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JEE Advanced 2019 — Paper 1 Solutions

Step-by-step mathematical proofs, multi-correct partial marking evaluation, and interactive practice for Paper 1 (9:00 AM – 12:00 PM).

All 3 Subjects (51 Qs)Physics SolutionsChemistry SolutionsMathematics Solutions
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Official JEE Adv Portal (2019)

Paper 1 Questions & Solutions (51 Questions)

Live Partial Marking Engine Enabled
Q1Paper 1PhysicsOne or More Correct (Partial Marking)Hard
Rotational Motion & Rigid Body Dynamics
Scoring Scheme: +4 Full Marks • -2 Negative Marks • Partial Marking Enabled
[JEE Advanced 2019 Paper 1 Physics Q1] A uniform solid circular disc of mass MM and radius RR is placed on a rough horizontal surface with coefficient of friction μ\mu. At time t=0t=0, a horizontal impulse JJ is applied at a height h=R2h = \frac{R}{2} above the center of mass. Which of the following statements is/are CORRECT?
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Q2Paper 1PhysicsOne or More Correct (Partial Marking)Tough / Advanced
Electrostatics & Electric Potential
Scoring Scheme: +4 Full Marks • -2 Negative Marks • Partial Marking Enabled
[JEE Advanced 2019 Paper 1 Physics Q2] A point charge +q+q is located at a distance d=1 cmd = 1\text{ cm} from the planar boundary of an infinite semi-infinite dielectric medium of relative permittivity K=5K = 5. Which of the following statements is/are CORRECT?
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Q3Paper 1PhysicsOne or More Correct (Partial Marking)Hard
Thermodynamics & Kinetic Theory
Scoring Scheme: +4 Full Marks • -2 Negative Marks • Partial Marking Enabled
[JEE Advanced 2019 Paper 1 Physics Q3] One mole of an ideal gas has a temperature-dependent molar heat capacity at constant volume CV=C0+αTC_V = C_0 + \alpha T, where C0=52RC_0 = \frac{5}{2}R and α=0.040 J/(mol K2)\alpha = 0.040\text{ J/(mol K}^2). The gas undergoes a quasi-static adiabatic expansion from (T1,V1)(T_1, V_1) to (T2,V2)(T_2, V_2). Which relation holds?
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Q4Paper 1PhysicsOne or More Correct (Partial Marking)Hard
Electromagnetic Induction & Alternating Current
Scoring Scheme: +4 Full Marks • -2 Negative Marks • Partial Marking Enabled
[JEE Advanced 2019 Paper 1 Physics Q4] A flat circular loop of radius r=0.1 mr = 0.1\text{ m} and resistance R=2.0 ΩR = 2.0\,\Omega has self-inductance L=0.04 HL = 0.04\text{ H}. It is placed in a normal magnetic field B(t)=1cos⁡(50t) TB(t) = 1 \cos(50 t)\text{ T}. Which of the following is/are CORRECT?
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Q5Paper 1PhysicsOne or More Correct (Partial Marking)Hard
Wave Optics & Interference
Scoring Scheme: +4 Full Marks • -2 Negative Marks • Partial Marking Enabled
[JEE Advanced 2019 Paper 1 Physics Q5] In a Young's Double Slit setup with slit separation dd and screen distance DD, a film of thickness t=13 μmt = 13\,\mu\text{m} is placed in front of slit S1S_1. The refractive index of the film varies linearly across thickness as μ(x)=1.5+0.35(xt)\mu(x) = 1.5 + 0.35 \left(\frac{x}{t}\right) for 0≤x≤t0 \le x \le t. Which of the following statements is/are CORRECT?
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Q6Paper 1PhysicsOne or More Correct (Partial Marking)Hard
Modern Physics & Quantum Mechanics
Scoring Scheme: +4 Full Marks • -2 Negative Marks • Partial Marking Enabled
[JEE Advanced 2019 Paper 1 Physics Q6] A hydrogen-like ion with atomic number Z=1Z = 1 consists of a nucleus of mass MM and an orbital electron of mass mem_e. Taking the finite nuclear mass into account with reduced mass μ=meMme+M\mu = \frac{m_e M}{m_e + M}, which of the following statements is/are CORRECT?
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Q7Paper 1PhysicsNumerical Value TypeMedium
Fluid Mechanics & Surface Tension
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2019 Paper 1 Physics Q7] A clean cylindrical glass capillary tube of internal radius r=0.50 mmr = 0.50\text{ mm} is held vertically and dipped into water of surface tension T=0.075 N/mT = 0.075\text{ N/m}, contact angle θ=0∘\theta = 0^\circ, and density ρ=1000 kg/m3\rho = 1000\text{ kg/m}^3. Taking acceleration due to gravity g=10 m/s2g = 10\text{ m/s}^2, calculate the equilibrium height hh (in mm) of the water column in the tube.
Q8Paper 1PhysicsNumerical Value TypeMedium
Gravitation & Satellite Motion
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2019 Paper 1 Physics Q8] A satellite orbits a spherically symmetric planet of mass M=2×1024 kgM = 2 \times 10^{24}\text{ kg} in a circular orbit of radius r=7×107 mr = 7 \times 10^7\text{ m}. Given universal gravitational constant G=6.67×10−11 N m2/kg2G = 6.67 \times 10^{-11}\text{ N m}^2/\text{kg}^2, determine the orbital speed v0v_0 of the satellite in km/s\text{km/s}. (Round off to two decimal places).
Q9Paper 1PhysicsNumerical Value TypeMedium
Wave Motion & Sound
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2019 Paper 1 Physics Q9] An organ pipe closed at one end and open at the other has length L=0.65 mL = 0.65\text{ m}. Taking the speed of sound in air as v=340 m/sv = 340\text{ m/s} and neglecting end correction, find the fundamental resonant frequency f1f_1 (in Hz).
Q10Paper 1PhysicsNumerical Value TypeMedium
Modern Physics & Quantum Mechanics
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2019 Paper 1 Physics Q10] Monochromatic ultraviolet light of wavelength λ=350 nm\lambda = 350\text{ nm} is incident on a clean metallic surface having work function ϕ=2.74 eV\phi = 2.74\text{ eV}. Using Planck's constant hc≈1240 eV⋅nmh c \approx 1240\text{ eV}\cdot\text{nm}, calculate the stopping potential V0V_0 (in Volts) required to reduce the photocurrent to zero.
Q11Paper 1PhysicsNumerical Value TypeMedium
Current Electricity & Transient Circuits
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2019 Paper 1 Physics Q11] A capacitor of capacitance C=35 μFC = 35\,\mu\text{F} is charged through a resistor R=80 kΩR = 80\text{ k}\Omega connected to a DC source of V0=12 VV_0 = 12\text{ V}. Find the capacitive time constant τ=RC\tau = R C of the charging circuit in seconds.
Q12Paper 1PhysicsNumerical Value TypeMedium
Magnetic Effects of Current & Magnetism
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2019 Paper 1 Physics Q12] A proton (m=1.67×10−27 kgm = 1.67 \times 10^{-27}\text{ kg}, q=1.6×10−19 Cq = 1.6 \times 10^{-19}\text{ C}) enters a uniform magnetic field B=0.50 TB = 0.50\text{ T} with speed v=2.0×106 m/sv = 2.0 \times 10^6\text{ m/s} perpendicular to the magnetic lines. Find the orbital radius rr of the resulting circular path in centimeters.
Q13Paper 1PhysicsSingle Correct OptionMedium
Ray Optics & Optical Instruments
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2019 Paper 1 Physics Q13] An equilateral glass prism (A=60∘A = 60^\circ) has refractive index μ=3\mu = \sqrt{3}. A monochromatic ray of light is incident at angle ii such that it suffers minimum deviation δmin⁡\delta_{\min}. What is the value of δmin⁡\delta_{\min}?
Q14Paper 1PhysicsSingle Correct OptionMedium
Oscillations & Simple Harmonic Motion
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2019 Paper 1 Physics Q14] A block of mass m=2 kgm = 2\text{ kg} is attached to a horizontal massless spring of force constant k=200 N/mk = 200\text{ N/m}. The block is pulled from equilibrium by x0=0.1 mx_0 = 0.1\text{ m} on a frictionless floor and released from rest. What is the maximum speed of the block?
Q15Paper 1PhysicsSingle Correct OptionMedium
Work, Energy and Power
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2019 Paper 1 Physics Q15] A small bob of mass mm tied to an inextensible string of length RR is whirled in a vertical circle. What is the minimum horizontal speed umin⁡u_{\min} that must be imparted to the bob at the lowest point so that it completes the full vertical loop without the string slacking?
Q16Paper 1PhysicsSingle Correct OptionMedium
Nuclear Physics & Radioactivity
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2019 Paper 1 Physics Q16] A radioactive sample contains N0N_0 active nuclei at time t=0t=0. Its half-life is T1/2=16 daysT_{1/2} = 16\text{ days}. What fraction of the original radioactive nuclei will remain undecayed after t=48 dayst = 48\text{ days}?
Q17Paper 1PhysicsSingle Correct OptionMedium
Fluid Mechanics & Properties of Matter
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2019 Paper 1 Physics Q17] A small spherical steel ball of radius rr falls vertically through a viscous liquid of density σ\sigma and viscosity η\eta. If the density of steel is ρ\rho, the terminal velocity vtv_t of the sphere is proportional to:
Q18Paper 1Chemistry (Inorganic)One or More Correct (Partial Marking)Medium
Coordination Compounds & Qualitative Inorganic Analysis
Scoring Scheme: +4 Full Marks • -2 Negative Marks • Partial Marking Enabled
[JEE Advanced 2019 Paper 1 Chemistry Q18] Among the following octahedral coordination complexes, which one is diamagnetic (n=0n=0 unpaired electrons) with the highest crystal field stabilization energy (CFSE) magnitude?
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Q19Paper 1Chemistry (Organic)One or More Correct (Partial Marking)Hard
Organic Chemistry & Reaction Mechanisms
Scoring Scheme: +4 Full Marks • -2 Negative Marks • Partial Marking Enabled
[JEE Advanced 2019 Paper 1 Chemistry Q19] Pinacol (2,3-dimethylbutane-2,3-diol) undergoes acid-catalyzed dehydration to form Pinacolone (3,3-dimethylbutan-2-one). Which of the following statements regarding this transformation is/are CORRECT?
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Q20Paper 1Chemistry (Physical)One or More Correct (Partial Marking)Hard
Chemical Kinetics & Radioactivity
Scoring Scheme: +4 Full Marks • -2 Negative Marks • Partial Marking Enabled
[JEE Advanced 2019 Paper 1 Chemistry Q20] Consider the consecutive first-order radioactive or chemical decay series A→k1B→k2CA \xrightarrow{k_1} B \xrightarrow{k_2} C, with initial concentration [A]0[A]_0 and [B]0=[C]0=0[B]_0 = [C]_0 = 0. Which of the following statements is/are CORRECT?
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Q21Paper 1Chemistry (Inorganic)One or More Correct (Partial Marking)Hard
Chemical Bonding & Molecular Structure
Scoring Scheme: +4 Full Marks • -2 Negative Marks • Partial Marking Enabled
[JEE Advanced 2019 Paper 1 Chemistry Q21] According to Molecular Orbital Theory (MOT), which of the following statements regarding O2,O2+,O2−,O_2, O_2^+, O_2^-, and O22−O_2^{2-} is/are CORRECT?
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Q22Paper 1Chemistry (Physical)One or More Correct (Partial Marking)Hard
Electrochemistry & Galvanic Cells
Scoring Scheme: +4 Full Marks • -2 Negative Marks • Partial Marking Enabled
[JEE Advanced 2019 Paper 1 Chemistry Q22] For the galvanic cell Zn(s)∣Zn2+(aq,0.01 M) ∣∣ Cu2+(aq,1.0 M)∣Cu(s)Zn(s) | Zn^{2+}(aq, 0.01\text{ M}) \,||\, Cu^{2+}(aq, 1.0\text{ M}) | Cu(s) at 298 K298\text{ K}, given standard reduction potentials EZn2+/Zn∘=−0.76 VE^\circ_{Zn^{2+}/Zn} = -0.76\text{ V} and ECu2+/Cu∘=+0.34 VE^\circ_{Cu^{2+}/Cu} = +0.34\text{ V}. Which of the following statements is/are CORRECT? (Take 2.303RTF=0.059 V\frac{2.303 RT}{F} = 0.059\text{ V}).
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Q23Paper 1Chemistry (Inorganic)One or More Correct (Partial Marking)Medium
p-Block Elements (Groups 13 to 18)
Scoring Scheme: +4 Full Marks • -2 Negative Marks • Partial Marking Enabled
[JEE Advanced 2019 Paper 1 Chemistry Q23] Which of the following statements regarding Diborane (B2H6B_2H_6) and Oxyacids of Phosphorus is/are CORRECT?
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Q24Paper 1Chemistry (Physical)Numerical Value TypeMedium
Solid State & Crystal Lattices
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2019 Paper 1 Chemistry Q24] An element crystallizes in a Face-Centered Cubic (FCC) unit cell with edge length a=430 pma = 430\text{ pm}. If the molar mass of the element is M=60 g/molM = 60\text{ g/mol} and Avogadro's number NA=6.022×1023 mol−1N_A = 6.022 \times 10^{23}\text{ mol}^{-1}, calculate the density of the crystal in g/cm3\text{g/cm}^3.
Q25Paper 1Chemistry (Physical)Numerical Value TypeMedium
Ionic & Chemical Equilibrium
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2019 Paper 1 Chemistry Q25] An acidic buffer is prepared by mixing sodium acetate (CH3COONaCH_3COONa) and acetic acid (CH3COOHCH_3COOH, pKa=4.76pK_a = 4.76) in the molar ratio [Salt][Acid]=2.50\frac{[\text{Salt}]}{[\text{Acid}]} = 2.50. Calculate the resulting pH of the buffer solution at 25∘C25^\circ\text{C}.
Q26Paper 1Chemistry (Physical)Numerical Value TypeMedium
Solutions & Colligative Properties
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2019 Paper 1 Chemistry Q26] What is the depression in freezing point ΔTf\Delta T_f (in K) of a solution prepared by dissolving 18.0 g18.0\text{ g} of glucose (C6H12O6C_6H_{12}O_6, molar mass =180 g/mol= 180\text{ g/mol}) in 1000 g1000\text{ g} of pure water? (Cryoscopic constant of water Kf=1.86 K kg/molK_f = 1.86\text{ K kg/mol}).
Q27Paper 1Chemistry (Physical)Numerical Value TypeMedium
Electrochemistry & Electrolysis
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2019 Paper 1 Chemistry Q27] An aqueous solution of CuSO4CuSO_4 is electrolyzed with a steady electric current of I=9.65 AI = 9.65\text{ A} for t=1000 secondst = 1000\text{ seconds}. How many moles of electrons are transferred through the cell? (Faraday constant F=96500 C/molF = 96500\text{ C/mol}).
Q28Paper 1Chemistry (Physical)Numerical Value TypeMedium
Chemical Kinetics
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2019 Paper 1 Chemistry Q28] The rate of a chemical reaction doubles when the temperature is raised from 300 K300\text{ K} to 310 K310\text{ K}. Given R=8.314 J/(mol K)R = 8.314\text{ J/(mol K)} and ln⁡2≈0.693\ln 2 \approx 0.693, determine the activation energy EaE_a (in kJ/mol). (Round off to one decimal place).
Q29Paper 1Chemistry (Inorganic)Numerical Value TypeMedium
Coordination Compounds
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2019 Paper 1 Chemistry Q29] Calculate the spin-only magnetic moment μs\mu_s (in Bohr Magnetons, BM) for the high-spin complex [Fe(H2O)6]2+[Fe(H_2O)_6]^{2+}. (Given: 24≈4.90\sqrt{24} \approx 4.90).
Q30Paper 1Chemistry (Organic)Single Correct OptionMedium
Haloalkanes & Haloarenes
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2019 Paper 1 Chemistry Q30] Reaction of (R)(R)-2-bromobutane with sodium hydroxide (NaOHNaOH) in acetone follows a bimolecular nucleophilic substitution (SN2S_N2) mechanism. The configuration of the major alcohol product formed is:
Q31Paper 1Chemistry (Organic)Single Correct OptionMedium
Aldehydes, Ketones and Carboxylic Acids
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2019 Paper 1 Chemistry Q31] When acetaldehyde (CH3CHOCH_3CHO) is warmed with dilute aqueous NaOHNaOH followed by heating with acid, the final major product formed is:
Q32Paper 1Chemistry (Organic)Single Correct OptionMedium
Aromatic Hydrocarbons
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2019 Paper 1 Chemistry Q32] Treatment of benzene with 1-chloropropane in the presence of anhydrous aluminium chloride (AlCl3AlCl_3) yields isopropylbenzene (Cumene) as the major product rather than n-propylbenzene because:
Q33Paper 1Chemistry (Inorganic)Single Correct OptionHard
Qualitative Inorganic Salt Analysis
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2019 Paper 1 Chemistry Q33] In the brown ring test for nitrate ion (NO3−NO_3^-), the brown ring formed at the junction of the two liquid layers is due to the formation of the complex:
Q34Paper 1Chemistry (Organic)Single Correct OptionMedium
Biomolecules & Carbohydrates
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2019 Paper 1 Chemistry Q34] α\alpha-D-glucopyranose and β\beta-D-glucopyranose are related to each other as:
Q35Paper 1MathematicsOne or More Correct (Partial Marking)Hard
Integral Calculus (Definite Integrals & Differential Equations)
Scoring Scheme: +4 Full Marks • -2 Negative Marks • Partial Marking Enabled
[JEE Advanced 2019 Paper 1 Mathematics Q35] Let f:(0,∞)→Rf: (0, \infty) \to \mathbb{R} be a differentiable function satisfying f(1)=1f(1) = 1 and x∫1xf(t)dt=(x+1)∫1xtf(t)dtx \int_1^x f(t) dt = (x+1) \int_1^x t f(t) dt for all x>0x > 0. Which of the following statements is/are CORRECT?
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Q36Paper 1MathematicsOne or More Correct (Partial Marking)Medium
Vectors and 3D Geometry
Scoring Scheme: +4 Full Marks • -2 Negative Marks • Partial Marking Enabled
[JEE Advanced 2019 Paper 1 Mathematics Q36] Find the square of the shortest distance (d2d^2) between the two skew lines L1:r⃗1=(i^+2j^+3k^)+λ(2i^+3j^+4k^)L_1: \vec{r}_1 = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda (2\hat{i} + 3\hat{j} + 4\hat{k}) and L2:r⃗2=(2i^+4j^+5k^)+μ(3i^+4j^+5k^)L_2: \vec{r}_2 = (2\hat{i} + 4\hat{j} + 5\hat{k}) + \mu (3\hat{i} + 4\hat{j} + 5\hat{k}).
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Q37Paper 1MathematicsOne or More Correct (Partial Marking)Hard
Complex Numbers & Quadratic Equations
Scoring Scheme: +4 Full Marks • -2 Negative Marks • Partial Marking Enabled
[JEE Advanced 2019 Paper 1 Mathematics Q37] Let zz be a complex number satisfying ∣z−3−4i∣=2|z - 3 - 4i| = 2. Which of the following statements is/are CORRECT?
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Q38Paper 1MathematicsOne or More Correct (Partial Marking)Medium
Matrices and Determinants
Scoring Scheme: +4 Full Marks • -2 Negative Marks • Partial Marking Enabled
[JEE Advanced 2019 Paper 1 Mathematics Q38] Let A=(1234)A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}. Which of the following statements regarding matrix AA is/are CORRECT?
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Q39Paper 1MathematicsOne or More Correct (Partial Marking)Medium
Probability & Statistics
Scoring Scheme: +4 Full Marks • -2 Negative Marks • Partial Marking Enabled
[JEE Advanced 2019 Paper 1 Mathematics Q39] An urn U1U_1 contains 3 white and 2 black balls, while urn U2U_2 contains 1 white and 4 black balls. A fair coin is tossed; if heads appears, a ball is drawn from U1U_1, otherwise from U2U_2. Given that the drawn ball is WHITE, which of the following statements is/are CORRECT?
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Q40Paper 1MathematicsOne or More Correct (Partial Marking)Hard
Permutations and Combinations
Scoring Scheme: +4 Full Marks • -2 Negative Marks • Partial Marking Enabled
[JEE Advanced 2019 Paper 1 Mathematics Q40] Let DnD_n denote the number of derangements of nn distinct objects (permutations where no object appears in its original position). Consider 5 letters addressed to 5 different recipients placed randomly into 5 corresponding envelopes. Which of the following statements is/are CORRECT?
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Q41Paper 1MathematicsOne or More Correct (Partial Marking)Hard
Differential Calculus (Continuity & AOD)
Scoring Scheme: +4 Full Marks • -2 Negative Marks • Partial Marking Enabled
[JEE Advanced 2019 Paper 1 Mathematics Q41] Let f:[0,2]→Rf: [0, 2] \to \mathbb{R} be a continuous function on [0,2][0, 2] and twice differentiable on (0,2)(0, 2) such that f(0)=0f(0) = 0, f(1)=1f(1) = 1, and f(2)=1f(2) = 1. Which of the following statements is/are necessarily TRUE?
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Q42Paper 1MathematicsNumerical Value TypeMedium
Definite Integration
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2019 Paper 1 Mathematics Q42] Evaluate the definite integral I=∫0π/2sin⁡2025xsin⁡2025x+cos⁡2025xdxI = \int_0^{\pi/2} \frac{\sin^{2025} x}{\sin^{2025} x + \cos^{2025} x} dx. Calculate the exact value of 4Iπ\frac{4 I}{\pi}.
Q43Paper 1MathematicsNumerical Value TypeMedium
Binomial Theorem
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2019 Paper 1 Mathematics Q43] For n=5n = 5, evaluate the double summation S=∑0≤i<j≤5(5i)(5j)S = \sum_{0 \le i < j \le 5} \binom{5}{i} \binom{5}{j}. What is the value of SS?
Q44Paper 1MathematicsNumerical Value TypeMedium
Coordinate Geometry - Parabola
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2019 Paper 1 Mathematics Q44] From a point P(h,k)P(h, k), three normal lines are drawn to the parabola y2=4xy^2 = 4x. If the sum of the ordinates of the three co-normal points is SyS_y, find the value of SyS_y.
Q45Paper 1MathematicsNumerical Value TypeMedium
Trigonometry & Inverse Trigonometric Functions
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2019 Paper 1 Mathematics Q45] Evaluate the infinite sum S=∑k=1∞tan⁡−1(11+k+k2)S = \sum_{k=1}^\infty \tan^{-1}\left(\frac{1}{1 + k + k^2}\right). Calculate the numerical value of 4Sπ\frac{4 S}{\pi}.
Q46Paper 1MathematicsNumerical Value TypeMedium
Limits, Continuity and Differentiability
Scoring Scheme: +4 Full Marks • 0 Negative Marks
[JEE Advanced 2019 Paper 1 Mathematics Q46] Evaluate the limit L=lim⁡x→0(cos⁡x)1/x2L = \lim_{x \to 0} (\cos x)^{1/x^2}. If L=e−kL = e^{-k}, find the value of kk.
Q47Paper 1MathematicsSingle Correct OptionMedium
Integral Calculus (Area Bounded by Curves)
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2019 Paper 1 Mathematics Q47] The area of the region bounded by the parabola y=x2y = x^2 and the line y=2xy = 2x is:
Q48Paper 1MathematicsSingle Correct OptionMedium
Differential Equations
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2019 Paper 1 Mathematics Q48] The integrating factor for the first order linear differential equation dydx+2xy=x3\frac{dy}{dx} + \frac{2}{x} y = x^3 (for x>0x > 0) is:
Q49Paper 1MathematicsSingle Correct OptionMedium
Functions & Relations
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2019 Paper 1 Mathematics Q49] If f(x)f(x) is a polynomial function satisfying f(x)f(1/x)=f(x)+f(1/x)f(x) f(1/x) = f(x) + f(1/x) for all x≠0x \ne 0, and f(3)=28f(3) = 28, then f(2)f(2) equals:
Q50Paper 1MathematicsSingle Correct OptionMedium
Coordinate Geometry - Ellipse & Hyperbola
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2019 Paper 1 Mathematics Q50] The locus of the point of intersection of two perpendicular tangents to the ellipse x216+y29=1\frac{x^2}{16} + \frac{y^2}{9} = 1 is a concentric circle (director circle) of radius:
Q51Paper 1MathematicsSingle Correct OptionMedium
Sequences and Series
Scoring Scheme: +3 Full Marks • -1 Negative Marks
[JEE Advanced 2019 Paper 1 Mathematics Q51] Find the sum of the infinite Arithmetico-Geometric Progression (AGP): S=1+22+34+48+516+…S = 1 + \frac{2}{2} + \frac{3}{4} + \frac{4}{8} + \frac{5}{16} + \dots