KVPY SB/SX Mock...

Three players play a total of 9 games. In each game, one person wins and the other two lose; the winner gets 2 points and the losers get -1 each. The number of ways in which they can play all the 9 games and finish each with a zero score is

If sin x + sin y = 7/5 and cos x + cos y = 1/5, then sin (x + y) equals

What is the value of? (Here, [t] denotes the integral part of the real number t.)

If m₁ and n are positive integers such that m n
(here dk means d is a positive divisor of k), then

Suppose two perpendicular tangents can be drawn from the origin to the circle x² + y² – 6x – 2py + 17 = 10, for some real p. Then |p| is equal to

The mid-point of the domain of the function f(x) = for real x is

Let [x] and {x} be the integer part and fractional part of a real number x, respectively. The value of the integralis

Suppose a₁, a₂, a₃, ....., a₂₀₁₂ are integers arranged in a circle. Each number is equal to the average of its two adjacent numbers. If the sum of all even indexed numbers is 3018, then what is the sum of all the numbers?

Let I, ω and ω²be the cube roots of unity. The least possible degree of a polynomial, with real coefficients, having 2ω², 3 + 4ω, 3 + 4ω²and 5 - ω - ω²as roots is:

The product (1 + tan1°)(1 + tan2°)(1 + tan3°)...(1 + tan45°) equals:

Let I_n = , where n is a non-negative integer. Then, I₂₀₁₁ + 2011 I₂₀₁₀ is equal to:

The number of natural numbers n in the interval [1005, 2010] for which the polynomial 1 + x + x² + x³ + ... + x^{n - 1} divides the polynomial 1 + x² + x³ + x⁴ + ... + x²⁰¹⁰ is:

Consider the cubic equation x³ + ax² + bx + c = 0, where a, b and c are real numbers. Which of the following statements is correct?

Let f (x) = cos 5x + A cos 4x + B cos 3x + C cos 2x + D cos x + E and T = f(0) - f() + f() - f() +..........+ f() - f(). Then, T

Let f: (2, ) N be defined by f(x) = The largest prime factor of [x]. Then, is equal to

The sum of (1² – 1 + 1)(1!) + (2² – 2 + 1)(2!) + …. + (n² – n + 1)(n!) is

If a and b are positive real numbers such that the lines ax + 9y = 5 and 4x + by = 3 are parallel, then the least possible value of a + b is

Let f: R R be a function such that f(x) = M > 0. Then, which of the following is false?

The area bounded by the curve y = cos x, the line joining and (0, 2), and the line joining and (0, 2) is:

For an integer n, let Sn = {n + 1, n + 2, ... n + 18}. Which of the following is true for all n 10?

The five sides of a regular pentagon are represented by vectors and , in cyclic order as shown. Corresponding vertices are represented by and , drawn from the centre of the pentagon.

Then, = ?

^{_{In the hydrogen spectrum, the ratio of the wavelength for Lyman -radiation to Balmer- radiation is}}

A point electric dipole placed at the origin has a potential given by , where θ is the angle made by the position vector with the direction of dipole. Then,

A stream of charged particles enters into the region with crossed electric and magnetic fields as shown in the figure. On the other side is a screen with a hole, that is right on the original path of the particles.

Then,

Seven identical coins are rigidly arranged on a flat table in the pattern shown below so that each coin touches its neighbours. Each coin is a thin disc of mass m and radius r. Note that the moment of inertia of an individual coin about an axis passing through centre and perpendicular to the plane of the coin is .


The moment of inertia of the system of seven coins about an axis that passes through the point P (the centre of the coin positioned directly to the right of the central coin) and perpendicular to the plane of the coins is

A steady current I flows through a wire of radius r, length L and resistivity . The current produces heat in the wire. The rate of heat loss in a wire is proportional to its surface area. The steady temperature of the wire is independent of

A particle is acted upon by a force given by F = -αx³ - βx⁴, where α and β are positive constants. At the point x = 0, the particle is

A parent nucleus X is decaying into daughter nucleus Y, which in turn decays to Z. The half lives of X and Y are 4000 years and 20 years, respectively. In a certain sample, it is found that the number of Y nuclei hardly changes with time. If the number of X nuclei in the sample is 4 × 10²⁰, the number of Y nuclei present in it is

A comet (assumed to be in an elliptical orbit around the sun) is at a distance of 0.4 AU from the sun at the perihelion. If the time period of the comet is 125 years, what is the arphelion distance? AU : Astronomical Unit

A point source of light is placed at the bottom of a vessel which is filled with water of refractive index μ to a height h. If a floating opaque disc has to be placed exactly above it so that the source is invisible from above, the radius of the disc should be

The flat face of a plano-convex lens of focal length 10 cm is silvered. A point source placed 30 cm in front of the curved surface will produce a

A charged particle of charge q and mass m gets deflected through an angle θ upon passing through a square region of side 'a' which contains a uniform magnetic field B normal to its plane. Assuming that the particle entered the square at right angles to one side, what is the speed of the particle?

A container with rigid walls is covered with perfectly insulating material. The container is divided into two parts by a partition. One part contains a gas while the other is fully evacuated (vacuum). The partition is suddenly removed. The gas rushes to fill the entire volume and comes to equilibrium after a little time. If the gas is not ideal, the initial

A plane polarised light passed through successive polarisers, which are rotated by 30° with respect to each other in the clockwise direction. Neglecting absorption by the polarisers and given that the first polariser's axis is parallel to the plane of polarisation of the incident light, the intensity of light at the exit of the fifth polariser is closest to

An electron collides with free molecules initially in its ground state. The collision leaves the molecules in an excited state that is metastable and does not decay to the ground state by radiation. Let K be the sum of the initial kinetic energies of the electron and the molecule and vector P the sum of their initial momenta. Let K' and vector P' represent the same physical quantities after the collision. Then,

Quantum Hall Resistance R_H is a fundamental constant with dimensions of resistance. If h is the Planck's constant and e is the electron charge, then the dimension of R_H is the same as

The equation of state of n moles of a non-ideal gas can be approximated by the equation , where a and b are constants characteristic of the gas.

Which of the following can represent the equation of a quasistatic adiabatic equation for this gas (Assume that C_V, the molar heat capacity at constant volume, is independent of temperature)?

The circuit shown has been connected for a long time. The voltage across the capacitor is

An electron enters a chamber in which a uniform magnetic field is present as shown. Ignore gravity.



During its motion inside the chamber,

A point particle of mass 0.5 kg is moving along the x-axis under a force described by the potential energy V shown below. It is projected towards the right from the origin with a speed v.



What is the minimum value of v for which the particle will escape infinitely far away from the origin?

In the following reactions

,

the major product X is

The solvent used for carrying out a Grignard reaction is

In one component second order reaction, if the concentration of the reactant is reduced to half, then the rate of the reaction

The half-life of a first order reaction is 30 min. The time required for 75% completion of the same reaction is

In the reaction of benzene with an electrophile E⁺, the structure of the intermediate σ-complex can be represented as

The isoelectronic pair of ions is:

For a tetrahedral complex [MCl4]^2-, the spin-only magnetic moment is 3.83 BM. The element M is:

The major product of the following reaction is:

The energy of a photon of wavelength λ = 1 metre is: (Planck's constant = 6.625 × 10^-34 Js, speed of light = 3 × 10⁸ m/s)

The most stable conformation of n-butane is:

From equations 1 and 2:

CO₂ CO + O₂ [K₁ = 9.1 × 10^-13 at 1000°C] (eq. 1)

H₂O H₂ + O₂ [K₂ = 7.1 × 10^-12 at 1000°C] (eq. 2),

The equilibrium constant for the reaction CO₂ + H₂ CO + H₂O at the same temperature is:

The aromatic carbocation among the following is:

At 298 K, the ratio of osmotic pressures of two solutions of a substance with concentrations of 0.01 M and 0.001 M, respectively is

When the size of a spherical nanoparticle decreases from 30 nm to 10 nm, the surface area/volume ratio becomes

The standard Gibbs free energy change (ΔG° in kJ mol^-1) in a Daniel cell (E°_cell = 1.1 V), when 2 moles of Zn(s) is oxidised at 298 K, is closest to

The reaction

is known as

Ethyl acetate reacts with NH₂NHCONH₂ to form

Hydrolysis of BCl₃ gives 'X', which on treatment with sodium carbonate produces 'Y'. 'X' and 'Y' respectively are

The number of peptide bonds in the compound is:

The order of S_N1 reactivity in aqueous acetic acid solution for the given compounds is:

Glycolysis is the

This cell organelle consists of two granule-like centrioles and is found in animal cells only. It helps in cell division. It is called

The natural source of Ti plasmid is

The probability of having a girl child with blood group O when parents have blood group A and B is

In which kind of rocks are fossils most commonly found?

Human chromosomes undergo structural changes during the cell cycle. Chromosomal structure can be best visualised if a chromosome is isolated from a cell at

DNA mutations that do not cause any functional change in the protein product are known as

A reflex action is a quick involuntary response to stimulus. Which of the following is an example of BOTH, unconditioned and conditioned reflex?

A horse has 64 chromosomes and a donkey has 62. Mules result from crossing a horse and a donkey. State which of the following is INCORRECT.

Fruit wrapped in paper ripens faster than when kept in open air because

Out of the following combinations of cell biological processes, which one is associated with embryogenesis?

The distance between two consecutive DNA base pairs is 0.34 nm. If the length of a chromosome is 1 mm, the number of base pairs in the chromosome is approximately

If the sequence of bases in DNA is 5'-ATGTATCTCAAT-3', then the sequence of bases in its transcript will be

Carbon dioxide in the blood is mostly carried

Modern evolutionary theory consists of the concepts of Darwin modified by knowledge concerning

Crossing over occurs at which of the following stages of meiosis I?

If DNA codons are ATG GAA, then insertion of thymine after the first codon results in

According to Mendel, ______________ segregate and _______________ assort independently.

Stroke could be prevented/treated with

In a large isolated population, alleles p and q at a locus are at Hardy-Weinberg equilibrium. The frequencies are p = 0.6 and q = 0.4. The proportion of the heterogeneous genotype in the population is

The lengths of the sides and the diagonal of an isosceles trapezium form a two-element set {a, b}. If a > b, then a/b equals

Consider all the natural numbers whose decimal expansion has only the even digits 0, 2, 4, 6, 8. Suppose these are arranged in an increasing order. If an denotes the n-th number in this sequence, then limn ➝ ∞ (log a_n / log n) equals

The sum of all x ∈ [0, π], which satisfy the equation sin x + cos x = sin² (x + ) is

A man tosses a coin 10 times, scoring 1 point for each heads and 2 points for each tails. Let P(K) be the probability of scoring at least K points. The largest value of K such that P(K) > is

The following figure shows the graph of a differentiable function y = f(x) on the interval [a, b] (not containing 0)



Let g(X) = f(x)/x, which of the following is a possible graph of y = g(x)?

Let ABC be a triangle and P be a point inside ABC such that . The ratio of the area of triangle ABC to that of APC is

The smallest possible positive slope of a line whose y-intercept is 5 and which has a common point with the ellipse 9x² + 16y² = 144 is

The maximum possible value of x² + y² - 4x - 6y, where x and y are real and are subject to the condition |x + y| + |x - y| = 4 is

In a triangle ABC, let G denote its centroid and let M and N be points in the interiors of the segments AB and AC, respectively such that M, G and N are collinear. If r denotes the ratio of the area of triangle AMN to the area of ABC, then

A box contains coupons labelled 1, 2, ..., 100. Five coupons are picked at random, one after another without replacement. Let the numbers on the coupons be x₁, x₂, ..., x₅. What is the probability that x₁ > x₂ > x₃ and x₃ < x₄ < x₅?

For what value of the resistor X, will the equivalent resistance of the two circuits shown be the same?

Two small blocks slide without losing contact with the surface along two frictionless tracks 1 and 2, starting at the same initial speed v. Track 1 is perfectly horizontal, while track 2 has a dip in the middle, as shown.



Which block reaches the finish line first?

[Hint: Use velocity time graph to solve]

A stream of photons having energy 3 eV each impinges on a potassium surface. The work function of potassium is 2.3 eV. The emerging photo-electrons are slowed down by a copper plate placed 5 mm away. If the potential difference between the two metal plates is 1 V, then the maximum distance the electrons can move away from the potassium surface before being turned back is

Two blocks (1 and 2) of equal mass m are connected by an ideal string (see figure below) over a frictionless pulley. The blocks are attached to the ground by springs having spring constants k₁ and k₂, such that k₁ > k₂.



Initially, both the springs are unstretched. The block 1 is slowly pulled down a distance x and released. Just after the release, the possible value of the magnitudes of the acceleration of the blocks a₁ and a₂ can be:

Consider the infinite ladder circuit shown below:



For what angular frequency 'ω' will the circuit behave like a pure inductance?

Two identical particles of mass m and charge q are shot at each other from a very great distance with an initial speed v. The distance of closest approach of these charges is

One mole of an ideal gas at initial temperature T undergoes a quasi-static process during which the volume V is doubled. During the process, the internal energy U obeys the equation U = aV³, where a is a constant. The work done during this process is

Two batteries V₁ and V₂ are connected to three resistors as shown below. If V₁ = 2 V and V₂ = 0 V, then the current I = 3 mA. If V₁ = 0 V and V₂ = 4 V, current I = 4 mA. Now, if V₁ = 10 V and V₂ = 10 V, then the current I will be

A singly ionised helium atom in an excited state (n = 4) emits a photon of energy 2.6 eV. Given that the ground state energy of hydrogen atom is –13.6 eV, the energy (E_t) and quantum number (n) of the resulting state are respectively,

A block of mass m slides from rest at a height H on a frictionless inclined plane as shown in the figure. It travels a distance d across a rough horizontal surface with coefficient of kinetic friction μ and compresses a spring k by a distance x before coming to rest momentarily. Then, the spring extends and the block travels back, attaining a final height h. Then,

Consider the following electrochemical cell:

Zn(s) + 2Ag⁺ (0.04 M) Zn²⁺ (0.28 M) + 2Ag(s)

If E°_cell = 2.57 V, then e.m.f of the cell at 298 K is

²³⁴Th₉₀ gets converted to ²⁰⁶Pb₈₂ through a series of radioactive decay processes. The number of alpha and beta particles lost in this transformation, respectively, are

Phenol on treatment with dil. HNO₃ gives two products P and Q. P is steam volatile but Q is not. P and Q are respectively

Three moles of an ideal gas expand reversibly under isothermal conditions from 2 L to 20 L at 300 K. The amount of heat-change (in kJ/mol) in the process is

For a reaction A B, ΔH° = 7.5 mol^-1 and ΔS° = 2.5 J mol^-1. The value of ΔG° and the temperature at which the reaction reaches equilibrium are respectively

The Crystal Field Stabilisation Energy (CPSE) and the spin-only magnetic moment in Bohr Magneton (BM) for the complex K₃[Fe(CN)₆] respectively are

The electron in hydrogen atom is in the first bohr orbit (n = 1). The ratio of transition energies, E(n = 1 n = 3) to E(n = 1 n = 2) is

The crystal field stabilisation energies(CFSE) of high spin and low spin d⁶ metal complexes in terms of Δ_o, respectively are

In 108 g of water, 18 g of a non-volatile compound is dissolved. At 100°C, the vapour pressure of the solution is 750 mm Hg. Assuming that the compound does not undergo association or dissociation, the molar mass of the compound in g mol^-1 is

The amount of Na₂S₂O₃.5H₂O required to completely reduce 100 mL of 0.25 N iodine solution is

Gregor Mendel showed that unit factors exist in pairs and exhibit a dominant-recessive relationship. These unit factors, in modern terminology, are called

Greatest proportion of photosynthesis in the world is carried out by

In cattle, the coat colours red and white are two dominant traits which express equally in F1 to produce roan (red and white colours in equal proportion). If F1 progeny are self-bred, the resulting progeny in F2 will have phenotypic ratio (red : roan : white)

The following sequence contains the open reading frame of polypeptide. How many amino acids will the polypeptide consist of?

5' AGCATATGATCTCGTTTCTGCTTTGAACT-3'

The amino acid sequences of a bacterial protein and a human protein carrying out similar functions are found to be 60% identical. However, the DNA sequences of the genes coding for these proteins are only 45% identical. This is possible because

Which of the following graphs accurately represents the insulin levels (Y-axis) in a body as a function of time (X-axis) after eating sugar and bread/roti?

Although blood flows through large arteries at high pressure, when the blood reaches small capillaries the pressure decreases because

Nocturnal animals have retinas that contain

From an early amphibian embryo, the cells that would give rise to skin in adults were transplanted into the developing brain region of another embryo. The transplanted cells developed into brain tissue in the recipient embryo. What do you infer from this experiment?

In some species, individuals forego reproduction and help bring up another individual's offspring. Such altruistic behaviour cannot be explained by which of the following?