JEE Advanced Past Paper Semi Major Test - 01 PDF

30-06-2024 9610ZJA801238240005 JA PART 1 : PHYSICS SE...

30-06-2024 9610ZJA801238240005 JA PART 1 : PHYSICS SECTION-I (i) 1) A collar of mass m is to be moved along a long fixed rough straight vertical rod by applying a force F which is constant in magnitude and whose direction is continuously changing such that θ is increased from 0° to 90° linearly with time. Initially the collar is at rest at ground and again comes to rest when the force becomes horizontal then (A) (B) (C) (D) 2) To verify Ohm's law, a student connects the voltmeter across the battery as, shown in the figure. The measured voltage is plotted as a function of the current, and the following graph is obtained: For More Material Join: @JEEAdvanced_2025  0 If V is almost zero, identify the correct statement: (A) The value of the resistance R is 1.5 Ω (B) The emf of the battery is 1.5 V and the value of R is 1.5 Ω (C) The emf of the battery is 1.5 V and its internal resistance is 1.5 Ω (D) The potential difference across the battery is 1.5 V when it sends a current of 1000 mA. 3) In the circuit shown when S is closed at t = 0,then : (A) Ratio of charge stored in capacitors C and 3C at any time t will be 1 : 3 (B) Steady state charge in capacitors C and 3C are in ratio of of 1 : 3 (C) Time constant of both capacitors are equal (D) Rate of increase of charge is same in both capacitors. 4) Two long straight wires, carrying currents I1 and I2, are placed perpendicular to each other. The line of shortest distance between them.PQ, has length r. The magnitude of the magnetic field at the centre O of PQ is (A) For More Material Join: @JEEAdvanced_2025  (B) (C) (D) SECTION-I (ii) 1) A charged particle moving in x-y plane enters a region of uniform magnetic field of intensity at P(0, 0) with velocity and leaves the magnetic field at Q(a, b) with velocity. Choose the correct statement(s) (A) The particle can't be positron (B) Deviation suffered by the particle as it moves from P to Q equals 60° (C) =b (D) 2) Consider a cube of side 'a' as shown. Eight point charges are placed at the corners. The cube is rotated about the axis with constant angular velocity 'ω': (A) Net magnetic field at the centre of cube is zero (B) Net magnetic field at the centre of cube is (C) Net magnetic field at the centre of cube is If polarity of any four charges are reversed, then magnetic field at the centre of cube will be (D) zero. 3) Two blocks of masses m1 and m2 are connected through a massless inextensible string. Block of mass m1 is placed at the fixed rigid inclined plane while the block of mass m2 hanging at the other For More Material Join: @JEEAdvanced_2025  end of the string, which is passing through a fixed massless frictionless pulley shown in figure. The coefficient of static friction between the block and the inclined plane is 0.8. The system of masses m1 and m2 is released from rest. (A) the tension in the string is 20 N after releasing the system (B) the normal force by the inclined surface on the block is along normal to the inclined surface (C) the magnitude of normal force by the inclined surface on the block m1 is (D) none of these 4) A parallel plate air capacitor has initial capacitance C. If plate separation is slowly decreased from d1 to d2, then mark the correct statement(s). [Throughout the process it remains connected to battery] (A) Work done by electric force = work done by external agent. (B) Work done by external force where is the electric force of attraction between the plates at plate separation x. (C) Work done by electric force ≠ Negative of work done by external agent. (D) Work done by battery = 2 times the change in electric potential energy stored in capacitor. 5) A cylindrical conductor has a resistance R. When the conductor is at a temperature (T) above its surrounding temperature (T0), the thermal power dissipated by the conductor is proportional to excess temperature (ΔT = T – T0) above surrounding and proportionalty constant k. The conductor is connected to ideal cell of emf V. Initially, the conductor was at room temperature T0. Mass and specific heat capacity of the conductor are m and s respectively. Assume no change in resistance due to temperature. The time (t) dependence of the temperature (T) of the conductor after it is connected to the cell (A) is. The time (t) dependence of the temperature (T) of the conductor after it is connected to the cell (B) is (C) the temperature of the conductor after a long time is (D) the temperature of the conductor after a long time is 6) ABCDFPA is a network of three batteries of the e.m.f's E, 12 V and 4 V respectively and three resistance 2 Ω, 4 Ω and 6 Ω connected as shown in the diagram. An ideal ammeter connected between F and P shows a current reading of 0.5 A from P to F. Then the value of the e.m.f. E is For More Material Join: @JEEAdvanced_2025  (A) 6 V (B) 9 V (C) 8 V (D) 11 V 7) A projectile is projected at an angle of elevation α. After t seconds, it appears to have an angle of elevation β as seen from the point of projection. Which of the following are correct : (A) the y-coordinate of the position of the projectile at time t is (v sin α) , where v is the velocity of projection the x-coordinate of the position of projectile at time t is (v cos α) t, where v is the velocity of (B) projection (C) tan β = tan α – , where v is the velocity of projection (D) velocity of projection is 8) A point mass is moving in the x-y plane. Its acceleration is a constant vector perpendicular to the x-axis. Which of the following do/does not change with time? (A) only y-component of its velocity vector (B) only x-component of its velocity vector (C) only y-component of its acceleration vector (D) only x-component of its acceleration vector SECTION-II 1) The velocity of a particle traveling in a straight line is given by m/s, where time t is in seconds and t ≥ 0. If the particle is observed at x=7m at the instant t=0, its position x is expressed as function of time. Find numerical value of. 2) James's bond is standing on a bridge above the road and his pursuers are getting too close for comfort. He spots a flatbed truck loaded with mattresses approaching at 30 m/s which he measures by knowing that the telephones pole ; the truck is passing are 20 m apart in this country. The bed of truck is 20m below the bridge and bond quickly calculates how many poles away the truck should be when he jumps down the bridge onto the truck making his get away. How many poles is it ? (Take g = 10 m/s2) For More Material Join: @JEEAdvanced_2025  3) The diagram shows combination of three cuboidal spaces (1), (2) and (3). Space 1 and 3 contain electric field E as shown while space 2 has magnetic field B. A particle of charge q and mass m is projected as shown with velocity. If the value of E is equal to then particle enters the magnetic field parallel to the x-axis and just passes through point P along the electric field at that point. (Neglect the effect of gravity). Find k. 4) A conducting rod of length meter and mass m = 4 kg lies on the horizontal table. Coefficient of friction between the rod and the table is. If the current in the conductor is 2 ampere, then find the minimum magnitude of magnetic field strength (in Tesla) such that conducting rod just starts to translate along x-axis. (take g = 10 m/s2). [neglect the radius of rod] 5) Find out the net emf of the three batteries between A & B when the point P is the null point, measured by the potentiometer arrangement shown in the figure. For More Material Join: @JEEAdvanced_2025  6) A non-conducting ring of mass m and radius (R = 20cm) is charged as shown. The charged density i.e. charge per unit length is (l = 2C/m) It is then placed on a rough non-conducting horizontal surface plane. At time t = 0, a uniform electric field is switched on and the ring start rolling without sliding. Determine the friction force in newton acting on the ring, when it starts 0 moving if E = 5N/C PART 2 : CHEMISTRY SECTION-I (i) 1) In an unielectronic species of unknown atomic number a transition of electron from a higher energy level to ground state resulted in emission of a photon having energy approximately 108.9 eV. If the change in wavelength of the electron in the two levels is two times the circumference of its first Bohr orbit then what would be the atomic number Z and orbit number of the higher energy shell involved ? (A) Z = 3, n = 3 (B) Z = 2, n = 10 (C) Z = 6, n = 6 (D) Z = 4, n = 3 2) Methyl acetate was hydrolyzed in HCl aqueous solution at 298K, samples of reaction mixture were withdrawn at different time intervals and titrated with 0.12N NaOH sample whose volumes were For More Material Join: @JEEAdvanced_2025  obtained as : Half- life for the reaction below is : CH3COOCH3(aq.) + H2O(ℓ) CH3COOH(aq.) + CH3OH(aq.) (A) 12.5 minutes (B) 25 minutes (C) 37.5 minutes (D) 50 minutes 3) Efficiency of a cell with cell reaction under standard condition is 85%. The is : (A) 1.26 V (B) 1.34 V (C) 2.96 V (D) 2.07 V 4) Which of the following statement is false? (Kf water = 1.86 K.Kg/mol) (A) C2H5Br + C2H5I forms an ideal solution. (B) A peeled egg when dipped in water swells, while in saturated solution it shrinks. On adding NaCl into water, boiling point increases while on adding CH3OH into water, its (C) vapour pressure increases. (D) 1 m NaCl in water have a freezing point equal to –1.86°C SECTION-I (ii) 1) For a solution formed by mixing liquids L and M, the vapour pressure of L plotted against the mole fraction of M in solution is shown in the following figure, here XL and XM represent mole fractions of L and M, respectively, in the solution. the correct statement(s) applicable to this system is(are) Attractive intramolecular interactions between L–L in pure liquid L and M–M in pure liquid M (A) are stronger than those between L–M when mixed in solution (B) The point Z represents vapour pressure of pure liquid M and Raoult's law is obeyed when XL → 0 (C) The point Z represents vapour pressure of pure liquid L and Raoult's law is obeyed when XL → 1 The point Z represents vapour pressure of pure liquid M and Raoult's law is obeyed from XL = 0 (D) to XL = 1 2) A photon of wavelength 124 nm is absorbed by a metal having work function 6 eV. If the maximum uncertainity in momentum of ejected photoelectron is For More Material Join: @JEEAdvanced_2025  then choose correct statements ? (A) Maximum K.E. of ejected photoelectron is 4J (B) Minimum uncertainity in position of electron is 1 nm (C) Maximum uncertainity in de-Broglie wavelength is 3 pm (D) Maximum uncertainity in de-Brogile wavelength is approximately 30 pm 3) 40 ml, 0.05 M solution of sodium sesquicarbonate dihydrate [Na2CO3.NaHCO3.2H2O] is titrated against 0.05 M HCl solution, x ml of acid is required to reach the phenolphthalein end point, while y ml. of same acid was required when methyl orange indicator was used in a separate titration. Which of the following statements are correct ? (A) y – x = 80 ml (B) y + x = 160 ml If the titration is started with phenolphthalein indicator and methyl orange is added at the end (C) point, 2x ml. HCl would be required further to reach end point. (D) y + x = 120 ml. 4) In a bimolecular reaction, the steric factor P was experimentally determined to be 4.5. The correct option(s) among the following is (are) The value of frequency factor predicted by Arrhenius equation is higher than that determined (A) experimentally (B) The activation energy of the reaction is unaffected by the value of the steric factor (C) Since P = 4.5, the reaction will not proceed unless an effective catalyst is used Experimentally determined value of frequency factor is higher than that predicted by Arrhenius (D) equation 5) Which of the following statements is / are correct ? (A) The conductance of one cm3 (or 1 unit3) of a solution is called conductivity. (B) Specific conductance increases while molar conductivity decreases on progressive dilution. The limiting equivalent conductivity of weak electrolyte cannot be determine exactly by (C) extraplotation of the plot of Λeq against. (D) The conductance of metals is due to the movement of free electrons. 6) The halogen form compounds among themselves with formula XX, XX'3, XX'5 and XX'7 where X is the heavier halogen. Which of the following pairs representing their structures and being polar and non-polar are correct? (A) XX' – Linear – polar (B) XX'3 – T-shaped – polar (C) XX'5 – square pyramidal – polar (D) XX'7 – Pentagonal bipyramidal – non-polar For More Material Join: @JEEAdvanced_2025  7) Which of the following are incorrect- (A) Cr2+ (g) ion has greater magnetic moment as compared to Co3+ (g) ion (B) Atomic size of B < Ga < Al < Tl (C) Lanthanide contraction is the cause of lower I.E. of Pb than Sn If successive ionization energy of an element are 332, 738, 849, 4080, 4958 (in kJ/mol). Then (D) this element can be of 15th group 8) Which of the following compound(s) has same no. of 1°, 2° & 3° carbons. (A) (B) (C) (D) SECTION-II 1) Find the number of different functional groups present in the given compound? 2) ICl3 is an orange colored solid that dimerizes in solid state as I2Cl6. Based on VSEPR theory, number of ~90 degree Cl – I – Cl bond angles is...................... in the dimeric species. Neglect any minor deviations from ideal bond angle. 3) For the reaction A (g) → 2B (g) when [A] = 0.1 M when [A] = 0.4 M Find the time taken (in minutes) for concentration of A to change from 0.6 M to 0.15 M. [ln2 = 0.693] For More Material Join: @JEEAdvanced_2025  (Express your answer to the nearest integer value). 4) A current of 3.7A is passed for 6hrs. between Ni electrodes in 0.5L of 2M solution of Ni(NO3)2. What will be the molarity of solution at the end of electrolysis ? 5) A solution of Na2S2O3 is standardized iodometrically against 125.25 mg of KBrO3 where BrO3– changes to Br–. This process requires 45 mL of the Na2S2O3 solution. What is the molarity of the Na2S2O3? [Mw. of KBrO3 = 167] 6) A 500 gm mixture of KHCO3 and silica on strong heating evolves gases which occupies 24 litre of volume at 2 atm & 300 K. Mass percentage of silica in original sample: [R = 0.08 atm-L/mole-K] Fill your answer as sum of digits (excluding decimal places) till you get the single digit answer. PART 3 : MATHEMATICS SECTION-I (i) 1) If graph of the expression y = x2 – 8x + 12 is shown in the figure then area of the square ABCD inscribed between parabola & x-axis is given by (A) (B) (C) (D) 2) Find domain of the function (A) (B) (C) (D) None of these 3) Match the lists: List-I List-II (I) (P) 1 If number of points where f(x) is continuous is For More Material Join: @JEEAdvanced_2025  (II) If where (Q) 3 , then (III) (R) 2 then a – b = (IV) (S) 4 Let be continuous & differentiable every where. Then b – a + 2 = (T) 0 The correct option is (A) I–T; II–S; III–R; IV–R (B) I–P; II–R; III–Q; IV–S (C) I–P; II–S; III–Q; IV–R (D) I–T; II–R; III–R; IV–S 4) The sum of series up to infinite terms is equal to (A) (B) (C) (D) SECTION-I (ii) 1) If log10(a – 2) + log10b = 0 and , then the value of (a + b) is - (A) (B) 4 (C) rational (D) irrational 2) The figure illustrates graph of the function y = f(x) defined in [–3, 2]. Identify the correct statement. For More Material Join: @JEEAdvanced_2025  (A) Range of y = f(–|x|) is [–2, 2] (B) Domain of y = f(|x|) is [–2, 2] (C) Domain of y = f(|x| + 1) is [–1, 1] (D) Range of y = f(|x| + 1) is [–1, 0] 3) Let and be two functions such that is defined. Which of the following statement(s) is/are INCORRECT? (A) If gof is into then f must be into. (B) If f is into and g is onto then gof must be onto. (C) If gof is one-one then g must be one-one. (D) If f is surjective and g is injective then gof must be surjective. 4) If 0 < x < 1, then is equal to (A) (B) (C) (D) 5) If has a finite limit L as x → 0, then (A) (B) (C) c = 0 For More Material Join: @JEEAdvanced_2025  (D) 6) Let and be function defined by ƒ(x) = [x2 – 3] and g(x) = |x| ƒ(x) + |4x – 7| ƒ(x), where [y] denotes the greatest integer less than or equal to y for y ∈. Then (A) ƒ is discontinuous exactly at three points in ƒ is discontinuous exactly at four points in (B) (C) g is NOT differentiable exactly at four points in (D) g is NOT differentiable exactly at five points in 7) Let a, b, c, d, e are 5 natural numbers (ranging from 1 to 9) are in AP then – (A) Arithmetic mean of 3a and 3e is b + c + d. (B) Number of possible values of 'a + e' is 9. (C) Sum of all possible values of b + c + d is 135. (D) Sum of all possible values of b + c + d is 150. 8) Let x,y,z ∈ R+ and x + y + z = 2, then choose the correct option(s) - (A) 2 2 maximum value of x y3z is (B) maximum value of x2y3z2 is 1. (C) minimum value of is 18 (D) maximum value of xyz is SECTION-II 1) Least positive integral value of k for which the equation e2x + (1 – 2k)ex + k2 – 1 = 0 does not have any real solution, is (where e is Napier's constant) 2) If where then find the least value of For More Material Join: @JEEAdvanced_2025  3) Let ' f ' be a function from the set of all non-negative integers into itself such that (f(2n + 1))2 – (f(2n))2 = 6f(n) + 1 and f(2n) ≥ f(n), then the value of f(4) – f(3) 4) The value of sin–1(sin5) + cos–1(cos5) + sin–1(sin8) is equal to nπ – 8, and if ; then S∞ = pπ/q, where n, p, q ∈ I then value of n + p + q is, where n ∈ I : 5) then equal to 6) If f(2016) = , f(x) = f(2x) ∀ x ∈ R and f(x) is a continuous function for all x ∈ R, and a = and b is number of points of g(x) = max{tanx, cotx} in x ∈ (–π,π) where it is non-differentiable, then is equal to For More Material Join: @JEEAdvanced_2025  ANSWER KEYS PART 1 : PHYSICS SECTION-I (i) Q. 1 2 3 4 A. A C B C SECTION-I (ii) Q. 5 6 7 8 9 10 11 12 A. B C,D A,B,C B,D A,C D A,B,C,D B,C,D SECTION-II Q. 13 14 15 16 17 18 A. 2.00 3.00 2.00 2.00 8.00 2.00 PART 2 : CHEMISTRY SECTION-I (i) Q. 19 20 21 22 A. A B A D SECTION-I (ii) Q. 23 24 25 26 27 28 29 30 A. A,C B,D A,B,C B,D A,C,D A,B,C,D A,C,D A,C SECTION-II Q. 31 32 33 34 35 36 A. 8.00 8.00 8.00 2.00 0.10 2.00 PART 3 : MATHEMATICS SECTION-I (i) Q. 37 38 39 40 A. D A B A SECTION-I (ii) Q. 41 42 43 44 45 46 47 48 A. A,D A,B,C,D A,B,C,D A,B,C A,B,C,D B,C A,B,C A,C,D For More Material Join: @JEEAdvanced_2025  SECTION-II Q. 49 50 51 52 53 54 A. 2.00 0.30 5.00 8.00 1.00 1.75 For More Material Join: @JEEAdvanced_2025  SOLUTIONS PART 1 : PHYSICS 1) θ = ωt (ω = constant) N = F sin ωt ∴ Also ∴ ∴ 2) V = E – Ir when V = V0 = 0 ⇒ 0 = E – Ir ∴ E=r when I = 0, V = E = 1.5V For More Material Join: @JEEAdvanced_2025  ∴ r = 1.5Ω. 3) q1 = q2 = In steady state 4) = 5) θ = 60 6) Current = for each face from which axis is passing For More Material Join: @JEEAdvanced_2025  Reversing any of the four charges mass the current either zero or in opposite sense at two surface. 7) If the tendency of relative motion along the common tangent does not exist, then component of contact force along common tangent will be zero. 8) Extra charge flown Work done by battery : Wb = V × charge flown Change in P.E. of capacitor 9) Power dissipated in resistance = Rate of heat dissipation to surrounding + Rate of heat absorption by conductor when 11) y-coordinate is y = v sin α t [Therefore A is correct] x –coordinate is x = v cos α t [Therefore B is correct] For More Material Join: @JEEAdvanced_2025  Equation of trajectory is [C is correct] Rearranging [∴ D is correct] 12) ax = 0 ay = –g vx = constant (B) only x-component of velocity vector. (C) only y-component of acceleration vector. 13) 14) 20 = ∴ t = 2s S = vt = 60 m ∴ no. of poles = =3 15) For More Material Join: @JEEAdvanced_2025  tA → B =....(1) t=....(2)....(3)....(4) From eq. (2) ∴ k=2 16) iℓBcosθ + N = mg N = mg – iℓBcosθ iℓBsinθ = µN = µmg – µiℓBcosθ 17) For balanced condition, current through galvanometer = 0. For More Material Join: @JEEAdvanced_2025  Thus equivalent EMF = = 4 + 4 = 8V 18) PART 2 : CHEMISTRY 19) Given En – E1 = 108.9 eV...(i) Given ∴ n–1=2 n=3 Putting n back in eq. (i) For More Material Join: @JEEAdvanced_2025  Z=3 20) 21) ∴ = –2 × 96500 × = 1.26 V 22) (B) Solvent flow from its higher concentration to its lower concentration. (C) NaCl is non volatile so decrease vapour pressure. While CH3OH is more volatile than water so vapour pressure increases. (D) ΔTf = iKf × m = 2 × 1.86 × 1 = 3.72 °C Tf = Tf° –ΔTf = 0–3.72 = –3.72 °C 23) → solution showing positive deviation from Raoult's law ∴ L-M interactions are weaker than L-L & M-M interactions. 24) 25) With HPh only : x = 40 ml With MeOH only : For More Material Join: @JEEAdvanced_2025  0.05 × y = (0.05 × 40 × 2) + (0.05 × 40 × 1) y = 120 ml 26) i.e. Aexp. > ATheoritical 27) On dilution specific conductance decreases while molar conductivity increases. 28) (A) I–Cl, linear and polar because of the difference in the electronegativities of iodine and chlorine. (B) (C) (D) 29) Theory based. 30) (A) (B) For More Material Join: @JEEAdvanced_2025  (C) (D) 31) 3° amine, 2° amine, 1° amide, anhydride, ether, alcohol, aldehyde, ketone. 32) 2ICl3 → I2Cl2 Number of 90° angle = 8. 33) Clearly order = 1 w.r.t. A ⇒ 3 × 10–4 = R (0.1) k = 3 × 10–3 sec–1 t1/2 = t3/4 = 462 sec. = = 7.7 min 34) With Ni electrode, cathode reaction, Ni2+ + 2e– → Ni, Anode reaction, Ni → Ni2+ + 2e No. Change in molarity of solution. 35) Eq. of Na2S2O3 ⇒ Eq. of KBrO3 N × 45 = N = 0.1 M= 36) 2 KHCO3(s) —→ K2CO3(s) + H2O(g) + CO2(g) For More Material Join: @JEEAdvanced_2025  Total moles of H2O & CO2 = = =2 moles of KHCO3 taken initially = 2 wt. of KHCO3 = 2 × 100 = 200 gm % wt of silica = × 100 = 60 % PART 3 : MATHEMATICS 37) Let the side length be 2k Now point ((k + 4), –2k) lies on parabola. y = x2 – 8x + 12 (–2k) = (k + 4)2 – 8(k + 4) + 12 ⇒ k2 + 2k – 4 = 0 ⇒ So area = 4k2 = 38) Case-1: & For More Material Join: @JEEAdvanced_2025  Case-2: 39) (P) Let x = a be the point at which f(x) is continuous. function is continuous at x = 0. (Q) Let then (R) 1 – a = 0, –b = 2 (S) Since function is continuous everywhere For More Material Join: @JEEAdvanced_2025  LHL = RHL at x = –1.....(A) Again, function is differentiable at everywhere. at x = –1 40) = = = = = 41) (a – 2)b = 1 squaring both side a+b–2+2 a(b – 2) = 1 a(1 – 2a + 4) = a – 2 5a – 2a2 = a – 2 2a2 – 4a – 2 = 0 a2 – 2a – 1 = 0 For More Material Join: @JEEAdvanced_2025  a= ⇒ a+b= 42) Use transformation of graphs 43) 44) Let...(i)...(ii)...(iii) 45) For More Material Join: @JEEAdvanced_2025  For limit to be exist coeff. of below x3 must be zero a+b=0...(1) 1 + a – b + c = 0...(2)...(3) by eq. (1) & (3) c = 0 by eq. (1) & (2) a + b = 0 1+a–b=0 46) f(x) = [x2 – 3] = [x2] – 3, x ∈ = at x = where n = 1, 2, 3, 4 so f(x) is discontinuous at x = , ∀ n = 1, 2, 3, 4. f (x) is discontinuous at 4 points in. g(x) = f(x) |x| + f(x) |4x – 7|, = f(x) Let S(x) = ' f(x) is discontinuous at x = 1, , and S(x) is non zero, x = 1, , ⇒ g(x) = f(x) × S(x) is discontinuons and non-differentiable at x = 1, ––(1) S(x) is non differentable at x= 0 & x = 7/4 At x = 0 , f (0) ≠ 0 ⇒ g(x) = f(x) × S(x) is non differentiable. ––(2) for x ∈ , f(x) = 0 so , g(x) = f(x) × S(x) = 0 ⇒ g(x) is differentiable at x = 7/4 from (1) and (2) g(x) is NOT diff at four points in For More Material Join: @JEEAdvanced_2025  47) 48) and ⇒ and 49) e2x + (1 – 2k)ex + k2 – 1 = 0 ex = t > 0 t2 + (1 – 2k)t + k2 – 1 = 0 Either D < 0 (1 – 2k)2 – 4(k2 – 1) < 0 or both the roots are ≤ 0 ⇒ Least integral value of k = 2 For More Material Join: @JEEAdvanced_2025  50) Let So, Hence, 51) Put n = 0 (f(1))2 – (f(0))2 = 6f(0) + 1 x2 – y2 = –8 Only possible solution x = 1, y = 3 f(1) = 1 f(0) = 0 Now, β – α = 2f(2n) β = 2f(2n) + α bmin = 2f(2n) + 1 ≥ 2f(n) + 1 βmin = 2f(n) + 1 α = 2 is not possible (RHS is odd) α=1 β = 6f(n) + 1 so f(2n + 1) = 3f(n) + 1 f(2n) = 3f(n) f(3) = 4 f(2) = 3 & f(4) = 9 f(4) – f(3) = 5 52) sin–1(sin5) = 5 – 2π; cos–1(cos5) = 2π – 5 sin–1(sin8) = 3π – 8. For More Material Join: @JEEAdvanced_2025  53) ⇒ 54) f(2x) = f(x) = if =0=a and g(x) is non-differentiable So, b = 7 For More Material Join: @JEEAdvanced_2025