IIT Ramaiah SAT April 30 2006 Question Paper

A triangle has integral sides. Its perimeter is 2006. One of the sides is sixteen times another side. The number of such triangles is [ ]

Given that $$\triangle ABC$$ is an arbitrary triangle. The number of points $$P$$ in the interior of the triangle such that $$\angle APB = \angle BPC = \angle CPA$$ is [ ]

The number of rhombi that can be inscribed in a parallelogram such that each side of the parallelogram contains precisely one vertex of the rhombus is [ ]

In this question a polynomial means a polynomial with real coefficients. $$P(x)$$ is a polynomial of degree 2006 and $$P(x) \ge 0$$ for all real $$x$$. Then [ ]

If $$0 < a \le x \le b$$, $$0 < a \le y \le b$$ and $$\frac{x}{b} = \frac{2x + y}{x + a + b}$$, then [ ]

$$p, q, r$$ are given real numbers. Consider the polynomials $$(q-r)(x-q)(x-r) + (r-p)(x-r)(x-p) + (p-q)(x-p)(x-q) + k$$ for all real $$k$$. Then [ ]

A lattice point is a point both of whose coordinates are integers. $$A$$ is $$(0, 0)$$, $$B$$ is $$(4012, 6018)$$. If $$n$$ is the number of lattice points lying on $$\overline{AB}$$, then $$n =$$ _________

Two circles of equal radius $$r$$ are inscribed in a semicircular sector of radius '1' such that they touch each other externally. Then $$r =$$ ___________

In a triangle $$ABC$$, $$D$$, $$E$$ are points on $$\overline{AC}$$, $$\overline{BC}$$ respectively such that $$\overline{DE} \parallel \overline{AB}$$ and $$DE = 5$$. If $$AB = 8$$ and $$\overline{AE}$$ bisects $$\angle DEB$$, then the length of $$\overline{CE} =$$ ____________

$$1 + \frac{1}{1+2} + \frac{1}{1+2+3} + \dots + \frac{1}{1+2+3+\dots+2006} =$$ ____________

The characteristic of $$\log_3 2006$$ is __________

If the positive integer $$m$$ is such that $$\frac{m}{2006}$$ is closest to $$\sqrt{2}$$, then $$m =$$ ________

$$A(0, 0)$$, $$B(4012, 0)$$, $$C(4012, 6018)$$, $$D(0, 6018)$$ determine a rectangle which is the union of $$4012 \times 6018$$ unit squares. Determine the number of these unit squares which have a nonempty intersection with $$\overline{AC}$$.

$$P$$ is a point not lying on a line $$l$$. Explain how you construct an equilateral triangle $$PQR$$ such that $$\overline{QR} \subset l$$.

Let $$E$$ be the midpoint of the side $$\overline{BC}$$ of $$\triangle ABC$$ and let $$F$$ be chosen in $$\overline{AC}$$ so that $$AC = 3FC$$. Determine the ratio of the areas, $$\triangle ABEF \colon \triangle EFC$$.

Does the equation $$2006 = (x+y)^2 + 3x + y$$ have a solution in nonnegative integers?

Without removing brackets at any stage factorize $$2y(y+z) - (x+y)(x+z)$$

Factorize the polynomial in $$x$$ given by $$(t^2+1)x^3 - (t^2-1)x^2 - (t^2+1)x + (t^2-1)$$

Treating the equalities $$\log_{10} 2 = 0.3010$$ and $$\log_{10} 3 = 0.4771$$ as exact, prove that the sum to 30 terms of the series $$2 + 6 + 18 + 54 + \dots$$ exceeds $$2 \times 10^{14}$$.

Factorize $$a^5(b-c) + b^5(c-a) + c^5(a-b)$$

Prove that $$(n+1)$$ is either itself a prime or a product of primes for every positive integer $$n$$.

A triangle $$ABC$$ is turned about the vertex $$A$$ in the plane of the triangle into the position $$AB'C'$$. If $$\overline{AC}$$ bisects $$\overline{BB'}$$, prove that $$\overline{AB'}$$ bisects $$\overline{CC'}$$.

$$P$$ is a point in the plane of parallel lines $$l$$, $$m$$. Explain how you locate a point $$Q$$ on $$l$$ and a point $$R$$ on $$m$$ such that $$\overline{PQ}$$ and $$\overline{PR}$$ are equal in length and perpendicular to each other. Discuss only the case when $$P$$ lies between $$l$$ and $$m$$.

$$A$$, $$B$$, $$C$$ are respectively $$(1002, 1005)$$, $$(2004, 2001)$$ and $$(1003, 1000)$$ and $$P_0$$ is $$(1, 4)$$. Points $$P_i$$, $$i \in \mathbb{N}$$ are such that<br>$$A$$ is the midpoint of $$\overline{P_0P_1}, \overline{P_3P_4}, \overline{P_6P_7}, \dots$$<br>$$B$$ is the midpoint of $$\overline{P_1P_2}, \overline{P_4P_5}, \overline{P_7P_8}, \dots$$<br>$$C$$ is the midpoint of $$\overline{P_2P_3}, \overline{P_5P_6}, \overline{P_8P_9}, \dots$$<br>Estimate the ratio of lengths $$\frac{P_{2005}P_{2006}}{P_1P_2}$$.

On a long journey if 10% of the distance is covered at an average speed of $$40\ \text{km/h}$$ and 90% of the distance is covered at an average speed of $$90\ \text{km/h}$$. What is the average speed for the entire journey? How does this answer change if the percentages relate to time rather than distance, that is, if 10% of the duration is covered at $$40\ \text{km/h}$$ and 90% of the duration is covered at average speed of $$90\ \text{km/h}$$.

In two glasses there is liquid material of the same type, the volume of one is $$V_1$$, its temperature is $$T_1$$, the volume of the other is $$V_2$$ and its temperature is $$T_2$$. We mix the liquids and wait while the temperature settles. What is the common temperature if we neglect any heat losses? What would be the common temperature if the volume of first liquid is halved and the second liquid doubled.

Three ice cubes each of side $$2\ \text{cm}$$ at $$0^\circ\text{C}$$ are dropped into a glass of $$18^\circ\text{C}$$ water. The volume of the water is $$200\ \text{cm}^3$$. What are the volume and the temperature of the water after the ice cubes have melted?<br>(Density of water $$= 1\ \text{g/cm}^3$$, density of ice $$= 0.9\ \text{g/cm}^3$$, latent heat of ice $$= 80\ \text{cal/g}$$, specific heat of water $$= 1\ \text{cal/g}^\circ\text{C}$$)

How much can the temperature of the water in a $$50\ \text{m} \times 25\ \text{m} \times 2\ \text{m}$$ size pool be raised with the energy released by a single gram of $$^{235}\text{U}$$ Uranium isotope in a nuclear power plant? Assume 47% of the energy ($$200\ \text{MeV}$$ of energy is liberated in the fission of one isotope) is used to heat up the water.<br>(Density of water $$= 1\ \text{g/cm}^3$$, specific heat of water $$= 4.2\ \text{J g}^{-1}\text{K}^{-1}$$)

Two cars are $$60\ \text{km}$$ away from each other and are moving towards each other at $$116\ \text{km/h}$$ and $$184\ \text{km/h}$$. How far are they from each other one and a half minutes before their collision?

A ray of light is incident on an isosceles right-angled prism made of a material of critical angle $$42^\circ$$. Find the distance of point of emergence of the ray from the prism if $$AB = d$$ and $$AP = L$$
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Large number of identical blocks are placed on a surface with adjacent blocks $$d$$ apart. The left most block is imparted a velocity $$V$$ to the right. On striking its neighbour it comes to rest and the struck block takes off at $$0.8V$$. This process repeats. What is the time interval between the third and fourth hits? Assume that the motion between the hits is uniform. Neglect the duration of collision.
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A stick of height $$h$$ is protruding out of a liquid contained in a large vessel with half of it inside the liquid. Rays of light from after noon sun are incident on the stick at $$45^\circ$$ as shown by the dotted lines. Find the length of the shadow on the bottom of the vessel. Would the shadow length increase or decrease as the sun goes down further?
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The wire mesh shown is made of a uniform wire. Resistance between any two points is $$R$$. When wires $$OH$$ and $$OD$$ are removed some current flows between the terminals when a potential difference $$V$$ is applied between them. If now the wires $$OB$$ and $$OF$$ are also removed, find the change in potential difference for same current to flow between the terminals.
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Ratio of specific heats $$\gamma$$ of a mixture of gases is given by $$\gamma = 1 + \frac{R(n_1 + n_2)}{n_1C_{v1} + n_2C_{v2}}$$ where $$C_{v1}$$ and $$C_{v2}$$ are the specific heats at constant volume of the two gases in the mixture and $$\gamma$$ is the ratio of specific heats, $$n_1$$ and $$n_2$$ are the number of moles of the gas in the given mixture.<br>Find the velocity of sound in a mixture by equal volumes Hydrogen and Helium at $$27^\circ\text{C}$$. Specific heats at constant volume for Hydrogen and Helium are $$\frac{5R}{2}$$ and $$\frac{3R}{2}$$ respectively.

Two pendulums are drawn aside and released at the same time. One has a time period of $$2\ \text{s}$$ and the other $$2.5\ \text{s}$$. What is the least time when they are at their extreme positions on opposite sides of equilibrium?

At a given time, the distance recorder of a car cruising at near constant speed on a road showed $$24942\ \text{km}$$. Two hours later the driver noticed that it showed a symmetric number again. What was the average speed of the car in this two-hour leg?

Explain why a glass of water quickly warms up when placed on a hot surface but does not cool as quickly when placed on a cold surface. Assume same temperature differences between the water and the surfaces in both the cases.

Two resistances, a 60 ohm and an unknown one are connected to a power source of $$120\ \text{V}$$ in a series arrangement. This way the power of the unknown resistance is $$60\ \text{W}$$. What is the Unknown resistance?

A hollow sphere made of material of density $$3\ \text{g/cm}^3$$ is floating with three fourth of its volume submerged in a liquid of density $$0.8\ \text{g/cm}^3$$. If the external volume of the sphere is $$200\ \text{cm}^3$$, find the volume of the cavity inside the sphere.

Chromatography is used by the 'Horse Racing Forensic Laboratory' to test for the presence of illegal drugs in racehorses. A concentrated sample of urine is spotted onto chromatography paper on the start line. Alongside this, known drugs are spotted. The chromatogram is run using methanol as the solvent. When finished, the paper is read by placing under ultra-violet light. A chromatogram of urine from four racehorses is shown.
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a) The results for known drugs are given as '$$R_f$$ values'. $$R_f\ \text{value} = \frac{\text{distance travelled by the substance}}{\text{distance travelled by the solvent}}$$<br>Calculate the $$R_f$$ value for caffeine.<br>b) Explain the meaning of the word 'solvent'.<br>c) State two factors which determine the distance a substance travels up the paper.<br>d) From the results the sample from one horse contains an illegal substance.<br>i) State the horse, and the drug present.<br>ii) Give a reason for the use of this drug.

Some solids dissolve easily, others do not.<br>a) i) What is the name given to liquids which dissolve solids?<br>ii) What is the name of the clear liquid formed when a solid dissolves in a liquid?<br>b) Temperature change affects the solubility of substances in water.<br>i) How does the solubility of most solids change as the temperature of the water increases?<br>ii) How does the solubility of a gas change as the temperature of the water increases?<br>c) Describe how dry solid can be obtained from the liquid in which it is dissolved.<br>d) The graph shows how the solubility of ammonium sulphate in water changes with temperature.
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100 g of ammonium sulphate was heated in 100 g of water until it all dissolved. Use the graph to answer the following.<br>i) What is the lowest temperature at which the water will dissolve all the ammonium sulphate?<br>ii) What mass of solid will crystallize from the liquid when it is cooled to room temperature?<br>iii) This solid is filtered off at $$20^\circ\text{C}$$. What will you see if the remaining liquid is then cooled to $$15^\circ\text{C}$$?<br>iv) What word is used to describe a liquid which cannot dissolve any more solid at a given temperature?

Write balanced chemical equations for the reactions between<br>i) Sodium thiosulphate and hydrochloric acid<br>ii) Magnesium and steam<br>iii) Calcium and oxygen<br>iv) Bromine and potassium iodide solution<br>v) Zinc and copper sulphate solution.

Magnesium is obtained by the electrolysis of magnesium chloride in a cell similar to the one shown below.
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a) i) What is the cathode made from?<br>ii) What is the anode made from?<br>iii) Why is the chlorine gas kept separate from the magnesium metal?<br>b) The magnesium chloride is made by the following reaction<br>$$\mathrm{MgO + C + Cl_2 \rightarrow MgCl_2 + CO}$$<br>Rewrite the equation in words.<br>c) Name two other metals that are obtained by electrolysis.

Reactions that give off carbon dioxide are used to raise dough when baking bread. On cooking, the bubbles of carbon dioxide become trapped in the dough and give the bread a pleasing texture. The ingredients are made into a dough which is left in a warm place for about 30 minutes. The volume of the dough increases as carbon dioxide is produced from the sugar by the yeast.<br>Recipe card, NAAN INDIAN BREAD, ingredients<br>$$75\ \text{cm}^3$$ of milk<br>1 teaspoon of sugar<br>1 teaspoon of yeast<br>200 g of plain flour<br>$$\frac{1}{2}$$ teaspoon of baking powder (sodium hydrogencarbonate and tartaric acid)<br>1 tablespoon of vegetable oil<br>$$75\ \text{cm}^3$$ of plain yogurt<br>1 egg<br>$$\frac{1}{4}$$ teaspoon of salt<br>a) There are two ionic compounds in the recipe. Name one of them.<br>The dough is then placed in a hot oven and the baking powder produces more carbon dioxide as the naan bread cooks. Sodium hydrogen carbonate decomposes when heated.<br>$$\mathrm{2NaHCO_3 \rightarrow Na_2CO_3 + CO_2 + H_2O}$$<br>Sodium carbonate has an unpleasant taste. It reacts with the tartaric acid to form sodium tartrate which has a pleasant flavour.<br>b) What type of compound is sodium tartrate?<br>c) Suggest two reasons why baking powder is a mixture of tartaric acid and sodium hydrogen carbonate, and not just sodium hydrogen carbonate.

A packet of washing soda crystals has been left open to the atmosphere for some time. The crystals have formed a white powder which may be represented by the formula $$\mathrm{Na_2CO_3 \cdot xH_2O}$$. The value of $$x$$ may be found by dissolving a known quantity of the powder in water and titrating the solution with acid of known concentration.<br>In one experiment a student weighed out 1.59 g of the powder and dissolved it in water. The volume of the solution was made up to exactly $$250\ \text{cm}^3$$. $$25\ \text{cm}^3$$ of this solution was titrated with hydrochloric acid of concentration $$0.10\ \text{mol/dm}^3$$ (mol per litre). $$25.6\ \text{cm}^3$$ of the acid was required for complete reaction. The equation for this reaction is<br>$$\mathrm{Na_2CO_3 + 2HCl \rightarrow 2NaCl + H_2O + CO_2}$$<br>a) Calculate the number of moles of hydrochloric acid used in the titration.<br>b) Calculate the number of moles of sodium carbonate in the $$25\ \text{cm}^3$$ sample of solution which reacted with the hydrochloric acid.<br>c) How many moles of sodium carbonate were in the $$250\ \text{cm}^3$$ of solution?<br>d) Calculate the mass of sodium carbonate in the $$250\ \text{cm}^3$$ of solution. (Relative atomic masses C = 12, O = 16, Na = 23)<br>e) Using your answer to part (d), what mass of water was in the original powder?<br>f) How many moles of water were present in the original powder? (Relative atomic masses H = 1, O = 16)<br>g) Using your answers to parts (c) and (f), calculate the value of $$x$$.

a) The Haber process is a reversible reaction. What does 'reversible reaction' mean?<br>b) The graphs show the amount of ammonia produced at different temperatures and pressures.
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i) What conditions favor the production of ammonia?<br>ii) In industry the reaction is carried out at $$450^\circ\text{C}$$ and 200 atmospheres pressure. Use the graph to explain why these conditions are used.<br>c) i) Some energy has to be put into the chemicals to start the reaction. Explain how this energy helps the reaction to start.<br>ii) Here is an energy level diagram for this reaction. Is this reaction exothermic or endothermic? Explain your answer in detail.
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This question is about alcohols.<br>a) The alcohols are members of a homologous series.<br>i) Write down the general formula for this homologous series.<br>ii) Write down the names and draw the structures of the two monohydric alcohols (alcohols that have only one $$-\mathrm{OH}$$ group) containing three carbon atoms per molecule.<br>b) On analysis, a 10.00 g sample of an alcohol was found to contain 3.75 g of carbon, 1.25 g of hydrogen, and 5.00 g of oxygen. Use this data to find the empirical formula of this alcohol. Suggest its molecular formula.<br>c) The dissociation energy of ethanol is shown below<br>$$\mathrm{C_2H_5OH(g) \rightarrow 2C(g) + 6H(g) + O(g)}$$, $$\Delta H = +3234\ \text{kJ/mol}$$ of ethanol<br>Using bond energy values (given below) calculate the bond energy of C-O per mole.<br>[Bond energies, H-H = 436 kJ/mol, C-H = 435 kJ/mol, O-H = 464 kJ/mol, C-C = 347 kJ/mol]

Red Lead, $$\mathrm{Pb_3O_4}$$ is used in paints. It protects metal surfaces against corrosion. Red lead is made by heating lead (II) oxide, PbO, in air.<br>a) Copy out and balance the following equation<br>$$\mathrm{PbO + O_2 \rightarrow Pb_3O_4}$$<br>b) Red lead paint may be used to protect iron from rusting. Name two other ways of stopping iron or steel from rusting.<br>c) On strong heating, red lead decomposes to form oxygen and lead (II) oxide.<br>i) How could you show that oxygen is given off in the decomposition?<br>ii) Suggest how you could show when the decomposition was finished.<br>d) Red lead reacts with concentrated hydrochloric acid to form lead (II) chloride, chlorine and water.<br>$$\mathrm{Pb_3O_4 + 8HCl \rightarrow 3PbCl_2 + Cl_2 + 4H_2O}$$<br>i) Describe how you could show that chlorine was given off during the reaction.<br>ii) Suggest how, using an electrical method, you could obtain lead from lead (II) chloride.

The rate of reaction between a metal and bromine can be studied using the apparatus shown below.
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A piece of metal, for example iron, was weighed and placed in the solution as shown above. The metal was removed at regular intervals and each time it was washed, dried and weighed. It was then replaced in the solution and the reaction continued.<br>From the results (mass of metal and time), the rate of the reaction can be calculated.<br>a) i) How would the rate change if more of the piece of metal was in the solution? Explain your answer.
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Explain the change in rate.<br>b) In a reaction between a solid and a solution, the rate may depend on the speed of stirring. Describe how you could find out if the rate of the reaction between solid iron and bromine solution depends on the speed of stirring.<br>c) Iron has two oxidation states so it can form two ions, $$\mathrm{Fe^{2+}}$$ and $$\mathrm{Fe^{3+}}$$. How could you test a solution to find out which ion is present? Outline the test, and give the results for both ions.<br>d) In the reaction between iron and bromine, iron atoms change into iron ions and bromine molecules change into bromide ions.<br>i) Which of these changes is a reduction? Explain your choice.<br>ii) If iodine was used instead of bromine, predict how the rate of the reaction would change (increase, decrease or stay the same). Give a reason for your answer.