ALLEN JEE Advanced 2025 Paper 1 (PDF)

(1001CJA101021240055) *1001CJA101021240055* Test Pattern English JEE(Advanced) CLASSROOM CONTACT PROGRAMME...

(1001CJA101021240055) *1001CJA101021240055* Test Pattern English JEE(Advanced) CLASSROOM CONTACT PROGRAMME FULL SYLLABUS (Academic Session : 2024 - 2025) 13-04-2025 JEE(Main+Advanced) : ENTHUSIAST & LEADER COURSE (SCORE ADVANCED) Time : 3 Hours PAPER-1 (OPTIONAL) Maximum Marks : 180 IMPORTANT NOTE : Students having 8 digits Form No. must fill two zero before their Form No. in OMR. For example, if your Form No. is 12345678, then you have to fill 0012345678. READ THE INSTRUCTIONS CAREFULLY GENERAL : 1. This sealed booklet is your Question Paper. Do not break the seal till you are told to do so. 2. Use the Optical Response Sheet (ORS) provided separately for answering the questions. DO NOT BREAK THE SEALS WITHOUT BEING INSTRUCTED TO DO SO BY THE INVIGILATOR 3. Blank spaces are provided within this booklet for rough work. 4. Write your name, form number and sign in the space provided on the back cover of this booklet. 5. After breaking the seal of the booklet, verify that the booklet contains 32 pages and that all the 17 questions in each subject and along with the options are legible. If not, contact the invigilator for replacement of the booklet. 6. You are allowed to take away the Question Paper at the end of the examination. OPTICAL RESPONSE SHEET : 7. The ORS will be collected by the invigilator at the end of the examination. 8. Do not tamper with or mutilate the ORS. Do not use the ORS for rough work. 9. Write your name, form number and sign with pen in the space provided for this purpose on the ORS. Do not write any of these details anywhere else on the ORS. Darken the appropriate bubble under each digit of your form number. DARKENING THE BUBBLES ON THE ORS : 10. Use a BLACK BALL POINT PEN to darken the bubbles on the ORS. 11. Darken the bubble COMPLETELY. 12. The correct way of darkening a bubble is as : 13. The ORS is machine-gradable. Ensure that the bubbles are darkened in the correct way. 14. Darken the bubbles ONLY IF you are sure of the answer. There is NO WAY to erase or "un-darken" a darkened bubble. 15. Take g = 10 m/s2 unless otherwise stated. QUESTION PAPER FORMAT : 16. The question paper has three parts : Physics, Chemistry and Mathematics. Please see the last page of this booklet for rest of the instructions h9O  Target : JEE (Main + Advanced) 2025/13-04-2025/Paper-1 B ® SOME USEFUL CONSTANTS Atomic No. : H = 1, B = 5, C = 6, N = 7, O = 8, F = 9, Al = 13, P = 15, S = 16, Cl = 17, Br = 35, Xe = 54, Ce = 58 Atomic masses : H = 1, Li = 7, B = 11, C = 12, N = 14, O = 16, F = 19, Na = 23, Mg = 24, Al = 27, P = 31, S = 32, Cl = 35.5, Ca = 40, Fe = 56, Br = 80, I = 127, Xe = 131, Ba=137, Ce = 140, Cu = 63.5, Ne = 20, K = 39, Mn = 55 Boltzmann constant k = 1.38 × 10–23 JK–1 1 Coulomb's law constant = 9 ×109 4 πε0 Universal gravitational constant G = 6.67259 × 10–11 N–m2 kg–2 Speed of light in vacuum c = 3 × 108 ms–1 Stefan–Boltzmann constant σ = 5.67 × 10–8 Wm–2–K–4 Wien's displacement law constant b = 2.89 × 10–3 m–K Permeability of vacuum µ0 = 4π × 10–7 NA–2 1 Permittivity of vacuum ∈0 = 2 µ0c Planck constant h = 6.63 × 10–34 J–s Space for Rough Work E-2/32 1001CJA101021240055 h9O  Enthusiast & Leader Course/Score Advanced/13-04-2025/Paper-1 BEWARE OF NEGATIVE MARKING PART-1 : PHYSICS SECTION-I (i) : (Maximum Marks: 12) This section contains FOUR (04) questions. Each question has FOUR options (A), (B), (C) and (D). ONLY ONE of these four options is the correct answer. For each question, choose the option corresponding to the correct answer. Answer to each question will be evaluated according to the following marking scheme : Full Marks : +3 If ONLY the correct option is chosen. Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered) Negative Marks : –1 In all other cases 1. A dimensionless quantity is constructed in terms of electronic charge e, permittivity of free space ε 0, Planck’s constant h, and speed of light c. If the dimensionless quantity is written as e α εβ0 h γ c δ and n is a non-zero integer, then ( α , β , γ , δ ) is given by (A) (n, – n, – n, – 2n) (B) (2n, – n, – 2n, – 2n) (C) (2n, – n, – n, – n) (D) (n, – n, – 2n, – n) 2. An infinitely long wire, located on the z-axis, carries a current I along the +z-direction and produces the → magnetic field B→. The magnitude of the line integral ∫ →. dl B along a straight line from the point (− √ 3a, a, 0) to ( 3a, a, 0) is given by : [ μ 0 is the magnetic permeability of free space.] √ (A) 7 μ 0I /24 (B) 7 μ 0I /12 (C) μ 0I /3 (D) μ 0I /6 3. Two beads, each with charge q and mass m, are on a horizontal, frictionless, non-conducting, circular hoop of radius R. One of the beads is glued to the hoop at some point, while the other one performs small oscillations about its equilibrium position along the hoop. The square of the angular frequency of the small oscillations is given by [ ε 0 is the permittivity of free space] (A) q2 / (8 π ε 0R3m) (B) q2 / (16 π ε 0R3m) (C) q2 / (32 π ε 0R3m) (D) q2 / (4 π ε 0R3m) 4. A block of mass 2.5 kg moves along the x-direction subject to the force F = (−10x + 5) N, with the value of x in metre. At time t = 0 s, it is at rest at position x = 1 m. The position and momentum of the block at t = ( π /4) s are (A) – 0.5 m, 2.5 kg m/s (B) 0.5 m, 0 kg m/s (C) 0.5 m, – 2.5 kg m/s (D) – 1 m, 5 kg m/s 1001CJA101021240055 E-3/32 h9O  Target : JEE(Main + Advanced) 2025/13-04-2025/Paper-1 SECTION-I (ii) : (Maximum Marks: 12) This section contains THREE (03) questions. Each question has FOUR options. ONE OR MORE THAN ONE of these four option(s) is (are) correct answer(s). For each question, choose the option(s) corresponding to (all ) the correct answer(s) Answer to each question will be evaluated according to the following marking scheme: Full Marks : +4 If only (all) the correct option(s) is (are) chosen. Partial Marks : +3 If all the four options are correct but ONLY three options are chosen. Partial Marks : +2 If three or more options are correct but ONLY two options are chosen and both of which are correct. Partial Marks : +1 If two or more options are correct but ONLY one option is chosen and it is a correct option. Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered). Negative Marks : –2 In all other cases. For Example : If first, third and fourth are the ONLY three correct options for a question with second option being an incorrect option; selecting only all the three correct options will result in +4 marks. Selecting only two of the three correct options (e.g. the first and fourth options), without selecting any incorrect option (second option in this case), will result in +2 marks. Selecting only one of the three correct options (either first or third or fourth option), without selecting any incorrect option (second option in this case), will result in +1 marks. Selecting any incorrect option(s) (second option in this case), with or without selection of any correct option(s) will result in –2 marks. 5. A particle of mass m is moving in a circular orbit under the influence of the central force F(r) = – kr, corresponding to the potential energy V(r) = kr2 / 2, where k is a positive force constant and r is the radial distance from the origin. According to the Bohr’s quantization rule, the angular momentum of the particle is given by L = nℏ, where ℏ = h/(2 π ), h is the Planck’s constant, and n a positive integer. If v and E are the speed and total energy of the particle, respectively, then which of the following expression(s) is(are) correct ? (A) L k =√ mr 2 m (B) nℏ k E= √ 2 m (C) k v2 = nℏ√ m3 (D) 1 r2 = nℏ√ mk E-4/32 1001CJA101021240055 h9O  Enthusiast & Leader Course/Score Advanced/13-04-2025/Paper-1 6. Two uniform string of mass per unit length μ and 4 μ , and length L and 2L, respectively, are joined at point O, and tied at two fixed ends P and Q, as shown in the figure. The strings are under a uniform tension T. If we 1 √T define the frequency v0 = , which of the following statement(s) is (are) correct ? 2L μ (A) No vibrational mode with an antinode at O is possible for the composite string. (B) When the composite string vibrates at the minimum frequency with a node at O, it has 6 nodes, including the end nodes. (C) With a node at O, the minimum frequency of vibration of the composite string is v0. (D) With an antinode at O, the minimum frequency of vibration of the composite string is 2v0. 7. A glass beaker has a solid, plano-convex base of refractive index 1.60, as shown in the figure. The radius of curvature of the convex surface (SPU) is 9 cm, while the planar surface (STU) acts as a mirror. This beaker is filled with a liquid of refractive index n up to the level QPR. If the image of a point object O at a height of h (OT in the figure) is formed onto itself, then, which of the following option(s) is(are) correct ? (A) For n = 1.42, h = 40 cm. (B) For n = 1.35, h = 52 cm. (C) For n = 1.45, h = 60 cm. (D) For n = 1.48, h = 75 cm. 1001CJA101021240055 E-5/32 h9O  Target : JEE(Main + Advanced) 2025/13-04-2025/Paper-1 SECTION-I (iii) : (Maximum Marks: 12) This section contains FOUR (04) Matching List Sets. Each set has ONE Multiple Choice Question. Each set has TWO lists : List-I and List-II. List-I has Four entries (P), (Q), (R) and (S) and List-II has Five entries (1), (2), (3), (4) and (5). FOUR options are given in each Multiple Choice Question based on List-I and List-II and ONLY ONE of these four options satisfies the condition asked in the Multiple Choice Question. Answer to each question will be evaluated according to the following marking scheme: Full Marks : +3 ONLY if the option corresponding to the correct combination is chosen; Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered); Negative Marks : –1 In all other cases. 8. One mole of a monatomic ideal gas undergoes the cyclic process J → K → L → M → J, as shown in the P-T diagram. Match the quantities mentioned in List-I with their values in List-II and choose the correct option. [R is the gas constant.] List-I List-II (P) Work done in the complete cyclic process (1) 0 (Q) Change in the internal energy of the gas in the process JK (2) – 3RT0 ln 2 (R) Heat given to the gas in the process KL (3) 3RT0 (S) Change in the internal energy of the gas in the process MJ (4) RT0 – 4RT0 ln 2 (5) – 2RT0 ln 2 (A) P → 1;Q → 3;R → 5;S → 4 (B) P → 5;Q → 3;R → 2;S → 1 (C) P → 5;Q → 4;R → 1;S → 1 (D) P → 2;Q → 5;R → 3;S → 4 E-6/32 1001CJA101021240055 h9O  Enthusiast & Leader Course/Score Advanced/13-04-2025/Paper-1 9. Four identical thin, square metal sheets, S1, S2, S3 and S4, each of side a are kept parallel to each other with equal distance d ( 0. Then for all x > 0, f(x) is equal to 1 (A) f (x) = x3 + x 1 (B) f (x) = x3 + x2 (C) 1 f (x) = 3x3 − x 1 (D) f (x) = 3x3 − x2 2. Let π < x < π and tanx = 2 −24 7 , then value of sin 13x 2 (sin 7x − cos 7x) + cos 13x 2 (sin 7x + cos7x) is equal to 7 (A) − 5 7 (B) 5 4 (C) 3 4 (D) − 3 1001CJA101021240055 E-23/32 h9O  Target : JEE(Main + Advanced) 2025/13-04-2025/Paper-1 3. A student appears for a quiz consisting of only, objective type questions, each question having 4 options of which exactly one option is correct and he answers all the questions. The student knows the answers of some questions and guesses the answers for the remaining questions. Whenever the student knows the answer of a question, he gives the correct answer. Assume that the probability of the student giving the correct answer for a 1 question, given that he has guessed it, is. Also assume that the probability of the answer for a question being 4 1 guessed, given that the student's answer is correct, is. Then the probability that the student knows the 6 answer of a randomly chosen question is : 4 5 (A) (B) 9 12 5 5 (C) (D) 7 9 4. Consider the ellipse x2 y 2 9 + 4 = 1. Let S(p, q) be a point in the first quadrant such that p2 q 2 9 + 4 > 1. Two tangents are drawn from S to the ellipse, of which one meets the ellipse at one end point of the minor axis and the other meets the ellipse at a point T in the fourth quadrant. Let R be the vertex of the ellipse with 9 positive x-coordinate and O be the center of the ellipse. If the area of the triangle Δ ORT is then which 5 of the following options is correct ? (A) p = 3, q = 2 (B) p = 6, q = 2 (C) p = 3, q = 1 (D) p = 6, q = 1 E-24/32 1001CJA101021240055 h9O  Enthusiast & Leader Course/Score Advanced/13-04-2025/Paper-1 SECTION-I (ii) : (Maximum Marks: 12) This section contains THREE (03) questions. Each question has FOUR options. ONE OR MORE THAN ONE of these four option(s) is (are) correct answer(s). For each question, choose the option(s) corresponding to (all ) the correct answer(s) Answer to each question will be evaluated according to the following marking scheme: Full Marks : +4 If only (all) the correct option(s) is (are) chosen. Partial Marks : +3 If all the four options are correct but ONLY three options are chosen. Partial Marks : +2 If three or more options are correct but ONLY two options are chosen and both of which are correct. Partial Marks : +1 If two or more options are correct but ONLY one option is chosen and it is a correct option. Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered). Negative Marks : –2 In all other cases. For Example : If first, third and fourth are the ONLY three correct options for a question with second option being an incorrect option; selecting only all the three correct options will result in +4 marks. Selecting only two of the three correct options (e.g. the first and fourth options), without selecting any incorrect option (second option in this case), will result in +2 marks. Selecting only one of the three correct options (either first or third or fourth option), without selecting any incorrect option (second option in this case), will result in +1 marks. Selecting any incorrect option(s) (second option in this case), with or without selection of any correct option(s) will result in –2 marks. 5. Let S = a + b 3 : a, b ∈ Z , T1 = { √ } {( n 2 − √3 ) : n ∈ N } and T2 = {( n 2 + √3 ) : n ∈ N } , then which of the following is (are) true ? (A) Z ∪ T1 ∪ T2 ⊂ S 1 (B) T2 ∩ (0, ) = ϕ 2025 (where ϕ denotes null set) 1 (C) T1 ∩ (0, ) ≠ ϕ 100 (D) For any a, b ∈ Z, cos π ( ( a + b√3)) + i sin( π (a + b√3)) ∈ Z if and only if b = 0, where i = √−1 1001CJA101021240055 E-25/32 h9O  Target : JEE(Main + Advanced) 2025/13-04-2025/Paper-1 6. Let R2 denote R × R. Let S = {(a, b, c) : a, b, c ∈ R and ax2 + 2bxy + cy2 < 0, ∀ (x, y) ∈ R2 – {(0, 0)}} then which of the following statements is(are) True ? (A) ( – 2, 3, – 4) ∈ S (B) If ( – 1, 2, c) ∈ S, then c < – 4 (C) For some given (a, b, c) ∈ S, the system of equation , may be inconsistent (D) For some given (a, b, c) ∈ S, the system of equation , has a unique solution 7. Let R 3 denote the three-dimensional space. Take two points P = (3, 4, 5) & Q(6, 4, 9). Let dist (X, Y) denote distance between two point X and Y in R3. Let S = {X ∈ R 3 : (dist (X, P))2 – (dist (X, Q))2 = 100} and T = {Y ∈ R 3 : (dist (Y, Q))2 – (dist (Y, P))2 = 100}, then which of the following is True ? (A) There is a triangle whose area is 2 sq. unit & all vertices are from S. (B) There are two distinct points L and M in T such that each point on the line segment LM is also in T. (C) There are infinitely many rectangles of perimeter 48, two of whose vertices are from S and the other two vertices are from T. (D) There is a square of perimeter 48, two of whose vertices are from S and the other two vertices are from T. E-26/32 1001CJA101021240055 h9O  Enthusiast & Leader Course/Score Advanced/13-04-2025/Paper-1 SECTION-I (iii) : (Maximum Marks: 12) This section contains FOUR (04) Matching List Sets. Each set has ONE Multiple Choice Question. Each set has TWO lists : List-I and List-II. List-I has Four entries (P), (Q), (R) and (S) and List-II has Five entries (1), (2), (3), (4) and (5). FOUR options are given in each Multiple Choice Question based on List-I and List-II and ONLY ONE of these four options satisfies the condition asked in the Multiple Choice Question. Answer to each question will be evaluated according to the following marking scheme: Full Marks : +3 ONLY if the option corresponding to the correct combination is chosen; Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered); Negative Marks : –1 In all other cases. 8. Let α , β and γ be the distinct roots of the equation x3 – 7x + 6 = 0. Consider the set T = { α , β , γ } for a 3 × 3 matrix M = (aij)3 × 3, define Ri = ai1 + ai2 + ai3 and Cj = a1j + a2j + a3j for i = 1, 2, 3 and j = 1, 2, 3. Match each entry in List-I to the correct entry in List-II. List-I List-II (P) The number of matrices M = (aij)3 × 3 with all entries in T such that Ri = Cj = 0 for all i, j, is (1) 1 The number of symmetric matrices M = (aij)3 × 3 with all entries in T such that Cj = 0 (Q) (2) 12 for all j, is Let M = (aij)3 × 3 be a skew symmetric matrix such that aij ∈ T for i > j. Then the number ⎧ ⎪⎛ x⎞ ⎛ x⎞ ⎛ a12 ⎞⎪ ⎫ ⎪ ⎪ (R) ⎪ ⎪ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎪ ⎪ (3) infinite of elements in the set ⎨⎜ ⎜ y⎟ ⎟ : x, y, z ∈ R, M ⎜ ⎜ y⎟ ⎟ =⎜ ⎜ 0 ⎟⎬ ⎟ is ⎪⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎝ ⎪ ⎠ ⎝ ⎠ ⎝ ⎠⎪⎭ z z −a23 Let M = (aij)3 × 3 be a matrix with all entries in T such that Ri = 0 for all i. Then the (S) (4) 6 absolute value of the determinant of M is (5) 0 (A) P → 2;Q → 4;R → 3;S → 5 (B) P → 3;Q → 2;R → 4;S → 2 (C) P → 2;Q → 2;R → 5;S → 2 (D) P → 3;Q → 4;R → 3;S → 1 1001CJA101021240055 E-27/32 h9O  Target : JEE(Main + Advanced) 2025/13-04-2025/Paper-1 9. Let α ∈ R be such that lines : x−1 y−1 z−4 L1 : = = and 2 3 α x y z−2 L2 : = = 3 4 4 intersect let R1 be the point of intersection of L1 & L2. Let O = (0, 0, 0) and n^ be a unit vector normal to plane containing lines L1 & L2 : List-I List-II 2 (P) α = (1) √ 21 (Q) A possible choice for n^ = (2) 4 ^ ^ ^ −−− → 4i 2j k (R) OR1 = (3) ( − − ) √ 21 √ 21 √ 21 −−− (S) OR→ ^ ∣ ∣ (4) 2 ∣1. n ∣ ∣ ∣ ^ ^ ^ (5) 3i + 4j + 6k (A) P → 2;Q → 3;R → 4;S → 2 (B) P → 4;Q → 1;R → 5;S → 3 (C) P → 2;Q → 1;R → 4;S → 1 (D) P → 4;Q → 3;R → 5;S → 1 10. Let line 2y = x + 3 touch a circle with centre ( α , 0), ( α > 0) and radius r at a point A1. Let B1 be the point on the circle such that the line segment A1B1 is a diameter of the circle. Let α + r = 2 + 5. Match List-I to List-II. √ List-I List-II (P) α = (1) (1, 2) (Q) r = (2) √ 5 (R) A1 = (3) 2 (S) B1 = (4) (3, – 2) (5) 1 (A) P → 3;Q → 5;R → 4;S → 1 (B) P → 5;Q → 3;R → 1;S → 2 (C) P → 3;Q → 2;R → 1;S → 4 (D) P → 5;Q → 2;R → 3;S → 4 E-28/32 1001CJA101021240055 h9O  Enthusiast & Leader Course/Score Advanced/13-04-2025/Paper-1 x2 sin 1x , 11. x≠0 1 − x, 0⩽x 0 के लिए f(x) बराबर है t→x t2 − x2 1 (A) f (x) = x3 + x 1 (B) f (x) = x3 + x2 1 (C) f (x) = 3x3 − x 1 (D) f (x) = 3x3 − x2 2. माना π 2 < x < π तथा tanx = −24 7 है, तो sin 13x 2 (sin 7x − cos 7x) + cos 13x 2 (sin 7x + cos7x) का मान है 7 (A) − 5 7 (B) 5 4 (C) 3 4 (D) − 3 1001CJA101021240055 H-23/32 h9O  Target : JEE(Main + Advanced) 2025/13-04-2025/Paper-1 3. एक छात्र के वल वस्तुनिष्ठ प्रकार के प्रश्नों वाली एक प्रश्नोत्तरी में शामिल होता है, प्रत्येक प्रश्न के 4 विकल्प होते हैं जिनमें से के वल एक विकल्प सही होता है तथा वह सभी प्रश्नों के उत्तर देता है। छात्र कु छ प्रश्नों के उत्तर जानता है तथा शेष प्रश्नों के उत्तर का अनुमान लगाता है। जब भी छात्र किसी प्रश्न का उत्तर जानता है, वह सही उत्तर देता है। माना छात्र द्वारा किसी प्रश्न का सही उत्तर देने की प्रयिकता, जब यह ज्ञात 1 है कि छात्र ने उत्तर अनुमान लगाया है, है। यह भी माना कि किसी प्रश्न के उत्तर का अनुमान लगाए जाने की प्रायिकता, जब यह ज्ञात है 4 1 कि छात्र का उत्तर सही है, है। तब छात्र के किसी यादृच्छया चुने गए प्रश्न का उत्तर जानने की प्रायिकता है: 6 4 5 (A) (B) 9 12 5 5 (C) (D) 7 9 4. माना दीर्घवृत्त x2 y 2 9 + 4 = 1 है। माना S(p, q) प्रथम चतुर्थाश में एक इस प्रकार का बिन्दु है कि p2 q 2 9 + 4 > 1 है। बिन्दु S से दीर्घवृत्त के लिए दो स्पर्श रेखाएं खींची गयी है, जिनमें से एक रेखा, दीर्घवृत्त पर लघुअक्ष के एक अंत्य बिन्दु (end point) पर मिलती है तथा दूसरी रेखा चौथे चतुर्थाश में दीर्घवृत्त के एक बिन्दु T पर मिलती है। माना R दीर्घवृत्त का वह शीर्ष है जिसका x-निर्देशांक धनात्मक है तथा दीर्घवृत्त 9 का के न्द्र O है। यदि त्रिभुज Δ ORT का क्षेत्रफल है, तब निम्नलिखित विकल्पों में से कौनसा सही है ? 5 (A) p = 3, q = 2 (B) p = 6, q = 2 (C) p = 3, q = 1 (D) p = 6, q = 1 H-24/32 1001CJA101021240055 h9O  Enthusiast & Leader Course/Score Advanced/13-04-2025/Paper-1 SECTION-I (ii) : (अधिकतम अंक: 12) इस खंड में तीन (03) प्रश्न हैं। प्रत्येक प्रश्न के सही उत्तर (उत्तरों) के लिए चार विकल्प दिए गए हैं। इस चार विकल्पों में से एक या एक से अधिक विकल्प सही है(हैं)। प्रत्येक प्रश्न के लिए, प्रश्न का (के ) उत्तर देने हेतु सही विकल्प (विकल्पों) को चुने। प्रत्येक प्रश्न के उत्तर का मूल्यांकन निम्न अंकन योजना के अनुसार होगा : पूर्ण अंक : +4 यदि के वल (सारे) सही विकल्प (विकल्पों) को चुना गया है। आंशिक अंक : +3 यदि चारों विकल्प सही हैं परन्तु के वल तीन विकल्पों को चुना गया है। आंशिक अंक : +2 यदि तीन या तीन से अधिक विकल्प सही हैं परन्तु के वल दो विकल्पों को चुना गया है और चुने हुए दोनों विकल्प सही विकल्प हैं। आंशिक अंक : +1 यदि दो या दो से अधिक विकल्प सही हैं परन्तु के वल एक विकल्प को चुना गया है और चुना हुआ विकल्प सही विकल्प हैं। शून्य अंक : 0 यदि किसी भी विकल्प को नहीं चुना गया है (अर्थात् प्रश्न अनुत्तरित है)। ऋण अंक : –2 अन्य सभी परिस्थितियों में। उदाहरण स्वरूप : यदि किसी प्रश्न के लिए के वल पहला, तीसरा एवं चौथा सही विकल्प हैं और दूसरा विकल्प गलत है; तो के वल सभी तीन सही विकल्पों का चयन करने पर ही +4 अंक मिलेंगें। बिना कोई गलत विकल्प चुने (इस उदाहरण में दूसरा विकल्प) तीन सही विकल्पों में से सिर्फ दो को चुनने पर (उदाहरणतः पहला तथा चौथा विकल्प) +2 अंक मिलेंगे। बिना कोई गलत विकल्प चुने (इस उदाहरण में दूसरा विकल्प), तीन सही विकल्पों में से सिर्फ एक को चुनने पर (पहला या तीसरा या चौथा विकल्प) +1 अंक मिलेंगे। कोई भी गलत विकल्प चुनने पर (इस उदाहरण में दूसरा विकल्प), –2 अंक मिलेंगे, चाहे सही विकल्प (विकल्पों) को चुना गया हो या न चुना गया हो। 5. माना S = a + b 3 : a, b ∈ Z , T1 = { √ } {( n 2 − √3 ) : n ∈ N } तथा T2 = {( n 2 + √3 ) : n ∈ N } है, तो निम्नलिखित कथनों में से कौनसा(से) सत्य है/हैं ? (A) Z ∪ T1 ∪ T2 ⊂ S 1 (B) T2 ∩ (0, ) = ϕ 2025 (जहाँ ϕ रिक्त समुच्चय को दर्शाता है) 1 (C) T1 ∩ (0, ) ≠ ϕ 100 (D) किसी a, b ∈ Z के लिए cos( π (a + b√3)) + i sin( π (a + b√3)) ∈ Z यदि और के वल यदि b = 0, जहाँ i = −1 है। √ 1001CJA101021240055 H-25/32 h9O  Target : JEE(Main + Advanced) 2025/13-04-2025/Paper-1 6. माना R2, R × R को दर्शाता है। माना S = {(a, b, c) : a, b, c ∈ R तथा ax2 + 2bxy + cy2 < 0, ∀ (x, y) ∈ R2 – {(0, 0)}} है; तब निम्नलिखित कथनों में से कौनसा(से) सत्य है(हैं) ? (A) ( – 2, 3, – 4) ∈ S (B) यदि ( – 1, 2, c) ∈ S हो, तो c < – 4 है। (C) किसी दिये गए (a, b, c) ∈ S के लिए रैखिक समीकरणों के निकाय असंगत हो सकते है। (D) किसी दिये गए (a, b, c) ∈ S के लिए रैखिक समीकरणों के निकाय का अद्वितीय हल है। 7. माना R 3 त्रिविमीय समष्टि (space) को दर्शाता है। दो बिन्दु P = (3, 4, 5) तथा Q(6, 4, 9) लीजिये। माना (X, Y), R3 में दो बिन्दुओं X तथा Y के बीच की दूरी को दर्शाता है। माना S = {X ∈ R 3 : (dist (X, P))2 – (dist (X, Q))2 = 100} तथा T = {Y ∈ R 3 : (dist (Y, Q))2 – (dist (Y, P))2 = 100} है, तब निम्नलिखित कथनों में से कौनसा(से) सत्य है(हैं) (A) एक ऐसा त्रिभुज है जिसका क्षेत्रफल 2 वर्ग इकाई है तथा जिसके सभी शीर्ष S से है। (B) T में दो ऐसे भिन्न बिन्दु L तथा M है कि रेखाखण्ड LM में स्थित प्रत्येक बिन्दु भी T में हैं। (C) परिमाप 48 के ऐसे अनंत आयत है जिनके दो शीर्ष S से तथा अन्य दो शीर्ष T से हैं। (D) परिमाप 48 का एक ऐसा वर्ग है जिनके दो शीर्ष S से तथा अन्य दो शीर्ष T से हैं। H-26/32 1001CJA101021240055 h9O  Enthusiast & Leader Course/Score Advanced/13-04-2025/Paper-1 SECTION-I (iii) : (अधिकतम अंक: 12) इस खंड में चार (04) सूची.सुमेलन (List-Match) सेट्स (sets) हैं। प्रत्येक सूची सुमेलन सेट (set) में एक एकाधिक विकल्प प्रश्न (Multiple Choice Questions) हैं। प्रत्येक सूची-सुमेलन सेट में दो सूचियाँ हैं : सूची-I और सूची-II सूची-I में चार प्रविष्टियाँ (P), (Q), (R) और (S) हैं एवं सूची-II में पाँच प्रविष्टियाँ (1), (2), (3), (4) और (5) हैं। प्रत्येक एकाधिक विकल्प प्रश्न में सूची-I और सूची-II पर आधारित चार विकल्प दिये गए हैं और इन विकल्पों में से केवल एक विकल्प ही एकाधिक विकल्प प्रश्न की शर्त पूरा करता है। प्रत्येक प्रश्न के उत्तर का मूल्यांकन निम्न योजना के अनुसार होगा । पूर्ण अंक : +3 यदि सिर्फ सही विकल्प को ही चुना गया है। शून्य अंक : 0 यदि किसी भी विकल्प को नहीं चुना गया है (अर्थात् प्रश्न अनुत्तरित है)। ऋण अंक : –1 अन्य सभी परिस्थितियों में। 8. माना α , β तथा γ समीकरण x3 – 7x + 6 = 0 के भिन्न मूल हैं। समुच्चय T = { α , β , γ } पर विचार कीजिए। एक 3 × 3 आव्यूह M = (aij)3 × 3 के लिए Ri = ai1 + ai2 + ai3 तथा Cj = a1j + a2j + a3j परिभाषित कीजिए जहाँ i = 1, 2, 3 तथा j = 1, 2, 3 है। सूची-I की प्रत्येक प्रविष्टि का सूची-II की सही प्रविष्टि से मिलान कीजिए। सूची-I सूची-II आव्यूहों M = (aij)3 × 3, जिनकी सभी प्रविष्टियाँ T से है तथा जिनमें सभी i, j के लिए Ri = Cj = 0 है, की (P) (1) 1 संख्या है सममित आव्यूहों M = (aij)3 × 3, जिनकी सभी प्रविष्टियाँ T से है तथा जिनमें सभी j के लिए Cj = 0 है, की (Q) (2) 12 संख्या है माना M = (aij)3 × 3 एक ऐसा विषम सममित आव्यूह है कि i > j के लिए aij ∈ T है। तब समुच्चय ⎧⎛ ⎪ x⎞ ⎛ x⎞ ⎛ a12 ⎞⎪ ⎫ ⎪ ⎪ (R) ⎪ ⎪ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎪ ⎪ (3) अनंत ⎨⎜ ⎜ y⎟ ⎟ : x, y, z ∈ R, M ⎜ ⎜ y⎟ ⎟ =⎜ ⎜ 0 ⎟⎬ ⎟ में अवयवों की संख्या है ⎪ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ ⎪⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎪ z z −a23 माना M = (aij)3 × 3 एक ऐसा आव्यूह है कि जिसकी सभी प्रविष्टियाँ T से है तथा जिसमें सभी i के लिए Ri = 0 (S) (4) 6 है। तब M के सारणिक का निरपेक्ष मान है (5) 0 (A) P → 2;Q → 4;R → 3;S → 5 (B) P → 3;Q → 2;R → 4;S → 2 (C) P → 2;Q → 2;R → 5;S → 2 (D) P → 3;Q → 4;R → 3;S → 1 1001CJA101021240055 H-27/32 h9O  Target : JEE(Main + Advanced) 2025/13-04-2025/Paper-1 9. माना कि α ∈ R इस प्रकार है कि रेखाएं : x−1 y−1 z−4 L1 : = = तथा 2 3 α x y z−2 L2 : = = 3 4 4 प्रतिच्छेदित करती हैं। माना L1 तथा L2 का प्रतिच्छेदन R1 है। माना O = (0, 0, 0) है तथा n^ , उस तल जिसमें L1 तथा L2 दोनों स्थित हैं, के एक मात्रक अभिलम्ब सदिश (unit normal vector) को दर्शाता है : सूची-I सूची-II 2 (P) α बराबर (1) √ 21 (Q) n^ का एक संभावित विकल्प है (2) 4 ^ ^ ^ −−− → 4i 2j k (R) OR1 बराबर (3) ( − − ) √ 21 √ 21 √ 21 −−− → OR1. n^ ∣ बराबर ∣ ∣ (S) ∣ (4) 2 ∣ ∣ ^ ^ ^ (5) 3i + 4j + 6k (A) P → 2;Q → 3;R → 4;S → 2 (B) P → 4;Q → 1;R → 5;S → 3 (C) P → 2;Q → 1;R → 4;S → 1 (D) P → 4;Q → 3;R → 5;S → 1 10. माना सरल रेखा 2y = x + 3, एक वृत्त जिसका के न्द्र ( α , 0), ( α > 0) है तथा जिसकी त्रिज्या r है, को एक बिन्द A1 पर स्पर्श करती है। माना B1 वृत्त पर वह बिन्दु है कि रेखाखण्ड A1B1 वृत्त का एक व्यास है। माना कि α + r = 2 + 5 है। सूची-I का सूची-II के साथ √ मिलान कीजिए सूची-I सूची-II (P) α बराबर है (1) (1, 2) (Q) r बराबर है (2) √ 5 (R) A1 बराबर है (3) 2 (S) B1 बराबर है (4) (3, – 2) (5) 1 (A) P → 3;Q → 5;R → 4;S → 1 (B) P → 5;Q → 3;R → 1;S → 2 (C) P → 3;Q → 2;R → 1;S → 4 (D) P → 5;Q → 2;R → 3;S → 4 H-28/32 1001CJA101021240055 h9O  Enthusiast & Leader Course/Score Advanced/13-04-2025/Paper-1 x2 sin 1x , 11. x≠0 1 − x, 0⩽x