH_2-CH_2-CH_2-Cl} \;\xrightarrow{\;\;HO^-\;\;} \mathrm{CH_3-CH_2-CH_2-CH_2-OH} + \mathrm{Cl^{(-)}}$$
Statement II: The conversion proceeds well in the more polar medium.

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

The product (A) formed in the following reaction sequence is

Aman has been asked to synthesise the molecule

(x).He thought of preparing the molecule using an aldol condensation reaction. He found a few cyclic alkenes in his laboratory. He thought of performing ozonolysis reaction on alkene to produce a dicarbonyl compound followed by aldol reaction to prepare " x ". Predict the suitable alkene that can lead to the formation of " x ".

Which of the following arrangements with respect to their reactivity in nucleophilic addition reaction is correct?

Let us consider an endothermic reaction which is non-spontaneous at the freezing point of water. However, the reaction is spontaneous at boiling point of water. Choose the correct option.

Preparation of potassium permanganate from $$MnO_2$$  involves two step process in which the 1st step is a reaction  with $$KOH$$ and  $$KNO_3$$ to produce:

For a reaction, $$N_2O_5(g) \rightarrow 2NO_2(g) + \frac{1}{2}O_2(g)$$ in a constant volume container, no products were present initially.  The final pressure of the system when $$50\%$$ of reaction gets completed is:

One mole of the octahedral complex compound  $$Co(NH_3)_5Cl_3$$  gives 3 moles of ions on dissolution in water. One mole of the same complex reacts with excess  $$AgNO_3$$ solution to yield two moles of  $$AgCl(s).$$  The structure of the complex is:

Which of the following ions is the strongest oxidizing agent? (Atomic Number of $$Ce = 58,\; Eu = 63,\; Tb = 65,\; Lu =  71$$)

$$ K_{sp} \text{ for } Cr(OH)_3 \text{ is } 1.6\times10^{-30}.$$ What is the molar solubility of this salt in water?

Which of the following statements are NOT true about the periodic table? A. The properties of elements are function of atomic weights. B. The properties of elements are function of atomic numbers. C. Elements having similar outer electronic configurations are arranged in same period. D. An element's location reflects the quantum numbers of the last filled orbital. E. The number of elements in a period is same as the number of atomic orbitals available in energy level that is being filled. Choose the correct answer from the options given below:

Given below are two statements I and II. Statement I: Dumas method is used for estimation of &amp;quot;Nitrogen&amp;quot; in an organic compound. Statement II: Dumas method involves the formation of ammonium sulphate by heating the organic compound with conc $$H_2 SO_4 $$. In the light of the above statements, choose the correct answer from the options given below

Which of the following statement is true with respect to  $$H_2O,\ NH_3$$ and  $$CH_4?$$ $$A.$$  The central atoms of all the molecules are  $$sp^3$$  hybridized.  $$B.$$ The  $$H-O-H,\ H-N-H$$ and $$H-C-H$$ angles in the above molecules  are $$104.5^\circ,\ 107.5^\circ$$  and $$109.5^\circ$$ respectively.  $$C.$$ The increasing order of dipole moment is $$CH_4 < NH_3 < H_2O.$$ $$D.$$ Both $$H_2O$$   and $$NH_3$$ are Lewis acids and  $$CH_4$$ is a Lewis base.  $$E.$$ A solution of  $$NH_3$$ in $$H_2O$$  is basic. In this solution  $$NH_3$$ and $$H_2O$$ act as Lowry-Bronsted acid and base respectively. Choose the correct answer from the options given below:

Which one of the carbocations from the following is most stable?

Following are the four molecules "P", "Q", "R" and "S". Which one among the four molecules will react with

$$H-Br_{(aq)}$$ at the fastest rate?

For the given cell $$Fe^{2+}_{(aq)} + Ag^+_{(aq)} \rightarrow Fe^{3+}_{(aq)} + Ag_{(s)},$$ the standard cell potential of the above reaction is Given: $$\begin{aligned}Ag^+ + e^- &\rightarrow Ag \qquad E^\circ = x\,V \\Fe^{2+} + 2e^- &\rightarrow Fe \qquad E^\circ = y\,V \\Fe^{3+} + 3e^- &\rightarrow Fe \qquad E^\circ = z\,V \end{aligned}$$

The large difference between the melting and boiling points of oxygen and sulphur may be explained on the basis of

Consider the given plots of vapour pressure (VP) vs temperature (T/K). Which amongst the following options is correct graphical representation  showing  $$\Delta T_f,$$  depression in the freezing point of a solvent in a solution?

Which of the following linear combination of atomic orbitals will lead to formation of molecular orbitals in homonuclear diatomic molecules [internuclear axis in $$z$$-direction] ? A. $$2p_z$$ and $$2p_x$$ B. 2 s and $$2p_x$$ C. 3 $$d_{xy}$$ and 3 $$d_{x^{2} - y^{2}}$$ D. 2 s and $$2p_z$$ E. $$2p_z$$ and $$3d_{x}^{2}- y^{2}$$ Choose the correct answer from the options given below:

$$ X\,g $$ of benzoic acid on reaction with aq. $$NaHCO_3$$ released $$CO_2$$ that occupied $$11.2\,L$$  volume at STP.  $$X$$  is  $$\underline{\hspace{2cm}}\,g. $$

Consider the following reaction occurring in the blast furnace: $$Fe_3O_4(s) + 4CO(g) \rightarrow 3Fe(l) + 4CO_2(g) x$$ kg of iron is produced when  $$2.32\times10^3\,kg\,Fe_3O_4$$ and $$2.8\times10^2\,kg\,CO$$ are brought together in the furnace. The value of  $$x$$  is  $$\underline{\hspace{2cm}}$$ (nearest integer). Given: $$M(Fe_3O_4)=232\,g\,mol^{-1},$$ molar mass of  $$CO=28\,g\,mol^{-1},$$ molar mass of $$(Fe)=56\,g\,mol^{-1}.$$

$$ 37.8\,g\,N_2O_5$$ was taken in a  $$1\,L$$ reaction vessel and allowed to undergo the following reaction at 500 K  $$2N_2O_5(g) \rightleftharpoons 2N_2O_4(g) + O_2(g)$$ The total pressure at equilibrium was found to be  $$18.65\,$$bar.  Then, $$K_p = \underline{\hspace{2cm}}\times10^{-2}$$  [nearest integer]. Assume  $$N_2O_5$$  to behave ideally under these conditions. Given:  $$R=0.082\,bar\,L\,mol^{-1}K^{-1}$$

Among the following cations, the number of cations which will give  characteristic precipitate in their identification tests with $$K_4[Fe(CN)_6]$$  is  $$\underline{\hspace{2cm}}$$. $$Cu^{2+},\; Fe^{3+},\; Ba^{2+},\; Ca^{2+},\; NH_4^{+},\; Mg^{2+},\; Zn^{2+}$$

Standard entropies of  $$X_2,\ Y_2$$  and  $$XY_5$$  are  $$70,\ 50$$  and  $$110\,J\,K^{-1}mol^{-1}$$ respectively. The temperature in Kelvin at which the reaction $$\frac{1}{2}X_2 + \frac{5}{2}Y_2 \rightleftharpoons XY_5 \Delta H^\ominus = -35\,kJ\,mol^{-1}$$ will be at equilibrium is $$\underline{\hspace{2cm}}$$  (Nearest integer).}