KVPY Math XII Class Stream SB/SX

Mathematics

• Simultaneous Equations: Solve simultaneous equations x + y2 = x2 + y = 12.
• Complex Numbers: Find the maximum possible value of |z + z^-1| given |z3 + z - 3| ≤ 2.
• Perfect Squares: Find the largest perfect square that divides 20143 - 20133 + ... + 23 - 13.
• Parabola: Find the equation of the curve on which point C lies given that OABC is a rectangle, A and B lie on the parabola y = x2, and C lies on the curve.

Calculus

• Limits: Evaluate the limit of (6/x) / sin(x) as x approaches 0.
• Polynomials: Find the value of p(0) where p(x) - p'(x) = xn.
• Integrals: Find the value of the integral of f(x) from 0 to 1, where f(x) = max{|x|, |x - 1|, ..., |x - 2n|}.
• Optimization: Maximize the area of a sector with fixed perimeter.

Probability

• Random Variables: Find the probability that a box contains at most one ball given that the probability of drawing a red, white, blue, or green marble is equally likely.

Physics

• Mechanics: Find the coefficient of kinetic friction between a box and a table given that the box slides 1m in 2 seconds.
• Nuclear Physics: Find the energy released when 1μg of Carbon-11 decays into Boron-11, given that the energy released per decay is 0.96 MeV.
• Electromagnetism: Find the angular velocity of a system consisting of two disks spinning in the same direction.

Chemistry

• Radioactivity: Find the half-life of Carbon-11 given that the decay formula is 11C → 11B + e+ + νe.

Others

• Electronics: Find the current-voltage characteristic of a diode connected in parallel with a resistor.
• Black Holes: Find the dimensions of the area of a black hole in terms of the universal gravitational constant G, its mass M, and the speed of light C.Here are the study notes for the provided text:

Chemistry

• Lewis acid strength of BBr3, BCI3, and BF3 is in the order of BF3 > BCI3 > BBr3.
• O2- is isoelectronic with Zn2+.
• Bond angles in methane, ammonia, and water are respectively 109.5, 104.5, and 107.1 degrees.
• The oxidation state of Mn in the reaction of hydrogen peroxide with potassium permanganate in alkaline medium is +2.
• The rate constant of a chemical reaction at a very high temperature will approach the Arrhenius frequency factor.
• The reducing strength of metals follows the order of Cr > Pb > Ag > Cu.
• Optical activity can be exhibited by 2-bromobutane.
• The structure of the polymer obtained by the reaction is III.
• The major product of the reaction between CH3CH2ONa and (CH3)3CCI in ethanol is CH3CH2OC(CH3)3.
• When H2S gas is passed through a hot acidic aqueous solution containing AI3+, Cu2+, Pb2+, and Ni2+, a precipitate is formed which consists of PbS and CuS.
• The electronic configuration of an element with the largest difference between the 1st and 2nd ionization energies is 1s2 2s2 2p6.
• The order of electronegativity of carbon is sp > sp2 > sp3 in hybridized states.
• The most abundant transition metal in the human body is iron.
• The moral conductivities of HCl, NaCl, CH3COOH, and CH3COONa at infinite dilution follow the order of HCl > CH3COOH > NaCl > CH3COONa.

Mathematics

• The value of x = 50 + 7 - (50 - 7) is a rational number, but not an integer.
• The coefficients of the equation 1 + x + x2 = a0 + a1x + a2x2 + ... follow a specific pattern.
• The slope of the incident beam in the given graph is 13/8.
• The series C(θ) = ∑(n=0 to ∞)cos(nθ)/n! has certain properties.
• The limit of a specific function as x approaches 0 is 2.
• The probability that 1 is an element of A, when A is a subset of Xn, is p, and the probability that 2 is an element of A is q, where p > q and p - q = 1/6.

Physics

• The height to which water should be poured in a cubical vessel so that an object at the bottom can be seen is 16 cm.
• The moments of inertia of a non-uniform circular disc about four mutually perpendicular tangents are related to the distance of the center of mass of the disc from its geometrical center.
• The stress on a horizontal steel railroad track on a hot summer day is 6.6 × 10^7 Pa.
• Electromagnetic waves can travel from air to a medium with a refractive index very nearly equal to zero.
• The ratio of the masses of two small metal balls connected by strings of equal length to a fixed point is close to 0.58.
• Diamagnetic particles scattered on a horizontal table accumulate around regions such as C.
• The distance between the vertex and the center of mass of a uniform solid planar circular segment of angular size θ and radius R is R(4sin(θ/2))/(3θ).
• The speed of an object when it returns to the surface of a planet is √(2GM/R).
• The currents through the three resistors in the given circuit are I downwards, I downwards, and I downwards.

Miscellaneous

• There are 60 questions in the provided text, divided into three parts: Chemistry, Mathematics, and Physics.### Cyclic Isomers
• The maximum number of cyclic isomers (positional and optical) of a compound having molecular formula C3H2CI2 is 4.

Ideal Gas

• The volume vs. temperature graph of 1 mole of an ideal gas is given, and the pressure of the gas at X, Y, and Z, respectively, are 0.328, 0.820, and 0.820 atm.

Oxidation of MnO2

• When fused with KOH and oxidized in air, MnO2 gives a dark green compound X.
• In acidic solution, X undergoes disproportionation to give an intense purple compound Y and MnO2.
• Compound X is K2MnO4, and compound Y is KMnO4.

Metal Reaction

• A metal (X) dissolves both in dilute HCl and dilute NaOH to liberate H2.
• Addition of NH4Cl and excess NH4OH to an HCl solution of X produces Y as a precipitate.
• Y is also produced by adding NH4Cl to the NaOH solution of X.
• The species X and Y, respectively, are Al and Al(OH)3.

FIITJEE Answer Keys

• The answer keys for KVPY 2015 are provided for questions 1-110.

KVPY – XII Class - Stream – SB/SX

Part I – Mathematics

• Question 1: Simultaneous equations x + y² = x² + y = 12
  • Number of ordered pairs (x, y) of real numbers that satisfy the equations
• Question 2: Complex number satisfying |z³ + z - 3| ≤ 2
  • Maximum possible value of |z + z⁻¹|
• Question 3: Largest perfect square that divides 20143 - 20133 + ... + 23 - 13
  • Largest perfect square that divides the given expression
• Question 4: Parabola y = x²
  • Coordinates of point C where the parabola intersects the line λ, which is parallel to the line joining the centres of C₁ and C₂
• Question 5: Circle C₁ and C₂ of radii r and R respectively, touch each other
  • Angle ∠AOB where λ is tangent to C₁ at P and intersects C₂ at A and B
• Question 6: Shortest distance from the origin to a variable point on the sphere (x - 2)² + (y - 3)² + (z - 6)² = 1
  • Shortest distance from the origin to the sphere
• Question 7:Equality sin(λα) cos(λα) = λ - 1, holds for all real α which are not integral multiples of π/2
  • Number of real numbers λ for which the equality holds
• Question 8: Hexagon ABCDEF with sides AB = BC = CD = 1 and DE = EF = FA = 2
  • Radius of the circle passing through the vertices A, B, C, D, E, F
• Question 9: Polynomial p(x) such that p(x) - p'(x) = xⁿ, where n is a positive integer
  • Value of p(0)
• Question 10: Limit lim (x → 0) (6/x²) sin x
  • Value of the limit
• Question 11: Maximum area sector
  • Angle at the centre of the sector with maximum area
• Question 12: Function f(x) = max{|x|, |x - 1|, ..., |x - 2n|}
  • Integral ∫ f(x) dx from 0 to 2n
• Question 13: Cubic polynomial p(x) with p(1) = 3, p(0) = 2 and p(-1) = 4
  • Integral ∫ p(x) dx from -1 to 1
• Question 14: Integral ∫ |x - t| dt from 0 to ∞
  • Value of the integral
• Question 15: Urn contains marbles of four colours
  • Smallest total number of marbles satisfying the given condition
• Question 16: Probability that B₁ contains at most one ball
  • Probability expression
• Question 17: Vectors a = 6i - 3j - 6k and d = i + j + k
  • Vector c, where a = b + c, b is parallel to d and c is perpendicular to d
• Question 18: Equality log(3x - 1) = log(9x² - 6x + 2)
  • Value of x
• Question 19: Positive integers a, b, c such that 2a + 4b + 8c = 328
  • Value of (a + 2b + 3c) / abc
• Question 20: Right-angled triangle with integer sides, one of which is 12
  • Largest possible radius of the incircle of such a triangle### Chemistry
• The Lewis acid strength of BBr3, BCI3, and BF3 is in the order of BF3 < BCI3 < BBr3.
• O2- is isoelectronic with Zn2+.
• The bond angles (in degrees) in methane, ammonia, and water are respectively closest to 109.5, 104.5, 107.1.
• In alkaline medium, the reaction of hydrogen peroxide with potassium permanganate produces a compound in which the oxidation state of Mn is +3.
• The rate constant of a chemical reaction at a very high temperature will approach the Arrhenius frequency factor.
• The reducing strength of the metals follows the order of Cr > Pb > Ag > Cu based on their standard reduction potentials.
• The molecule that can exhibit optical activity is 2-bromobutane.
• The structure of the polymer obtained by a reaction is a branched polymer.
• The major product of the reaction between CH3CH2ONa and (CH3)3CCI in ethanol is CH3CH2OC(CH3)3.

Mathematics

• The limit of a function involving the series expansion of cos(θ) is related to the limit of a function involving the series expansion of sin(θ).
• The smallest α for which f(x) ≥ 0 for all x > 0 is 3.
• The graph of y = 2x - 4x^3 has a line y = c such that the areas of the regions marked I and II are equal.
• The probability that 1 ∈ A is greater than the probability that 2 ∈ A, and the difference between the two probabilities is 1/6.
• The remainder when the determinant of a 3x3 matrix is divided by 5 is 1.

Physics

• The moments of inertia of a non-uniform circular disc about four mutually perpendicular tangents are related to the distance of the center of mass of the disc from its geometrical center.
• The stress on a horizontal steel railroad track on a hot summer day is 6.6 × 10^7 Pa.
• Electromagnetic waves emanating from a point A in air are incident on a rectangular block of material M and emerge from the other side.
• The ratio of the masses of two small metal balls connected by strings of equal length to a fixed point is close to 3.0.
• The distance between the vertex and the center of mass of a uniform solid planar circular segment of angular size θ and radius R is given by R sin(θ/2) / θ.

Let me know if you would like me to clarify or expand on any of these points!### Questions and Answers

• The maximum number of cyclic isomers (positional and optical) of a compound having molecular formula C3H2CI2 is 4.

Ideal Gas Volume vs Temperature Graph

• The pressure of the gas at X, Y, and Z, respectively, are 3.28, 8.20, 3.28 atm.

Compound X and Y

• MnO2 when fused with KOH and oxidized in air gives a dark green compound X.
• In acidic solution, X undergoes disproportionation to give an intense purple compound Y and MnO2.
• The compound X and Y, respectively, are K2MnO4 and KMnO4.

Metal X and Y

• Metal X dissolves both in dilute HCI and dilute NaOH to liberate H2.
• Addition of NH4CI and excess NH4O to an HCI solution of X produces Y as a precipitate.
• Y is also produced by adding NH4CI to the NaOH solution of X.
• The species X and Y, respectively, are Zn and Zn(OH)2.

KVPY 2015 Official Answer Keys

• The answer keys for questions 1-110 are provided, with the correct answers labeled A, B, C, or D.

KVPY – XII Class - Stream – SB/SX

Mathematics

• There are 15 questions in the mathematics section, each with 4 options.
• Questions cover various topics, including algebra, geometry, and calculus.
• Question 1 involves solving a system of equations, while question 2 deals with complex numbers.
• Question 3 asks for the largest perfect square that divides a given expression.
• Question 4 involves a parabola and a rectangle, while question 5 deals with circles and tangents.
• Question 6 asks for the shortest distance from the origin to a point on a sphere.
• Question 7 involves trigonometry and equalities.
• Question 8 deals with a hexagon and circles.
• Question 9 involves polynomials and derivatives.
• Question 10 asks for the value of a limit.
• Question 11 deals with the maximum area of a sector.
• Question 12 involves a function and an integral.
• Question 13 asks for the value of an integral.
• Question 14 deals with an exponential function and an integral.
• Question 15 involves probability and marbles.

Physics

• There are 25 questions in the physics section, each with 4 options.
• Questions cover various topics, including mechanics, electromagnetism, and thermodynamics.
• Question 21 involves friction and a box sliding off a table.
• Question 22 deals with nuclear physics and energy release.
• Question 23 involves a composite body and center of mass.
• Question 24 deals with two spherical objects and strings.
• Question 25 involves a disk and angular momentum.
• Question 26 involves a ball thrown upward from a tower.
• Question 27 deals with a diode and current-voltage characteristics.
• Question 28 involves an interference pattern and electrons.
• Question 29 deals with a point charge and a magnetic field.
• Question 30 involves a bottle of water on the moon.
• Question 31 deals with a simple pendulum and its time period.
• Question 32 involves a circuit and an ac voltmeter.
• Question 33 deals with a donor atom in a semiconductor.
• Question 34 involves an ideal gas and isobaric lines.
• Question 35 deals with a metallic ring and a magnetic field.
• Question 36 involves a black hole and its area.
• Question 37 deals with an infrared source and photon flux.
• Question 38 involves a wire bent in a polygonal shape and a magnetic field.
• Question 39 deals with sound intensity and decibel levels.
• Question 40 involves a wire, a block, and a rigid wall.### Chemistry Questions
• Question 41: The Lewis acid strength of BBr3, BCI3, and BF3 is in the order of BF3 < BCI3 < BBr3.
• Question 42: O2– is isoelectronic with Mg2+.
• Question 43: The H-C-H, H-N-H, and H-O-H bond angles in methane, ammonia, and water are closest to 109.5°, 107.1°, and 104.5°, respectively.
• Question 44: In alkaline medium, the reaction of hydrogen peroxide with potassium permanganate produces a compound in which the oxidation state of Mn is +3.

Optics and Magnetism

• Question 91: A cubical vessel has opaque walls. An observer can see only the wall CD but not the bottom. Nearly to what height should water be poured so that she can see an object placed at the bottom at a distance of 10 cm from the corner C?

Thermodynamics

• Question 88: The figure shows a portion of the graph y = 2x – 4x3. The line y = c is such that the areas of the regions marked I and II are equal. If a, b are the x-coordinates of A, B respectively, then a + b equals 3/7.

Physics

• Question 92: The moments of inertia of a non-uniform circular disc (of mass M and radius R) about four mutually perpendicular tangents AB, BC, CD, DA are I1, I2, I3, and I4, respectively.
• Question 93: A horizontal steel railroad track has a length of 100 m when the temperature is 25°C. The track is constrained from expanding or bending. The stress on the track on a hot summer day, when the temperature is 40°C, is 6.6 × 10^7 Pa.

Mathematics

• Question 81: Let x = (50 + 7)^(-1/3). Then x = 2.
• Question 82: Let 1 + x + x2 = a0 + a1x + a2x2 + ... + a4028x4028. Then A = a0 - a3 + a6 - ... + a4026, B = a1 - a4 + a7 - ... - a4027, C = a2 - a5 + a8 - ... + a4028.### Molecular Formula and Isomers
• The maximum number of cyclic isomers (positional and optical) of a compound having molecular formula C3H2CI2 is 4.

Ideal Gas and Pressure

• The volume vs. temperature graph of 1 mole of an ideal gas is given, with points X, Y, and Z.
• The pressure of the gas at X, Y, and Z, respectively, are 0.328 atm, 0.820 atm, and 0.820 atm.

Manganese Compounds

• MnO2 when fused with KOH and oxidized in air gives a dark green compound X.
• In acidic solution, X undergoes disproportionation to give an intense purple compound Y and MnO2.
• The compound X is K2MnO4 and the compound Y is KMnO4.

Metal Reactivity

• A metal (X) dissolves both in dilute HCl and dilute NaOH to liberate H2.
• Addition of NH4CI and excess NH4O to an HCl solution of X produces Y as a precipitate.
• Y is also produced by adding NH4CI to the NaOH solution of X.
• The species X is Al and the species Y is Al(OH)3.

KVPY 2015 Official Answer Keys

• The answer keys for the KVPY 2015 examination are provided, with answers for questions 1-110.