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JEE Main 2025 January 28, Shift 2 Question Paper with Solutions

All 75 questions from the JEE Main 2025 (January 28, Shift 2) shift — Physics (25), Chemistry (25) and Mathematics (25) — with the correct answer and a step-by-step solution for every question.

Physics25 questions

Q26Single correctElectromagnetic Induction and Alternating Currents
A uniform magnetic field of 0.40.4 T acts perpendicular to a circular copper disc 2020 cm in radius. The disc is having a uniform angular velocity of 10π10\pi rad s1s^{-1} about an axis through its centre and perpendicular to the disc. What is the potential difference developed between the axis of the disc and the rim? (π=3.14\pi = 3.14)
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(B)
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(D)
Q27Single correctElectrostatics
A parallel plate capacitor of capacitance 11 μF is charged to a potential difference of 2020 V. The distance between plates is 11 m. The energy density between plates of capacitor is:
(A)
(B)
(C)
(D)
Q28Single correctUnits and Measurements
Match List-I with List-II
List-I (Physical Quantity)List-II (Dimensional Formula)
(A) Angular Impulse(I) [M0L2T2][M^0L^2T^{-2}]
(B) Latent Heat(II) [ML2T3A1][ML^2T^{-3}A^{-1}]
(C) Electrical resistivity(III) [ML2T1][ML^2T^{-1}]
(D) Electromotive force(IV) [ML3T3A2][ML^3T^{-3}A^{-2}]
Choose the correct answer from the options given below:
(A)
(B)
(C)
(D)
Q29Single correctKinetic Theory of Gases
The ratio of vapour densities of two gases at the same temperature is 425\frac{4}{25}, then the ratio of r.m.s. velocities will be:
(A)
(B)
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(D)
Q30Single correctKinetic Theory of Gases
The kinetic energy of translation of the molecules in 5050 g of CO2O_2 gas at 17°17°C is:
(A)
(B)
(C)
(D)
Q31Single correctOptics
In a long glass tube, mixture of two liquids A and B with refractive indices 1.31.3 and 1.41.4 respectively, forms a convex refractive meniscus towards A. If an object placed at 1313 cm from the vertex of the meniscus in A forms an image with a magnification of '2-2' then the radius of curvature of meniscus is:
(A)
(B)
(C)
(D)
Q32Single correctAtoms and Nuclei
The frequency of revolution of the electron in Bohr's orbit varies with nn, the principal quantum number as
(A)
(B)
(C)
(D)
Q33Single correctDual Nature of Matter and Radiation
Which of the following phenomena can not be explained by wave theory of light?
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(B)
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(D)
Q34Single correctKinematics
The velocity-time graph of an object moving along a straight line is shown in figure. What is the distance covered by the object between t=0t = 0 to t=4st = 4s?
Question diagram for Q34
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Q35Single correctMagnetic Effects of Current and Magnetism
A bar magnet has total length 2l=202l = 20 units and the field point P is at a distance d=10d = 10 units from the centre of the magnet. If the relative uncertainty of length measurement is 1%, then uncertainty of the magnetic field at point P is:
Question diagram for Q35
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Q36Single correctGravitation
Earth has mass 8 times and radius 2 times that of a planet. If the escape velocity from the earth is 11.2 km/s, the escape velocity in km/s from the planet will be:
(A)
(B)
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(D)
Q37Single correctOscillations and Waves
Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R). **Assertion (A):** Knowing initial position x0x_0 and initial momentum p0p_0 is enough to determine the position and momentum at any time t for a simple harmonic motion with a given angular frequency ω\omega. **Reason (R):** The amplitude and phase can be expressed in terms of x0x_0 and p0p_0. In the light of the above statements, choose the correct answer from the options given below:
(A)
(B)
(C)
(D)
Q38Single correctOptics
A concave mirror produces an image of an object such that the distance between the object and image is 20 cm. If the magnification of the image is '−3', then the magnitude of the radius of curvature of the mirror is:
(A)
(B)
(C)
(D)
Q39Single correctWork, Energy and Power
A body of mass 4 kg is placed on a plane at a point P having coordinate (3, 4) m. Under the action of force F=(2i^+3j^)\vec{F} = (2\hat{i} + 3\hat{j}) N, it moves to a new point Q having coordinates (6, 10)m in 4 sec. The average power and instantaneous power at the end of 4 sec are in the ratio of:
(A)
(B)
(C)
(D)
Q40Single correctElectronic Devices
In the circuit shown here, assuming threshold voltage of diode is negligibly small, then voltage VABV_{AB} is correctly represented by:
Question diagram for Q40
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Q41Single correctMagnetic Effects of Current and Magnetism
An infinite wire has a circular bend of radius aa, and carrying a current II as shown in figure. The magnitude of magnetic field at the origin O of the arc is given by:
(A)
(B)
(C)
(D)
Q42Single correctRotational Motion
A uniform rod of mass 250 g having length 100 cm is balanced on a sharp edge at 40 cm mark. A mass of 400 g is suspended at 10 cm mark. To maintain the balance of the rod, the mass to be suspended at 90 cm mark, is
(A)
(B)
(C)
(D)
Q43Single correctProperties of Solids and Liquids
A 400 g solid cube having an edge of length 10 cm floats in water. How much volume of the cube is outside the water? (Given: density of water = 1000 kg m3m^{-3})
(A)
(B)
(C)
(D)
Q44Single correctElectromagnetic Waves
The magnetic field of an E.M. wave is given by B=(32i^+12j^)30sin(ωtzc)\vec{B} = \left(\frac{\sqrt{3}}{2}\hat{i} + \frac{1}{2}\hat{j}\right) 30\sin\left(\omega t - \frac{z}{c}\right) (S.I. Units). The corresponding electric field in S.I. units is:
(A)
(B)
(C)
(D)
Q45Single correctLaws of Motion
A balloon and its content having mass M is moving up with an acceleration 'a'. The mass that must be released from the content so that the balloon starts moving up with an acceleration '3a' will be: (Take 'g' as acceleration due to gravity)
(A)
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Q46NumericalElectromagnetic Induction and Alternating Currents
A conducting bar moves on two conducting rails as shown in the figure. A constant magnetic field B exists into the page. The bar starts to move from the vertex at time t = 0 with a constant velocity. If the induced EMFisEEMF is E \propto tnt^n, then value of n is ___.
Question diagram for Q46
Q47NumericalElectrostatics
An electric dipole of dipole moment 6×6 \times 10610^{-6}  C-m\text{ C-m} is placed in uniform electric field of magnitude 10610^6  V/m\text{ V/m}. Initially, the dipole moment is parallel to electric field. The work that needs to be done on the dipole to make its dipole moment opposite to the field, will be _____ J.
Q48NumericalProperties of Solids and Liquids
The volume contraction of a solid copper cube of edge length 10 cm, when subjected to a hydraulic pressure of7×of 7 \times 10610^6  Pa\text{ Pa}, would be _____ mm3mm^3. (Given bulk modulus of copper =1.4×= 1.4 \times 101110^{11}  N m\text{ N m}2)^{-2})
Q49NumericalCurrent Electricity
The value of current I in the electrical circuit as given below, when potential at A is equal to the potential at B, will be __________ A.
Question diagram for Q49
Q50NumericalOptics
A thin transparent film with refractive index 1.4, is held on circular ring of radius 1.8 cm. The fluid in the film evaporates such that transmission through the film at wavelength 560 nm goes to a minimum every 12 seconds. Assuming that the film is flat on its two sides, the rate of evaporation is _______ ×1013 m3/s\times 10^{-13} \text{ m}^3/\text{s}.

Chemistry25 questions

Q51Single correctChemical Kinetics
Consider the elementary reaction A(g)+B(g)C(g)+D(g)A(g) + B(g) \rightarrow C(g) + D(g). If the volume of reaction mixture is suddenly reduced to 13\frac{1}{3} of its initial volume, the reaction rate will become 'x' times of the original reaction rate. The value of x is:
(A)
(B)
(C)
(D)
Q52Single correctd- and f-Block Elements
The amphoteric oxide among V2O3V_2O_3, V2O4V_2O_4 and V2O5V_2O_5 upon reaction with alkali leads to formation of an oxide anion. The oxidation state of V in the oxide anion is:
(A)
(B)
(C)
(D)
Q53Single correctBiomolecules
Match List-I with List-II
List-I (Saccharides)List-II (Glycosidic linkages found)
(A) Sucrose(I) α\alpha 1-4
(B) Maltose(II) α\alpha 1-4 and α\alpha 1-6
(C) Lactose(III) α\alpha 1-β\beta 2
(D) Amylopectin(IV) β\beta 1-4
Choose the correct answer from the options given below:
(A)
(B)
(C)
(D)
Q54Single correctHydrocarbons
Identify product [A], [B] and [C] in the following reaction sequence:

CH3CCHH2,Pd/C[A](i)O3,(ii)Zn,H2O[B]+[C]CH_3-C\equiv CH \xrightarrow{H_2, Pd/C} [A] \xrightarrow{(i) O_3, (ii) Zn, H_2O} [B] + [C]
(A)
(B)
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(D)
Q55Single correctEquilibrium
Arrange the following in increasing order of solubility product: Ca(OH)2Ca(OH)_2, AgBr, PbS, HgS
(A)
(B)
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Q56Single correctPurification and Characterisation of Organic Compounds
The purification method based on the following physical transformation is:

Solid (X)HeatVapour (X)CoolSolid (X)\text{Solid (X)} \xrightarrow{\text{Heat}} \text{Vapour (X)} \xrightarrow{\text{Cool}} \text{Solid (X)}
(A)
(B)
(C)
(D)
Q57Single correctBiomolecules
Identify correct conversion during acidic hydrolysis from the following:

(A) starch gives galactose
(B) cane sugar gives equal amount of glucose and fructose
(C) milk sugar gives glucose and galactose
(D) amylopectin gives glucose and fructose
(E) amylose gives only glucose

Choose the correct answer from the options given below:
(A)
(B)
(C)
(D)
Q58Single correctChemical Thermodynamics
An ideal gas undergoes a cyclic transformation starting from the point A and coming back to the same point by tracing the path A\rightarrowBCDAasB \rightarrow C \rightarrow D \rightarrow A as shown in the three cases. Choose the correct option regarding ΔU\Delta U.
Question diagram for Q58
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Q59Single correctOrganic Compounds Containing Nitrogen
The product B formed in the following reaction sequence is:

Styrene (Ph-CH=CH2)HCl(A) MajorAgCN(B) Major\text{Styrene (Ph-CH=CH}_2) \xrightarrow{\text{HCl}} \text{(A) Major} \xrightarrow{\text{AgCN}} \text{(B) Major}
Question diagram for Q59
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Q60Single correctSolutions
Concentrated nitric acid is labelled as 75% by mass. The volume in mL of the solution which contains 30 g of nitric acid is ___________. Given: Density of nitric acid solution is 1.25 g/mL
(A)
(B)
(C)
(D)
Q61Single correctCoordination Compounds
Match List-I with List-II:
List-I (Complex)List-II (Hybridization)
(A) [CoF6]3[\text{CoF}_6]^{3-}(I) d2sp3\text{d}^2\text{sp}^3
(B) [NiCl4]2[\text{NiCl}_4]^{2-}(II) sp3\text{sp}^3
(C) [Co(NH3)6]3+[\text{Co(NH}_3)_6]^{3+}(III) sp3d2\text{sp}^3\text{d}^2
(D) [Ni(CN)4]2[\text{Ni(CN)}_4]^{2-}(IV) dsp2\text{dsp}^2
Choose the correct answer from the options given below:
(A)
(B)
(C)
(D)
Q62Single correctHydrocarbons
The total number of compounds from below when treated with hot KMnO4O_4 giving benzoic acid is:

Toluene, Ethylbenzene, Isopropylbenzene, tert-Butylbenzene, Benzyl alcohol, Styrene
Question diagram for Q62
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Q63Single correctOrganic Compounds Containing Halogens
The major product of the following reaction is:

Ph-CH(Br)-CH2-CH2-CH(Br)-CH3KOH/EtOH (excess)/ΔMajor product\text{Ph-CH(Br)-CH}_2\text{-CH}_2\text{-CH(Br)-CH}_3 \xrightarrow{\text{KOH/EtOH (excess)}/\Delta} \text{Major product}
Question diagram for Q63
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Q64Single correctClassification of Elements and Periodicity in Properties
Given below are two statements:

**Statement (I):** According to the Law of Octaves, the elements were arranged in the increasing order of their atomic number.

**Statement (II):** Meyer observed a periodically repeated pattern upon plotting physical properties of certain elements against their respective atomic numbers.
(A)
(B)
(C)
(D)
Q65Single correctChemical Kinetics
For bacterial growth in a cell culture, growth law is very similar to the law of radioactive decay. Which of the following graphs is most suitable to represent bacterial colony growth?
(A)
(B)
(C)
(D)
Q66Single correctAtomic Structure
Which of the following is/are not correct with respect to energy of atomic orbitals of hydrogen atom?

(A) 1s < 2p < 3d < 4s
(B) 1s < 2s = 2p < 3s = 3p
(C) 1s < 2s < 2p < 3s < 3p
(D) 1s < 2s < 4s < 3d
(A)
(B)
(C)
(D)
Q67Single correctSolutions
Assume a living cell with 0.90.9% (ω\omega/ω\omega) of glucose solution (aqueous). This cell is immersed in another solution having equal mole fraction of glucose and water. (Consider the data upto first decimal place only). The cell will:
(A)
(B)
(C)
(D)
Q68Single correctOrganic Compounds Containing Nitrogen
Identify correct statements:
(A) Primary amines do not give diazonium salts when treated with NaNO2NaNO_2 in acidic condition.
(B) Aliphatic and aromatic primary amines on heating with CHCl3CHCl_3 and ethanolic KOH form carbylamines.
(C) Secondary and tertiary amines also give carbylamine test.
(D) Benzenesulfonyl chloride is known as Hinsberg's reagent.
(E) Tertiary amines react with benzenesulfonyl chloride very easily.
Choose the correct answer from the options given below:
(A)
(B)
(C)
(D)
Q69Single correctSome Basic Principles of Organic Chemistry
Given below are two statements: In the light of the these statements, choose the correct answer from the options given below:
Question diagram for Q69
(A)
(B)
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Q70Single correctPrinciples Related to Practical Chemistry
Identify the inorganic sulphides that are yellow in colour:

(A) (NH4)2S(\text{NH}_4)_2\text{S} (B) PbS\text{PbS} (C) CuS\text{CuS} (D) As2S3\text{As}_2\text{S}_3 (E) As2S5\text{As}_2\text{S}_5

Choose the correct answer from the options given below:
(A)
(B)
(C)
(D)
Q71Numericald- and f-Block Elements
The spin only magnetic moment (μ\mu) value (B.M.) of the compound with strongest oxidising power among Mn2O3\text{Mn}_2\text{O}_3, TiO\text{TiO} and VO\text{VO} is _____ B.M. (Nearest integer).
Q72NumericalChemical Thermodynamics
Consider the following data:
Heat of formation of \text{Heat of formation of } CO2CO_2(g) = -393.5 kJ mol\text{ kJ mol}1^{-1}
Heat of formation of \text{Heat of formation of } H2H_2O(l) = -286.0 kJ mol\text{ kJ mol}1^{-1}
Heat of combustion of benzene\text{Heat of combustion of benzene} = -3267.0 kJ mol\text{ kJ mol}1^{-1}
The heat of formation of benzene is _____\text{The heat of formation of benzene is \_\_\_\_\_}  kJ mol\text{ kJ mol}1^{-1}. (Nearest integer)\text{(Nearest integer)}
Q73NumericalRedox Reactions and Electrochemistry
Electrolysis of 600 mL aqueous solution of NaCl for 5 min changes the pH of the solution to 12. The current in Amperes used for the given electrolysis is ____. (Nearest integer).
Q74Numericalp-Block Elements
A group 15 element forms dπ\pi-dπ\pi bond with transition metals. It also forms hydride, which is a strongest base among the hydrides of other group members that form dπ\pi-dπ\pi bond. The atomic number of the element is _____.
Q75NumericalChemical Bonding and Molecular Structure
Total number of molecules/species from following which will be paramagnetic is ______. O2O_2, O2O_2^+, O2O_2^-, NO, NO2NO_2, CO, K2K_2[NiCl4NiCl_4], [Co((NH_3)6)_6]Cl3Cl_3, K2K_2[Ni(CN)4(CN)_4]

Mathematics25 questions

Q1Single correctStatistics and Probability
Bag B1B_1 contains 6 white and 4 blue balls, Bag B2B_2 contains 4 white and 6 blue balls, and Bag B3B_3 contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability, that the ball is drawn from Bag B2B_2, is:
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Q2Single correctCo-ordinate Geometry
Let A, B, C be three points in xy-plane, whose position vectors are given by 3i^+j^\sqrt{3}\hat{i} + \hat{j}, i^+3j^\hat{i} + \sqrt{3}\hat{j} and ai^+(1a)j^a\hat{i} + (1-a)\hat{j} respectively with respect to the origin O. If the distance of the point C from the line bisecting the angle between the vectors OA\vec{OA} and OB\vec{OB} is 92\frac{9}{\sqrt{2}}, then the sum of all the possible values of a is:
(A)
(B)
(C)
(D)
Q3Single correctVector Algebra
If the components of a=αi^+βj^+γk^\vec{a} = \alpha\hat{i} + \beta\hat{j} + \gamma\hat{k} along and perpendicular to b=3i^+j^k^\vec{b} = 3\hat{i} + \hat{j} - \hat{k} respectively, are 1611(3i^+j^k^)\frac{16}{11}(3\hat{i} + \hat{j} - \hat{k}) and 111(4i^5j^17k^)\frac{1}{11}(-4\hat{i} - 5\hat{j} - 17\hat{k}), then α2+β2+γ2\alpha^2 + \beta^2 + \gamma^2 is equal to:
(A)
(B)
(C)
(D)
Q4Single correctComplex Numbers and Quadratic Equations
If α+iβ\alpha + i\beta and γ+iδ\gamma + i\delta are the roots of x2(32i)x(2i2)=0x^2 - (3-2i)x - (2i-2) = 0, i=1i = \sqrt{-1}, then αγ+βδ\alpha\gamma + \beta\delta is equal to:
(A)
(B)
(C)
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Q5Single correctCo-ordinate Geometry
If the midpoint of a chord of the ellipse x29+y24=1\frac{x^2}{9} + \frac{y^2}{4} = 1 is (2,43)(\sqrt{2}, \frac{4}{\sqrt{3}}), and the length of the chord is 2α3\frac{2\alpha}{3}, then α\alpha is:
(A)
(B)
(C)
(D)
Q6Single correctPermutations and Combinations
Let S be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set S, one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is:
(A)
(B)
(C)
(D)
Q7Single correctIntegral Calculus
Let f be a real valued continuous function defined on the positive real axis such that g(x)=0xtf(t)dtg(x) = \int_0^x tf(t)dt. If g(x3)=x6+x7g(x^3) = x^6 + x^7, then value of r=115f(r3)\sum_{r=1}^{15} f(r^3) is:
(A)
(B)
(C)
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Q8Single correctThree Dimensional Geometry
The square of the distance of the point (157,327,7)\left(\frac{15}{7}, \frac{32}{7}, 7\right) from the line x13=y35=z57\frac{x-1}{3} = \frac{y-3}{5} = \frac{z-5}{7} in the direction of the vector i^+4j^+7k^\hat{i} + 4\hat{j} + 7\hat{k} is:
(A)
(B)
(C)
(D)
Q9Single correctIntegral Calculus
The area of the region bounded by the curves x(1+y2)=1x(1 + y^2) = 1 and y2=2xy^2 = 2x is:
(A)
(B)
(C)
(D)
Q10Single correctMatrices and Determinants
Let A=[12201]A = \begin{bmatrix} \frac{1}{\sqrt{2}} & -2 \\ 0 & 1 \end{bmatrix} and P=[cosθsinθsinθcosθ]P = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}, θ>0\theta > 0. If B=PAPTB = \text{PAP}^T, C=PTB10PC = P^TB^{10}P and the sum of the diagonal elements of C is mn\frac{m}{n}, where gcd(m,n)=1\gcd(m, n) = 1, then m+nm + n is:
(A)
(B)
(C)
(D)
Q11Single correctIntegral Calculus
If f(x)=1x1/4(1+x1/4)dxf(x) = \int \frac{1}{x^{1/4}(1+x^{1/4})} dx, f(0)=6f(0) = -6, then f(1)f(1) is equal to:
(A)
(B)
(C)
(D)
Q12Single correctIntegral Calculus
Let f:RRf : \mathbb{R} \to \mathbb{R} be a twice differentiable function such that f(2)=1f(2) = 1. If F(x)=xf(x)F(x) = xf(x) for all xRx \in \mathbb{R}, 02xF(x)dx=6\int_0^2 xF'(x)dx = 6 and 02x2F"(x)dx=40\int_0^2 x^2F"(x)dx = 40, then F(2)+02F(x)dxF'(2) + \int_0^2 F(x)dx is equal to:
(A)
(B)
(C)
(D)
Q13Single correctSequence and Series
For positive integers n, if 4an=(n2+5n+6)4a_n = (n^2 + 5n + 6) and Sn=k=1n1akS_n = \sum_{k=1}^{n} \frac{1}{a_k}, then the value of 507S2025507 S_{2025} is:
(A)
(B)
(C)
(D)
Q14Single correctLimit, Continuity and Differentiability
Let f:[0,3]Af : [0, 3] \to A be defined by f(x)=2x315x2+36x+7f(x) = 2x^3 - 15x^2 + 36x + 7 and g:[0,)Bg : [0, \infty) \to B be defined by g(x)=x2025x2025+1g(x) = \frac{x^{2025}}{x^{2025} + 1}. If both the functions are onto and S={xZ:xA or xB}S = \{x \in \mathbb{Z} : x \in A \text{ or } x \in B\}, then n(S) is equal to:
(A)
(B)
(C)
(D)
Q15Single correctTrigonometry
Let [x] denote the greatest integer less than or equal to x. Then domain of f(x)=sec1(2[x]+1)f(x) = \sec^{-1}(2[x]+1) is:
(A)
(B)
(C)
(D)
Q16Single correctTrigonometry
If r=1131sin(π4+(r1)π6)sin(π4+rπ6)=a3+b\sum_{r=1}^{13} \frac{1}{\sin\left(\frac{\pi}{4} + (r-1)\frac{\pi}{6}\right)\sin\left(\frac{\pi}{4} + r\frac{\pi}{6}\right)} = a\sqrt{3} + b, a,bZa, b \in \mathbb{Z}, then a2+b2a^2 + b^2 is equal to:
(A)
(B)
(C)
(D)
Q17Single correctCo-ordinate Geometry
Two equal sides of an isosceles triangle are along x+2y=4 -x + 2y = 4 and x+y=4 x + y = 4 . If m m is the slope of its third side, then the sum, of all possible distinct values of m m , is:
(A)
(B)
(C)
(D)
Q18Single correctBinomial Theorem and its Simple Applications
Let the coefficients of three consecutive terms Tr T_r , Tr+1 T_{r+1} and Tr+2 T_{r+2} in the binomial expansion of (a+b)12 (a + b)^{12} be in a G.P. and let p be the number of all possible values of r. Let q be the sum of all rational terms in the binomial expansion of (334+443)12 \left(3\sqrt[4]{3} + 4\sqrt[3]{4}\right)^{12} . Then p+q p + q is equal to:
(A)
(B)
(C)
(D)
Q19Single correctCo-ordinate Geometry
If A and B are the points of intersection of the circle x2+y28x=0 x^2 + y^2 - 8x = 0 and the hyperbola x29y24=1 \frac{x^2}{9} - \frac{y^2}{4} = 1 and a point P moves on the line 2x3y+4=0 2x - 3y + 4 = 0 , then the centroid of PAB \triangle \text{PAB} lies on the line:
(A)
(B)
(C)
(D)
Q20Single correctLimit, Continuity and Differentiability
Let f:R{0}(,1) f : \mathbb{R} - \{0\} \to (-\infty, 1) be a polynomial of degree 2, satisfying f(x)f(1x)=f(x)+f(1x) f(x)f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{x}\right) . If f(K)=2K f(K) = -2K , then the sum of squares of all possible values of K is:
(A)
(B)
(C)
(D)
Q21NumericalPermutations and Combinations
The number of natural numbers, between 212 and 999, such that the sum of their digits is 15, is _____.
Q22NumericalLimit, Continuity and Differentiability
Let f(x)=limnr=0ntan(x/2r+1)+tan3(x/2r+1)1tan2(x/2r+1) f(x) = \lim_{n \to \infty} \sum_{r=0}^{n} \frac{\tan(x/2^{r+1}) + \tan^3(x/2^{r+1})}{1 - \tan^2(x/2^{r+1})} . Then limx0exef(x)xf(x) \lim_{x \to 0} \frac{e^x - e^{f(x)}}{x - f(x)} is equal to ____.
Q23NumericalSequence and Series
The interior angles of a polygon with n sides, are in an A.P. with common difference 6 6^\circ . If the largest interior angle of the polygon is 219 219^\circ , then n is equal to _____.
Q24NumericalCo-ordinate Geometry
Let A and B be the two points of intersection of the line y+5=0 y + 5 = 0 and the mirror image of the parabola y2=4x y^2 = 4x with respect to the line x+y+4=0 x + y + 4 = 0 . If d denotes the distance between A and B, and a denotes the area of SAB \triangle \text{SAB} , where S is the focus of the parabola y2=4x y^2 = 4x , then the value of (a+d) (a + d) is ____.
Q25NumericalDifferential Equations
If y=y(x) y = y(x) is the solution of the differential equation, 4x2dydx=[(sin1x2)3ysin1x2] \sqrt{4-x^2}\frac{dy}{dx} = \left[\left(\sin^{-1}\frac{x}{2}\right)^3 - y\sin^{-1}\frac{x}{2}\right] , 2x2 -2 \leq x \leq 2 , y(2)=π284 y(2) = \frac{\pi^2 - 8}{4} , then y2(0) y^2(0) is equal to ____.

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