IIT Ramaiah SAT May 6 2007 Question Paper

Real numbers $$x_1, x_2, \ldots, x_{2007}$$ are chosen such that $$(x_1, x_2), (x_2, x_3), \ldots, (x_{2006}, x_{2007})$$ are all points on the graph of $$y = \frac{1}{x - 1}$$.

The consecutive sides of an equiangular hexagon measure $$x$$, $$y$$, 2, 2006, 3, 2007 units

ABCD is a convex quadrilateral

A lattice point in a plane is one both of whose coordinates are integers. Let O be $$(\sqrt{2}, 1)$$ and P any given lattice point. Then the number of lattice points Q, distinct from P, such that $$OP = OQ$$ is

i) $$f(x, y)$$ is the polynomial $$f_0(x)y^{2007} + f_1(x)y^{2006} + f_2(x)y^{2005} + \cdots + f_{2006}(x)y + f_{2007}(x)$$, where each $$f_i(x)$$ is a polynomial in $$x$$ with real coefficients, and ii) $$(x - \alpha)$$ is a factor of $$f(x, y)$$, where $$\alpha$$ is a real number. Then

$$(b - c)(x - a)(y - a) + (c - a)(x - b)(y - b) + (a - b)(x - c)(y - c)$$ is

For the purpose of this question, a square is considered a kind of rectangle. Given the rectangle with vertices $$(0, 0)$$, $$(0, 223)$$, $$(9, 223)$$, $$(9, 0)$$, divided into 2007 unit squares by horizontal and vertical lines. By cutting off a rectangle from the given rectangle, we mean making cuts along horizontal and (or) vertical lines to produce a smaller rectangle. Let $$m$$ be the smallest positive integer such that a rectangle of area $$m$$ cannot be cut off from the given rectangle. Then $$m =$$ _________

A line has an acute angled inclination and does not pass through the origin. If it makes intercepts $$a$$ and $$b$$ on x-, y-axes respectively, then $$\frac{|ab|}{ab} =$$ _________

If $$k$$ is a positive integer, let $$D_k$$ denote the ultimate sum of digits of $$k$$. That is, if $$k$$ is a digit, then $$D_k = k$$. If not, take the sum of digits of $$k$$. If this sum is not a single digit, take the sum of its digits. Continue this process until you obtain a single digit number. By $$D_k$$ we mean this single digit number. $$\{D_p \mid p \text{ is a positive multiple of } 2007\}$$ = _________, in roster form. (The set has a single element; give it.)

The digits of a positive integer $$m$$ can be rearranged to form the positive integer $$n$$ such that $$m + n$$ is the 2007-digited number, each digit of which is 9. The number of such positive integers $$m$$ is ________.

$$\overline{AB}$$ and $$\overline{CD}$$ are chords of a circle such that $$\overrightarrow{BA}$$ and $$\overrightarrow{DC}$$ intersect in a point E outside the circle. F is a point on the minor arc BD such that $$\angle FAB = 22^\circ$$, $$\angle FCD = 18^\circ$$. Then $$\angle AEC + \angle AFC$$ (in degrees) = _________.

The quadratic $$ax^2 + bx + a = 0$$ has a positive coincident root $$\alpha$$. Then $$\alpha =$$ _________.

Explain a way of subdividing a $$102 \times 102$$ square into 2007 non-overlapping squares of integral sides.

ABC is a triangle. Explain how you inscribe a rhombus BDEF in the triangle such that $$D \in \overline{BC}$$, $$E \in \overline{CA}$$ and $$F \in \overline{AB}$$.

Equilateral triangle $$\triangle ABC$$ has centroid G. $$A_1$$, $$B_1$$, $$C_1$$ are points on $$\overline{AG}$$, $$\overline{BG}$$, $$\overline{CG}$$ such that $$\overleftrightarrow{A_1B_1}$$, $$\overleftrightarrow{B_1C_1}$$, $$\overleftrightarrow{C_1A_1}$$ are respectively parallel to $$\overleftrightarrow{AB}$$, $$\overleftrightarrow{BC}$$, $$\overleftrightarrow{CA}$$. If the distance between $$\overleftrightarrow{BC}$$ and $$\overleftrightarrow{B_1C_1}$$ is one-sixth of the altitude of $$\triangle ABC$$, determine the ratio of areas $$\frac{\triangle A_1B_1C_1}{\triangle ABC}$$.

$$P(x)$$ is a polynomial in $$x$$ with real coefficients. Given that the polynomial $$P^2(x) + (9x - 2007)^2$$ has a real root $$\alpha$$, determine $$\alpha$$ and also the multiplicity of $$\alpha$$.

Find the homogeneous function of $$2^{\text{nd}}$$ degree in $$x$$, $$y$$, which shall vanish when $$x = y$$ and also when $$x = 4$$, $$y = 3$$ and have value 2 when $$x = 2$$, $$y = 1$$.

If $$3yz + 2y + z + 1 = 0$$ and $$3zx + 2z + x + 1 = 0$$, then prove that $$3xy + 2x + y + 1 = 0$$.

$$x_3$$ is the $$753^{\text{rd}}$$ AM of 2007 AM's inserted between $$x_1$$ and $$x_2$$. $$y_3$$ is the $$753^{\text{rd}}$$ AM of 2007 AM's inserted between $$y_1$$ and $$y_2$$. Show that $$A(x_1, y_1)$$, $$P(x_3, y_3)$$, $$B(x_2, y_2)$$ are collinear. Determine also the ratio $$AP \colon PB$$.

Lines $$l$$ and $$m$$ intersect in O. Explain how you will construct a triangle OPQ such that $$P \in l$$, $$Q \in m$$, $$\overline{OP}$$ and $$\overline{OQ}$$ are equal in length and $$\overline{PQ}$$ is of given length $$a$$.

i) $$\angle ABC = 120^\circ$$. ii) $$\triangle ACD$$ is equilateral, iii) B and D are on opposite sides of $$\overleftrightarrow{AC}$$. Prove that a) $$\overleftrightarrow{BD}$$ bisects $$\angle ABC$$ and b) $$\overline{BD}$$ is in length equal to the sum of lengths of $$\overline{AB}$$ and $$\overline{BC}$$.

$$a_1, a_2, \ldots, a_{2007}$$ are $$1, 2, \ldots, 2007$$ in some order. If $$x$$ is the greatest of $$1 \cdot a_1, 2 \cdot a_2, \ldots, 2007 \cdot a_{2007}$$, prove that $$x \ge (1004)^2$$.

Prove that for all integers $$n \ge 2$$, $$2^{n-1}(3^n + 4^n) > 7^n$$.

Resolve $$x^8 + y^8$$ into real quadratic factors.

Two liquids of densities $$1\text{ g/cm}^3$$ and $$3\text{ g/cm}^3$$ are mixed in equal volumes. A cube of external dimension 20 cm has an inner cubical portion of side 10 cm whose density is twice that of the outer portion. If this cube is just floating in the liquid mixture, find the density of the inner portion. If the density of the inner portion is half of this value, what fraction of the volume of the cube will be submerged?

A particle P is located at A. Another particle Q is located at B 200 m south of A. P starts moving at 2 m/s towards east at some moment and simultaneously Q starts moving at 4 m/s. (i) If Q moved first towards A and on reaching A quickly changed direction and moved towards east, find the time it takes to catch up with P. (ii) If instead Q aimed at a point C east of A at which it arrives at the same time as P, find the elapsed time. (iii) What should be the minimum speed of Q to catch up with P before it moves 50 m from its initial position? (Assume Q catches up by aiming at the position P would reach at the same time as it does)

A lens $$L_1$$ produces a twice-magnified image of an object on a screen located 90 cm from the object. Without disturbing the object and the screen, another lens $$L_2$$ of equal focal length is placed in contact with $$L_1$$. This combination is adjusted to produce the image of the object on the screen again. What is the new magnification? (The two lenses together act as a single lens. The reciprocal of the focal length of the equivalent lens is equal to the sum of the reciprocals of the focal lengths of the individual lenses)

Rate at which energy is received by a square meter of area facing the sun is $$1400\text{ W m}^{-2}$$. Assume $$50\%$$ efficiency for conversion of this energy in to heat and find the amount of time it takes for the temperature of 10 kg of water to rise through $$10^\circ\text{C}$$ due to energy received by a $$2\text{ m} \times 2\text{ m}$$ area. Assume the surface always faces the sun. How many kilowatt-hours of energy is this equal to? How long would a 40 W lamp glow if this energy were converted in to electrical energy at $$25\%$$ efficiency.

Fission of a nucleus of $$\text{U}^{235}$$ releases about 250 MeV of energy. If 235 g of $$\text{U}^{235}$$ is subjected to fission, find the energy (in joule) evolved in the process. Find the fraction of the mass of uranium that disappears in the fission process. (Assume 1 amu is 900 MeV for the purpose of calculation.)

Waves reaching a point from two different paths from two different sources interfere. If the path lengths differ by integral multiples of a wavelength the interference will be constructive and if the path lengths differ by odd multiples of half wavelengths interference will be destructive. In the given situation waves travel from A to B along path 1 and path 2. If the frequency of the waves is $$n$$ and the velocity of the waves is $$v$$, find the least difference between length of paths 2 and 1 that produces constructive interference at B. If the velocity along path 1 is $$v_1$$ and along path 2 is $$v_2$$, find the least difference between length of paths 2 and 1 that produces destructive interference at B.
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Two particles are moving round concentric circular paths of radii $$r_1$$ and $$r_2$$ ($$r_2 = 3r_1$$) and are closest to each other at a certain moment. The particle on the outer path has slower speed $$v_2$$ as compared to the inner one which has a speed $$v_1$$ ($$v_1 = 2v_2$$). Find the least time interval between their successive appearances at their initial positions at the same time.

A metal rod R tightly fits in to a ring as shown. Will there be any distortion in the ring on heating or cooling it? Discuss possible cases.
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An open room contains air (Assume air behaves as an ideal gas). Does the air in the room obey Charles law ($$P \propto T$$)

It is not as hot at some distance at the side of a flame as the same distance above it. Why is this so?

The bob of a simple pendulum is made of metal. If a wooden bob replaces the metallic bob, how do the time periods of oscillations compare?

A metallic vessel is placed on a non-conducting stand. The inner and outer surfaces of the vessel are connected to two gold leaf electroscopes A and B respectively. A charged sphere suspended by a non-conducting string is slowly lowered in to the vessel. How do the leaves of the electroscopes A and B behave? How will the behaviour change if the sphere makes contact with the inner surface of the vessel?

A particle starts from rest and accelerates along a straight line at $$a_1$$ for some time and then at $$a_2$$ for an equal time covering a distance $$d_1$$. If the acceleration for the first interval were $$a_2$$ and for the second $$a_1$$ the distance travelled is $$d_2$$. How do the distances $$d_1$$ and $$d_2$$ compare? Support your answer either qualitatively or quantitatively.

A thin bar magnet of magnetic moment $$M$$ is bent to form an arc of a circle making an angle $$\theta$$ (radians). Find its new magnetic moment.

The reaction between dichromate ions and iron (II) ions can be carried in the presence of either dilute HCl (or) sulphuric acids. But the dilute HCl cannot be used to acidify the solution for titration between Iron (II) and permanganate ion system. The reason for this is ________.

Acetic acid is a weak acid and hydrochloric acid is a strong acid it follows that the<br>(A) pH of the 0.1M HCl will be 1<br>(B) Solution containing 0.1 mole acetic acid and 0.1 mole of sodium acetate will be a good buffer<br>(C) pH of the 0.1M HCl will be less than that of 0.1M acetic acid<br>(D) pH of a solution formed by mixing equimolar quantities of sodium hydroxide and HCl will be greater that of a similar solution formed from sodium hydroxide and acetic acid.<br>The correct statement(s) is/are ________________

A metal oxide has formula $$\text{X}_2\text{O}_3$$ it can be reduced by hydrogen to give free metal and water. 0.1596 g of metal oxide requires 6 mg of hydrogen for complete reduction. The atomic weight of the metal is __________.

Helium is mixed with the oxygen gas for deep sea divers. The percent by volume of oxygen gas in the mixture of the diver has to submerge to a depth where the total pressure is 4.2 atm and partial pressure of oxygen is maintained at 0.2 atm at this depth is ___________.

In the reaction $$A + B \rightarrow$$ products, if the concentration of B is kept constant and concentration of A is doubled, the rate becomes four times of initial rate and when A is kept constant and concentration of B is doubled rate remains the same. This indicates that reaction rate is proportional to the __________ of concentration of ___________ and rate equation is _____________.

One ________ of a substance is equal to its atomic or molecular mass expressed in __________ and contains $$6.023 \times 10^{23}$$ chemical particles, which is known as ______________ of particles. Since 1 mole of any chemical substance contains same number __________ as 1 mole of any other chemical substance, we can calculate the weight of one chemical that will react with a given weight of other by using their _________ in other calculations.

15 moles of $$\text{H}_2$$ and 5.2 moles of $$\text{I}_2$$ are mixed and allowed to attain equilibrium. Concentration of HI is found to be 10 moles the equilibrium constant for the formation of HI is ____________.

600 ml of a mixture of $$\text{O}_2$$ and $$\text{O}_3$$ weighs 1 gram at NTP. The volume of ozone (in ml) in the mixture _______.

At $$27^\circ\text{C}$$ 4.6 grams of $$\text{N}_2\text{O}_4$$ undergoes decomposition and exerts an equilibrium pressure of 1.6 atmosphere when vapourised in a 1 lit flask. The fraction of $$\text{N}_2\text{O}_4$$ dissociated is ________. [$$\text{N}_2\text{O}_4(g) \rightleftharpoons 2\text{NO}_2(g)$$]

A compound contains $$28\%$$ nitrogen and $$72\%$$ metal by mass. 3 atoms of the metal combine with 2 atoms of the nitrogen. The atomic weight of the metal is __________

The V.D of Gas is 15. 15 g of this gas will occupy a volume of ______ cc at $$273^\circ\text{C}$$ and 1 atm.

You are given 500 ml of 2N HCl and 500 ml of 5N HCl. The maximum volume (in ml) of the 3 M that you can make from these two solutions is ___________.

The equilibrium constant for the reaction $$2A + 2B \rightleftharpoons 2C + 2D$$ is 100. The equilibrium constant for the $$A + B \rightleftharpoons C + D$$ at same temperature is _________.

The solubility product of sparingly soluble salt $$\text{Ag}_2\text{CrO}_4$$ is $$4 \times 10^{-12}$$. The molar solubility of the salt (in mol per litre) is _____________

For the preparation of HBr in the lab the corresponding bromide salt should be treated with __________ acid.

Differentiate the following<br>(a) Redox reaction - disproportionation reaction<br>(b) Primary standard substance - Secondary standard substance<br>(c) Electrolysis - Hydrolysis<br>(d) Gangue - Slag<br>(e) Lattice energy - Hydration energy<br>(f) Equilibrium constant - Rate constant<br>(g) Ionic product - Solubility product<br>(h) Activation energy - Threshold energy<br>(i) Valence - Oxidation number<br>(j) Allotropy - Enantiotropy

A compound of oxygen and nitrogen has the composition 1.52 g N and 3.47 g of oxygen. The molar mass of this compound is known to be between 90 and 95 g. Determine the molecular formula and molecular mass of compound.

A transition metal X forms an oxide of formula $$\text{X}_2\text{O}_3$$. It is found that only $$50\%$$ of X atoms in this compound are in the $$+3$$ oxidation state. The only other stable oxidation states of X are $$+2$$ and $$+5$$. What percentage of X atoms are in the $$+2$$ oxidation state in this compound?

Many diabetics must determine their blood sugar content daily. The measurements provided by the devices used for such purpose are in milligrams per deciliter.<br>(a) What is the concentration of glucose [$$\text{C}_6\text{H}_{12}\text{O}_6$$] in grams per liter when a reading of 95 is observed?<br>(b) What is the concentration in moles per litre?

A student was given a sample of colourless solution containing three cations and was asked to identify the cations. The student carried out a series of reactions as shown in the flow chart below.
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(i) From the information above, suggest three cations which could be in the sample and would form the precipitates A, B and C. Give the formula for each precipitate,<br>(ii) Give an additional test which would help to confirm the presence of each cation,<br>(iii) Name any other cations that would fit the requirements for precipitates A, B and C.