KVPY SB/SX Mock...

Let p(x) = a₀ + a₁ x + ….... + a_n x⁽ⁿ⁾ be a non-zero polynomial with integer coefficients. If p() = 0, then the smallest possible value of n is

If cosec²(α + β) - sin²(β - α) + sin²(2α - β) = cos²(α - β) where α, β ∈ (0, π/2), then sin (α - β) is equal to

Ten trucks numbered 1 to 10, are carrying packets of sugar. Each packet weighs either 999 gm or 1000 gm and each truck carries only the packets of equal weight. The combined weight of 1 packet selected from the first truck, 2 packets from the second, 4 packets from the third, and so on, and 2⁹ packets from the tenth truck is 1,022,870 gm. The trucks that have the lighter bags are

An envelope has space for at most 3 stamps. If you are given three stamps of denomination 1 and 3 of denomination a, (a > 1), the least positive integer for which there is no stamp value is

The maximum value M of 3^x + 5^x – 9^x + 15^x – 25^x, as x varies over reals, satisfies

The value of tan 81° – tan 63° – tan 27° + tan 9° is

Find the value of the following function:

The number of ordered pairs (m, n), where m, n ∈ {1, 2, 3, ......, 50}, such that 6^m + 9ⁿ is a multiple of 5 is

The number of rectangles that can be obtained by joining four of the twelve vertices of a 12-sided regular polygon is:

The number of roots of the equation cos⁷θ - sin⁴θ = 1 that lie in the interval [0, 2π] is:

The following figure shows the graph of a continuous function y = f(x) on the interval [1, 3]. The points A, B and C have coordinates (1, 1), (3, 2) and (2, 3), respectively, and the lines L₁ and L₂ are parallel, with L₁ being tangent to the curve at C. If the area under the graph of y = f(x) from x = 1 to x = 3 is 4 square units, then the area of the shaded region is:

How many six-digit numbers are there in which no digit is repeated, even digits appear at even places, odd digits appear at odd places and the number is divisible by 4?

Let P be an m × m matrix such that P² = P. Then, (I + P)ⁿ equals

In the xy-plane, three distinct lines l₁, l₂, l₃ concur at a point (λ , 0). Further, the lines l₁, l₂, l₃ are normal to the parabola y² = 6x at the points A = (x₁, y₁), B = (x₂, y₂) and C = (x₃, y₃), respectively. Then, we have

The minimum distance between a point on the curve y = e^x and a point on the curve y = log_ex is

Let be three vectors in the xyz space such that = = ≠ 0. If A, B and C are points with position vectors respectively, then the number of possible positions of the centroid of triangle ABC is

Let ω be a cube root of unity not equal to 1. Then, the maximum possible value of |a + bω + cω²|, where a, b, c ∈ {+1, -1}, is

The angles , and of a triangle satisfy the equations 2 sin + 3 cos = 3 and 3 sin + 2 cos = 1. Then, angle equals?

For a real number x, let [x] denote the largest integer less than or equal to x and {x} = x - [x]. The smallest possible integer value of n for which exceeds 2013 is

The number of integers n with 100 ≤ n ≤ 999 and containing at most two distinct digits is

A parallel plate capacitor is charged fully by using a battery. Without disconnecting the battery, the plates are moved further apart. Then, the

An ideal diatomic gas is heated at constant pressure. The ratio of the work done to the heat supplied is

A plane electromagnetic wave propagating in the direction of the unit vector with a speed c is described by electric and magnetic field vectors and , respectively. Which of the following relations (in SI units) between and can be ruled out on dimensional grounds alone?

A block of mass m is stationary on a rough plane of mass M inclined at an angle to the horizontal while the whole set up is accelerating upwards at an acceleration a. If the coefficient of friction between the block and the plane is μ, then the force that the plane exerts on the block is

A liquid drop placed on a horizontal plane has a near spherical shape (slightly flattened due to gravity). Let R be the radius of its largest horizontal section. A small disturbance causes the drop to vibrate with frequency v about its equilibrium shape. By dimensional analysis, the ratio can be: (Here, σ is surface tension and ρ is density.)

A conducting rod of mass m and length l is free to move without friction on two parallel long conducting rails, as shown below. There is a resistance R across the the rails. In the entire space around, there is a uniform magnetic field B normal to the plane of the rod and rails. The rod is given an impulsive velocity V₀.


Finally, the initial energy

Two identical bodies are made of a material for which the heat capacity increases with temperature. One of these is held at a temperature of 100°C, while the other one is kept at 0°C. If the two are brought into contact, then, assuming no heat loss to the environment, the final temperature that they will reach is

A cylindrical steel rod of length 0.10 m and thermal conductivity 50 W. m^-1.K^-1 is welded end to end to copper rod of thermal conductivity 400 W.m^-1.K^-1 and of the same area of cross section but 0.20 m long. The free end of the steel rod is maintained at 100°C and that of the copper rod at 0°C. Assuming that the rods are perfectly insulated from the surroundings, the temperature at the junction of the two rods is

A bead of mass m is attached to the mid-point of a taut, weightless string of length l and is placed on a frictionless horizontal table.


Under a small transverse displacement x, as shown, if the tension in the string is T, then the frequency of oscillation is

The following travelling electromagnetic wave E_x = 0, E_y = E₀sin(kx + ωt), E_z = 2E₀sin(kx - ωt) is

A van der Waals gas obeys the equation of state (V - nb) = nRT. Its internal energy is given by U = CT - . The equation of a quasi-static adiabat for this gas is given by:

Two masses m₁ and m₂ are connected by a massless spring of spring constant k and unstretched length l. The masses are placed on a frictionless straight channel which we consider our x-axis. They are initially at rest at x = 0 and x = l, respectively. At t = 0, a velocity of v₀ is suddenly imparted to the first particle. At a later time t₀, the centre of mass of the two masses is at:

If the speed (v) of the bob in a simple pendulum is plotted against the tangential acceleration (a), then the correct graph will be represented by

A speaker emits a sound wave of frequency f₀. When it moves towards a stationary observer with speed u, the observer measures a frequency f₁. If the speaker is stationary and the observer moves towards it with speed u, the measured frequency is f₂. Then,

In the circuit shown below, the resistances are given in ohms and the battery is assumed ideal with emf equal to 3.0 volts. The resistor that dissipates the most power is

The total radiative power emitted by a spherical blackbody with radius R and temperature T is P. If the radius is doubled and the temperature is halved, then the radiative power will be

A nuclear decay is possible if the mass of the parent nucleus exceeds the total mass of the decay particles.

If M(A, Z) denotes the mass of a single neutral atom of an element with mass number A and atomic number Z, then the minimal condition that the β-decay will occur is (m_e denotes the mass of the b-particle and the neutrino mass m_v can be neglected):

An arrangement with a pair of quarter circular coils of radii r and R with a common centre C and carrying a current I is shown.

The permeability of free space is . The magnetic field at C is

Photons of energy 7 eV are incident on two metals A and B with work functions 6 eV and 3 eV, respectively. The minimum de Broglie wavelengths of the emitted photoelectrons with maximum energies respectively are _A and _B, where _A/_B is nearly

The average distance between molecules of an ideal gas at STP is approximately of the order of

The IUPAC name for the given compound is

Which of the following has the highest bond angle?

The equilibrium constant for the reaction N₂ + 3H₂ 2NH₃ at 400 K is 41. The equilibrium constant for the reaction N₂ + H₂ NH₃ at the same temperature will be close to

The newman projection of



is known as the

For a one-electron atom, the set of allowed quantum numbers is:

A zero order reaction, A Product, with an initial concentration [A]_o has a half life of 0.2 s. If one starts with the concentration 2[A]_o, then the half life is

The pKa of a weak acid is 5.85. The concentrations of the acid and its conjugate base are equal at a pH of

The equilibrium constant for the following reactions are K₁ and K₂, respectively.
2P(g) + 3Cl2(g) 2PCl3(g)
PCl3(g) + Cl2(g) PCl5(g)

Then, the equilibrium constant for the reaction given below is:
2P(g) + 5Cl2(g) 2PCl5(g)

The bond order of is

The order of acidity of compounds I-IV is:

In metallic solids, the number of atoms for the face centred and the body centred cubic unit cells, respectively are:

Consider the reaction:

2 NO₂(g) 2 NO(g) + O₂ (g).

In the given figure, identify the curves X, Y and Z associated with the three species in the reaction:

Assuming ideal behaviour, the enthalpy and volume of mixing of two liquids, respectively are

For a process to occur spontaneously,

To obtain a diffraction peak, for a crystalline solid with interplane distance equal to the wavelength of incident X-ray radiation, the angle of incidence should be

The adsorption isotherm for a gas is given by the relation x = ap/(1 + bp), where x is the number of moles of gas adsorbed per gram of the adsorbent, p is the pressure of the gas and a and b are constants. Then, x

The compound which reacts with excess bromine to produce 2,4,6-tribromophenol is:

The compound that readily tautomerizes is:

In structure of Borax, the number of boron atoms and B-O-B units respectively are

The spin-only magnetic moments of [Fe(NH₃)₆]³⁺ and [FeF₆]^3- in BM respectively are

The function of enzymes is to

Which of the following diseases is not sexually transmitted?

Sickle cell anaemia is caused by

Mitochondria are not associated with which of the following functions?

Diabetes insipidus is due to

A ripe mango, kept with unripe mangoes, causes their ripening. This is due to the release of a gaseous plant hormone called

A hormone molecule binds to a specific protein on the plasma membrane inducing a signal. The protein that binds to it is called a/an

If the sequence of bases in sense strand of DNA is 5'-GTTCATCG-3', then the sequence of the bases in its RNA transcript would be

Bacteria can survive by absorbing soluble nutrients via their outer body surface, but animals cannot, because

If you compare adults of two herbivore species of different sizes, but from the same geographical area, the amount of faeces produced per kg body weight would be

A couple went to a doctor and reported that both of them were "carriers" of a particular disorder, their first child was suffering from that disorder and that they were expecting their second child. What is the probability that the new child would be affected by the same disorder?

The fluid part of blood flows in and out of capillaries in tissue to exchange nutrients and waste materials. Under which of the following conditions will fluid flow out from the capillaries into the surrounding tissue?

In humans, the composition of a zygote that will develop into a female is

In which of the following sets of fractions will you find nucleic acids if you fractionate all the organelles from the cytoplasm of a plant cell?

Circadian rhythm is an endogenously driven cycle for biochemical, physiological and behavioural processes. In humans, the approximate duration of this 'biological clock' is

As proposed by Watson and Crick in 1953, the original model of DNA is a

The sequences of four DNA molecules are given below.



Which of these DNA molecules will have the highest melting temperature (Tm)?

In green leaves, the light and dark reactions occur in

Which of the following organs is/are capable of producing fructose in humans?

Species that are most effective at colonising new habitats show

The area of the region bounded by y = ||x - 3| - 4| - 5 and the x-axis is

Two points are randomly chosen on the circumference of a circle of radius r. The probability that the distance between the two points is at least r is equal to

Let (x, y) be a variable point on the curve 4x² + 9y² - 8x - 36y + 15 = 0. Then, min (x² - 2x + y² - 4y + 5) + max (x² - 2x + y² - 4y + 5) is

Consider

L = + +......+

R = + +......+

and l = dx

Suppose a, b are real numbers such that ab ≠ 0. Which of the following four figures represents the curve (y - ax - b)(bx² + ay² - ab) = 0?

The value of min {|x - π|, cos^-1(cos x)} dx min is

The locus of the point P = (a, b), where a and b are real numbers such that the roots of x³ + ax² + bx + a = 0 are in arithmetic progression is a/an

For each positive integer n, define f_n(x) = minimum for 0 ≤ x ≤ 1. Let I_n = f_n (x)dx, n ≥ 1. Then, I_n = I_n is equal to

The vertices of the base of an isosceles triangle lie on a parabola y² = 4x and the base is a part of the line y = 2x – 4. If the third vertex of the triangle lies on the x-axis, its coordinates are

Let n be a positive integer. For a real number x, let [x] denote the largest integer not exceeding x and {x} = x – [x]. Then, dx is equal to

n moles of a van der Waals gas obeying the equation of state (P + )(V - nb) = nRT, where a and b are gas dependent constants, is made to undergo a cyclic progress that is depicted by a rectangle in the P-V diagram as shown in the figure. What is the hear absorbed by the gas in one cycle?

Two small identical speakers are connected in phase to the same source. The speakers are 3 m apart and at ear level. An observer stands at P, 4 m in front of one speaker as shown. The sound she hears is least intense when the wavelength is and most intense when the wavelength is .



Then, possible values of and are:

A radioactive nucleus A has a single decay mode with half-life τ_A. Another radioactive nucleus B has two decay modes 1 and 2. If decay mode 2 were absent, the half-life of B would have been τ_A/2. If decay mode 1 were absent, the half-life of B would have been 3T_A. If the actual half-life of B is τ_B, then the ratiois

A standing wave in a pipe with a length L = 1.2 m is described as

y(x, t) = y₀ sin sin

Based on the above information, which one of the following statements is incorrect?
(Speed of sound in air is 300 ms^-1)

Three equal charges +q are placed at the three vertices of an equilateral triangle centred at the origin. They are held in equilibrium by a restoring force of magnitude F(r) = kr directed towards the origin, where k is a constant. What is the distance of the three charges from the origin?

An ideal gas with constant heat capacity C_v = nR is made to carry out a cycle that is depicted by a triangle in the figure given below:



The following statement is true about the cycle:

Figure given below shows a small mass connected to a string, which is attached to a vertical post. If the ball is released when the string is horizontal as shown, the magnitude of the total acceleration (including radial and tangential) of the mass as a function of the angle θ is

A rectangular loop of wire shown below is coplanar with a long wire carrying current I.

The loop is pulled to the right as indicated. What are the directions of the induced current in the loop and the magnetic forces on the left and the right sides of the loop?

In the P-V diagram below, the dashed curved line is an adiabat.



For a process that is described by a straight line joining two points X and Y on the adiabat (solid line in the diagram), heat is
(Hint: Consider the variations in temperature from X to Y along the straight line)

The three processes in a thermodynamic cycle shown in the figure are : Process 1 2 is isothermal, Process 2 3 is isochoric (volume remains constant) and Process 3 1 is adiabatic. The total work done by the ideal gas in this cycle is 10 J. The internal energy decreases by 20 J in the isochoric process. The work done by the gas in the adiabatic process is –20 J. The heat added to the system in the isothermal process is

The addition of 0.643 g of a compound to 50 ml of benzene (density = 0.879 g ml^-1) lowers the freezing point from 5.51°C to 5.03°C. If the freezing point constant (K_f) for benzene is 5.12 K kg mol^-1, then the molar mass of the compound is approximately

In the above sequence of reactions, the major products X and Y are

(R)-2-bromobutane upon treatment with aq. NaOH gives

Among the following, the species that is both tetrahedral and diamagnetic is

The numbers of possible enatiomeric pairs from the bromination of I and II, respectively, are

Diborane is formed by the elements as shown in equation (1):

2B(s) + 3H₂(g) B₂H₆(g) ----------(1)

Given that:



The ΔH° for the reaction (1) is

If = -0.440 V and = 0.770 V, then is

At 300 K, the vapour pressure of two pure liquids, A and B is 100 and 500 mm Hg, respectively. If in a mixture of A and B, the vapour pressure is 300 mm Hg, then the mole fractions of A in the liquid and in the vapour phase, respectively are

The compound X (C₇H₉N) reacts with benzensulphonyl chloride to give Y (C₁₃H₁₃NO₂S), which is insoluble inalkali. The compound X is

The density and equivalent weight of a metal are 10.5 g cm^–3 and 100, respectively. The time required for a current of 3 amp to deposit a 0.005 mm thick layer of the same metal on an area of 80 cm² is closest to

Instead of 3, if 2 bases coded for an amino acid, degeneracy of codons coding for the same amino acid would have

The condition varicose veins is swelling of veins, that occurs due to

Insectivorous plants that mostly grow on swampy soil use insects as a source of

When a person begins to fast, after some time, glycogen stored in the liver is mobilised as a source of glucose. Which of the following graphs best represents the change of glucose level (y-axis) in his blood, starting from the time (x-axis) when he begins to fast?

Two students are given two different double-stranded DNA molecules of equal lengths. They are asked to denature the DNA molecule by heating. The DNA given to student A has the following composition of bases (A:G:T:C:35:15:35:15) while that given to student B is (A:G:T:C:12:38:12:38). Which of the following statements is true?

Consider a locus with two alleles, A and a. If the frequency of AA is 0.25, then what will be the frequency of A under Hardy-Weinberg equilibrium?

Rodents can distinguish between many different types of odours. The basis for odour discrimination is that

For a human male, what is the probability that all the maternal chromosomes will end up in the same gamete?

Eco RI and Rsa I restriction endonucleases require 6 bp and 4 bp sequences, respectively for cleavage. In a 10 kb DNA fragment, how many probable cleavage sites are present for these enzymes.

Brown fat is specialised adipose tissue with abundant mitochondria and rich blood supply. Brown fat