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Considering only the principal values of the inverse trigonometric functions, the value of 1132tan sin 2cos5 5−− − is
Let 22( , ) : 0, 0, 4 , 12 2 and 3 8 5 8 . S x y x y y x y x y x= − + If the area of the region S is 2, then is equal to
Let . k If ( )2 6 0lim , sin(sin ) cosx xe kx x x →+= ++ then the value of k is
Let :f→ be a function defined by 2 2sin , if 0,() 0, if 0.xxfx x x = = Then which of the following statements is TRUE?
Let S be the set of all ( , ) such that 2 21sin( )(log ) sin lim 0.(log (1 ))e xexxx xx →=+ Then which of the following is (are) correct?
A straight line drawn from the point (1,3, 2),P parallel to the line 2 4 6,1 2 1x y z− − −== intersects the plane 1: 3 6L x y z− + = at the point .Q Another straight line which passes through Q and is perpendicular to the plane 1L intersects the plane 2: 2 4L x y z− + =− at the point .R Then which of the following statements is (are) TRUE?
Let 1 1 1,,A B C be three points in the xy -plane. Suppose that the lines 11AC and 11BC are tangents to the curve 28yx= at 1A and 1,B respectively. If (0,0)O= and 1( 4,0), C=− then which of the following statements is (are) TRUE?
Let :f→ be a function such that ( ) ( ) ( ) for all , ,f x y f x f y x y+ = + and : (0, )g→ be a function such that ( ) ( ) ( ) for all , . g x y g x g y x y+ = If 3112 and 2,53fg−− == then the value of ()1 + 2 8 (0)4f g g−− is ________.
Numerical answer type
A bag contains N balls out of which 3 balls are white, 6 balls are green, and the remaining balls are blue. Assume that the balls are identical otherwise. Three balls are drawn randomly one after the other without replacement. For 1,2,3,i= let , , and i i iW G B denote the events that t he ball drawn in the thi draw is a white ball, green ball, and blue ball, respectively. If the probability 1 2 32()5P W G BN= and the conditional probability 3 1 22( ) ,9P B W G = then N equals ______ .
Let the function :f→ be defined by ( ) ()( ) ()2023 2023 222024 2025 2024 2025 sin 2() 33xxx x x xxfxee x x x x+ + + + =+ − + − + . Then the number of solutions of ( ) 0fx= in is ______.
Numerical answer type
Let ˆ ˆˆ23p i j k= + + and ˆ ˆˆ . q i j k= − + If for some real numbers ,, and , we have ()()()ˆ ˆˆ15 10 6 2 2 ,i j k p q p q p q + + = + + − + then the value of is _______.
Numerical answer type
A normal with slope 1 6 is drawn from the point (0, )− to the parabola 24, x ay=− where 0.a Let L be the line passing through (0, )− and parallel to the directrix of the parabola. Suppose that L intersects the parabola at two points A and .B Let r denote the length of the latus rectum and s denote the square of the length of the line segment .AB If : 1:16,rs= then the value of 24a is __________.
Numerical answer type
Let the function :[1, )f→ be defined by () ()1( 1) 2, if 2 1, , () 2 1 (2 1)(2 1) (2 1), if 2 1 2 1, .22nt n n ft n t t nf n f n n t n n+ − = − =+ − − −− + + − + Define 1( ) ( ) , (1, ).x g x f t dt x= Let denote the number of solutions of the equation ( ) 0gx= in the interval (1,8] and 1()lim .1xgx x →+=− Then the value of + is equal to _____.
If 6 ( ) ,mn X C= then the value of m is __________. PARAGRAP H “I” Let 1,2,3,4,5,6S= and X be the set of all relations R from S to S that satisfy both the following properties: i. R has exactly 6 elements. ii. For each ( , ) ,a b R we have 2. ab− Let : The range of has exactly one element Y R X R= and : is a function from to . Z R X R S S= Let ()nA denote the number of elements in a set .A (There are two questions based on PARAGRAP H “I”, the question given below is one of them) PARAGRAP H “I” Let 1,2,3,4,5,6S= and X be the set of all relations R from S to S that satisfy both the following properties: i. R has exactly 6 elements. ii. For each ( , ) ,a b R we have 2. ab− Let : The range of has exactly one element Y R X R= and : is a function from to . Z R X R S S= Let ()nA denote the number of elements in a set .A (There are two questions based on PARAGRAP H “I”, the question given below is one of them)
Numerical answer type
If the value of ( ) ( )n Y n Z+ is 2,k then ||k is _____________. PARAGRAP H “II” Let : 0, [0,1]2f→ be the function defined by 2( ) sinf x x= and let : 0, [0, )2g→ be the function defined by 2( ) .2xg x x=− (There are two questions based on PARAGRAP H “II”, the question given below is one of them)
Numerical answer type
The value of 22 002 ( ) ( ) ( )f x g x dx g x dx − is __________. PARAGRAP H “II” Let : 0, [0,1]2f→ be the function defined by 2( ) sinf x x= and let : 0, [0, )2g→ be the function defined by 2( ) .2xg x x=− (There are two questions based on PARAGRAP H “II”, the question given below is one of them)
Numerical answer type
The value of 2 3 016( ) ( )f x g x dx is ____________.
Numerical answer type
A region in the form of an equilateral triangle (in x-y plane) of height L has a uniform magnetic field 𝐵⃗ pointing in the +z-direction. A conducting loop PQR, in the form of an equilateral triangle of the same height 𝐿, is placed in the x-y plane with its vertex P at x = 0 in the orientation shown in the figure . At 𝑡=0, the loop starts entering the region of the magnetic field with a uniform velocity 𝑣 along the +x-direction. The plane of the loop and its orientation remain unchanged throughout its motion . Which of the following graph best depicts the variation of the induced emf ( E) in the loop as a function of the distance (𝑥) starting from 𝑥 = 0?
A particle of mass 𝑚 is under the influence of the gravitational field of a body of mass 𝑀 (≫𝑚). The particle is moving i n a circular orbit of radius 𝑟0 with time period 𝑇0 around the mass 𝑀. Then, the particle is subjected to an additional central force, corresponding to the potential energy 𝑉c(𝑟) =𝑚𝛼/𝑟3, where 𝛼 is a positive constant of suitable dimensions and 𝑟 is the distance from the center of the orbit. If t he particle move s in the same circular orbit of radius 𝑟0 in the combined gravitational potential due to 𝑀 and 𝑉c(𝑟), but with a new time period 𝑇1, then (𝑇12−𝑇02)/𝑇12 is given by [𝐺 is the gravitational constant.]
A metal target with atomic number 𝑍=46 is bombarded with a high energy electron beam. The emission of X-rays from the target is analyzed. The ratio 𝑟 of the wavelengths of the 𝐾𝛼-line and the cut -off is found to be 𝑟 = 2. If the same electron beam bombards another metal target with 𝑍 =41, the value of 𝑟 will be
A thin stiff insulated metal wire is bent into a circular loop with its two ends extending tangentially from the same point of the loop. The wire loop has mass 𝑚 and radius 𝑟 and it is in a uniform vertical magnetic field 𝐵0, as shown in the figure. Initially, it hangs vertically down wards , because of acceleration due to gravity 𝑔, on two conducting supports at P and Q. When a current 𝐼 is passed through the loop , the loop turns about the line PQ by an angle 𝜃 given by
A small electric dipole 𝑝 0, having a moment of inertia I about its center, is kept at a distance r from the center of a spherical shell of radius R. The surface charge density is uniformly distributed on the spherical shell. The dipole is initially oriented at a small angle 𝜃 as shown in the figure. While staying at a distance r, the dipole is free to rotate about its center. If released from rest, then which o f the following statement(s) is (are) correct? [𝜀0 is the permittivity of free space. ]
A table tennis ball has radius (3/2)×10−2 m and mass (22/7)×10⁻³ kg. It is slowly pushed down into a swimming pool to a depth of 𝑑=0.7 m below the water surface and then released from rest. It emerges from the water surface at speed 𝑣, without getting we t, and rises up to a height 𝐻. Which of the following option(s) is( are) correct? [Given: 𝜋=22/7, 𝑔=10 m s−2, density of water =1×103 kg m−3, viscos ity of water =1×10⁻³ Pa-s.]
A positive, singly ionized atom of mass number 𝐴M is accelerated from rest by the voltage 192 V. There after, it enters a rectangular region of width 𝑤 with magnetic field 𝐵⃗ 0=0.1𝑘̂ Tesla, as shown in the figure . The ion finally hits a detector at the distance 𝑥 below its starting trajectory . [Given: Mass of neutron/proton =(5/3)×10−27 kg, charge of the electron =1.6× 10−19 °C.] Which of the following option(s) is (are) correct?
The dimensions of a cone are measured using a scale with a least count of 2 mm. The diameter of the base and the height are both measured to be 20.0 cm . The maximum percentage error in the determin ation of the volume is ______.
Numerical answer type
A ball is thrown from the location (𝑥0,𝑦0)=(0,0) of a horizontal playground with an initial speed 𝑣0 at an angle 𝜃0 from the +𝑥-direction. The ball is to be hit by a stone, which is thrown at the same time from the location (𝑥1,𝑦1)=(𝐿,0). The stone is thrown at an angle (180−𝜃1) from the +𝑥-direction with a suitable initial speed . For a fixed 𝑣0, when (𝜃0,𝜃1)=(45°,45°), the stone hits the ball after time 𝑇1, and when (𝜃0 ,𝜃1)=(60°,30°), it hits the ball after time 𝑇2. In such a case, (𝑇1/𝑇2)2 is ______.
Numerical answer type
A charge is kept at the central point P of a cylindrical region . The two edges subtend a half -angle at P, as shown in the figure. When = 30∘, then the electric flux through the curved surface of the cylinder is Φ. If = 60∘, then the electric flux through the curved surface becomes Φ/√𝑛, where the value of n is______.
Numerical answer type
Two equilateral -triangular prisms P 1 and P 2 are kept with their sides parallel to each other , in vacuum, as shown in the figure. A light ray enters prism P1 at an angle of incidence 𝜃 such that the outgoing ray undergoes minimum deviation in prism P2. If the respective refractive indices of P 1 and P 2 are √3 2 and √3, then 𝜃=sin−1[√3 2sin(𝜋 𝛽)], wherethe value of is ______.
Numerical answer type
An infinitely long thin wire, having a uniform charge density per unit length of 5 nC/m, is passing through a spherical shell of radius 1 m, as shown in the figure. A 10 nC charge is distributed uniformly over the spherical shell. If the configuration of the charges remain s static, t he magnit ude of the potential difference between points P and R, in Volt , is ______. [Given: In SI units 1 4πϵ0=9×109, ln2=0.7. Ignore the area pierced by the wire .]
Numerical answer type
A spherical soap bubble inside an air chamber at pressure 𝑃0=105 Pa has a certain radius so that the excess pressure inside the bubble is Δ𝑃=144 Pa. Now , the chamber pressure is reduced to 8𝑃0/27 so that the bubble radius and its excess pressure change. In this process, all the temperature s remain unchanged . Assume air to be an ideal gas and the excess pressure Δ𝑃 in both the cases to be much smaller than the chamber pressure . The new excess pressure Δ𝑃 in Pa is ______. PARAGRAPH I In a Young’s double slit experiment, each of the two slits A and B, as shown in the figure, are oscillating about their fixed center and with a m ean separation of 0.8 mm. The distance between the slits at time t is given by 𝑑=(0.8+0.04sin𝜔𝑡) mm, where 𝜔=0.08 rad s⁻¹. The distance of the screen from the slits is 1 m and t he wavelength of the light used to illuminate the slits is 60 00 Å. The interference pattern on the screen changes with time , while the central bright fringe (zeroth frin ge) remains fixed at point O.
Numerical answer type
The 8th bright fringe above the point O oscillates with time between two extreme positions. The separation between these two extreme positions, in micro meter (𝜇m), is ______ . PARAGRAPH I In a Young’s double slit experiment, each of the two slits A and B, as shown in the figure, are oscillating about their fixed center and with a m ean separation of 0.8 mm. The distance between the slits at time t is given by 𝑑=(0.8+0.04sin𝜔𝑡) mm, where 𝜔=0.08 rad s⁻¹. The distance of the screen from the slits is 1 m and t he wavelength of the light used to illuminate the slits is 60 00 Å. The interference pattern on the screen changes with time , while the central bright fringe (zeroth fringe) r emains fixed at point O.
Numerical answer type
The maximum speed in 𝜇m/s at which the 8th bright fringe will move is __________ . PARAGRAPH I I Two particles, 1 and 2, each of mass 𝑚, are connected by a massless spring , and are on a horizontal frictionless plane , as shown in the figure. Initially, t he two particles , with their center of mass at 𝑥0, are oscillating with amplitude 𝑎 and angular frequency 𝜔. Thus, the ir positions at time 𝑡 are given by 𝑥1(𝑡)=(𝑥0+𝑑)+𝑎 sin𝜔𝑡 and 𝑥2(𝑡)=(𝑥0−𝑑)−𝑎 sin𝜔𝑡, respectively , where 𝑑>2𝑎. Particle 3 of mass 𝑚 moves towa rds this system with speed 𝑢0=𝑎𝜔/2, and undergoes instantaneous el astic collision with particle 2 , at time 𝑡0. Finally, particles 1 and 2 acquire a center of mass speed 𝑣cm and oscillate with amplitude 𝑏 and the same angular frequency 𝜔.
Numerical answer type
If the collision occurs at time 𝑡0=0, the value of 𝑣cm/(𝑎𝜔) will be __________ . PARAGRAPH II Two particles, 1 and 2, each of mass 𝑚, are connected by a massless spring , and are on a horizontal frictionless plane , as shown in the figure. Initially, t he two particles, with their center of mass at 𝑥0, are oscillating with amplitude 𝑎 and angular frequency 𝜔. Thus, the ir positions at time 𝑡 are given by 𝑥1(𝑡)=(𝑥0+𝑑)+𝑎 sin𝜔𝑡 and 𝑥2(𝑡)=(𝑥0−𝑑)−𝑎 sin𝜔𝑡, respectively, where 𝑑>2𝑎. Particle 3 of mass 𝑚 moves towa rds this system with speed 𝑢0=𝑎𝜔/2, and undergoes instantaneous elastic collision with particle 2, at time 𝑡0. Finally, particles 1 and 2 acquire a center o f mass speed 𝑣cm and oscillate with amplitude 𝑏 and the same angular frequency 𝜔.
If the collision occurs at time 𝑡0=𝜋/(2𝜔), then the value of 4𝑏2/𝑎2 will be ________.
According to Bohr’s model, the highest kinetic energy is associated with the electron in the
In a metal deficient oxide sample, MXY2O4 (M and Y are metals) , M is present in both +2 and +3 oxidation states and Y is in +3 oxidation state. If the fraction of M2+ ions present in M is 1 3 , the value of X is _____.
In the following reaction sequence, the major product Q is
The species formed on fluorination of phosphorus pentachloride in a polar organic solvent are
An aqueous solution of hydrazine (N 2H4) is electrochemically oxidized by O 2, thereby releasing chemical energy in the form of electrical energy. One of the products generated from the electrochemical reaction is N 2(g). Choose the correct statement(s) about the above process
The option(s) with correct sequence of reagents for the conversion of P to Q is(are)
The compound(s) having peroxide linkage is(are)
To form a complete monolayer of acetic acid on 1g of charcoal, 100 mL of 0.5 M acetic acid was used. Some of the acetic acid remained unadsorbed. To neutralize the unadsorbed acetic acid, 40 mL of 1 M NaOH solution was required. If each molecule of acetic acid occupies P x 10−23 m² surface area on charcoal, the value of P is _____. [Use given data: Surface area of charcoal = 1.5 x 102 m²g−1; Avogadro’s number (NA) = 6.0 x 1023 mol⁻¹]
Numerical answer type
Vessel -1 contains w2 g of a non -volatile solute X dissolved in w1 g of water. Vessel -2 contains w2 g of another non -volatile solute Y dissolved in w1 g of water. Both the vessels are at the same temperature and pressure. The molar mass of X is 80% of that of Y. The van’t Hoff factor for X is 1.2 times of that of Y for their respective concentrations. The elevation of boiling point for solution in Vessel -1 is _____ % of the solution in Vessel -2.
Numerical answer type
For a double strand DNA, one strand is given below : The amount of energy required to split the double strand DNA into two single strands is _____ kcal mol⁻¹. [Given : Average energy per H-bond for A -T base pair = 1.0 kcal mol⁻¹, G-C base pair = 1.5 kcal mol⁻¹, and A -U base pair = 1.25 kcal mol⁻¹. Ignore electrostatic repulsion between the phosphate groups.] JEE (Advanced) 202 4 Paper 2 6/8
Numerical answer type
A sample initially contains only U -238 isotope of uranium. With time, some of the U -238 radioactively decays into Pb -206 while the rest of it remains undisintegrated. When the age of the sample is P x108 years, the ratio of mass of Pb -206 to that of U -238 in the sample is found to be 7. The value of P is _____. [Given: Half -life of U -238 is 4.5 x109 years; log e2 = 0.693]
Numerical answer type
Among [Co(CN)4]4−,[Co(CO)3(NO)],XeF₄, [PCl4]+,[PdC l4]2− ,[ICl4]− ,[Cu(CN)4]3− and P₄ the total number of species with tetrahedral geometry is _____.
Numerical answer type
An organic compound P having molecular formula C 6H6O3 gives ferric chloride test and does not have intramolecular hydrogen bond. The compound P reacts with 3 equivalents of NH 2OH to produce oxime Q. Treatment of P with excess methyl iodide in the presence of KOH produces compound R as the major product . Reaction of R with excess iso-butylmagnesium bromide followed by treatment with H 3O+ gives compound S as the major product. The total number of methyl (−CH 3) group(s) in compound S is _____. JEE (Advanced) 202 4 Paper 2 7/8 “PARAGRAPH I” An organic compound P with molecular formula C 9H18O₂ decolorizes bromine water and also shows positive iodoform test. P on ozonolysis followed by treatment with H 2O₂ gives Q and R. While compound Q shows positive iodoform test, compound R does not give positive iodoform test. Q and R on oxidation with pyridinium chlorochromate (PCC) followed by heating give S and T, respectively. Both S and T show positive iodoform test. Complete c opolymerization of 500 moles of Q and 500 moles of R gives one mole of a single acyclic copolymer U. [Given, atomic mass: H =1, C = 12, O =16]
Numerical answer type
Sum of number of oxygen atoms in S and T is _____. “PARAGRAPH I” An organic compound P with molecular formula C 9H18O₂ decolorizes bromine water and also shows positive iodoform test. P on ozonolysis followed by treatment with H 2O₂ gives Q and R. While compound Q shows positive iodoform test, compound R does not give positive iodoform test. Q and R on oxidation with pyridinium chlorochromate (PCC) followed by heating give S and T, respectively. Both S and T show positive iodoform test. Complete c opolymerization of 500 moles of Q and 500 moles of R gives one mole of a single acyclic copolymer U. [Given, atomic mass: H =1, C = 12, O =16]
Numerical answer type
The molecular weight of U is _____. JEE (Advanced) 202 4 Paper 2 8/8 “PARAGRAPH II” When potassium iodide is added to an aqueous solution of potassium ferricyanide, a reversible reaction is observed in which a complex P is formed. In a strong acidic medium, the equilibrium shifts completely towards P. Addition of zinc chloride to P in a slightly acidic medium results in a sparingly soluble complex Q.
Numerical answer type
The number of moles of potassium iodide required to produce two moles of P is _____. “PARAGRAPH II” When potassium iodide is added to an aqueous solution of potassium ferricyanide, a reversible reaction is observed in which a complex P is formed. In a strong acidic medium, the equilibrium shifts completely towards P. Addition of zinc chloride to P in a slightly acidic medium results in a sparingly soluble complex Q.
Numerical answer type
The number of zinc ions present in the molecular formula of Q is _____.
Numerical answer type
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