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The positive integer k for which is a maximum is

Let the line 2x + 3y = 18 intersect the y-axis at B. Suppose C (. ≠ B) with coordinates (a, b) is a point on the line such that PB = PC, where P = (10, 10). Then, 8a + 2b equals

The shortest distance from (0, 3) to the parabola y² - 4x is

A vector which bisects the angle between and is

In the real number system, the equation has

Let f(x) = , then

The value is

Let H be the orthocentre of an acute-angled triangle ABC and O be its circumcentre. Then,

Suppose the sides of a triangle form a geometric progression with common ratio r. Then, r lies in the interval of:

Let P be an arbitrary point on the ellipse = 1, a > b > 0. Suppose F₁ and F₂ are the foci of the ellipse. The locus of the centroid of the triangle PF₁F₂ as P moves on the ellipse is

The figure shown below is the graph of the derivative of some function y = f'(x):



Then,

Let and be vectors in R³ and be a unit vector in the xy-plane. Then, the maximum possible value of || is:

Suppose n is a natural number such that = 18, where i is the square root of -1. Then, n is

In a rectangle ABCD, the coordinates of A and B are (1, 2) and (3, 6), respectively and some diameter of the circumscribing circle of ABCD has equation 2x – y + 4 = 0. Then, the area of rectangle is

If f(x) = (2011 + x)ⁿ, where x is a real variable and n is a positive integer, then the value of f(0) + f'(0) + + ........ + is

Two players play the following game: A writes 3, 5 and 6 on three different cards, B writes 8, 9 and 10 on three different cards. Both draw randomly two cards from their collections. Then, A computes the product of two numbers he/she has drawn and B computes the sum of two numbers he/she has drawn. The player getting the larger number wins. What is the probability that A wins?

Let n be a positive integer such that

log₂log₂log₂log₂log₂(n) < 0 < log₂log₂log₂log₂(n).

Let l be the number of digits in the binary expansion of n. Then, the minimum and the maximum possible values of l are:

Let ABC be an acute-angled triangle and let D be the midpoint of BC. If AB = AD, then equals

For real X with -10 ≤ X ≤ 10, define, where for a real number r, we denote by [r] the largest integer less than or equal to r. The number of points of discontinuity of f in the interval (-10, 10) is

Let be unit vectors in the xy-plane, one each in the interior of the four quadrants. Which of the following statements is necessarily true?

The molecules of air in the room that you are sitting in are all experiencing the force of gravity trending to bring them down. The molecules are also frequently and randomly undergoing collisions, which tend to oppose the effect of fall under gravity. The density of air is nearly uniform throughout the room because

A charge Q is spread non-uniformly on the surface of a hollow sphere of radius R such that the charge density is given by , where is the usual polar angle. The potential at the centre of the sphere is

The moment of inertia of a solid disc made of a thin metal of radius R and mass M about one of its diameters is given by . What will be the moment of inertia about this axis if the disc is folded in half about the diameter?

A source of frequency f is emitting sound waves. If temperature of the medium increases, then

The capacitor of capacitance C in the circuit shown is fully charged initially. Resistance is R.



After the switch S is closed, the time taken to reduce the stored energy in the capacitor to half its initial value is:

A loop carrying current I has the shape of a regular polygon of n sides. If R is the distance from the centre to any vertex, then the magnitude of the magnetic induction vector B at the centre of the loop is:

A body is executing simple harmonic motion of amplitude a and period T about the equilibrium position x = 0. Large numbers of snapshots are taken at random of this body in motion. The probability of the body being found in a very small interval x to x + |dx| is highest at

A particle released from rest is falling through a thick fluid under gravity. The fluid exerts a resistive force on the particle proportional to the square of its speed. Which one of the following graphs best depicts the variation of its speed v with time t?

Two masses m₁ and m₂ connected by a spring of spring constant k rest on a frictionless surface. If the masses are pulled apart and let go, the time period of oscillation is

A charge +q is placed somewhere inside the cavity of a thick conducting spherical shell of inner radius R₁ and outer radius R₂. If a charge Q is placed at a distance r > R₂ from the centre of the shell, then the electric field in the hollow cavity

A book is resting on shelf that is undergoing vertical simple harmonic oscillations with an amplitude of 2.5 cm. What is the minimum frequency of oscillation of the shelf from which the book will lose contact with the shelf? (Assume that g = 10 m/s²)

In Young's double slit experiment, the distance between the two slits is 0.1 mm, the distance between the slits and the screen is 1 m and the wavelength of the light used is 600 nm. The intensity at a point on the screen is 75% of the maximum intensity. What is the smallest distance of this point from the central fringe?

Two plates, each of the mass m, are connected by a mass-less spring as shown.



A weight W is put on the upper plate, which compresses the spring further. When W is removed, the entire assembly jumps up. The minimum weight W needed for the assembly to jump up when the weight is removed is just more than

Jet aircrafts fly at altitudes above 30,000 ft, where the air is very cold at -40°C and the pressure is 0.28 atm. The cabin is maintained at 1 atm pressure by means of a compressor, which exchanges air from outside adiabatically. In order to have a comfortable cabin temperature of 25°C, we will require in addition

For a domestic AC supply of 220 V at 50 cycles per second, the potential difference between the terminals of a two pin electric outlet in a room is given by

Due to transitions among its first three energy levels, hydrogenic atom emits radiation at three discrete wavelengths λ₁, λ₂ and λ₃, where λ₁ < λ₂ < λ₃. Then,

Consider the circuit shown in the figure below:



All the resistors are identical. The charge stored in the capacitor, once it is fully charged, is

An engine moving away from a vertical cliff blows a horn at a frequency f. Its speed is 0.5% of the speed of sound in air. The frequency of the reflected sound received at the engine is

The bulk modulus of a gas is defined as B = –VdP/dV. For an adiabatic process, the variation of B is proportional to Pⁿ. For an idea gas, n is

A plank is resting on a horizontal ground in the northern hemisphere of the Earth at 45° latitude. Let the angular speed of the Earth be and its radius r_e. The magnitude of the frictional force on the plank will be:

Assuring ideal behaviour, the ratio of kinetic energies of 3 g of H₂ and 4 g of O₂ at any temperature is

The ion that is isoelectronic with CO is

Oxalic acid, when treated with potassium permanganate in the presence of an acid, produces

The order of basicity of the following compounds is:

The molecule with non-zero dipole moment is:

The major final product in the following reaction is:

CH3CH2CN

The oxidation state of cobalt in the following molecule is:

The C-O bond length in CO, CO2 and CO₃^2- follows the order:

Among the following, the π-acid ligand is

Friedel-Crafts Acylation is

Increasing the temperature increases the rate of reaction but does not increase the

The enantiomeric pair among the following four structures is:

Extraction of silver is achieved by initial complexation of the ore (argentite) with 'X', followed by reduction with 'Y'. X and Y, respectively are

Two possible stereoisomers forare which of the following?

The value of the limiting molar conductivity (∧°) for NaCl, HCl and NaOAc are 126.4, 425.9 and 91.0 S cm² mol^-1, respectively. For HOAc, ∧° in S cm² mol^-1 is

In a zero-order reaction, if the initial concentration of the reactant is doubled, the time required for half the reactant to be consumed

For a zero order reaction with rate constant k, the slope of the plot of reactant concentration against time is

The reaction of methyl ketone with Cl₂/excess OH^- gives the following major product:

D-Glucose upon treatment with bromine water gives

The number of ions produced in water by dissolution of the complex having the empirical formula COCl₃.4NH₃ is

In the organism muscle, oxygen is carried by

The chromosomal attachment of the spindle microtubule is

ELISA, the standard screening test for HIV, detects

The abnormal development of which of the following lymphoid organs results in the most severe immunodeficiency?

Which of the following classes of immunoglobulins can trigger the complement cascade?

Some animals excrete uric acid in urine (uricotelic) as it requires very little water. This is an adaptation to conserve water loss. Which animals among the following are most likely to be uricotelic?

Which of the following is true for TCA cycle in eukaryotes?

In Dengue virus infection, patients often develop haemorrhagic fever due to internal bleeding. This happens due to the reduction of

In 1670, Robert Boyle conducted an experiment wherein he placed a viper (a poisonous snake) in a chamber and rapidly reduced the pressure in that chamber. Which of the following would be true?

Sometimes urea is fed to ruminants to improve their health. It works by

Athletes often experience muscle cramps. Which of the following statements is true about muscle cramps?

Restriction endonucleases are enzymes that cleave DNA molecules into smaller fragments. Which type of bond do they act on?

A person has difficulty in breathing at higher altitudes because oxygen

A protein with 100 amino acid residues has been translated based on triplet genetic code. Had the genetic code been quadruplet, the gene that codes for the protein would have been

A water molecule can be formed from a maximum of _____ hydrogen bonds.

The process of cell death involving DNA cleavage in cells is known as

Nucleolus is an organelle that is responsible for the production of

Fission yeasts are

Morphogenetic movements take place predominantly during which of the following embryonic stages?

Part of epidermis that keeps out unwanted particles is called

Let a, b, c be the sides of a triangle. If t denotes the expression (a² + b² + c²)/(ab + bc + ca), then the set of all possible values of t is

Let A denote the area bounded by the curve y = 1/x and the lines y = 0, x = 1, x = 10, let B = 1 + + ...... + , and let C = + ...... + . Then

The coefficient of x²⁰¹² in is

The value of the integral , where a > 0, is

Let r be a real number and n ∈ N be such that the polynomial 2x² + 2x + 1 divides the polynomial (x + 1)ⁿ - r. Then (n, r) can be

Let f: R R be a continuous function satisfying f(x) = x +f(t) dt, for all x ∈ R. The number of elements in the set S = { x ∈ R ; f(x) = 0 }, is

The minimum value of n for which < 1.01

Let f(x) = x¹² – x⁹ + x⁴ – x + 1. Which of the following is true?

Let a, b, c and d be real numbers such that



for every natural number n. Then, |a| + |b| + |c| + |d| is equal to

Suppose that the Earth is a sphere of radius 6400 kilometers. What is the height from the Earth's surface from where exactly a fourth of the Earth's surface is visible?

A plano-convex lens made of material of refractive index with radius of curvature R is silvered on the curved side. How far away from the lens-mirror must you place a point object so that the image coincides with the object?

A vehicle is moving with speed v on a curved road of radius r. The coefficient of friction between the vehicle and the road is μ. The angle θ of banking needed is given by

A small asteroid is orbiting around the sun in a circular orbit of radius r₀ with speed V₀. A rocket is launched from the asteroid with speed V = α V₀, where V is the speed relative to the sun. The highest value of α for which the rocket will remain bound to the solar system is (ignoring gravity due to the asteroid and effect of other planets)

As shown in the figure below, a cube is formed with ten identical resistances R (thick lines) and two shorting wires (dotted lines) along the arms AC and BD.



Resistance between points A and B is

A certain planet completes one rotation about its axis in time T. The weight of an object placed at the equator on the planet's surface is a fraction f (f is close to unity) of its weight recorded at a latitude of 60⁰. The density of the planet (assumed to be a uniform sphere) is given by

An ideal gas with heat capacity at constant volume C_v undergoes a quasistatic process described by PV^α in a P-V diagram, where α is a constant. The heat capacity of the gas during this process is given by

The surface of a planet is found to be uniformly charged. When a particle of mass m and no charge is thrown at an angle from the surface of the planet, it has a parabolic trajectory as in projectile motion with horizontal range L. A particle of mass m and charge q, with the same initial conditions has a range L/2. The range of particle of mass m and charge 2q with the same initial conditions is

At a distance L from a uniformly charged long wire, a charged particle is thrown radially outward with a velocity u in the direction perpendicular to the wire. When the particle reaches a distance 2L from the wire, its speed is found to be . The magnitude of the velocity, when it is a distance 4L away from the wire, is (ignore gravity)

A small boy is throwing a ball towards a wall 6 m in front of him. He releases the ball at a height of 1.4 m from the ground. The ball bounces from the wall at a height of 3 m, rebounds from the ground and reaches the boy's hand exactly at the point of release. Assuming the two bounces (one from the wall and the other from the ground) to be perfectly elastic, how far ahead of the boy did the ball bounce from the ground?

A solid sphere rolls without slipping, first horizontally and then, up to a point X at height h on an inclined plane before rolling down, as shown.

The initial horizontal speed of the sphere is

A cylinder of cooking gas in a household contains 11.6 kg of butane. The thermochemical reaction for the combustion of butane is:
2C₄H₁₀(g) + 13O₂(g) 8CO₂(g) + 10H₂O(l); ΔH = -2658 kJ/mol
If the household needs 15,000 KJ of energy per day, the cooking gas cylinder will last for about

Reactions and their equilibrium constants are given below:



The equilibrium constant, K for the reaction is

In the DNA of E. Coli, the mole ratio of adenine to cytosine is 0.7. If the number of moles of adenine in the DNA is 3,50,000, then the number of moles of guanine is equal to

The H for vaporisation of a liquid is 20 kJ/mol. Assuming ideal behaviour, the change in internal energy for the vaporisation of 1 mole of the liquid at 60°C and 1 bar is close to

Two isomeric compounds I and II are heated with HBr:



The products obtained are:

Identify the cyclic silicate ion given in the figure below:

oxidises

(i) oxalate ion in acidic medium at 333 K and
(ii) HCl.

For balanced chemical equations, the ratios [] in (i) and [] in (ii), respectively are

Optically active (S) -α – methoxypropionaldehyde on reaction with MeMgX gave a mixture of alcohols. The major diastereomer 'P' on treatment with Mel/K₂CO₃ gave an optically inactive compound. P is

The maximum number of isomers that can result from monobromination of 2-methyl-2-pentene with N-bromosuccinimide in boiling CCl₄ is

Aqueous solution of metallic nitrate X reacts with NH₄OH to form Y, which dissolves in excess NH₄OH. The resulting complex is reduced by acetaldehyde to deposit the metal. X and Y, respectively are

Which of the following statements is true for meiosis?

The effect of consumption of excess protein by normal individual would result in

Persons suffering from hypertension (high blood pressure) are advised a low-salt diet because

Vibrio cholerae causes cholera in humans. Ganga water was once used successfully to combat the infection. The possible reason could be

In a diploid organism, there are three different alleles for a particular gene. Out of these three alleles, one is recessive and the other two alleles exhibit co-dominance. How many phenotypes are possible with this set of alleles?

Four species of birds have different egg colours: [1] White with no markings, [2] Pale brown with no markings, [3] Grey-brown with dark streaks and spots, [4] Pale blue with dark blue-green spots. Based on egg colour, which species is most likely to nest in a deep tree hole?

Rice has a diploid genome with 2n = 24. If crossing-over is stopped in a rice plant and then selfed seeds are collected, will all the offspring be genetically identical to the parent plant?

Increasing the number of measurements of an experimental variable will

Watson and Crick model of DNA is

A scientist has cloned an 8 kb fragment of a mouse gene into the Eco RI site of a vector of 6 kb size. The cloned DNA has no other Eco RI site within. Digestion of the cloned DNA is shown below.



Which of the following sets of DNA fragments generated by digestion with both Eco RI and Bam HI as shown in (iii) is from the gene?