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In a triangle, two vertices are (2, 3) and (4, 0), and its circumcentre is (2, z) for some real number z. The circumradius is

The number of solutions to sin x = 6/x with 0 ≤ x ≤ 12 is

The value of equals

The number of relations R from an m-element set A to an n-element set B satisfying the condition (a₁b₁) ε R₁ (a₁b₂) ε R b₁ = b₂ for a ε A, b₁, b₂ ε B is

Let a, b, c, d be the numbers in the set {1, 2, 3, 4, 5, 6} such that the curves y = 2x³ + ax + b and 2x³ + cx + d have no point in common. The maximum possible value of (a – c)² + b – d is

Let n be a natural number and let a be a real number. The number of zeroes of x^{2n + 1} – (2n + 1)x + a = 0 in the interval [-1, 1] is

Let S_n = k denote the sum of the first n positive integers. The numbers S₁, S₂, S₃, _.... S₉₉ are written on 99 cards. The probability of drawing a card with an even number written on it is

Let S = {1, 2, 3, ...., n} and A = {(a, b) | 1 a, b n} = S S. A subset B of A is said to be a good subset if (x, x) ∈ B for every x ∈ S. Then, the number of good subsets of A is

A circle touches the parabola y² = 4x at (1, 2) and also touches its directrix. The y-coordinate of the point of contact of the circle and the directrix is:

Let f: R R be a differentiable function such that f(a) = 0 = f(b) and f'(a)f'(b) > 0 for some a < b. The minimum number of roots of f'(x) = 0 in the interval (a, b) is:

Consider the regions A = {(x, y) | x² + y² ≤ 100} and B = {(x, y) | sin (x + y) > 0 | in the plane. Then, the area of the region A ∩ B is:

Let a_o = 0 and a_n = 3a_{n - 1} + 1 for n ≥ 1. Then, the remainder obtained after dividing a₂₀₁₀by 11 is:

All the points (x, y) in the plane satisfying the equation x² + 2x sin(xy) + 1 = 0 lie on a/an

In triangle ABC, we are given that 3 sin A + 4 cos B = 6 and 4 sin B + 3 cos A = 1. Then, the measure of the angle C is

Let [x] denote the largest integer not exceeding x and {x} = x – [x]. Then, is equal to

Let X be a non-empty set and let P(X) denote the collection of all subsets of X. Define f: X × P(X) R by

f(x, A ∪ B)

Then, f(x, A ∪ B) equals

Two line segments AB and CD are constrained to move along the x and y axes, respectively, in such a way that the points A, B, C and D are concyclic. If AB = a and CD = b, then the locus of the centre of the circle passing through A, B, C and D in polar coordinates is:

For x, t ∈ R, let p_t(x) = (sin t)x² - (2cos t)x + sin t be a family of quadratic polynomials in x with variable coefficients.
Let A(t) =

Which of the following statements are true?

1. A(t) < 0 for all t
2. A(t) has infinitely many critical points
3. A(t) = 0 for infinitely many t
4. A'(t) < 0 for all t

A box contains coupons labelled 1, 2, 3 ... n. A coupon is picked at random and the number x is noted. The coupon is put back into the box and a new coupon is picked at random. The new number is y. Then, the probability that one of the numbers x and y divides the other is (In the options below, [r] denotes the largest integer less than or equal to r.)

Let P be a closed polygon with 10 sides and 10 vertices (assume that the sides do not intersect except at the vertices). Let k be the number of interior angles of P that are greater than 180°. The maximum possible value of k is

Four metallic plates, each of surface area (of one side) A, are placed at a distance d apart from each other. The two outer plates are connected to a point P and the two inner plates to another point Q as shown in the figure.



Then, the capacitance of the system is

Two identical conducting spheres carry identical charges. If the spheres are set at a certain distance apart, they repel each other with a force F. A third conducting sphere, identical to the other two, but initially unchanged, is then touched to the one sphere and then to the other before being removed. The force between the original two spheres is now

A uniform non-deformable cylinder of mass m and radius R is rolling without slipping on a horizontal rough surface. The force of friction

A small body is released from a height H of an inclined plane. At the bottom of the plane is a loop of radius R as shown.

Ignoring friction, the minimum H required for the body to just complete the loop (that is, reach the point 0) is

A planet orbits in an elliptical path of eccentricity 'e' around a massive star considered fixed at one of the foci. The point in space where it is closest to the star is denoted by P and the point where it is farthest is denoted by A. Let V_P and V_A be the respective speeds at P and A. Then,

The ratio of the speed of sound to the average speed of an air molecule at 300 K and 1 atmospheric pressure is close to

The potential energy of a point particle is given by the expression . A dimensionless combination of the constants , and is:

An unpolarised beam of light of intensity I₀ passes through two linear polarisers making an angle of 30° with respect to each other. The emergent beam will have an intensity of

The circuit shown consists of a switch (S), a battery (B) of emf E, a resistance R, and an inductor L.



The current in the circuit at the instant the switch is closed, is

Three transparent media of refractive indices μ₁, μ₂ and μ₃ are stacked as shown. A ray of light follows the path shown. No light enters the third medium.



Then,

A bus driving along at 39.6 kmph is approaching a person who is standing at the bus stop, while honking repeatedly at an interval of 30 seconds. If the speed of the sound is 330 m/s, at what interval will the person hear the horn?

A piece of hot copper at 100°C is plunged into a pond at 30°C. The copper cools down to 30°C, while the pond, being huge, stays at its initial temperature. Then,

Two bulbs of identical volumes are connected by a small capillary and are initially filled with an ideal gas at temperature T. Bulb 2 is heated to maintain a temperature 2T while bulb 1 remains at temperature T. Assuming throughout that the heat conduction by the capillary is negligible, the ratio of final mass of the gas in bulb 2 to the initial mass of the gas in the same bulb is close to

At 23°C, a pipe open at both ends resonates at a frequency of 450 hertz. At what frequency does the same pipe resonate on a hot day when the speed of sound is 4 percent higher than it would be at 23°C?

In the circuit shown, the switch is closed at time t = 0.



Which of the graphs shown below best represents the voltage across the inductor, as seen on an oscilloscope?

Four students measure the height of a tower. Each student uses a different method and each measures the height many times. The data for each are plotted below. The measurement with highest precision is

A blackbox (BB) which may contain a combination of electrical circuit elements (resistor, capacitor or inductor) is connected with other external circuit elements as shown below in the figure (a). After the switch (S) is closed at time t = 0, the current (I) as a function of time (t) is shown in the figure (b).



From this, we can infer that the blackbox contains

A wheel of radius R with an axle of radius R/2 is shown in the figure and is free to rotate about a frictionless axis through its centre and perpendicular to the page. Three forces (F, F, 2F) are exerted tangentially to the respective rims as shown in the figure.



The magnitude of the net torque acting on the system is nearly

A ray of light incident on a glass sphere (refractive index ) suffers total internal reflection before emerging out exactly parallel to the incident ray. The angle of incidence was

The figure below shows pressure variation in two different sound waves in air with time at a given position. Both the figures are drawn to the same scale.



Which of the following statements is true?

Among , the species that acts as a Bronsted acid as well as a Bronsted base is

The reaction of butanal with n-propylmagnesium bromide followed by hydrolysis gives a/an

The conjugate bases for HCO₃^- and NH₃, respectively are

The hydrogen ion concentration in a mixture of 10 ml of 0.1 M H₂SO₄ and 10 ml of 0.1 M KOH solution in water is

The most stable conformation of 2,3-dibromobutane is:

The major product in the following reaction is:

Among the following graphs showing the variation of rate (K) with temperature (T) for a reaction, the one that exhibits Arrhenius behaviour over the entire temperature range is:

Doping silicon with boron produces a/an

The concentration of a substance undergoing a chemical reaction becomes one-half of its original value after time t regardless of the initial concentration. The reaction is an example of a

In the nuclear reaction:

+ X,

'X' is:

For a first order reaction R P, the rate constant is k. If the initial concentration of R is [R_o], the concentration of R at any time 't' is given by the expression:

Cyclohexene is reacted with bromine in CCl₄ in the dark. The product of the reaction will be

The rate of gas phase chemical reactions generally increases rapidly with the rise in temperature. This is mainly because the

The major product of the following reaction is

The products formed in the oxidation of NaBH₄ by I₂ are

Boiling points in i – iii follows the order

The variation of solubility of four different gases (G₁, G₂, etc.) in a given solvent with pressure at a constant temperature is shown in the plot.



The gas with the highest value of Henry's law constant is:

The numbers of lone pairs on Xe in XeF₂ and XeF₄ respectively are:

For isothermal reversible expansion of an ideal gas,

An ionic compound is formed between a metal M and a non metal Y. If M occupies half the octahedral voids in the cubic close packed arrangement formed by Y, the chemical formula of the ionic compound is

Plants are attracted to light through the hormonal action of which of the following?

Nucleotides are monomers of DNA. Each nucleotide consists of a

Earthworms are bisexual but still cross-fertilisation is common. This is because

Wooden doors and windows swell up in the rainy season by

Peptic ulcers are caused by

By which of the following mechanisms is glucose reabsorbed from the glomerular filtrate by the kidney tubule?

Plant roots are usually devoid of chlorophyll and cannot perform photosynthesis. However, three are exceptions. Which of the following plant roots can perform photosynthesis?

In a food chain such as grass deer lion, the energy cost of respiration as a proportion of total assimilated energy at each level would be

If the total number of photons falling per unit area of a leaf per minute is kept constant, then which of the following will result in maximum photosynthesis?

When a person is suffering from high fever, it is sometimes observed that the skin has a reddish tinge. Why does this happen?

Conversion of the Bt protoxin produced by Bacillus thuringiensis to its active form in the gut of the insects is mediated by

Estimate the order of the speed of propagation of an action potential or nerve impulse.

The Na⁺/K⁺ pump is present in the plasma membrane of mammalian cells where it

Patients who have undergone organ transplants are given anti-rejection medications to

The process of organogenesis starts as soon as the three germ layers are formed in a developing embryo. The human brain is formed from the

An electrode is placed in the axoplasm of a mammalian axon and another electrode is placed just outside the axon. The potential difference measured will be

Genetic content of a cell reduces to half during

The two enzymatic activities associated with RuBisCo are

In orange and lemon, the edible part of the fruit is

In vertebrates, 'glycogen' is stored chiefly in

Define the sequence {a_n}, _{n > 0, where:}

a_n = for

Then, equals

The sum of all the absolute values of the differences of the numbers 1, 2, 3,..., n, taken two at a time, i.e. equals

A polynomial P(x) with real coefficients has the property that P''(x) ≠ 0 for all x. Suppose P(0) = 1 and P'(0) = -1. What can you say about P(1)?

Let f(x) = for all x ≠ 1. Let f¹(x) = f(x), f²(x) = f(f(x)) and generally fⁿ(x) = f(f^{n - 1}(x)) for n > 1. Let P = f¹(2)f²(3)f³(4)f⁴(5), then which of the following is a multiple of P?

Among all cyclic quadrilaterals inscribed in a circle of radius R with one of its angles equal to 120°, consider the one with maximum possible area. Its area is

Suppose m, n are positive integers such that 6^m + 2^m+n.3^W + 2ⁿ = 332. The value of the expression m² + mn + n² is

Let A = {θ ∈ R | cos² (sin θ) + sin² (cosθ) = 1} and B = {θ ∈ R | cos(sin θ) sin(cos θ) = 0}. Then, A ∩ B

The arithmetic mean and the geometric mean of two distinct 2-digit numbers x and y are two integers one of which can be obtained by reversing the digits of the other (in base 10 representation). Then, x + y equals

Let XY be the diameter of a semicircle with centre O. Let A be a variable point on the semicircle and B another point on the semicircle such that AB is parallel to XY. The value of ÐBOY for which the inradius of triangle AOB is maximum, is

In a tournament with five teams, each team plays against every other team exactly once. Each game is won by one of the playing teams and the winning team scores one point, while the losing team scores zero. Which of the following is not necessarily true?

A solid uniform sphere having a mass M, radius R, and moment of inertia MR² rolls down a plane inclined at an angle θ to the horizontal starting from rest. The coefficient of static friction between the sphere and the plane is μ. Then,

Consider 1 kg of liquid water undergoing change in phase to water vapour at 100°C, the vapour pressure is 1.01 × 10⁵ N-m² and the latent heat of vaporisation is 22.6 × 10⁵ J kg². The density of liquid water is 10³ kg-m^-3 and that of vapour is kg-m^-3. The change in internal energy in this phase change is nearly

Consider three concentric metallic spheres A, B and C of radii a, b, c respectively where a is

A simple pendulum is released from rest at the horizontally stretched position. When the string makes an angle θ with the vertical, the angle which the acceleration vector of the bob makes with the string is given by

A narrow parallel beam of light falls on a glass sphere of radius R and refractive index μ at normal incidence. The distance of the image from the outer edge is given by

At time t = 0, a container has N₀ radioactive atoms with a decay constant λ. In addition, c numbers of atoms of the same type are being added to the container per unit time. How many atoms of this type are there at t = T?

A constant amount of an ideal gas undergoes the cyclic process ABCA in the PV diagram shown below.



The path BC is isothermal. The work done by the gas during one complete cycle, beginning and ending at A, is nearly

A particle moves in a plane along an elliptic path given by . At point (0, b), the x-component of velocity is u. The y-component of acceleration at this point is

The figure below shows a circuit and its input voltage v_i as a function of time t.


Assuming the diodes to be ideal, which of the following graphs depicts the output voltage v₀ as a function of time t?

A metallic prong consists of 4 rods made of the same material, cross-section and same lengths as shown. The three forked ends are kept at 100°C and the handle end is at 0°C. The temperature of the junction is

When Co(II) chloride is dissolved in concentrated HCl, a blue solution is obtained. Upon dilution with water, the colour changes to pink because

In the following transformation



Reagents 1 and 2, respectively, are

A metal is irradiated with light of wavelength 660 nm. Given that the work function of the metal is 1.0 eV, the de Broglie wavelength of the ejected electron is close to

The following data is obtained for a reaction, X + Y Products.



The overall order of the reaction is

The solubility product of Mg(OH)₂ is 1.0 × 10^-12. If concentrated NaOH solution is added to a 0.01 M aqueous solution of MgCl₂, then pH at which precipitation of Mg²⁺occurs is:

A solution containing 8.0 g of nicotine in 92 g of water freezes 0.925 degrees below the normal freezing point of water. If the molal freezing depression constant K_f = 1.85°C kg mol^-1, then the molar mass of nicotine is

In the following conversion,


the major products X and Y, respectively are

Emulsification of 10 ml of oil in water produces 2.4 × 10¹⁸ droplets. If the surface tension at the oil – water interface is 0.03 Jm^-2 and the area of each droplet is 12.5 × 10 ^-16 m², then the energy spent in the formation of oil droplets is

The standard electrode potential of Zn²⁺/Zn is –0.76 V and that of Cu²⁺/Cu is 0.34 V. The emf (V) and the free energy change (kJ/mol), respectively for a Daniel cell will be

In aqueous solution, [Co(H₂O)₆]²⁺(X) reacts with molecular oxygen in the presence of excess liquor NH₃ to give a new complex Y. The number of unpaired electrons in X and Y, respectively are

E. coli has an optimal temperature of 37°C. Which of the following is an INCORRECT explanation for this?

Energetically unfavourable reactions occur in human cells through

The restriction endonuclease EcoR-I recognises and cleaves DNA sequence as shown below:

5' - G A A T T C – 3'
3' – C T T A A G – 5'

What is the probable number of cleavage sites that can occur in a 10 kb long random DNA sequence?

Insects constitute the largest animal group on earth. About 25-30% of the insect species are known to be herbivores. In spite of such huge herbivore pressure, globally, green plants have persisted. One possible reason for this persistence is that

The following DNA sequence (5 ' 3') specifies part of a protein coding sequence, starting from position I. Which of the following mutations will give rise to a protein that is shorter than the full-length protein?

You mark two ink-spots along the height at the base of a coconut tree and also at the top of the tree. When you examine the spots next year when the tree has grown taller, you will see that

E.coli was pulsed with tritiated thymidine for 5 min and then transferred to normal medium. Which of the following observations would be correct after one cell division?

The length of one complete turn of a DNA double helix is

Presence of plastids in plasmodium suggests it is a

Lions in India are currently restricted to Gir, Gujarat. Efforts are being made to move them to other parts of the country. This is because they are most susceptible to extinction due to infectious diseases under the following conditions when present as