KVPY SB/SX Mock...

Suppose the sequence a₁, a₂, a₃ ....... is an arithmetic progression of distinct numbers such that the sequence a₁, a₂, a₄, a₈ ........ is a geometric progression. The common ratio of the geometric progression is

Consider an ellipse with foci at (5, 15) and (21, 15). If the x-axis is a tangent to the ellipse, then the length of its major axis equals

Define a function f : R R by



Suppose f(x) is differentiable on R. Then

The area bounded by the parabolas y = x² and y = 1 - x² equals

Three children, each accompanied by a guardian, seek admission in a school. The principal wants to interview all the 6 persons one after the other, subject to the condition that no child is interviewed before its guardian. In how many ways can this be done?

Consider the conic ex² + πy² - 2e²x - 2π²y + e³ + π³ = πe. Suppose P is any point on the conic and S₁, S₂ are the foci of the conic, then the maximum value of (PS₁ + PS₂) is

Let f : R R be the function f(x) = (x – a₁)(x - a₂) + (x – a₂)(x – a₃) + (x – x₃)(x – x₁) with a₁, a₂, a₃ ∈ R. Then f(x) ≥ 0, if and only if

A purse contains 4 copper coins and 3 silver coins. A second purse contains 6 copper coins and 4 silver coins. A purse is chosen randomly and a coin is taken out of it. What is the probability that it is a copper coin?

Let A denote the matrix, where i² = 1, and let I denote the identity matrix.

Then, I + A + A² + ... + A²⁰¹⁰ is equal to

Let ABC be an equilateral triangle. Let KLMN be a rectangle with K and L on BC, M on AC, and N on AB. Suppose AN/NB = 2 and the area of triangle BKN is 6. Then, the area of the triangle ABC is:

The roots of (x - 41)⁴⁹ + (x - 49)⁴¹ + (x - 2009)²⁰⁰⁹ = 0 are

Three vertices are chosen randomly from the seven vertices of a regular 7-sided polygon. The probability that they form the vertices of an isosceles triangle is:

Suppose log_ab + log_ba = c. The smallest possible integer value of c for all a, b > 1 is

Let A = (4, 0), B = (0, 12) be two points in the plane. The locus of a point C such that the area of triangle ABC is 18 sq. units is

Which of the following intervals is a possible domain of the function f(x) = log_{x}[x] + log_[x]{x}, where [x] is the greatest integer not exceeding x and {x} = x – [x]?

The value of is

The sum of non-real roots of the polynomial equation x³ + 3x² + 3x + 3 = 0

Consider a triangle ABC in the xy-plane with vertices A = (0, 0), B = (1, 1) and C = (9, 1). If the line x = a divides the triangle into two parts of equal area, then a equals

Let f(x) = and g(x) = cos x. Which of the following statements is/are true?

1. Domain of f((g(x))²) = Domain of f(g(x))
2. Domain of f(g(x)) + g(f(x)) = Domain of g(f(x))
3. Domain of f(g(x)) = Domain of g(f(x))
4. Domain of g((f(x))³) = Domain of f(g(x))

Let n ≥ 3. A list of numbers 0 < X₁ < X₂ < ... X_n has mean and standard deviation . A new list of numbers is made as follows:

Y₁ = 0, Y₂ = X₍₂₎ ..... X_{(n - 1)}, y_n = X₁ + X_n

The mean and the standard deviation of the new list are and . Which of the following is necessarily true?

The relation Cp - C_v = R (Cp, C_v; Molar specific heats at constant pressure, volume) is exactly true for

A progressive wave travelling in positive x-direction given by y = a cos(kx - ωt) meets a denser surface at x = 0 and t = 0. The reflected waves are then given by

A small rectangular loop of wire in the plane of the paper is moved with uniform speed across a limited region of uniform speed across a limited region of uniform magnetic field, perpendicular to the plane of the paper.



Which of the following graphs would best represent the variation of the electric current I in the wire with time t?

Consider a one dimensional potential V(x) as shown in the figure below. A classical particle of mass m moves under its influence and has total energy E as shown.


The motion is

An ideal monatomic gas expands to twice its volume. If the process is isothermal, the magnitude of work done by the gas is W_i. If the process is adiabatic, the magnitude of work done by the gas is W_e. Then, which of the following is true?

In a Young's double slit experiment the intensity of light at each slit is I₀. Interference pattern is observed along a direction parallel to the line S₁ S₂ on screen S.


The minimum, maximum, and the intensity averaged over the entire screen are respectively:

In one model of the electron, the electron of mass m_e is thought to be a uniformly charged shell of radius R and total charge e, whose electrostatic energy E is equivalent to its mass m_e via Einstein's mass energy relation E = m_eC². In this model, R is approximately (m_e = 9.1 × 10^-31 kg, c = 3 × 10⁸ m/s^-1, πε₀ = 9 × 10⁹ Farads m^-1, magnitude of the electron charge = 16 × 10^-19 C)

A ball of mass m suspended from a rigid support by an inextensible massless string is released from a height h above its lowest point. At its lowest point, it collides elastically with a block of mass 2m at rest on a frictionless surface. Neglect the dimensions of the ball and the block. After the collision, the ball rises to a maximum height of

A pen of mass 'm' is lying on a piece of paper of mass M placed on a rough table. If the coefficient of friction between the pen and paper, and the paper and table are μ₁ and μ₂, respectively, then the minimum horizontal force with which the paper has to be pulled for the pen to start slipping is given by

Consider a uniform spherical volume charge distribution of radius R. Which of the following graphs correctly represents the magnitude of the electric field E at a distance r from the centre of the sphere?

A nucleus has a half life of 30 minutes. At 3 p.m, its decay rate was measured as 1,20,000 counts/sec. What will be the decay rate at 5 p.m?

Velocity of sound measured at a given temperature in oxygen and hydrogen is in the ratio:

A narrow but tall cabin is falling freely near the Earth's surface. Inside the cabin, two small stones A and B are released from rest (relative to the cabin). Initially, A is much above the centre of mass and B is much below the centre of mass of the cabin. A close observation of the motion of A and B will reveal that

Two rods, one made of copper and the other steel of the same length and cross sectional area are joined together. (The thermal conductivity of copper is 385 Js^-1.m^-1. K^-1 and steel is 50 J.s^-1.m^-1.K^-1). If the copper end is held at 100ºC and the steel end is held at 0ºC, what is the junction temperature (assuming no other heat losses)?

In a Young's double slit set-up, light from a laser source falls on a pair of very narrow slits separated by 1.0 micrometre and bright fringes separated by 1.0 millimeter are observed on a distant screen. If the frequency of the laser light is doubled, what will be the separation of the bright fringes?

Given below are three schematic graphs of potential energy V(r) versus distance r for three atomic particles: electron (e), proton (p+) and neutron (n), in the presence of a nucleus at the origin O. The radius of the nucleus is r_o. The scale on the V-axis may not be the same for all figures. The correct pairing of each graph with the corresponding atomic particle is

Consider an initially neutral hollow conducting spherical shell with inner radius r and outer radius 2r. A point charge +Q is now placed inside the shell at a distance r/2 from the centre. The shell is then grounded by connecting the outer surface to the earth. P is an external point at a distance 2r from the point charge +Q on the line passing through the centre and the point charge +Q as shown in the figure.



The magnitude of the force on a test charge +q placed at P will be

In a photocell circuit, the stopping potential, V₀, is a measure of the maximum kinetic energy of the photoelectrons. The following graph shows experimentally measured values of stopping potential versus frequency v of incident light:



The values of Planck's constant and the work function as determined from the graph are (Taking the magnitude of electronic charge to be e = 1.6 × 10^-19C)

Two species of radioactive atoms are mixed in equal number. The disintegration of the first species is and of the second is /3. After a long time, the mixture will behave as a species with mean life of approximately

Young-Laplace law states that the excess pressure inside a soap bubble of radius R is given by P = 4/R, where is the coefficient of surface tension of the soap. The number E₀ is a dimensionless number that is used to describe the shape of bubbles rising through a surrounding fluid. It is a combination of g, the acceleration due to gravity, , the density of the surrounding fluid, and a characteristic length scale L which could be the radius of the bubble. A possible expression for E₀ is

Which of the following gases has the slowest rate of diffusion?

If the ratio of the heat capacities C_p/C_v for one mole of a gas is 1.67, then the gas is

What are the respective hybridisations of Ni centre in [Ni(PPh₃)₂Cl₂] and [NiCl₄]^2-?

Which of the following compounds are aromatic?

Among the following, the species with the highest bond order is:

Typical electronic energy gaps in molecules are about 1.0 eV. In terms of temperature, the gap is closest to

The major product of the following reaction is:

The reaction that gives the above molecule as the major product is

The number of isomers of [Co(dien)Cl3] is

The shape of the molecule ClF₃ is:

A concentrated solution of copper sulphate, which is dark blue in colour, is mixed at room temperature with a dilute solution of copper sulphate, which is light blue. For this process,

The correct structure of PCl₃F₂ is:

The hybridisations of [Ni(CO)₄] and [Cr(H₂O)₆]³⁺, respectively are

In i - iv,



the compound that does not undergo polymerisation under radical initiation is

For transforming
,
the reagent used is

The spin-only magnetic moments of [Mn(CN)

Among the following, the set of isoelectronic ions is:

For the reaction , the concentration of A decreases from 0.06 to 0.03 mol/L and that of B rises from 0 to 0.06 mol/L at equilibrium. The values of n and equilibrium constant for the reaction respectively are

The entropy change in the isothermal reversible expansion of 2 moles of an ideal gas from 10 to 100 L at 300 K is

The angle of incidence of X-ray of wavelength 3 which produces a second offer diffraction beam from the (100) plane in a simple cubic lattice with interlayer spacing d = 6 is 30°. The angle of incidence that produces a first order diffracted beam from the (200) plane is

During photosynthesis, the chemical conversion of water is called

During development, unspecified cells become cells having unique functions. This process is called

Fertilisation in humans usually takes place in

One difference between blood and lymph is that blood

The disorders that arise when the immune system destroys self cells are called autoimmune disorders. Which of the following would be classified under this?

Transfer RNA (tRNA)

In mammals, the hormones secreted by the pituitary, the master gland, are regulated by

Vitamin A deficiency leads to night-blindness. Which of the following is the reason for the disease?

Ribonucleic Acids (RNA) that catalyse enzymatic reactions are called ribozymes. Which one of the following acts as a ribozyme?

Pathfinding by ants is by means of

Bacteriochlorophylls are photosynthetic pigments found in phototrophic bacteria. Their function is distinct from the plant chlorophyll in that they

If you dip a sack full of paddy seeds in water overnight and then keep it out for a couple of days, it feels warm. What generates this heat?

The major constituents of neurofilaments are

In which of the following phases of the cell cycle are sister chromatids available as template for repair?

Saline drip is given to a cholera patient because

Puffs in the polytene chromosomes of Drosophila melanogaster salivary glands represent

Human foetal haemoglobin differs from the adult haemoglobin in that it

Which of the following techniques is used for the detection of proteins?

Chlorofluorocarbons (CFCs) are believed to be associated with cancer because

Which of the following statements about nitrogenase is correct?

Let p(x) = a₀ + a₁x + . . . + a_nxⁿ. If p(-2) = -15, p(-1) = 1, p(0) = 7, p(1) = 9, p(2) = 13 and p(3) = 25, then the smallest possible value of n is

The range of the function f(x) = (sin x)^{sin x} defined on (0, ) is

Suppose a, b, c are real numbers and each of the equations x² + 2ax + b² = 0 and x² + 2bx + c² = 0 has two distinct real roots. Then, the equation x² + 2cx + a² = 0 has

Define a sequence (a_n) by a₁ = 5, a_n = a₁a₂ .... a_{n =} 1 + 4 for n > 1. Then,

Arrange the expansion of in decreasing powers of x. Suppose the coefficient of the first three terms form an arithmetic progression. Find the number of terms in the expansion having integer powers of x.

Let V₁ be the volume of a given right circular cone with O as the centre of the base and A as its apex. Let V₂ be the maximum volume of the right circular cone inscribed in the given cone whose apex is O and whose base is parallel to the base of the given code. The ratio V₂/V₁ is

Let A and B be any two n × n matrices such that the following conditions hold: AB = BA and there exists a positive integers k and l such that A^k = I (the identity matrix) and B^l = 0 (the zero matrix). Then,

Let f(x) = x³ + ax² + bx + c, where a, b and c are real numbers. If f(x) has a local minimum at x = 1 and a local maximum at x = - and f(2) = 0, then f(x) dx equals

Let f(x) be a non-constant polynomial with real coefficients such that f() = 100 and f(x) ≤ 100 for all real x. Which of the following statements is not necessarily true?

Let f(x) = 1 + . How many real roots does f(x) = 0 have?

A spherical cavity of radius r is carved out of a uniform solid sphere of radius R as shown in the figure.

The distance of the centre of mass of the resulting body from that of the solid sphere is given by

A cubical box of side a sitting on a rough table-top is pushed horizontally with a gradually increasing force until the box moves. If the force is applied at a height from the table-top which is grater than critical height H, the box topples first. If it is applied at a height less than H, the box starts sliding first. Then the coefficient of friction between the box and the table-top is

The total energy of a black body radiation source is collected for five minutes and used to heat water. The temperature of the water increases from 10.0°C to 11.0°C. The absolute temperature of the black body is doubled and its surface area halved, and the experiment is repeated for the same time. Which of the following statements would be most nearly correct?

On a bright sunny day, a diver of height h stands at the bottom of a lake of depth H. Looking upward, he can see objects outside the lake in a circular region of radius R. Beyond this circle, he sees the images of objects lying on the floor of the lake. If refractive index of water is 4/3, then the value of R is

A ball is dropped vertically from a height of h onto a hard surface. If the ball rebounds from the surface with a fraction r of the speed with which it strikes the latter on each impact, what is the net distance travelled by the ball up to the 10^th impact?

A particle of mass m undergoes oscillations about x = 0 in a potential given by , where V₀, k and a are constants. If the amplitude of oscillation is much smaller than a, the time period is given by

An isolated sphere of radius R contains uniform volume distribution of positive charge. Which of the curves shown below correctly illustrate the dependence of the magnitude of the electric field of the sphere as a function of the distance r from its centre?

A material is embedded between two glass plates. Refractive index n of the material varies with thickness as shown below. The maximum incident angle (in degrees) on the material for which beam will pass through the material is

A bullet of mass m is fired horizontally into a large sphere of mass M and radius R resting on a smooth horizontal table.



The bullet hits the sphere at a height h from the table and sticks to its surface. If the sphere starts rolling without slipping immediately on impact, then

A ball is rolling without slipping in a spherical shallow bowl (radius R) as shown in the figure and is executing simple harmonic motion. If the radius of the ball is doubled, the period of oscillation

If the pH of a mixture of 10 ml of 0.1 M NH₄OH and 10 ml of 1 M NH₄Cl solution is 8, then the pK_b value of NH₄OH is closest to

The rate constant for the reaction COCI₂(g) CO(g) + CI₂(g) is given by

In k (min^-1) = + 31.33. The temperature at which the rate of this reaction will be doubled from that at 25°C is

The final major product obtained in the following sequence of reactions is:

The inter-planar spacing between the (2 2 1) planes of a cubic lattice of length 450 pm is

2.52 g of oxalic acid dihydrate was dissolved in 100 mL of water, 10 mL of this solution was diluted to 500 mL. The normality of the final solution and the amount of oxalic acid (in g) in the solution respectively are

A metal with an atomic radius of 141.4 pm crystallises in the face centred cubic structure. The volume of the unit in pm³ is

XeF₆ hydrolysis to give an oxide. The structure of XeF₆ and the oxide, respectively is

In the reaction sequence,



The major products X and Y respectively, are

The major product obtained in the reaction of aniline with acetic anhydride is

Consider the equilibria (1) and (2) with equilibrium constants K₁ and K₂, respectively.





K₁ and K₂ are related as

The mode of action of penicillin is as follows:

Male offspring of which of the following couples has the highest chance of haemophilia?

When hydrogen peroxide is applied on the wound as a disinfectant, there is frothing at the site of injury, which is due to the presence of an enzyme in the skin that uses hydrogen peroxide as a substrate to produce

Which one of the following is true about enzyme catalysis?

Which of the following statements is true about bacterial symbionts in insects?

Which of the following correctly represents the results of an enzymatic reaction?
Enzyme is E, substrate is S and products are P1 and P2.

Which of the following sequence of events gives rise to flaccid guard cells and stomatal closure at night?

Selection of lysine auxotroph from a mixed population of bacteria can be done by growing the bacterial population in the presence of

10⁹ bacteria were spread on an agar plate containing penicillin. After incubation overnight at 37⁰ > C, 10 bacterial colonies were observed on the plate. The colonies that are likely to be resistant to penicillin can be tested by

The figure below demonstrates the growth curves of two organisms A and B growing in the same area. What kind of relation exists between A and B?