KVPY (SA) Mock ...

The number 1 + 3 + 9 + … + 3²³ is divisible by exactly two numbers between 724 and 734. Find those two numbers.

If the third and fifth terms of a geometric progression are 12 and 48 respectively, then the second term in it is

Which of the following is the solution of x + = ^?

Let (1 + 2x)²⁰ = a₀ + a₁x + a₂x² + ........... + a₂₀x²⁰. Then, 3a₀ + 2a₁ + 3a₂ + 2a₃ + 3a₄ + 2a₅ + ........ + 2a₁₉ + 3a₂₀ equals

One of the factors of x² – y² + z² – a² + 2(xz + ay) is

A company situated at (2, 0) in the xy-plane charges Rs. 2 per km for delivery. A second company at (0, 3) charges Rs. 3 per km for delivery. The region of the plane where it is cheaper to use the first company is

At what time between 10 o'clock and 11 o'clock are the two hands of a clock symmetric with respect to the vertical line (give the answer to the nearest second)?

Given below are four equations in terms of x. Assume that 0 < r < 4. Which of the following equations has the largest solution for x?

_{If x and y are real numbers such that then the value of (x + y)/(x - y) is}

A closed conical vessel is filled with water fully and is placed with its vertex down. The water is let out at a constant speed. After 21 minutes, it is found that the height of the water column is half of the original height. How much more time in minutes does it require to empty the vessel?

Directions: A question is followed by data in the form of two statements labeled as I and II. You must decide whether the data given in the statements are sufficient to answer the question.

What is the time taken by a train to cross a platform of length 50 metres?

I. The speed of the train is 60 kmph.
II. The length of the train is 30 meters.

The sides of a triangle are distinct positive integers in an arithmetic progression. If the smallest side is 10, the number of such triangles is

Let a, b, x, y be real numbers such that a² + b² = 81, x² + y² = 121 and ax + by = 99. Then, the set of all possible values of ay - bx is

Let a, b, c, d be real numbers such that |a - b| = 2, |b - c| = 3, |c - d| = 4. Then, the sum of all possible values of |a - d| is

A frog is presently located at the origin (0, 0) in the xy-plane. It always jumps from a point with integer coordinates to a point with integer coordinates moving a distance of 5 units in each jump. What is the minimum number of jumps required for the frog to go from (0, 0) to (0, 15)?

Consider two spherical planets of same average density. Planet 2 is 8 times as massive as planet 1. The ratio of the acceleration due to gravity of the second planet to that of the first is

Two balls are projected simultaneously with the same speed from the top of a tower, one vertically upwards and the other vertically downwards. They reach the ground in 9 s and 4 s, respectively. What is the height of the tower?
(g = 10 m/s²)

If the pressure exerted by a liquid column of height 7 m on the bottom of the container is 77,000 Nm^–2 and g is taken as 10 ms^-2, then the density of the liquid is

A block of mass 'M' is at rest on a plane surface inclined at an angle to the horizontal. The magnitude of force exerted by the plane on the block is

What should be the speeds of two persons such that when they move uniformly towards each other, they get 3.0 m closer each second and when they move uniformly in the same direction with the original speeds, they get 3.0 m closer each 10 sec?

atom disintegrates to with a half-life of 10⁹ years. In the process, it emits 7 alpha particles and 'n' particles. Here, 'n' is

A block of mass M rests on a rough horizontal table. A steadily increasing horizontal force is applied such that the block starts to slide on the table without toppling. The force is continued even after sliding has started. Assume the coefficients of static and kinetic friction between the table and the block to be equal. The correct representation of the variation of the frictional force, f₁ exerted by the table on the block with time t is given by:

White light is split into a spectrum by a prism and it is seen on a screen. If we put another identical inverted prism behind it in contact, what will be seen on the screen?

A new temperature scale uses X as a unit of temperature, where the numerical value of the temperature t_x in this scale is related to the absolute temperature T by t_x = 3T + 300. If the specific heat of a material using this unit is 1400 J kg^-1 X^-1, its specific heat in the S.I. system of units is

A small child tries to move a large rubber toy placed on the ground. The toy does not move but gets deformed under her pushing force () which is obliquely upward as shown. Then

The Kirchoff's first and second law, where the symbols have their usual meanings, are respectively based on

The following three objects (1) a metal tray, (2) a block of wood, and (3) a woollen cap are left in a closed room overnight. Next day the temperature of each is recorded as T₁, T₂ and T₃ respectively. The likely situation is

A girl standing at point P on a beach wishes to reach a point Q in the sea as quickly as possible. She can run at 6 kmh^-1 on the beach and swim at 4 kmh^-1 in the sea. She should take the path

Light enters an isosceles right triangular prism at normal incidence through face AB and undergoes total internal reflection at face BC as shown below.



^{The minimum value of the refractive index of the prism is close to}

A photon of energy 8 eV is incident on metal surface of threshold frequency 1.6 × 10¹⁵ Hz. The magnitude of kinetic energy of the photo electrons emitted (in eV) is (Take h = 6 × 10^-34 Js)

The IUPAC name of acetic acid (CH₃COOH) is

In the following reaction, what is X?

In which of the following states is carbon most electronegative?

The symbol of the element, whose atoms have the outer most electronic configuration 2s², 2p³ is

Two balloons, A and B, containing 0.2 moles and 0.1 moles, respectively, of helium at room temperature and 2.0 atm are connected. When equilibrium is established, the final pressure of He in the system is

Which of the following are arranged in the decreasing order of dipole moment?

Hydrogen bond is the strongest in

Two cylinders containing carbon dioxide are connected together by a tube fitted with a tap. The capacity of one cylinder is 3 dm³ and that of the other cylinder is 2 dm³; the pressure of the first cylinder is 550 mm Hg and that of the second cylinder is 800 mm Hg. What will be the final pressure in the cylinders on opening the tap? (Temperature remains constant.)

What are the products formed when sodium reacts with acetic acid?

For reaction PCl₃ (g) + Cl₂ (g) ⇌ PCl₅ (g), the value of K_C at 250°C is 26 mole^-1 litre^-1. The value of K_P at this temperature will be

What is the weight of oxygen required for the complete combustion of 2.8 kg of ethylene?

Which of the following elements is least reactive to oxygen present in air at room temperature?

Which of the following orders of hydrocarbon compounds is correct in terms of decrease in the acidic character?

In the following set of aromatic compounds, the correct order of reactivity towards Friedel-Crafts alkylation is

The energy of second Bohr orbit of the hydrogen atom is - 328 kJ mol^-1; hence the energy of fourth Bohr orbit would be

The series of reactions, which results in the production of urea, is

Direction of growth of roots in plants is an example of which tropic movement?

Which of the following is the cause of Kwashiorkor?

If tall (TT) is cross bred with short (tt), then what will be the characteristic(s) of the F1 plants?

Which one of the following pairs is wrongly matched?

What is the name of an antibiotic that works against bacteria?

Total number of blood groups in the human body are

If birds are moved from 30°C to 10°C, their body temperature

Which of the following organelles has its own DNA and ribosome like mitochondria?

'On the origin of species, by means of Natural selection' was written by

A fish has evolved many features to help it survive in its environment. Which part of a fish protects the body?

What is the difference between flora and fauna?

Ram's uncle was suffering from 'diabetes' because his pancreas was not producing a hormone in a sufficient quantity. Identify the hormone.

In warmer weather, curd from milk forms faster because

What is common among snakes, turtles and lizards?

(1) Presence of scales
(2) Presence of lungs
(3) Presence of tough covering over the egg
(4) Presence of a notochord

In a triangle ABC, the altitudes from B and C on to the opposite sides are not shorter than their respective opposite sides. Then, one of the angles of ABC is

If area of a circle is decreased by 20%, its diameter, which is supposed to be 2r, is reduced to

Working together, Bill and Tom painted a fence in 8 hours. Last year, Tom painted the fence by himself. The year before, Bill painted it by himself, but took 12 hours less than what Tom took. How long did Bill and Tom take (in hours) when each was painting alone?

If we have two cans of cold drinks as shown in the figure. Find out which of the two packing can hold large amount of drink.

The sum of the first n terms of the series 1² + 2.2² + 3² + 2.4² + 5² + 2.6² + … is when n is even. When n is odd the sum is:

A mass m is moving with a constant velocity along a line parallel to the x-axis, away from the origin. Its angular momentum with respect to the origin

If equal volumes of two liquids with densities ₁ and ₂ are mixed, the new density will be _______.

A cylindrical copper rod has length L and resistance R. If it is melted and formed into another rod of length 2L, the resistance will be

A 2 kg block slides on a horizontal floor with a speed of 4 m/s. It strikes an uncompressed spring and compresses it till the block is motionless. The kinetic friction force is 15 N and spring constant is 10,000 N/m. The spring compresses by

If R₁ and R₂ are resistances of the filaments of 200 W and 100 W electric bulbs operating at 220 v, then

Grignard reagents are highly reactive organic compounds with general formula RMgX. When acetone is treated with RMgX, it produces a/an

Consider the following sequence of reactions:



The end product C is

How many isomers does C₇H₇Cl have?

A naturally occurring element Y is 80.99% ³⁴Y, 8.00% ³⁵Y and 11.01% ³⁶Y. What is the atomic weight of Y?

On ozonolysis, an alkene having the molecular formula C₉H₁₈ gives 2, 2-dimethyl propanal and 2-butanone. The alkene is

Independent assortment of Mendel was proved by __________.

The primary function of DNA polymerase is to

Which of the following pairs of nitrogen bases consititues the purines of DNA?

A regular check on the unborn baby of a lady towards the end of her pregnancy showed a heart rate of 80 beats per minute. What would the doctor infer about the baby`s heart condition from this?

What is the systolic and diastolic pressure of a normal human being respectively?