Let $$A = \{1, 2, 3, \ldots, 2008\}$$, $$B = \{1, 2, 3, \ldots, 1004\}$$.
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Let $$A = \{1, 2, 3, \ldots, 2008\}$$, $$B = \{1, 2, 3, \ldots, 1004\}$$.
Both sets are finite, so there are exactly $$1004^{2008}$$ functions from A to B, a finite number; this rules out a and d. Since A has more elements than B, onto functions exist (so b is false) but no function can be one-one (so c is false). Hence the answer is "None of these".
The number of equiangular octagons fixing 6 consecutive sides is
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All the numbers between 1947 and 2008 are written, including 1947 and 2008. From the list, all the multiples of 3 and 5 are struck off. The sum of the remaining numbers is
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A square ABCD is inscribed in a quarter circle where B is on the circumference of the circle and D is the center of the circle. The length of diagonal AC of the square, if the circle's radius is 5, is
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$$50 \times 50 \times 50 \times \cdots$$ (where there are a hundred 50s) is how many times $$100 \times 100 \times 100 \times \cdots$$ (where there are fifty 100s)?
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$$a$$, $$b$$ are positive integers. A is the set of all divisors of $$a$$ except for $$a$$. B is the set of all divisors of $$b$$. If $$A = B$$ then which of the following is a wrong statement?
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The domain of the real function $$f(x) = \sum_{k=1}^{2008} \frac{1}{\sqrt{|x - k|}}$$ is __________
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The number of points P strictly lying inside an equilateral triangle ABC such that the sum of the perpendicular distances from P to the three sides of the triangle is minimum is________
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Positive integers $$a$$, $$b$$ are such that both are relatively prime and less than or equal to 2008, $$a^2 + b^2$$ is a perfect square and that $$b$$ has the same digits as $$a$$ in the reverse order. The number of such ordered pairs $$(a, b)$$ is _________
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Let ABCD be a square. E, F, G, H be the mid points of its sides $$\overline{AB}$$, $$\overline{BC}$$, $$\overline{CD}$$, $$\overline{DA}$$ respectively. Let P, Q, R, S be the points of intersection of the line segments $$\overline{AF}$$, $$\overline{BG}$$, $$\overline{CH}$$, $$\overline{DE}$$ inside the square. The ratio of the areas $$\triangle PQRS \colon \triangle ABCD$$ is _________ (give the value of the ratio as a decimal)
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The number of elements in the set $$\{(a, b, c) \mid a, b, c \text{ are three consecutive integers in some order}, a + b + c = abc\}$$ is __________
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The sum of all positive integers for which the quotient and remainder are equal if the number is divided by 2008 is __________
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Is $$y$$ a real function of $$x$$ in the equation $$|y + 2008| = x^2 + 2$$?
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The people living on Street 'S' of Y-City all decide to buy new house numbers so they line up at the only Hardware store in order of their addresses 1, 2, 3, ... If the store has 100 of each digit, what is the first address that won't be able to buy the digits for its house number?
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Let ABCD be a quadrilateral such that AB is perpendicular to BC, AD is perpendicular to BD and $$AB = BC$$, $$BD = a$$, $$AD = c$$, $$CD = x$$. Find $$x$$ in terms of $$a$$ and $$c$$.
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For what pair wise different positive integers is the value of $$\frac{a}{a + 1} + \frac{b}{b + 1} + \frac{c}{c + 1} + \frac{d}{d + 1}$$ an integer?
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Side AB of rectangle ABCD is 2 units long and side AD is 3 units. E is a point on the line AC such that C is the mid point of the line segment AE. What is the length of line segment BE?
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How many number of integers are there between 2008 and 20082008 including 2008 and 20082008 such that the sum of the digits in the square is 42?
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Let $$S = \{1, 2, 3, \ldots, 2008\}$$. Find the number of elements in the set $$\{(A, B) \mid A \cup B = S\}$$
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A square is drawn in side a triangle with sides 3, 4 and 5 such that one corner of the square touches the side 3 of the triangle, another corner touches the side 4 of the triangle, and the base of the square rests on the longest side of the triangle. What is the side of the square?
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State and prove the test of divisibility of a positive integer $$a$$ by 11.
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A square cake 6 in by 6 in and 3 in tall was cut into four pieces of equal volume as shown in the figure. Determine how far in (in inches) from the side of the cake the cuts should be made? (i.e., $$x = $$ ?)
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Solve the following simultaneous equations for $$a$$ and $$b$$<br>$$a\sqrt{a} + b\sqrt{b} = 183$$, $$a\sqrt{b} + b\sqrt{a} = 182$$
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$$f$$ and $$g$$ are two real variable real valued functions defined by $$f(x) = x + 2$$ if $$x \ge 0$$, $$f(x) = 2x - 3$$ if $$x < 0$$, and $$g(x) = 3x + 7$$ if $$x \ge 2$$, $$g(x) = x - 1$$ if $$-2 < x < 2$$, $$g(x) = x + 1$$ if $$x \le -2$$. Find $$gof$$.
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Two liquids of masses 600 g and 500 g and densities $$0.6\text{ g cm}^{-3}$$ and $$1.5\text{ g cm}^{-3}$$ are combined to form a homogenous mixture. A solid block of mass 420 g just floats in this mixture. Find the density of the block material. If the equal volumes of the two liquids were mixed instead, would this block float in that mixture? If it floats find the fraction of the block that is visible above the mixture.
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A lens of focal length $$f$$ produces an image of an object located 30 cm on one side of it at a distance of 60 cm on the other side. If the lens is replaced by another of $$\frac{3f}{4}$$, where would the image form? If the image has to form at the earlier position by what distance should the object be shifted?
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The devices $$D_1$$ and $$D_2$$ shown (below left) are called diodes and have an interesting property. When current flows in the direction of the arrow the resistance is low and is called forward bias resistance. If the current were to flow in the opposite direction the resistance is high and it is called reverse bias resistance. For $$D_1$$ forward bias resistance is 20 Ohm and reverse bias resistance is 600 Ohm. For $$D_2$$ forward bias resistance is 30 Ohm and reverse bias resistance is 200 Ohm. Find the equivalent resistance between A and B if (i) A is at a higher potential (ii) B is at a higher potential
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Rays of light are incident symmetrically on a sphere of radius $$R$$ as shown (below middle). Determine the refractive index of the sphere material if they meet at P (You need to write the correct expression for refractive index). Can the rays be made to meet at C, the center of the sphere. If not, how should the rays be incident to meet at C.
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If the measured potential difference between A and B is 5.4 V, Find the rate of heat production in the ring (above right). Also find the rate of heat production in sections ACB and ADB.
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Two identical rectangular tanks are connected by a small tube at the bottom. In the left tank contains liquid of density $$d_1 = 800\text{ kg/m}^3$$ to a height $$h_1 = 30\text{ cm}$$. The right tank contains liquid of density $$d_2 = 1600\text{ kg/m}^3$$, to a height $$h_2 = 20\text{ cm}$$. The two liquids don't mix. a) Find the heights of the respective columns, if a small tube connects them at the bottom. b) Where should a horizontal tube be connected, without causing any flow between the tanks?
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Two candles of identical lengths and different thicknesses were lit. Half of the thin candle and one third of the thick candle could be seen to burn down in 15 minutes. How long (in minutes) after the candles had been lit was the ratio of their lengths 4.
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The average daily loss of water of a human body is 0.8 kg, evaporated by the body. How much food (in kg) has to be consumed in order to compensate for the loss of energy due to the evaporation if human body can use $$70\%$$ of the energy provided by the decomposition of food. The heat of combustion of food is 6000 kJ/kg, the heat of vaporisation of water is 2440 kJ/kg
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A train of length 300 m passes through two tunnels with a gap of 400 m between them. The tunnels are of length 800 m and 600 m respectively. From the moment it just emerges from the first tunnel to the moment it just disappears in to the second tunnel time elapsed is 60 s. Find the time elapsed (in s) between the moments the train just enters the first tunnel to the moment it just leaves the second tunnel.
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Fiber optic cable of length $$L$$ has an inner core surrounded by an outer jacket. Light can travel along the length of the cable through the inner and outer portions. The refractive indices of the material of the inner and outer portions are $$\mu_1$$ and $$\mu_2$$. Find the time interval between the moments when a light pulse entering one end of the cable arrives at the other end traveling through the inner and outer portions.
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A person's heart does approximately a work of 1.6 J during a beat. How much work does it do during a day? By how much does the temperature of 10 kg of water rise if it absorbs this much of energy?
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Find the ratio $$\frac{E_i}{E_f}$$, where $$E_i$$ and $$E_f$$ are the energies of a neutron before and after interaction with the nuclei of the moderator in a nuclear reactor.
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A stone projected vertically upward on some planet reaches a maximum height $$h$$. Find the ratio of times (earlier to later, as a decimal) after projection when it is at a height of $$\frac{5h}{9}$$.
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A horizontal conductor is oriented north south and carries some current. A positively charged particle located vertically above it and having a velocity directed northward experiences an upward force. What is the direction of the force if this charged particle were located to the east of the conductor and had a velocity directed towards the conductor.
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Explain the following. The value of K (equilibrium constant, rate constant, solubility constant) changes with temperature. But radioactive decay constant is independent of temperature.
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Explain the following. On expose to air, caustic soda becomes liquid and after some time it changes to white powder.
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Explain the following. Aqueous solution of $$\text{AlCl}_3$$ behaves acidic towards litmus while that of NaCl not.
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Explain the following. Both coke and flux are used in smelting process of metallurgy.
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Explain the following. Bottle containing NaOH solution should not be closed by glass lid.
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Differentiate the following. An equivalent in acid-base reaction and an equivalent in oxidation-reduction reaction
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Differentiate the following. Acid strength and concentration
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Differentiate the following. Mixed salt and Double salt
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Differentiate the following. Complex salt and Simple salt
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Contrast the action of heat on the following (with equations)<br>(a) $$\text{Na}_2\text{CO}_3$$ and $$\text{CaCO}_3$$<br>(b) $$\text{MgCl}_2 \cdot 6\text{H}_2\text{O}$$ and $$\text{CaCl}_2 \cdot 6\text{H}_2\text{O}$$<br>(c) $$\text{Ca(NO}_3)_2$$ and $$\text{NaNO}_3$$
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A fuel gas contains carbon and hydrogen only. Burning a small sample of it in oxygen gives 3.38 gm carbon dioxide, 0.69 gm of water and no other products. A volume of 10 lit (measured at STP) of this gas is found to weigh 11.6 gm.<br>(a) Calculate (i) empirical formula (ii) molecular mass of the gas (iii) molecular formula<br>(b) Write one single step preparation of the gas<br>(c) Does it react with bromine? If so what colour change do you expect?<br>(d) What do you expect the calorific value of the gas with reference to marsh gas?
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At a given temperature, the degree of ionization (fraction dissociated) of water is found to be $$1.8 \times 10^{-9}$$. Calculate the ionization constant and the ionic product of water at this temperature.
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Suppose the gas-phase isomerisation reactions $$A \rightleftharpoons B$$ ($$K_1$$); $$A \rightleftharpoons C$$ ($$K_2$$); $$B \rightleftharpoons C$$ ($$K_3$$) reach equilibrium at a fixed temperature. Express the equilibrium mole fractions of A, B and C in terms of equilibrium constants, $$K_1$$, $$K_2$$ and $$K_3$$.
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A sample of $$\text{P}_2\text{O}_5$$ contains some impurity. 0.405 gm sample is reacted with water and the resulting solution requires 42.5 ml of 0.25 M NaOH solution. The salt resulted is monobasic acid salt. Calculate the percent impurity.
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What will be the resultant pH when 200 ml of an aqueous solution of HCl (pH = 2) is mixed with 300 ml of an aqueous solution of NaOH (pH = 12).
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$$2\text{COF}_2(g) \rightleftharpoons \text{CO}_2(g) + \text{CF}_4(g)$$, with forward and backward rate constants $$k_f$$ and $$k_b$$. $$K = 2$$ for the reaction at $$1000^\circ\text{C}$$. If a 5L mixture contains 0.105 mole $$\text{COF}_2$$, 0.22 mol $$\text{CO}_2$$ and 0.055 mol $$\text{CF}_4$$ at $$1000^\circ\text{C}$$.<br>(a) Will the mixture be at equilibrium?<br>(b) If the gases are not at equilibrium, in which direction will a net reaction occur?<br>(c) What is the amount of each gas present when equilibrium is established.<br>(d) What are the rate equations for the forward and backward reactions?<br>(e) Compare the rate constants at (i) initial stage and (ii) equilibrium stage
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For the reaction<br>(i) $$A \rightleftharpoons B$$; $$K_c = 2$$<br>(ii) $$B \rightleftharpoons C$$; $$K_c = 4$$<br>(iii) $$C \rightleftharpoons D$$; $$K_c = 6$$<br>For the reaction $$A \rightleftharpoons D$$ what is the $$K_c$$?
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Four elements coded A, B, C and D form a series of substances e.g. AB, $$\text{B}_2$$, $$\text{CB}_3$$, $$\text{DB}_2$$ and $$\text{DB}_3$$. If the atomic number of these elements are not necessarily in the order are 13, 19, 26 and 35. Write down extra nuclear electronic structures of these elements. From this information and the formula of the compounds, allocate A, B, C and D with appropriate atomic numbers. Mention the nature of bonding in each of them.
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Calculate the molecular weight of hydrogen fluoride if density of this gas is 3.17 g/L at 300K and 1.0 atm pressure. What information can you deduce from the result?
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Anhydrous copper sulphate turns to blue from colourless on expose to atomosphere and again make it colourless by some substance (X). When copper sulphate is added to water a blue colour solution is resulted with some turbidity. The turbidity disappears by the addition of a drop of HCl solution. On placing Iron rod in it, a greenish solution is resulted. A chocolate like precipitate is resulted when a drop of potassium ferrocyanide solution is added to the blue colour solution of copper sulphate. Black precipitate is resulted by passing $$\text{H}_2\text{S}$$ gas through copper sulphate solution but the precipitate is soluble in conc. $$\text{HNO}_3$$.<br>(a) What is X?<br>(b) Write the chemical reactions to explain the observations.
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