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If $$x$$ and $$y$$ are two positive integers such that $$xy = 2015$$ then the number of distinct values $$x$$ can take is
$$x$$ can be any positive divisor of 2015, with $$y = \frac{2015}{x}$$. Since $$2015 = 5 \times 13 \times 31$$, the number of divisors is $$2 \times 2 \times 2 = 8$$.
The number of values of $$a$$ such that the nine digit number $$8544a5894$$ is divisible by 7 is
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A table is filled with integers as shown below<br>ODD EVEN ODD<br>EVEN ODD EVEN<br>ODD EVEN ODD
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If $$x > y > 0$$, $$x^2 + y^2 = 6xy$$ then $$\frac{x + y}{x - y}$$ is equal to
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$$\sqrt{144} + \sqrt[3]{216} = \sqrt{a} = 6\sqrt{b}$$ then $$\frac{a}{4b}$$ is equal to ______
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A is a two digit positive integer. B is the two digit number obtained by reversing (inter-changing) the digits of A. If A and B are such that $$A \colon B = 4 \colon 7$$ then the number of such A is _______
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In a trapezium $$PQRS$$, the side $$PQ$$ is perpendicular to the parallel sides $$QR$$, $$PS$$ whose lengths are $$x$$ and $$y$$. If the diagonals $$PR$$ and $$QS$$ intersect in $$M$$, then the perpendicular distance from $$M$$ to the side $$PQ$$ is ________
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$$ABCD$$ is a convex quadrilateral. Angles $$BAC$$, $$CAD$$, $$CBD$$, $$BDC$$ are $$50^\circ$$, $$60^\circ$$, $$30^\circ$$, $$25^\circ$$ respectively. If $$E$$ is the point of intersection of the line segments $$AC$$, $$BD$$ then angle $$AEB$$ (in degrees) is ________
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State True or False. Coordinates of three points are $$A(-3, 2)$$, $$B(25, 10)$$, $$C(k, k)$$. If the value of $$AC + BC$$ is minimum then $$k = 6$$.
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State True or False. If $$a$$, $$b$$ are real numbers such that $$(a - b) \colon (a + b) \colon ab = 1 \colon 5 \colon 24$$ then $$(a, b) = (12, 8)$$.
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State True or False. In a circle with Centre $$O$$, $$AB$$, $$BC$$ are two chords each of length 3 units. If the area of the quadrilateral $$OABC$$ is 6 sq. units then the radius of the circle is 5 units.
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State True or False. The function $$f \colon (-\infty, 2] \rightarrow [1, \infty)$$ defined as $$f(x) = x^2 - 4x + 5$$ is a bijection.
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In a square $$ABCD$$ of side 6 units $$P$$, $$Q$$ are mid points of $$BC$$, $$CD$$ respectively. The line segments $$BQ$$, $$DP$$ intersect in $$R$$ then find the area of the quadrilateral $$ABRD$$.
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Solve the system of equations in $$x$$, $$y$$, $$x^2 + xy = p^2 + pq$$, $$y^2 + xy = p^2 - pq$$, where $$p$$, $$q$$ are real numbers, $$p \ne 0$$.
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$$ABCD$$ is a square of side length 1 unit. Two circles $$C_1$$, $$C_2$$ of equal radii are drawn inside the square such that they are touching each other and $$C_1$$ touching the sides $$AB$$, $$BC$$ and $$C_2$$ touching the sides $$CD$$, $$DA$$. Then find the radius.
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Solve $$3^{2x + 1} - 3^{x + 3} - 3^x + 3^2 = 0$$
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27 spherical drops of a liquid are merged to form a single large spherical drop. If the radius of the smaller drop is $$1\text{ mm}$$, find the radius of the larger drop. Also find the ratio surface area of the larger drop to the total surface area of all the drops.
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Equal volumes of two miscible liquids of relative densities 6 and 2 are mixed to form a homogenous mixture. A cube floats on this liquid mixture with half its volume submerged. If instead equal masses of these liquids were mixed, find the fraction of the volume of the cube that would submerge in the mixture.
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A car travels between places A and B covering part of the distance at $$40\text{ kmph}$$ in 1 hour and the remaining distance at $$30\text{ kmph}$$ in 2 hours. In the return journey from B to A along a different path which is $$\frac{3}{2}$$ times longer the speed is $$75\text{ kmph}$$ for the entire journey. Find the average speed (in kmph) of the car for the round trip.
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An electric train is moving at $$36\text{ kmph}$$. When its power is turned off, it stops in $$20\text{ s}$$. If the train were moving at $$54\text{ kmph}$$, find how far ahead (in m) of the station should the power be turned off, to halt the train at the station. Assume same uniform retardation in both the cases.
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A ray of light passing from air in to a medium is incident at $$53^\circ$$. Angle of refraction in the medium is $$37^\circ$$. Find the sine of angle of refraction if the angle of incidence is changed to $$37^\circ$$. (Use $$\sin 37^\circ = \frac{3}{5}$$ and $$\sin 53^\circ = \frac{4}{5}$$)
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A uniform wire is made in to a circular ring. Equivalent resistance between points A and B on the ring which subtend an angle of $$60^\circ$$ at the center of the ring is $$R$$. Find the equivalent resistance between A and B if the angle is increased to $$120^\circ$$ (as a multiple of $$R$$).
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Space above mercury in a barometric tube has a water column of $$27.2\text{ cm}$$. If the atmospheric pressure is $$76\text{ cm}$$ of mercury, find the length of the mercury column above the mercury level in the trough of the barometer. What is the pressure at the top of the mercury column? Assume that space above water is vacuum. Atmospheric pressure is $$76\text{ cm}$$ of Hg and densities of mercury and water are $$13.6\text{ g/cm}^3$$ and $$1\text{ g/cm}^3$$.
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Potential difference between A and B is $$16\text{ V}$$. Find the currents in the $$6\,\Omega$$ and $$8\,\Omega$$ resistances.
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A car is moving on a highway at $$34\text{ m/s}$$. A man is standing by the side of the highway a long distance ahead of the car. Car sounds its horn for a $$5\text{ s}$$ duration. The car passes the man sometime after the horn is sounded. Find the length of time (in s) during which the man hears the sound of the horn. (Speed of sound is $$340\text{ m/s}$$.)
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Energy released per fission of one uranium nucleus $$\text{U}^{235}$$ is $$200\text{ MeV}$$. Find the number of reactions taking place in one minute in a nuclear reactor producing a power of $$800\text{ MW}$$. Assume $$50\%$$ efficiency in the conversion process. Also find the mass lost in 1 hour.
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$$\text{NO}_3^-$$ ion is detected by a popular test known as brown ring test. During this test $$\text{NO}_3^-$$ ion acts as
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One of the phosphorus compound is popularly used as rat poison. The oxidation number of the phosphorus in this compound is
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If there were 9-periods in the periodic table then how many elements would $$9^{\text{th}}$$ period can maximum comprise of
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Energy is emitted when electron jumps from higher energy level to lower energy. In which transition the wave length emitted is minimum
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Which of the following is the correct set of quantum numbers for differentiating electron between calcium and scandium
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As safety measure in public places fire extinguishers are arranged. What compounds are there in fire extinguishers? _______, ________
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On Dewali day while burning a cracker a brilliant green light emitted. Which metal you expect in the cracker ____________
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1 mole of radioactive substance X disintegrates by $$\alpha$$-emission. Its half life time is 10 days. After 20 days the amount of Helium gas liberated at STP (in litres) is _____________
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Silica on reaction with sodium hydroxide produces a compound. The common name of the compound formed is ______________ and its chemical formula is ________________
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In metallurgy in the isolation of metals, in calcinations/roasting process usually ore is finally converted into ____________ form
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Among $$\text{Ni}^{4+}$$, $$\text{Cu}^{2+}$$, $$\text{Mn}^{4+}$$, $$\text{Co}^{3+}$$, $$\text{Cr}^+$$ the species having same number of unpaired electrons ____________
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The concentration of the hydrogen ion in a sample of soft drink is $$2 \times 10^{-3}\text{ M}$$. The pOH of the solution is ____________ and its $$\text{p}K_w$$ is ________
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Arrange in the increasing order of ionic radii for isoelectronic species $$\text{K}^+$$, $$\text{S}^{2-}$$, $$\text{Ca}^{2+}$$, $$\text{Cl}^-$$ ____________
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What is the starting material for the preparation of propane by decarboxylation method ___________
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The solubility product of an electrolyte $$\text{A}_3\text{B}_2$$ if its solubility is $$S$$ ____________
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In the extraction of Mg metal from $$\text{MgCl}_2$$, the NaCl and KCl are added. This is to decrease ______ and increase ______.
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The pH of the pure water at $$25^\circ\text{C}$$ is 7. The pH of the solution after addition of 1 gm of KCl per liter of water is _________
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Hot and conc. NaOH is treated with chlorine gas. The gas undergoes disproportionation. The balanced equation for this reaction is ____________
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The valencies of carbon in alkane, alkene and alkyne respectively are ___, ____, ____
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Ammonia and air mixture is passed over heated platinum gauze at $$800^\circ\text{C}$$ under high pressure conditions. Write balanced equation ________________
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