IIT Ramaiah SAT May 3 2009 Question Paper

$$\log_{\frac{1}{2009}} 2009 =$$

Let $$a$$, $$b$$, $$c$$ be the lengths of the sides of a triangle $$ABC$$. Let $$\alpha$$, $$\beta$$, $$\gamma$$ be positive integers and $$p = \beta a + \gamma b + \alpha c$$, $$q = \gamma a + \alpha b + \beta c$$ and $$r = \alpha a + \beta b + \gamma c$$. Then $$p$$, $$q$$, $$r$$ are lengths of the sides of a triangle

Let $$f \colon \mathbb{R} \rightarrow \mathbb{R}$$ be defined by $$f(x) = x^2$$ for all $$x$$ in $$\mathbb{R}$$. Let $$A = \{4, 3, 0, -1\}$$. Then

If $$1216451{\ast}0408832000$$ is equal to $$19!$$, where $$\ast$$ denotes a digit, then the missing digit represented by $$\ast$$ is

A polynomial $$f(x)$$ is said to be reducible if it can be written as $$f(x) = g(x)h(x)$$ where $$\deg(g(x)) > 1$$ and $$\deg(h(x)) > 1$$. Otherwise it is said to be irreducible. If $$f(x)$$ is a polynomial of degree 2009, Choose the correct statement from the following.

The number of positive integers less than 2009 that have exactly three prime factors, not necessarily distinct, such that each prime factor is greater than 10 is _____________

If the odd natural numbers are arranged as shown, then the extremities of the row containing the number 2009 are _____<br>1<br>3 5<br>7 9 11<br>13 15 17 19<br>21 23 ...

The number of triangles having a fixed base of $$a$$ units and area 2009 square units is _________

The coefficient of $$x$$ in the expansion of $$\left(\sqrt{7}\,x^4 + \frac{13}{4}x^3 + x^2 + 2x - 1\right)^{2009}$$ is ___________

There are four circles in the figure. Two of them have radius 2 and the smallest one has radius 1. The radius of the largest circle, if the area of the dashed part is equal to the area of the dotted part, is __________
[image]

How many positive integers less than 2009 are divisible by any one perfect square greater than 1678?

The square ABCD of side $$a$$ and the square BEFG of side $$b$$ are drawn next to each other as shown in the figure. Express in terms of $$a$$ and $$b$$ the area of the quadrilateral formed by the mid points of the line segments AB, BE, FC and DG.
[image]

Find out the unequal pairs of sets, if any, among the following sets.<br>i) $$(A - B) \cup (B - A)$$ ii) $$(A \cup B) - (A \cap B)$$ iii) $$(A \cup B) \cap (A \cap B)'$$<br>iv) $$(A \cap B') \cup (B \cap A')$$ v) $$(A \cup B) \cap (A' \cup B')$$

If ABC is a triangle with AB, BC measuring $$p$$ and $$q$$ units respectively, find the maximum area possible for triangle ABC.

Two real numbers $$x$$ and $$y$$ are varying such that their sum is 2009. Find the values of $$x$$ and $$y$$ such that their product is maximum. (Both values are equal; give that value.)

Solve $$2x^2 y - 3x^2 + 4y = 2009$$ for $$x$$ and $$y$$ in positive integers.

The lower part of the house (see the Figure) in the circle is a square, and the top is an equilateral triangle. Find a relation between the length of the side of the house and the radius of the circle
[image]

A function $$f \colon \mathbb{N} \rightarrow \mathbb{Z}$$ is defined by $$f(n) = \frac{n}{2}$$ if $$n$$ is even, and $$f(n) = -\frac{n - 1}{2}$$ if $$n$$ is odd. Is $$f$$ a bijection?

Solve the system of equations $$ax + 2y = b$$ and $$2x - y = 3$$ for all real values of $$a$$ and $$b$$.

Is it possible to insert exactly 125 Harmonic Means between $$-\frac{1}{49}$$ and $$\frac{1}{2009}$$? If so, what is the $$6^{\text{th}}$$ HM?

A solid sphere and a hollow sphere of same external radius and made of same material are floating in water. Solid sphere has three fourths of its volume submerged and the hollow sphere is half submerged. What fraction of the volume of hollow sphere is hollow?

Two men are running on a circular track. They start from the same position. If they take 3 minutes and 5 minutes to run once around the track, find the time interval between successive crossings and the number of times they cross each other in an hour if they travel the path in (i) opposite sense (ii) same sense

A lens of focal length 20 cm forms the image of a bright object kept 30 cm from it on a screen. If the lens is replaced by another lens, the object has to be moved by 5 cm to obtain a sharp image on the screen. Find the possible values of the focal length of the second lens.

A wire of resistance $$R$$ is bent to form a ring. Contacts can be made at points A and B on the ring. A is a fixed contact and B can not be closer to A than a quarter of the perimeter of the ring. Find the maximum and minimum possible resistances between various choices of the contact point B.
[image]

A row-boat travels downstream on a river at a speed of 8 m/s with respect to the shore. A motor-boat comes from the opposite direction at a speed of 10 m/s with respect to the water. 10 seconds after meeting each other they are at 160 metres from each other. At what speed (in m/s) does the river flow

A piece of ice at $$-10^\circ\text{C}$$ is heated to $$-1^\circ\text{C}$$ using a certain quantity of energy. Then another 20 times as much energy is necessary to finally obtain water. Using that the specific heat of ice is half of the specific heat of $$4.2\text{ kJ}/(\text{kg}\,^\circ\text{C})$$ of water, determine the heat of fusion of ice (in kJ/kg) from the above measurement data.

The electrical power of a vacuum cleaner is 1 kW. It expels 50 litre of air per second with a speed of 80 m/s. What percent of the total power is in the form of kinetic energy of air? (Density of the flowing air is $$1.2 \times 10^{-3}\text{ g cm}^{-3}$$)

A stone is projected vertically upward from some height at $$2v$$. Some $$\Delta t$$ later another stone is projected vertically upward at $$v$$ from the same position. If they reach the initial position at the same time, find $$v$$.

A nuclear reactor uses 1 kg of uranium enriched to $$1\%$$ in one hour. Assuming $$60\%$$ efficiency find the electrical power generated. Assume each fission of uranium nucleus produces 200 MeV of energy and 238 g of Uranium contains $$6.0 \times 10^{23}$$ atoms. What fraction of mass has disappeared in the process?

An alpha particle moving vertically downward at right angles to a uniform magnetic field experiences a northward force. What is the direction of the force on a beta particle moving northward?

A ray of light is incident at E at $$45^\circ$$ as it passes from air in to a medium across boundary AB. It then is incident at F on boundary CD and emerges in to air. Sum of the angles at E and F in the medium is $$60^\circ$$. Find the deviation (in degrees) of the ray at F. Refractive index of the medium with respect to air is $$\sqrt{2}$$.
[image]

Nucleons in a nucleus are held together by attractive short range forces. These forces are stronger than columbic forces and act between neighbouring nucleons. These forces are charge independent.<br>Explain in one are two sentences why<br>(i) A large nucleus can not consist only of protons<br>(ii) All matter is not nuclear.

X rays are produced when a beam of electrons strikes a target. If $$n$$ electrons strike the target in one second, find (i) the beam current in an x ray tube. If electrons are accelerated by $$V$$ before striking the target, find the rate at which heat is produced in the target if $$90\%$$ of their kinetic energy is converted to heat.

To the mixture of aqueous solution of a Nitrate salt and Ferrous sulphate, con. $$\text{H}_2\text{SO}_4$$ is slowly added through the slanted walls of the test tube. A brown ring is formed little above (nearly middle) the bottom of the test tube.<br>a) What is the significance of this test?<br>b) Which acts as reductant?<br>c) What is the change of oxidation state of oxidant?<br>d) What is the chemical composition of the Brown's ring?<br>e) Why con. $$\text{H}_2\text{SO}_4$$ is added only through the slanted walls of the test tube?<br>f) Write the related chemical equations for above chemical changes.

It is observed that a solid mixture of chloride salt and potassium dichromate is heated with con. $$\text{H}_2\text{SO}_4$$ red vapours are formed. When these vapours are passed into aqueous caustic soda solution yellow solution is formed. This on further treatment with lead acetate solution, a yellow precipitate is formed.<br>a) What is significance of the sequence of reactions?<br>b) What is the name and formula of the red vapours?<br>c) Why red vapours are changed to yellow solution? And write its formula<br>d) What is the yellow precipitate formed in the last step and its formula?<br>e) Write the related chemical equations for above chemical changes.

a) What is carbohydrate?<br>b) How atmospheric $$\text{CO}_2$$ is transformed into carbohydrate in green plants? Indicate through chemical equation?<br>c) What is Tollen's test?<br>d) What is significance of Tollen's test in carbohydrates?<br>e) What chemical changes takes place during the test?<br>f) Glucose gives Tollen's test but not sucrose why?<br>g) What is oxidant in this reaction?<br>h) What is the formula of the red precipitate formed in Benedict's test conducted on carbohydrates?

0.1 moles/lit of a weak acid solution contains 0.001 moles/lit of the conjugate base ($$\text{B}^-$$).<br>a) What is the dissociation constant ($$K_a$$) of the acid?<br>b) Calculate the pH of 0.1 M solution of weak acid (AH)?<br>c) This acid is neutralized by adding required quantity of aqueous NaOH solution. What is the approximate the range of pH of solution?<br>d) To the original solution HCl is added. What happens to the ratio of conjugate base to the acid and how $$K_a$$ of the acid is affected?<br>e) To the above original solution sodium salt of same weak acid is added. What happens to the pH of the solution?

In each sub question below a statement S and an explanation E is given. Choose the correct answer from the codes A, B, C, D, E given below for each question.<br>(A) S is correct but E is incorrect<br>(B) S is incorrect but E is correct<br>(C) Both S and E are correct and E is correct explanation of S<br>(D) Both S and E are correct but E is not correct explanation of S<br>(E) Both S and E are incorrect<br>i) S - The third period contains only 8 elements and not 18 like $$4^{\text{th}}$$ period. E - In III period filling starts from $$3s^1$$ and complete at $$3p^6$$ whereas in IV period it starts from $$4s^1$$ and complete after $$3d^{10}$$ and $$4s^2$$<br>ii) S - Transition elements show horizontal as well as vertical relationship. E - This is due to shielding effect and similar electronic configuration<br>iii) S - Cs and $$\text{F}_2$$ combines violently to form CsF. E - Cs is most electropositive and F is most electronegative<br>iv) S - A jump in $$3^{\text{rd}}$$ ionisation energy is noticed in case of alkaline earth metals. E - The jump in ionisation energy is due to change in major energy shell during successive removal of electron<br>v) S - First ionisation energy of Be is more than that $$\text{IE}_1$$ of B. E - Removal of electron in Be occurs from 2s subshell whereas in B from 2p subshell<br>vi) S - Formation of $$\text{Cl}^-$$ is exothermic whereas formation of $$\text{O}^{2-}$$ endothermic. E - $$\text{EA}_2$$ of oxygen is endothermic and greater than its exothermic value $$\text{EA}_1$$ of oxygen<br>vii) S - $$\text{BiCl}_5$$ does not exist. E - In Bi inert pair effect is more predominant<br>viii) S - Second EA for halogens is zero. E - Fluorine has maximum value of electron affinity

Match each of the molecules in Column I with its property / properties given in corresponding Column II.<br>Column I - a) $$\text{NH}_3$$, b) $$\text{CH}_4$$, c) $$\text{PCl}_5$$, d) $$\text{C}_2\text{H}_4$$, e) $$\text{BeCl}_2$$, f) $$\text{PCl}_3$$, g) $$\text{CO}_2$$, h) $$\text{H}_2\text{O}$$, i) $$\text{SO}_2$$, j) $$\text{SO}_3$$<br>Column II - 1. $$sp^3$$ hybridisation, 2. $$sp^3d$$ hybridisation, 3. $$sp^2$$ hybridisation, 4. $$sp$$ hybridisation, 5. $$sp^3d^2$$ hybridisation, 6. Tetrahedral, 7. Trigonal Bipyramidal, 8. Trigonal pyramidal, 9. Planar, 10. Linear, 11. Collected by downward displacement of air, 12. Refrigerant, 13. V-shape, 14. Bleaching agent, 15. All bond angles are not equivalent, 16. Lighter than air, 17. On burning $$\text{CO}_2$$ and $$\text{H}_2\text{O}$$ are formed