KVPY (SA) Mock ...

Let S be the set of all positive integers, none of whose prime divisors is greater than 3. Thus 1, 2, 3, 4, 6, 8, 9 and 12 are the smallest elements of S. What is the sum of the reciprocals of the elements of S?

The number of solutions of x² – 7|x| + 12 = 0 is

How many positive real numbers 'x' satisfy the equation x³ - 3|x| + 2 = 0?

Let P₁, P₂, P₃, P₄, P₅ be the five equally spaced points on the circumference of a circle of radius 1, centered at O. Let R be the set of points in the plane of the circle that are closer to 0 than any of P₁, P₂, P₃, P₄, P₅. Then, R is a

The expression ax² + bx + c, a > 0 is positive for all real x only if

If the decimal 0.d25d25d25…… is expressible in the form n/27, then d + n must be

A student notices that the roots of the equation x² + bx + a = 0 are each 1 less than the roots of the equation x² + ax + b = 0. Then, a + b is

Let ABC be a triangle with B = 90°. Let AD be the bisector of A with D on BC. Suppose AC = 6 cm and the area of the triangle ADC is 10 cm². Then, the length of BD (in cm) is equal to

I carried 1000 kg of watermelon in summer by train. In the beginning, the water content was 99%. By the time I reached the destination, the water content had dropped to 98%. The reduction in the weight of the watermelon was

What will be the area of given shape?

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

What is the perimeter of the square?

I. The square is within a circle.
II. A circle of radius 4 cm passes through the four vertices of the square.

The solution of |x – 1| + |x – 2| + |x – 3| ≥ 6 is:

If x + x² + then x³ + is

Suppose a, b and c are three distinct real numbers. Let . When simplified, P(x) becomes

Consider a triangle drawn on the X - Y plane with its three vertices at (41, 0), (0, 41) and (0, 0), each vertex being represented by its (X, Y) coordinates. What is the number of points with integer coordinates inside the triangle (excluding all the points on the boundary)?

A boy standing on the footpath tosses a ball straight up and catches it. The driver of a car passing by moving with uniform velocity sees this.



^{The trajectory of the ball as seen by the driver will be}

Two charges +q and -q are placed at a distance 'b' apart as shown in the figure below.



^{The electric field at a point P on the perpendicular bisector as shown is}

The given diagram shows a cubical block completely submerged in water. Which of the following relations among the pressures acting on its four faces (as shown) is true?

The note "Saa" on the Sarod and the Sitar has the same pitch. The property of sound that is most important in distinguishing between the two instruments is

Two charges +Q and -2Q are located at points A and B on a horizontal line as shown below:



^{The electric field is zero at a point which is located at a finite distance}

A hollow pendulum bob filled with water has a small hole at the bottom through which water escapes at a constant rate. Which of the following statements describes the variation of the time period (T) of the pendulum as the water flows out?

Which of the following colours cannot be distinguished by a person suffering from colour blindness?

Two identical blocks of metal are at 20°C and 80°C, respectively. The specific heat of the material of the two blocks increases with temperature. Which of the following is true about the final temperature T_f when the two blocks are bought into contact (assuming that no heat is lost to the surroundings)?

Seven resistances are connected as shown in the given figure. The effective resistance between the points A and B is

A simple pendulum oscillates freely between points A and B.



^{We now put a peg (nail) at some point C as shown. As the pendulum moves from A to the right, the string will bend at C and the pendulum will go to its extreme point D. Ignoring friction, the point D}

In an arrangement of resistances shown in the diagram, the potential difference between B and D will be zero, when the unknown resistance X is

The mass of body A is twice that of B, and the speed of A is half that of B. The kinetic energies of A and B are in the ratio

We sit in the room with windows open. Then,

A 750 W motor drives a pump which lifts 300 litres of water per minute to a height of 6 metres. The efficiency of the motor is nearly (take acceleration due to gravity to be 10 m/s²)

The figure below shows a portion of an electric circuit with the currents in amperes and their directions. The magnitude and direction of the current in the portion PQ is

The IUPAC name of –C₂H₅OH is

Which of the following options was compared with the dry fruits in a spherical Christmas pudding in the Thomson's model of atom?

Order of hybridisation (of carbon) in methane, ethylene and acetylene will be, respectively

The electronic configuration of 1s², 2s² 2p⁶, 3s², 3p³ corresponds to

The electronic configuration of a dipositive ion M is 2, 8, 14 and its atomic weight is 56. The number of neutrons in the nucleus would be ____________.

The geometry & the dipole moment of H₂S is

The value of univeral gas constant 'R' depends only on the

A gas occupies 800 cm³ under a pressure of 760 mm Hg. Under what pressure will the volume of the gas be reduced by 20% of its original volume? (Temperature remains constant.)

What is the colour of the coating formed on copper when it reacts with moist carbon dioxide?

The equilibrium constant for the reaction NH₄NO₂(s) N₂(g) + 2H₂O(g) is given by:

What is molarity of K⁺ in aqueous solution that contains 17.4 ppm of K₂SO₄ (174 g mol^-1)?

Which one the following sets of ions represents the collection of isoelectronic species?

How many and bonds are present in toluene?

Which of the following metals will not be able to displace lead from its salt solution?

Excitation energy of a hydrogen like ion in its first excitation state is 40.8 eV. The energy needed to remove the electron from the ion in the ground state is:

The plasma resembles in its composition with the filtrate found in the Bowman's capsule except for the presence of

Which of the following is the master gland of the body?

Which of the following diseases cannot be prevented by vaccines?

What is the gene ratio of F₂ generation in a monohybrid cross?

Identify from the following, the disease that enters the human body through mosquito bite and affects the brain.

A disease transmitted by sexual contact is

An individual has O blood group. His/her blood sample

Which base is present in RNA but not in DNA?

Which of the following is/are the smallest in size?

The connecting link between a mother's blood and foetal blood is termed as

What is the scientific name of housefly?

Animals like tigers, wild buffaloes, barasingha, etc. are categorised as

Dietary fibres are composed of

The structural adaptation in flowers that prevents self pollination is known as

The avian heart is __________.

Let f(x) = 1 + x + x² + x³ + x⁴ + x⁵. What is the remainder when f(x¹²) is divided by f(x)?

Let log_a b = 4 and log_c d = 2, where a, b, c and d are natural numbers. Given that b - d = 7, the value of c - a is

If a² = by + cz, b² = cz + ax and c² = ax + by, then is equal to

A solid cube with 6 m side is given, having a hole exactly at its centre with radius of 2 m. Find the surface area of the remaining cube.

If a rectangle and an equilateral triangle with equal area are inscribed in a circle of radius 2 units, then one of the dimensions of the rectangle is

Two particles A and B, initially at rest, move towards each other under a mutual force of attraction. At the instant when the speed of A is V and that of B is 2V, the speed of the centre of mass of the system is

A spring balance 'A' reads 2 kg with a block of mass 'm' suspended from it. Another balance 'B' reads 3 kg when a beaker with a liquid is put on its pan. The two balances are now so arranged that the hanging mass 'm' is fully immersed inside the liquid in the beaker as shown in the figure. In this situation,

A metal rod joins three iron balls having different temperatures as shown below:



What will be the direction of the flow of heat?

In a military training exercise, three teams are stationed at points A, B and C, as shown below, with a hillock between teams A and C. Team A sends a light signal to B, which has to reflect it to team C, so that all the three teams start the exercise simultaneously. What should be the angle of the reflecting mirror, held by team B?

Two springs, of force constants k₁ and k₂, are connected to a mass m as shown. The frequency of oscillation of the mass is f. If both k₁ and k₂ are made four times their original values, the frequency of oscillation becomes

1, 3-Butadiene on reaction with HBr at 313 K gives the formation of _______ as the major product.

Ethyl chloride on reduction with LiAlH₄ gives compound 'X' as a major product. 'X' on chlorination with one mole of Cl₂ in the presence of light at ordinary temperature gives 'Y'. What is 'Y'?

Benzene when treated with ₂ in the presence of sunlight, primarily gives

What is the ratio of the occurrence of Cl - 37 and Cl - 35 isotopes of chlorine in nature?

The nucleus ₆C¹² absorbs an energetic neutron and emits a beta particle (β). The resulting nucleus is

The nucleolus, golgi complex and ER reform in which of the following phases of the cell cycle?

Match the following:

When do two chromatids of a chromosome separate from each other?

What is the triangular cartilaginous structure located between the tongue and the trachea called?

What do you mean by Residual volume?