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JEE Main 2025 January 29, Shift 1 Question Paper with Solutions

All 75 questions from the JEE Main 2025 (January 29, Shift 1) 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
Given below are two statements : one is labelled as **Assertion (A)** and the other is labelled as **Reason (R)**.

**Assertion (A):** Choke coil is simply a coil having a large inductance but a small resistance. Choke coils are used with fluorescent mercury-tube fittings. If household electric power is directly connected to a mercury tube, the tube will be damaged.

**Reason (R):** By using the choke coil, the voltage across the tube is reduced by a factor RR2+ω2L2\frac{R}{\sqrt{R^2 + \omega^2 L^2}}, where ω\omega is frequency of the supply across resistor R and inductor L. If the choke coil were not used, the voltage across the resistor would be the same as the applied voltage.

In the light of the above statements, choose the most appropriate answer from the options given below:
(A)
(B)
(C)
(D)
Q27Single correctKinematics
Two projectiles are fired with same initial speed from same point on ground at angles of (45α)(45^\circ - \alpha) and (45+α)(45^\circ + \alpha), respectively, with the horizontal direction. The ratio of their maximum heights attained is:
(A)
(B)
(C)
(D)
Q28Single correctElectrostatics
An electric dipole of mass m, charge q, and length \ell is placed in a uniform electric field E=E0i^\vec{E} = E_0 \hat{i}. When the dipole is rotated slightly from its equilibrium position and released, the time period of its oscillations will be:
(A)
(B)
(C)
(D)
Q29Single correctUnits and Measurements
The pair of physical quantities not having same dimensions is:
(A)
(B)
(C)
(D)
Q30Single correctOscillations and Waves
Given below are two statements : one is labelled as **Assertion (A)** and the other is labelled as **Reason (R)**.

**Assertion (A):** Time period of a simple pendulum is longer at the top of a mountain than that at the base of the mountain.

**Reason (R):** Time period of a simple pendulum decreases with increasing value of acceleration due to gravity and vice-versa.

In the light of the above statements, choose the most appropriate answer from the options given below:
(A)
(B)
(C)
(D)
Q31Single correctUnits and Measurements
The expression given below shows the variation of velocity (v) with time (t), v=At2+BtC+tv = At^2 + \frac{Bt}{C+t}. The dimension of ABC is:
(A)
(B)
(C)
(D)
Q32Single correctElectromagnetic Induction and Alternating Currents
Consider I1I_1 and I2I_2 are the currents flowing simultaneously in two nearby coils 1 & 2, respectively. If L1L_1 = self inductance of coil 1, M12M_{12} = mutual inductance of coil 1 with respect to coil 2, then the value of induced emf in coil 1 will be
(A)
(B)
(C)
(D)
Q33Single correctOptics
At the interface between two materials having refractive indices n1n_1 and n2n_2, the critical angle for reflection of an em wave is θ1C\theta_{1C}. The n2n_2 material is replaced by another material having refractive index n3n_3, such that the critical angle at the interface between n1n_1 and n3n_3 materials is θ2C\theta_{2C}. If n3>n2>n1n_3 > n_2 > n_1; n2n3=25\frac{n_2}{n_3} = \frac{2}{5} and sinθ2Csinθ1C=12\sin\theta_{2C} - \sin\theta_{1C} = \frac{1}{2}, then θ1C\theta_{1C} is
(A)
(B)
(C)
(D)
Q34Single correctMagnetic Effects of Current and Magnetism
Consider a long straight wire of a circular cross-section (radius aa) carrying a steady current II. The current is uniformly distributed across this cross-section. The distances from the centre of the wire's cross-section at which the magnetic field [inside the wire, outside the wire] is half of the maximum possible magnetic field, any where due to the wire, will be
(A)
(B)
(C)
(D)
Q35Single correctWork, Energy and Power
As shown below, bob A of a pendulum having massless string of length 'R' is released from 6060^\circ to the vertical. It hits another bob B of half the mass that is at rest on a friction less table in the centre. Assuming elastic collision, the magnitude of the velocity of bob A after the collision will be (take g as acceleration due to gravity)
Extracted diagram from question paper
(A)
(B)
(C)
(D)
Q36Single correctDual Nature of Matter and Radiation
Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A): Emission of electrons in photoelectric effect can be suppressed by applying a sufficiently negative electron potential to the photoemissive substance.

Reason (R): A negative electric potential, which stops the emission of electrons from the surface of a photoemissive substance, varies linearly with frequency of incident radiation.

In the light of the above statements, choose the most appropriate answer from the options given below:
(A)
(B)
(C)
(D)
Q37Single correctElectromagnetic Induction and Alternating Currents
A coil of area A and N turns is rotating with angular velocity ω\omega in a uniform magnetic field B\vec{B} about an axis perpendicular to B\vec{B}. Magnetic flux ϕ\phi and induced emf ε\varepsilon across it, at an instant when B\vec{B} is parallel to the plane of coil, are:
(A)
(B)
(C)
(D)
Q38Single correctProperties of Solids and Liquids
The fractional compression (ΔVV)\left(\frac{\Delta V}{V}\right) of water at the depth of 2.5 km below the sea level is ________ %. Given, the Bulk modulus of water = 2×1092 \times 10^9 Nm2m^{-2}, density of water = 10310^3 kg m3m^{-3}, acceleration due to gravity = g=10g = 10 ms2s^{-2}.
(A)
(B)
(C)
(D)
Q39Single correctDual Nature of Matter and Radiation
If λ\lambda and K are de Broglie wavelength and kinetic energy, respectively, of a particle with constant mass. The correct graphical representation for the particle will be:
(A)
(B)
(C)
(D)
Q40Single correctElectronic Devices
For the circuit shown above, equivalent GATE is:
Extracted diagram from question paper
(A)
(B)
(C)
(D)
Q41Single correctWork, Energy and Power
A body of mass 'm' connected to a massless and unstretchable string goes in vertical circle of radius 'R' under gravity g. The other end of the string is fixed at the center of circle. If velocity at top of circular path is ngR\sqrt{n gR}, where n1n \geq 1, then ratio of kinetic energy of the body at bottom to that at top of the circle is:
(A)
(B)
(C)
(D)
Q42Single correctOptics
Let uu and vv be the distances of the object and the image from a lens of focal length ff. The correct graphical representation of uu and vv for a convex lens when u>f|u| > f, is:
(A)
(B)
(C)
(D)
Q43Single correctElectrostatics
Match List-I with List-II.
List-IList-II
A. Electric field inside (distance r>0r > 0 from center) of a uniformly charged spherical shell with surface charge density σ\sigma, and radius R.I. σ/ε0\sigma / \varepsilon_0
B. Electric field at distance r>0r > 0 from a uniformly charged infinite plane sheet with surface charge density σ\sigmaII. σ/2ε0\sigma / 2\varepsilon_0
C. Electric field outside (distance r>0r > 0 from center) of a uniformly charged spherical shell with surface charge density σ\sigma, and radius RIII. 00
D. Electric field between 2 oppositely charged infinite plane parallel sheets with uniform surface charge density σ\sigma.IV. σR2ε0r2\frac{\sigma R^2}{\varepsilon_0 r^2}
Choose the correct answer from the options given below:
(A)
(B)
(C)
(D)
Q44Single correctThermodynamics
The work done in an adiabatic change in an ideal gas depends upon only:
(A)
(B)
(C)
(D)
Q45Single correctElectromagnetic Waves
Given below are two statements: one is labelled as Assertion (A) and other is labelled as Reason (R).

Assertion (A): Electromagnetic waves carry energy but not momentum.

Reason (R): Mass of a photon is zero.

In the light of the above statements, choose the most appropriate answer from the options given below:
(A)
(B)
(C)
(D)
Q46NumericalRotational Motion
The coordinates of a particle with respect to origin in a given reference frame is (1,1,1)(1, 1, 1) meters. If a force of F=i^+j^+k^\vec{F} = \hat{i} + \hat{j} + \hat{k} acts on the particle, then the magnitude of torque (with respect to origin) in z-direction is _________.
Q47NumericalKinetic Theory of Gases
A container of fixed volume contains a gas at 2727^\circC. To double the pressure of the gas, the temperature of gas should be raised to _______ ^\circC.
Q48NumericalOptics
Two light beams fall on a transparent material block at point 1 and 2 with angle θ1\theta_1 and θ2\theta_2, respectively, as shown in figure. After refraction, the beams intersect at point 3 which is exactly on the interface at other end of the block. Given: the distance between 1 and 2, d=43d = 4\sqrt{3} cm and θ1=θ2=cos1(n22n1)\theta_1 = \theta_2 = \cos^{-1}\left(\frac{n_2}{2n_1}\right), where refractive index of the block n2>n_2 > refractive index of the outside medium n1n_1, then the thickness of the block is ________ cm.
Extracted diagram from question paper
Q49NumericalProperties of Solids and Liquids
In a hydraulic lift, the surface area of the input piston is 66 cm2m^2 and that of the output piston is 15001500 cm2m^2. If 100100 N force is applied to the input piston to raise the output piston by 2020 cm, then the work done is _________ kJ.
Q50NumericalKinematics
The maximum speed of a boat in still water is 2727 km/h. Now this boat is moving downstream in a river flowing at 99 km/h. A man in the boat throws a ball vertically upwards with speed of 1010 m/s. Range of the ball as observed by an observer at rest on the river bank, is _________ cm. (Take g=10g = 10 m/s2s^2)

Chemistry25 questions

Q51Single correctSome Basic Principles of Organic Chemistry
Total number of nucleophiles from the following is:-
NH3NH_3, PhSH, (H3C)2S(H_3C)_2S, H2C=CH2H_2C=CH_2, OH\overline{OH}, H3OH_3O^{\oplus}, (CH3)2CO(CH_3)_2CO, NCH3\gtrdot \text{NCH}_3
(A)
(B)
(C)
(D)
Q52Single correctRedox Reactions and Electrochemistry
The standard reduction potential values of some of the p-block ions are given below. Predict the one with the strongest oxidising capacity.
(A)
(B)
(C)
(D)
Q53Single correctRedox Reactions and Electrochemistry
The molar conductivity of a weak electrolyte when plotted against the square root of its concentration, which of the following is expected to be observed?
Extracted diagram from question paper
(A)
(B)
(C)
(D)
Q54Single correctEquilibrium
At temperature T, compound AB2(g)AB_{2(g)} dissociates as AB2(g)AB(g)+12B2(g)AB_2(g) \rightleftharpoons AB(g) + \frac{1}{2}B_2(g) having degree of dissociation x (small compared to unity). The correct expression for x in terms of KpK_p and p is
(A)
(B)
(C)
(D)
Q55Single correctHydrocarbons
Match List-I with List-II.
Choose the correct answer from the options given below:
Extracted diagram from question paper
(A)
(B)
(C)
(D)
Q56Single correctSome Basic Concepts in Chemistry
Choose the correct statements.

(A) Weight of a substance is the amount of matter present in it.

(B) Mass is the force exerted by gravity on an object.

(C) Volume is the amount of space occupied by a substance.

(D) Temperatures below 0°C are possible in Celsius scale, but in Kelvin scale negative temperature is not possible.

(E) Precision refers to the closeness of various measurements for the same quantity.
(A)
(B)
(C)
(D)
Q57Single correctCoordination Compounds
The correct increasing order of stability of the complexes based on Δo\Delta_o value is:

(I) [Mn(CN)6]3[Mn(CN)_6]^{3-}

(II) [Co(CN)6]4[Co(CN)_6]^{4-}

(III) [Fe(CN)6]4[Fe(CN)_6]^{4-}

(IV) [Fe(CN)6]3[Fe(CN)_6]^{3-}
(A)
(B)
(C)
(D)
Q58Single correctCoordination Compounds
Match List-I with List-II.
List-I (Complex)List-II (Hybridisation & Magnetic character)
(A) [MnBr4]2[\text{MnBr}_4]^{2-}(I) d2sp3d^2sp^3 & diamagnetic
(B) [FeF6]3[\text{FeF}_6]^{3-}(II) sp3d2sp^3d^2 & paramagnetic
(C) [Co(C2O4)3]3[Co(C_2O_4)_3]^{3-}(III) sp3sp^3 & diamagnetic
(D) [Ni(CO)4][Ni(CO)_4](IV) sp3sp^3 & paramagnetic
Choose the correct answer from the options given below:
(A)
(B)
(C)
(D)
Q59Single correctOrganic Compounds Containing Halogens
In the following substitution reaction:
Product 'P' formed is:
Extracted diagram from question paper
(A)
(B)
(C)
(D)
Q60Single correctRedox Reactions and Electrochemistry
For a MgMg2+(aq)Ag+(aq)AgMg | Mg^{2+}(aq) || Ag^+(aq) | Ag the correct Nernst Equation is:
(A)
(B)
(C)
(D)
Q61Single correctd- and f-Block Elements
The correct option with order of melting points of the pairs (Mn, Fe), (Tc, Ru) and (Re, Os) is:
(A)
(B)
(C)
(D)
Q62Single correctSolutions
1.24 g of AX2AX_2 (molar mass 124 g mol1l^{-1}) is dissolved in 1 kg of water to form a solution with boiling point of 100.0156°C, while 25.4 g of AY2AY_2 (molar mass 250 g mol1l^{-1}) in 2 kg of water constitutes a solution with a boiling point of 100.0260°C.

Kb(H2O)=0.52K_b(H_2O) = 0.52 K kg mol1l^{-1}

Which of the following is correct?
(A)
(B)
(C)
(D)
Q63Single correctChemical Thermodynamics
500 J of energy is transferred as heat to 0.5 mol of Argon gas at 298 K and 1.00 atm. The final temperature and the change in internal energy respectively are:

Given: R=8.3 J K1 mol1R = 8.3 \text{ J K}^{-1} \text{ mol}^{-1}
(A)
(B)
(C)
(D)
Q64Single correctChemical Kinetics
The reaction A2+B22AB\text{A}_2 + \text{B}_2 \rightarrow 2\text{AB} follows the mechanism:

A2k1k1A+A\text{A}_2 \xrightleftharpoons[k_{-1}]{k_1} \text{A} + \text{A} (fast)

A+B2k2AB+B\text{A} + \text{B}_2 \xrightarrow{k_2} \text{AB} + \text{B} (slow)

A+BAB\text{A} + \text{B} \rightarrow \text{AB} (fast)

The overall order of the reaction is:
(A)
(B)
(C)
(D)
Q65Single correctAtomic Structure
If a0a_0 is denoted as the Bohr radius of hydrogen atom, then what is the de-Broglie wavelength (λ\lambda) of the electron present in the second orbit of hydrogen atom? [n : any integer]
(A)
(B)
(C)
(D)
Q66Single correctOrganic Compounds Containing Oxygen
The product (P) formed in the following reaction is:
Extracted diagram from question paper
(A)
(B)
(C)
(D)
Q67Single correctChemical Bonding and Molecular Structure
An element 'E' has the ionisation enthalpy value of 374 kJ mol1l^{-1}. 'E' reacts with elements A, B, C and D with electron gain enthalpy values of 328-328, 349-349, 325-325 and 295-295 kJ mol1l^{-1}, respectively.
The correct order of the products EA, EB, EC and ED in terms of ionic character is:
(A)
(B)
(C)
(D)
Q68Single correctBiomolecules
Match List-I with List-II.
List-I (Carbohydrate)List-II (Linkage, Source)
(A) Amylose(I) β\beta-C1C_1-C4C_4, plant
(B) Cellulose(II) α\alpha-C1C_1-C4C_4, animal
(C) Glycogen(III) α\alpha-C1C_1-C4C_4, α\alpha-C1C_1-C6C_6, plant
(D) Amylopectin(IV) α\alpha-C1C_1-C4C_4, plant
Choose the correct answer from the options given below:
(A)
(B)
(C)
(D)
Q69Single correctPurification and Characterisation of Organic Compounds
The steam volatile compounds among the following are:

(A) o-Nitrophenol
(B) o-Nitroaniline
(C) o-Aminophenol
(D) p-Aminophenol

Choose the correct answer from the options given below:
(A)
(B)
(C)
(D)
Q70Single correctClassification of Elements and Periodicity in Properties
Given below are two statements:

**Statement (I):** The radii of isoelectronic species increases in the order: Mg2+g^{2+} < Na+^+ < F^- < O2O^{2-}

**Statement (II):** The magnitude of electron gain enthalpy of halogen decreases in the order: Cl > F > Br > I

In the light of the above statements, choose the most appropriate answer from the options given below:
(A)
(B)
(C)
(D)
Q71NumericalOrganic Compounds Containing Nitrogen
Given below are some nitrogen containing compounds.

Each of them is treated with HCl separately. 1.0 g of the most basic compound will consume ______ mg of HCl.

(Given molar mass in g mol1l^{-1} C:12, H:1, O:16, Cl:35.5)
Extracted diagram from question paper
Q72Numericald- and f-Block Elements
The molar mass of the water insoluble product formed from the fusion of chromite ore (FeCr2O4r_2O_4) with Na2CO3a_2CO_3 in presence of O2O_2 is ______ g mol1l^{-1}.
Q73NumericalChemical Bonding and Molecular Structure
The sum of sigma (σ\sigma) and pi (π\pi) bonds in Hex-1,3-dien-5-yne is _______.
Q74NumericalSolutions
If A2A_2B is 30% ionised in an aqueous solution, then the value of van't Hoff factor (i) is _____ ×101\times 10^{-1}.
Q75NumericalOrganic Compounds Containing Oxygen
A cyclic compound with OH group undergoes the following reactions:

0.1 mole of compound 'S' will weigh _____ g.

(Given molar mass in g mol1l^{-1} C:12, H:1, O:16)
Extracted diagram from question paper

Mathematics25 questions

Q1Single correctCo-ordinate Geometry
Let the line x+y=1x + y = 1 meet the circle x2+y2=4x^2 + y^2 = 4 at the points A and B. If the line perpendicular to AB and passing through the mid point of the chord AB intersects the circle at C and D, then the area of the quadrilateral ADBC is equal to
(A)
(B)
(C)
(D)
Q2Single correctMatrices and Determinants
Let M and m respectively be the maximum and the minimum values of f(x)=1+sin2xcos2x4sin4xsin2x1+cos2x4sin4xsin2xcos2x1+4sin4x,xRf(x) = \begin{vmatrix}1+\sin^2 x & \cos^2 x & 4\sin 4x \\ \sin^2 x & 1+\cos^2 x & 4\sin 4x \\ \sin^2 x & \cos^2 x & 1+4\sin 4x\end{vmatrix}, x \in \mathbb{R}. Then M4m4M^4 - m^4 is equal to:
(A)
(B)
(C)
(D)
Q3Single correctCo-ordinate Geometry
Two parabolas have the same focus (4,3)(4,3) and their directrices are the x-axis and the y-axis, respectively. If these parabolas intersects at the points A and B, then (AB)2(AB)^2 is equal to
(A)
(B)
(C)
(D)
Q4Single correctCo-ordinate Geometry
Let ABC be a triangle formed by the lines 7x6y+3=07x - 6y + 3 = 0, x+2y31=0x + 2y - 31 = 0 and 9x2y19=09x - 2y - 19 = 0. Let the point (h,k) be the image of the centroid of ABC\triangle \text{ABC} in the line 3x+6y53=03x + 6y - 53 = 0. Then h2+k2+hkh^2 + k^2 + hk is equal to
(A)
(B)
(C)
(D)
Q5Single correctVector Algebra
Let a=2i^j^+3k^\vec{a} = 2\hat{i} - \hat{j} + 3\hat{k}, b=3i^5j^+k^\vec{b} = 3\hat{i} - 5\hat{j} + \hat{k} and c\vec{c} be a vector such that a×c=c×b\vec{a} \times \vec{c} = \vec{c} \times \vec{b} and (a+c)(b+c)=168(\vec{a} + \vec{c}) \cdot (\vec{b} + \vec{c}) = 168. Then the maximum value of c2|\vec{c}|^2 is:
(A)
(B)
(C)
(D)
Q6Single correctPermutations and Combinations
Let P be the set of seven digit numbers with sum of their digits equal to 11. If the numbers in P are formed by using the digits 1, 2 and 3 only, then the number of elements in the set P is:
(A)
(B)
(C)
(D)
Q7Single correctIntegral Calculus
Let the area of the region {(x,y):2yx2+3,y+x3,yx1}\{(x,y): 2y \leq x^2 + 3, y + |x| \leq 3, y \geq |x| - 1\} be A. Then 6A is equal to:
(A)
(B)
(C)
(D)
Q8Single correctBinomial Theorem and its Simple Applications
The least value of n for which the number of integral terms in the Binomial expansion of (73+1112)n\left(\sqrt[3]{7} + \sqrt[12]{11}\right)^n is 183, is:
(A)
(B)
(C)
(D)
Q9Single correctComplex Numbers and Quadratic Equations
The number of solutions of the equation (9x9x+2)(2x7x+3)=0\left(\frac{9}{x} - \frac{9}{\sqrt{x}} + 2\right)\left(\frac{2}{x} - \frac{7}{\sqrt{x}} + 3\right) = 0 is:
(A)
(B)
(C)
(D)
Q10Single correctDifferential Equations
Let y=y(x)y = y(x) be the solution of the differential equation cosx(loge(cosx))2dy+(sinx3ysinxloge(cosx))dx=0\cos x (\log_e(\cos x))^2 dy + (\sin x - 3y\sin x \log_e(\cos x))dx = 0, x(0,π2)x \in \left(0, \frac{\pi}{2}\right). If y(π4)=1loge2y\left(\frac{\pi}{4}\right) = \frac{-1}{\log_e 2}, then y(π6)y\left(\frac{\pi}{6}\right) is:
(A)
(B)
(C)
(D)
Q11Single correctSets, Relations and Functions
Define a relation R on the interval [0,π2)\left[0, \frac{\pi}{2}\right) by x R y if and only if sec2xtan2y=1\sec^2 x - \tan^2 y = 1. Then R is:
(A)
(B)
(C)
(D)
Q12Single correctCo-ordinate Geometry
Let the ellipse, E1:x2a2+y2b2=1E_1: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a>ba > b and E2:x2A2+y2B2=1E_2: \frac{x^2}{A^2} + \frac{y^2}{B^2} = 1, A<BA < B have same eccentricity 13\frac{1}{\sqrt{3}}. Let the product of their lengths of latus rectums be 323\frac{32}{3}, and the distance between the foci of E1E_1 be 4. If E1E_1 and E2E_2 meet at A, B, C and D, then the area of the quadrilateral ABCD equals:
(A)
(B)
(C)
(D)
Q13Single correctSequence and Series
Consider an A.P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its 11th term is:
(A)
(B)
(C)
(D)
Q14Single correctThree Dimensional Geometry
Let a=i^+2j^+k^\vec{a} = \hat{i} + 2\hat{j} + \hat{k} and b=2i^+7j^+3k^\vec{b} = 2\hat{i} + 7\hat{j} + 3\hat{k}. Let L1:r=(i^+2j^+k^)+λaL_1: \vec{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda\vec{a}, λR\lambda \in \mathbb{R} and L2:r=(j^+k^)+μbL_2: \vec{r} = (\hat{j} + \hat{k}) + \mu\vec{b}, μR\mu \in \mathbb{R} be two lines. If the line L3L_3 passes through the point of intersection of L1L_1 and L2L_2, and is parallel to a+b\vec{a} + \vec{b}, then L3L_3 passes through the point:
(A)
(B)
(C)
(D)
Q15Single correctSequence and Series
The value of limnk=1nk3+6k2+11k+5(k+3)!\lim_{n \to \infty} \sum_{k=1}^{n} \frac{k^3 + 6k^2 + 11k + 5}{(k+3)!} is:
(A)
(B)
(C)
(D)
Q16Single correctIntegral Calculus
The integral 800π/4sinθ+cosθ9+16sin2θdθ80\int_0^{\pi/4} \frac{\sin\theta + \cos\theta}{9 + 16\sin 2\theta} d\theta is equal to:
(A)
(B)
(C)
(D)
Q17Single correctThree Dimensional Geometry
Let L1:x11=y21=z12L_1: \frac{x-1}{1} = \frac{y-2}{-1} = \frac{z-1}{2} and L2:x+11=y22=z1L_2: \frac{x+1}{-1} = \frac{y-2}{2} = \frac{z}{1} be two lines. Let L3L_3 be a line passing through the point (α,β,γ)(\alpha, \beta, \gamma) and be perpendicular to both L1L_1 and L2L_2. If L3L_3 intersects L1L_1, then 5α11β8γ|5\alpha - 11\beta - 8\gamma| equals:
(A)
(B)
(C)
(D)
Q18Single correctStatistics and Probability
Let x1,x2,,x10x_1, x_2, \ldots, x_{10} be ten observations such that i=110(xi2)=30\sum_{i=1}^{10}(x_i - 2) = 30, i=110(xiβ)2=98\sum_{i=1}^{10}(x_i - \beta)^2 = 98, β>2\beta > 2 and their variance is 45\frac{4}{5}. If μ\mu and σ2\sigma^2 are respectively the mean and the variance of 2(x11)+4,2(x21)+4,,2(x101)+42(x_1-1)+4, 2(x_2-1)+4, \ldots, 2(x_{10}-1)+4, then βμσ2\frac{\beta\mu}{\sigma^2} is equal to:
(A)
(B)
(C)
(D)
Q19Single correctComplex Numbers and Quadratic Equations
Let z182i1|z_1 - 8 - 2i| \leq 1 and z22+6i2|z_2 - 2 + 6i| \leq 2, z1,z2Cz_1, z_2 \in \mathbb{C}. Then the minimum value of z1z2|z_1 - z_2| is:
(A)
(B)
(C)
(D)
Q20Single correctMatrices and Determinants
Let A=[aij]5×4=[log5128log45log58log425]A = [a_{ij}]_{5 \times 4} = \begin{bmatrix} \log_5 128 & \log_4 5 \\ \log_5 8 & \log_4 25 \end{bmatrix}. If AijA_{ij} is the cofactor of aija_{ij}, Cij=k=12aikAjkC_{ij} = \sum_{k=1}^{2} a_{ik}A_{jk}, 1i,j21 \leq i, j \leq 2, and C=[Cij]C = [C_{ij}], then 8C8|C| is equal to:
(A)
(B)
(C)
(D)
Q21NumericalLimit, Continuity and Differentiability
Let f:(0,)Rf : (0,\infty) \rightarrow \mathbb{R} be a twice differentiable function. If for some a0a \neq 0, 01f(λx)dλ=af(x)\int_{0}^{1} f(\lambda x) d\lambda = af(x), f(1)=1f(1) = 1 and f(16)=18f(16) = \frac{1}{8}, then 16f(116)16 - f'\left(\frac{1}{16}\right) is equal to ______.
Q22NumericalMatrices and Determinants
Let S={mZ:Am2+Am=3IA6}S = \left\{m \in \mathbb{Z} : A^{m^2} + A^m = 3I - A^{-6}\right\}, where A=[2110]A = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix}. Then n(S) is equal to ____.
Q23NumericalLimit, Continuity and Differentiability
Let [t] be the greatest integer less than or equal to t. Then the least value of pNp \in \mathbb{N} for which limx0+(x([1x]+[2x]++[px])x2([1x2]+[22x2]++[92x2]))1\lim_{x \to 0^+} \left(x\left(\left[\frac{1}{x}\right] + \left[\frac{2}{x}\right] + \ldots + \left[\frac{p}{x}\right]\right) - x^2\left(\left[\frac{1}{x^2}\right] + \left[\frac{2^2}{x^2}\right] + \ldots + \left[\frac{9^2}{x^2}\right]\right)\right) \geq 1 is equal to ______.
Q24NumericalPermutations and Combinations
The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is 44 ______.
Q25NumericalTrigonometry
Let S={x:cos1x=π+sin1x+sin1(2x+1)}S = \left\{x : \cos^{-1}x = \pi + \sin^{-1}x + \sin^{-1}(2x+1)\right\}. Then xS(2x1)2\sum_{x \in S} (2x-1)^2 is equal to ______.

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