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JEE Main 27 Jan Shift 1 2024 question paper with solutions and answers pdf is available here. NTA conducted JEE Main 2024 Jan 27 Shift 1 exam from 9 AM to 12 PM. The question paper for JEE Main 2024 Jan 27 Shift 1 includes 90 questions equally divided into Physics, Chemistry and Maths. Candidates must attempt 75 questions in a 3-hour time duration. More questions will soon be added to the memory-based JEE Main 2024 paper 1 question paper for Jan 27 Shift 1 provided in the article below.
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JEE Main 27 Jan Shift 1 2024 Questions
| Question | Answer | Solution |
|---|---|---|
| Question 1: n−1Cr = (k^2 −8) nCr+1 if and only if: (1) 2√2 < k ≤ 3 (2) 2√3 < k ≤ 3√2 (3) 2√3 < k < 3√3 (4) 2√2 < k < 2√3 |
(1) 2√2 < k ≤ 3 | We know n−1Cr = (k^2 −8) nCr+1. For this expression to hold, k^2 − 8 must be positive: k^2 − 8 > 0 => k > 2√2 or k < -2√2. Checking the range -3 ≤ k ≤ 3 to satisfy the constraint, we find k ∈ [2√2, 3]. |
| Question 2: The distance of the point (7,−2,11) from the line x−6/1 = y−4/0 = z−8/3 is: (1) 12 (2) 14 (3) 18 (4) 21 |
(2) 14 | To find the distance, we first determine coordinates of a point B on the line, substituting values to find λ. The shortest distance AB between points A = (7,−2,11) and B = (3,4,−1) is calculated, resulting in √196 = 14. |
| Question 3: Let x=x(t) and y=y(t) be solutions of dx/dt +ax=0 and dy/dt +by=0 respectively. Given x(0)=2, y(0)=1, and 3y(1)=2x(1), find t for which x(t)=y(t). (1) log2 3/2 (2) log4 3 (3) log3 4 (4) log4 2/3 |
(4) log4 2/3 | Differential equations solutions give x(t) = 2e^(-at) and y(t) = e^(-bt). Given conditions yield b = a + ln(4/3). Solving for t, we obtain t = log4(2/3). |
| Question 4: If (a,b) is the orthocenter of a triangle with vertices (1,2), (2,3), (3,1), then 36I1/I2 is equal to: (1) 72 (2) 88 (3) 80 (4) 66 |
(1) 72 | Given triangle vertices, the orthocenter lies on x+y=4. Using King’s rule, we derive I1 = 2 I2, giving 36I1/I2 = 72. |
| Question 5: Let A be the sum of all coefficients in the expansion of (1 −3x +10x^2)^n and B for (1 +x^2)^n. Then: (1) A = B^3 (2) 3A = B (3) B = A^3 (4) A = 3B |
(1) A = B^3 | Substituting x = 1, A = 8^n and B = 2^n, thus A = B^3. |
| Question 6: Number of common terms in sequences 4, 9, 14... up to 25th term and 3, 6, 9... up to 37th term: (1) 9 (2) 5 (3) 7 (4) 8 |
(3) 7 | Using LCM(5,3) = 15, common terms form 9, 24, 39... with seven terms. |
| Question 7: The shortest distance of parabola y^2=4x from circle x^2 + y^2 - 4x -16y +64 = 0 is d. Find d^2: (1) 16 (2) 24 (3) 20 (4) 36 |
(3) 20 | Rewriting circle’s equation and calculating for parabola’s shortest distance using normal, we find d^2 = 20. |
| Question 8: The shortest distance between lines x −4/1 = y+1/2 = z−3 and x−λ/2 = y+1/4 = z−2/−5 is 6√5. The sum of all possible values of λ is: (1) 5 (2) 8 (3) 7 (4) 10 |
(2) 8 | Finding the required conditions on λ, the answer is 8. |
| Question 9: Evaluate integral ∫(1 to 0) 1/(√3+x + √1+x) dx in form a + b√2 + c√3; find 2a + 3b - 4c. (1) 4 (2) 10 (3) 7 (4) 8 |
(4) 8 | Rationalizing the integral and simplifying, a=4/3, b=-4/3, c=-1. Thus, 2a+3b−4c=8. |
| Question 10: Let S = {1,2,3...10}. M is the set of all subsets, and relation R = {(A,B): A∩B=∅}. R is: (1) symmetric and reflexive only (2) reflexive only (3) symmetric and transitive only (4) symmetric only |
(4) symmetric only | R is symmetric because (A,B) ∈ R implies (B,A) ∈ R but is not reflexive or transitive. |
| Question 11: If S = {z ∈ C : |z−i| = |z+i| = |z−1|}, then n(S) is: (1) 1 (2) 0 (3) 3 (4) 2 |
(1) 1 | Given symmetry around three points, only z = 0 satisfies, so n(S) = 1. |
| Question 12: Four points (2k,3k), (1,0), (0,0), and (0,1) lie on a circle for k: (1) 2/13 (2) 3/13 (3) 5/13 (4) 1/13 |
(3) 5/13 | For points to lie on the circle’s circumference, k = 5/13. |
| Question 13: Consider f(x)= a(7x−12−x^2), b |x^2−7x+12| for continuity. Find n(S). (1) 2 (2) Infinitely many (3) 4 (4) 1 |
(4) 1 | Solving for continuity at x=3 gives unique solution (a,b) = (-4,2). |
| Question 14: Let a₁, a₂, ..., a₁₀ be 10 observations with ∑aₖ=50 and ∑(aₖ · aⱼ)=1100. Find the standard deviation. (1) 5 (2) √5 (3) 10 (4) √115 |
(2) √5 | Using standard deviation formula σ = √[(∑aₖ²/n) - (∑aₖ/n)²], we calculate values for ∑aₖ and ∑aₖ · aⱼ, resulting in σ = √5. |
| Question 15: The length of the chord of the ellipse x²/25 + y²/16 = 1, with midpoint (1, 2/5), is: (1) √169/5 (2) √200/9 (3) √174/5 (4) √154/5 |
(1) √169/5 | Applying the chord midpoint formula for ellipses and substituting values, we derive the length as √169/5. |
| Question 16: The portion of the line 4x + 5y = 20 in the first quadrant is trisected by lines L₁ and L₂ passing through the origin. The tangent of the angle between L₁ and L₂ is: (1) 8/5 (2) 25/41 (3) 2/5 (4) 30/41 |
(4) 30/41 | Using trisection points, we find slopes m₁ and m₂ for L₁ and L₂. Calculating tanθ between these slopes yields 30/41. |
| Question 17: Let a = i + 2j + k, b = 3(i - j + k). Let c satisfy a × c = b and a · c = 3. Then a · (c × b) - b · c is: (1) 32 (2) 24 (3) 20 (4) 36 |
(2) 24 | Given conditions a × c = b and a · c = 3, we simplify using vector identities to find a · (c × b) - b · c = 24. |
| Question 18: If a = lim(x→0) [1+√(1+x⁴)-√2]/x⁴ and b = lim(x→0) [sin2x/√(2 - √(1+cosx))], then ab³ is: (1) 36 (2) 32 (3) 25 (4) 30 |
(2) 32 | Simplifying limits a and b by rationalizing and substituting values, we find ab³ = 32. |
| Question 19: Given f(x)= cos(x) -sin(x) 0 sin(x) cos(x) 0 0 0 1 Evaluate: Statement I: f(-x) is the inverse of f(x). Statement II: f(x)·f(y)=f(x+y). (1) I is false, II is true (2) Both I and II are false (3) I is true, II is false (4) Both I and II are true |
(4) Both I and II are true | Verifying matrix properties, both statements hold true. Thus, both I and II are correct. |
| Question 20: The function f: N-{1}→N defined by f(n)=highest prime factor of n, is: (1) one-one and onto (2) one-one only (3) onto only (4) neither one-one nor onto |
(4) neither one-one nor onto | Since different values of n can share the same highest prime factor and not all natural numbers are primes, f(n) is neither one-one nor onto. |
| Question 21: Least positive integral value of α, for which the angle between vectors αi−2j+2k and αi+2αj−2k is acute, is: (1) 3 (2) 4 (3) 5 (4) 6 |
(3) 5 | Condition for an acute angle between vectors yields α² - 4α - 4 > 0. Solving, we find α > 4.828, so minimum integral α is 5. |
| Question 22: For differentiable function f: (0, ∞) → R, if f(x) − f(y) ≥ log(x/y) + x − y, find ∑f'(1/n) from n=1 to 20. (1) 2890 |
(2890) | Simplifying conditions given for continuity, we derive f’(x) = 1/x + 1. Summing up from n=1 to 20, we get 2890. |
| Question 23: Differential equation (2x + 3y − 2)dx + (4x + 6y − 7)dy = 0, y(0) = 3, has solution αx + βy + 3log|2x + 3y − γ| = 6. Find α+2β+3γ. (1) 29 |
(29) | Solving the differential equation and applying the initial condition, we find α+2β+3γ = 29. |
| Question 24: Let the area of region {(x,y): x − 2y + 4 ≥ 0, x + 2y² ≥ 0, x+4y² ≤ 8, y ≥ 0} be m/n with m and n coprime. Find m+n. (1) 119 |
(119) | Setting up the integration limits and solving the area, we express A in coprime form m/n, resulting in m+n=119. |
| Question 25: If (1+α)^7 = A + Bα + Cα², with A, B, C ≥ 0, find 5(3A − 2B − C). (1) 5 |
(5) | Expanding using binomial theorem and matching terms, we find A, B, C values. Then 5(3A − 2B − C) = 5. |
| 25. If 8 = 3 + 1/4 (3 + p) + 1/4^2 (3 + 2p) + 1/4^3 (3 + 3p) + ..., then the value of p is: (1) 9 (2) 8 (3) 7 (4) 6 |
9 | The series sum was calculated to find p, leading to the solution. |
| 26. A fair die is tossed repeatedly until a six is obtained. Let X denote the number of tosses required, and let a = P(X = 3), b = P(X ≥ 3), and c = P(X ≥ 6 | X > 3). Then (b + c)/a is equal to: (1) 12 (2) 8 (3) 10 (4) 6 |
12 | Probability calculations were used with the geometric progression, yielding the solution. |
| 27. Let the set of all a in ℝ such that the equation cos(2x) + a sin(x) = 2a - 7 has a solution be [p, q], and r = tan(9°) - tan(27°) - 1/(cot(63°) + tan(81°)). Then pqr is equal to: (1) 36 (2) 48 (3) 24 (4) 16 |
48 | Solution involves using trigonometric identities to find the solution intervals for a. |
| 28. Given f(x) = x^3 + x^2 f'(1) + x f''(2) + f'''(3), x in ℝ. Then f'(10) is equal to: (1) 202 (2) 204 (3) 206 (4) 208 |
202 | The solution uses given derivatives and substitution of values to find f'(10). |
| 29. Let A denote the matrix [[2, 0, 1], [1, 1, 0], [1, 0, 1]], and B = [B1, B2, B3] where B1, B2, and B3 are column matrices such that: AB1 = [1, 0, 0], AB2 = [2, 3, 0], AB3 = [3, 2, 1]. If α = |B| and β is the sum of all diagonal elements of B, then α^3 + β^3 is equal to: (1) 16 (2) 20 (3) 24 (4) 28 |
28 | Using matrix multiplication and properties to calculate determinants and trace. |
| 30. If α satisfies x^2 + x + 1 = 0 and (1 + α)^7 = A + Bα + Cα^2, where A, B, C ≥ 0, then 5(3A - 2B - C) is equal to: (1) 5 (2) 10 (3) 15 (4) 20 |
5 | Using properties of cube roots of unity and binomial expansion for simplification. |
| 31. A particle starts from the origin at t = 0 with a velocity 5i m/s and moves in the xy-plane under a force producing constant acceleration of (3i + 2j) m/s^2. If the x-coordinate of the particle is 84 m at that instant, the speed of the particle at this time is √α m/s. Find α: (1) 625 (2) 673 (3) 600 (4) 720 |
673 | Calculations involved in applying kinematic equations for both x and y components. |
| 32. Consider the function f : ℕ - {1} → ℕ defined by f(n) as the highest prime factor of n. Which of the following describes this function? (1) Both one-one and onto (2) One-one only (3) Onto only (4) Neither one-one nor onto |
Neither one-one nor onto | Analyzing prime factor properties and multiple mappings to show non-injectivity and non-surjectivity. |
| 33. The distance of the point (7, −2, 11) from the line (x-6)/1 = (y-4)/0 = (z-8)/3 is: (1) 12 (2) 14 (3) 18 (4) 21 |
14 | Distance formula applied between a point and a line in 3D. |
| 34. A particle executes simple harmonic motion with amplitude 4 cm. At the mean position, the particle's speed is 10 cm/s. The distance of the particle from the mean position when its speed becomes 5 cm/s is √α cm, where α is: (1) 3 (2) 4 (3) 12 (4) 8 |
12 | Applying SHM velocity equations and solving for displacement. |
| 35. The least positive integral value of α for which the angle between the vectors α i - 2 j + 2 k and α i + 2α j - 2 k is acute is: (1) 3 (2) 4 (3) 5 (4) 6 |
5 | Using dot product conditions for an acute angle and solving for minimum α. |
| 36. A rectangular loop of length 2.5 m and width 2 m is placed at 60° to a magnetic field of 4 T. The loop is removed from the field in 10 s. The average emf induced in the loop during this time is: (1) -2V (2) +2V (3) +1V (4) -1V |
+1V | Using Faraday's law to calculate induced emf. |
| 37. The refractive index of the material of a prism is cot(A/2), where A is the angle of the prism. The angle of minimum deviation δm is: (1) A (2) 2A (3) A/2 (4) 3A |
A | Application of prism formula for minimum deviation. |
| 38. A spherometer has a circular base of radius 3.5 cm. The central screw moves 2 mm for every complete rotation. How many rotations should be given to the central screw to raise it from the base by 4.2 mm? (1) 3 (2) 5 (3) 6 (4) 7 |
3 | Calculating required rotations based on height increase per rotation. |
| 39. A charged particle of mass m and charge q is projected perpendicular to a magnetic field B with speed v. The pitch of the helical path followed by the particle is: (1) 2πmv/qB (2) 2mv/qB (3) 2πqB/mv (4) 2qB/πmv |
2πmv/qB | Using the formula for pitch in a helical path under magnetic influence. |
| 40. For a reaction, if the equilibrium constant at 500 K is 4, then the value of the standard Gibbs free energy ΔG° at this temperature is: (1) -1155 J (2) 1386 J (3) -1386 J (4) 1155 J |
-1155 J | Applying ΔG° = -RT ln(K) to find the Gibbs free energy. |
| 41. The element with the highest first ionization enthalpy among the following is: (1) B (2) Al (3) Ga (4) In |
B | Based on periodic trends in ionization enthalpy. |
| 42. For a given reaction, the rate of appearance of B is four times the rate of disappearance of A. The balanced reaction is: (1) A → 4B (2) 4A → B (3) 2A → 2B (4) 4A → 4B |
A → 4B | Analyzing the rate relation to deduce the balanced reaction. |
| 43. Among the following, the most acidic compound is: (1) Benzene (2) Phenol (3) Ethanol (4) Acetylene |
Phenol | Considering the acidity of functional groups and resonance stability. |
| 44. The correct order of bond angle for NH3, PH3, and AsH3 is: (1) NH3 > PH3 > AsH3 (2) PH3 > NH3 > AsH3 (3) AsH3 > PH3 > NH3 (4) All have the same bond angle |
NH3 > PH3 > AsH3 | Using VSEPR theory and electronegativity differences. |
| 45. Which one of the following complex ions is diamagnetic? (1) [Fe(CN)6]3- (2) [Co(NH3)6]3+ (3) [NiCl4]2- (4) [CuCl4]2- |
[Co(NH3)6]3+ | Using electron configurations and ligand field theory. |
| 46. In a hypothetical reaction A → B, the rate of formation of B is 0.04 mol L^-1 s^-1. The rate of disappearance of A is: (1) 0.02 mol L^-1 s^-1 (2) 0.04 mol L^-1 s^-1 (3) 0.08 mol L^-1 s^-1 (4) 0.01 mol L^-1 s^-1 |
0.04 mol L^-1 s^-1 | Direct correlation as the reaction rate remains the same for the disappearance of A and formation of B. |
| 47. In which of the following molecules/ions does the central atom obey the octet rule? (1) BeCl2 (2) BF3 (3) SO2 (4) NO2 |
SO2 | Analyzing each structure for octet fulfillment around the central atom. |
| 48. The reaction Zn + H2SO4 → ZnSO4 + H2 is an example of: (1) Combination reaction (2) Decomposition reaction (3) Displacement reaction (4) Redox reaction |
Redox reaction | Classified based on electron transfer, indicating oxidation and reduction. |
| 49. If the boiling point of a solution containing 1 mole of glucose in 1000 g of water is 100.52°C, the ebullioscopic constant (Kb) of water is: (1) 0.52 K kg/mol (2) 1.52 K kg/mol (3) 2.52 K kg/mol (4) 3.52 K kg/mol |
0.52 K kg/mol | Using the boiling point elevation formula ΔTb = i Kb m to solve for Kb. |
| 50. The IUPAC name of the compound CH3CH2CH(OH)CH3 is: (1) 1-Butanol (2) 2-Butanol (3) tert-Butanol (4) Isobutanol |
2-Butanol | Naming according to IUPAC conventions, identifying the hydroxyl group position. |
Also Check:
| JEE Main 2024 Paper Analysis | JEE Main 2024 Answer Key |
| JEE Main 2024 Cutoff | JEE Main 2024 Marks vs Rank |
JEE Main 2024 Jan 27 Shift 1 Question Paper by Coaching Institute
| Coaching Institutes | Question Paper with Solutions PDF |
|---|---|
| Aakash BYJUs | Download PDF |
| Reliable Institute | Physics Chemistry Maths |
| Resonance | Physics Chemistry Maths |
| Vedantu | Download PDF |
| Sri Chaitanya | To be updated |
| FIIT JEE | To be updated |
JEE Main 27 Jan Shift 1 2024 Paper Analysis
JEE Main 2024 Jan 27 Shift 1 paper analysis for B.E./ B.Tech is available here with details on the difficulty level of the exam, topics with the highest weightage in the exam, section-wise difficulty level, etc. after the conclusion of the exam.
JEE Main 2024 Question Paper Pattern
| Feature | Question Paper Pattern |
|---|---|
| Examination Mode | Computer-based Test |
| Exam Language | 13 languages (English, Hindi, Assamese, Bengali, Gujarati, Kannada, Malayalam, Marathi, Odia, Punjabi, Tamil, Telugu, and Urdu) |
| Number of Sections | Three- Physics, Chemistry, Mathematics |
| Exam Duration | 3 hours |
| Sectional Time Limit | None |
| Total Marks | 300 marks |
| Total Number of Questions Asked | 90 Questions |
| Total Number of Questions to be Answered | 75 questions |
| Type of Questions | MCQs and Numerical Answer Type Questions |
| Section-wise Number of Questions | Physics- 20 MCQs and 10 numerical type, Chemistry- 20 MCQs and 10 numerical type, Mathematics- 20 MCQs and 10 numerical type |
| Marking Scheme | +4 for each correct answer |
| Negative Marking | -1 for each incorrect answer |
Read More:
- JEE Main 2024 question paper pattern and marking scheme
- Most important chapters in JEE Mains 2024, Check chapter wise weightage here
JEE Main 2024 Question Paper Session 1 (January)
Those appearing for JEE Main 2024 can use the links below to practice and keep track of their exam preparation level by attempting the shift-wise JEE Main 2024 question paper provided below.
| Exam Date and Shift | Question Paper PDF |
|---|---|
| JEE Main 24 Jan Shift 2 2024 Question Paper | Check Here |
| JEE Main 27 Jan Shift 2 2024 Question Paper | Check Here |
| JEE Main 29 Jan Shift 1 2024 Question Paper | Check Here |
| JEE Main 29 Jan Shift 2 2024 Question Paper | Check Here |
| JEE Main 30 Jan Shift 1 2024 Question Paper | Check Here |
| JEE Main 30 Jan Shift 2 2024 Question Paper | Check Here |
| JEE Main 31 Jan Shift 1 2024 Question Paper | Check Here |
| JEE Main 31 Jan Shift 2 2024 Question Paper | Check Here |
| JEE Main 1 Feb Shift 1 2024 Question Paper | Check Here |
| JEE Main 1 Feb Shift 2 2024 Question Paper | Check Here |
JEE Main Previous Year Question Paper
JEE Main Questions
1. Following gates section is connected in a complete suitable circuit.
For which of the following combination, bulb will glow (ON):
Following gates section is connected in a complete suitable circuit.
For which of the following combination, bulb will glow (ON):
For which of the following combination, bulb will glow (ON):
- A = 0, B = 1, C = 1, D = 1
- A = 1, B = 0, C = 0, D = 0
- A = 0, B = 0, C = 0, D = 1
- A = 1, B = 1, C = 1, D = 0
2. If G be the gravitational constant and u be thee nergy density then which of the following quantity have the dimension as that the \(\sqrt{UG}\)
If G be the gravitational constant and u be thee nergy density then which of the following quantity have the dimension as that the \(\sqrt{UG}\)
- Pressure gradient per unit mass
- Force per unit mass
- Gravitational potential
- Energy per unit mass
4. Suppose AB is a focal chord of the parabola \( y^2 = 12x \) of length \( l \) and slope \( m<\sqrt{3} \). If the distance of the chord AB from the origin is \( d \), then \( ld^2 \) is equal to _________.
Suppose AB is a focal chord of the parabola \( y^2 = 12x \) of length \( l \) and slope \( m<\sqrt{3} \). If the distance of the chord AB from the origin is \( d \), then \( ld^2 \) is equal to _________.
5. Let \(\vec{a} = \hat{i} - 3\hat{j} + 7\hat{k}, \quad \vec{b} = 2\hat{i} - \hat{j} + \hat{k}, \quad \text{and} \quad \vec{c} \text{ be a vector such that}\) \((\vec{a} + 2\vec{b}) \times \vec{c} = 3(\vec{c} \times \vec{a}).\)
If \(\vec{a} \cdot \vec{c} = 130\), then \(\vec{b} \cdot \vec{c}\) is equal to \(\_\_\_\_\_\_\_\_ .\)
Let \(\vec{a} = \hat{i} - 3\hat{j} + 7\hat{k}, \quad \vec{b} = 2\hat{i} - \hat{j} + \hat{k}, \quad \text{and} \quad \vec{c} \text{ be a vector such that}\) \((\vec{a} + 2\vec{b}) \times \vec{c} = 3(\vec{c} \times \vec{a}).\)
If \(\vec{a} \cdot \vec{c} = 130\), then \(\vec{b} \cdot \vec{c}\) is equal to \(\_\_\_\_\_\_\_\_ .\)
If \(\vec{a} \cdot \vec{c} = 130\), then \(\vec{b} \cdot \vec{c}\) is equal to \(\_\_\_\_\_\_\_\_ .\)
6. Let \( a_1, a_2, a_3, \dots \) be in an arithmetic progression of positive terms.
Let \( A_k = a_1^2 - a_2^2 + a_3^2 - a_4^2 + \dots + a_{2k-1}^2 - a_{2k}^2 \).
If \( A_3 = -153 \), \( A_5 = -435 \), and \( a_1^2 + a_2^2 + a_3^2 = 66 \), then \( a_{17} - A_7 \) is equal to _________.
Let \( a_1, a_2, a_3, \dots \) be in an arithmetic progression of positive terms.
Let \( A_k = a_1^2 - a_2^2 + a_3^2 - a_4^2 + \dots + a_{2k-1}^2 - a_{2k}^2 \).
If \( A_3 = -153 \), \( A_5 = -435 \), and \( a_1^2 + a_2^2 + a_3^2 = 66 \), then \( a_{17} - A_7 \) is equal to _________.
Let \( A_k = a_1^2 - a_2^2 + a_3^2 - a_4^2 + \dots + a_{2k-1}^2 - a_{2k}^2 \).
If \( A_3 = -153 \), \( A_5 = -435 \), and \( a_1^2 + a_2^2 + a_3^2 = 66 \), then \( a_{17} - A_7 \) is equal to _________.
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