KVPY Class-XI (SA) - 2009

T₂₀ = T₁₉ + (T₂₀ with largest side 20)
Largest side is 20 and another side is a and b.

∴ a + b > 20

Suppose a = 2, then the number of possible values of b is 1, b has only b = 19
and for a = 3, the number of possible values of b is 2, and b = 18, 19
for a = 4, the number of possible values of b is 3, and b = 17, 18, 19
for a = 5, the number of possible values of b is 4, and b = 16, 17, 18, 19
for a = 6, the number of possible values of b is 5, and b = 15, 16, 17, 18, 19
for a = 7, the number of possible values of b is 6, and b = 14, 15, 16, 17, 18, 19
for a = 8, the number of possible values of b is 7, and b = 13, 14, 15, 16, 17, 18, 19
for a = 9, the number of possible values of b is 8, and b = 12, 13, 14, 15, 16, 17, 18, 19
for a = 10, the number of possible values of b is 9, and b = 11, 12, 13, 14, 15, 16, 17, 18, 19
for a = 11, the number of possible values of b is 8, and b = 12, 13, 14, 15, 16, 17, 18, 19
for a = 12, the number of possible values of b is 7, and b = 13, 14, 15, 16, 17, 18, 19
for a = 13, the number of possible values of b is 6, and b = 14, 15, 16, 17, 18, 19
for a = 14, the number of possible values of b is 5, and b = 15, 16, 17, 18, 19
for a = 15, the number of possible values of b is 4, and b = 16, 17, 18, 19
for a = 16, the number of possible values of b is 3, and b = 17, 18, 19
for a = 17, the number of possible values of b is 2, and b = 18, 19
for a = 18, the number of possible values of b is 1, and b = 19
Hence, the number of possible triangles with largest side = 2 (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8) + 9 = 81
Hence, T₂₀ - T₁₉ = 81

The magnitude of force exerted by the plane on the block will be Mg and components of force are Mg cosθ and Mg sinθ.

We are able to squeeze snow and make balls out of it due to water's unusual behaviour.

Hence, answer is anomalous behaviour of water.

Sound waves cannot be polarised. The reason is that sound waves are longitudinal waves, which means molecules vibration takes place in forward and backward direction, but in light waves which are transverse waves, molecules vibration takes place up and down and side by side. This is the reason why light waves can be polarised and sound waves cannot be polarised.

Waveform is the most important property in distinguishing between the two instruments.
We cannot distinguish between the two instruments on the basis of displacement amplitude or fundamental frequency or intensity.

Silicon has a valency of 4. So, it will form a stable SiCl₄ (general formula: XCl₄) molecule to acquire stable noble gas configuration.
Aluminium is trivalent and forms AlCl₃ (general formula: XCl₃).
Sodium being monovalent forms NaCl (general formula: XCl).
Calcium being divalent forms CaCl₂ (general formula: XCl₂).

A mixture of NH₄Cl and NaCl can be separated by sublimation. This is due to the fact that ammonium chloride changes directly from solid state to gaseous state on heating. So, to separate such mixtures that contain a sublimable volatile component (NH₄Cl) from a non-sublimable impurity (NaCl, in this case), sublimation method can be used.

Among the given options, Fe(OH)₂, Cu(OH)₂ and Ca(OH)₂ are ionic, while CH₃OH and CH₃CH₂OH are covalent.

We know that the metallic oxides are basic in nature, and on dissolving in water, they give their hydroxides, which are also basic in nature.

According to pH scale,

pH < 7 - acidic solution (lower the pH value, greater the acidic nature)

pH = 7 - neutral solution

pH > 7 - basic solution (higher the pH value, greater the basic nature)

The pH value can be calculated as:

pH = -log [H₃O⁺]

Thus, the pH values of the given solutions are:

From the above table, it is clear that the correct increasing order of pH values is: (i) < (iii) < (iv) < (ii)

As we move from top to bottom in a group, the atomic radii increase due to an increase in the number of shells and a decrease in the effective nuclear charge.
Therefore, the correct order of atomic radii of alkali metals is: Cs > K > Na > Li

The chemical reaction between any fatty acid (ester) and sodium hydroxide is a saponification reaction. For example:
Triglyceride + Sodium hydroxide (or potassium hydroxide) → Glycerol + Three soap molecules

Concentrated PbNO₃ solution can be stored in a copper vessel due to higher reduction potential of copper as compared to that of lead. Therefore, the redox reaction will not take place and it is safe to store PbNO₃ in a copper vessel.

Viruses are the smallest. Specific kinds of viruses range in size from a few dozen nanometres up to about 1,000 nanometres or 1 micrometre.

Birds are warm-blooded. Their body temperature remains the same irrespective of the temperature outside.

Vitamin C is known as ascorbic acid.

Bile salts emulsify fats and solubilise them. This aids in digestion and absorption of fats.

Fibre consists of non-starch polysaccharides, such as cellulose, dextrins, inulin, lignin, chitins, pectins, beta-glucans, waxes, and oligosaccharides.

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

In dogs, panting is a way to release excess metabolic heat. The more overheated a dog becomes, the heavier he will pant.

Haemodialysis, commonly called kidney dialysis or simply dialysis, is a process of purifying the blood of a person whose kidneys are not working normally.

If the blood cells clump (agglutinate), the antibody has bound to the appropriate antigen on the cells. Blood group O does not clump when either antiserum A or antiserum B is added because it has no antigens present on its surface.

The rate of bacterial multiplication increases in warm weather. Bacteria need warmth to multiply. In summers, the temperature is high which helps in the conversion of milk to curd.

Plants grow fast in the dark, and do so because they operate on circadian cycles. Tall spindly plant is a byproduct of "Shade Avoidance Syndrome," a scientific term for a plant's tendency to increase its production of the growth hormone auxin, allowing the plant to grow and stretch more rapidly toward sunlight to improve its conditions.

Rhesus condition or Erythroblastosis fetalis, also called hemolytic disease of the newborn, commonly happens when a woman with Rh-negative blood becomes pregnant by a man with Rh-positive blood and conceives a baby with Rh-positive blood.

The cerebrum is the major structure in the brain, composed of a right and left hemispheres. It is the part of the brain that controls intelligence and memory.

Diet rich in saturated fats increase the risk of heart diseases because these get deposited in the arteries. This leads to narrowing of the arteries.

Crop rotation is the practice of growing a series of different types of crops in the same area in sequential seasons. Crop rotation gives various nutrients to the soil. It is specially useful in maintaining the nitrogen content of the soil.

Distance between centre of mass and inertial axes is the largest. So, torque will be high and kinetic energy will be high.

The chemical equation of reaction of aluminium with manganese dioxide is:
4Al + 3MnO₂ → 2Al₂O₃ + 3Mn

Now, it is clear from the above reaction:
To reduce 3 moles of MnO₂ = 4 g moles of Al are required
To reduce 1 mol of MnO₂ = 4/3 g moles of Al are required

As we know, on dilution, milliequivalent of the solute remains constant.
Initially, pH of HCl = 4
So, normality of HCl = 10^-4 N
After dilution, pH of HCl = 5
So, normality of HCl will be 10^-5 N.
Apply normality equation:
N₁V₁ = N₂V₂
10^-4 × 10 = 10^-5 × V
V = 100 mL
So, 90 mL of water should be added for this pH change.

Ribosomes are the protein synthesising factory of the cell. Ribosomes are present on the rough endoplasmic reticulum.

The cornea acts as the eye's outermost lens. It functions like a window that controls and focuses the entry of light into the eye.

This is a case of mitotic cell division. In this type of cell division, the number of chromosomes remains the same in the daughter cell.

Same type of vegetation is observed in the same type of latitude.

Tree-Insect-Bird does not represent an erect pyramid of numbers. A numbers pyramid shows the relevant number of organisms that each trophic level occupies in an ecosystem. There are more primary consumers than producers.