text:
# Solved papers for J & K CET Engineering J and K - CET Engineering Solved Paper-2003

### done J and K - CET Engineering Solved Paper-2003

• question_answer1) The following is the truth table for

 A B Y 0 0 1 1 0 1 0 1 1 1 1 0

A) NAND

B) AND

C) XOR

D) NOT

• question_answer2) In a full wave rectifier circuit operating from $50\text{ }Hz$main frequency, the fundamental frequency in the ripple would be

A) $\text{25 }Hz$

B) $50\text{ }Hz$

C) $70.7\text{ }Hz$

D) $100\text{ }Hz$

• question_answer3) The current gain a of a transistor is 0.9. The transistor is connected to common base configuration. What would be the change in collector current when base current changes by $4mA$?

A) $1.2\,\,mA$

B) $12\,\,mA$

C) $24\,\,mA$

D) $36\,\,mA$

• question_answer4) What is the name of the level formed due to impurity atom in p-type in the forbidden gap?

A) Donor level

B) Acceptor level

C) Conduction level

D) Forbidden level

• question_answer5) Which of the following is not true?

A) All unit cells are primitive

B) FCC structure is a closed packed structure

C) A unit cell is primitive if it contains lattice points only at its corner

D) A lattice does not contain any atom or molecule

• question_answer6) A radioactive element A decays into B with a half-life of 2 days. A fresh prepared sample of A has a mass of $12\text{ }g$. What mass of A and B are there in the sample after 4 days?

A) $~A=3g,B=9g$

B) $A=6g,B=6g$

C) $A=12g,B=0g$

D) $A=9g,B=3g$

• question_answer7) How many $\alpha$-particle and $\beta$-particles are emitted when uranium nucleus $_{92}^{238}U$ decays to lead nucleus $_{82}^{206}Pb$?

A) $\alpha =6,\beta =8$

B) $\alpha =10,\beta =8$

C) $\alpha =8,\beta =10$

D) $\alpha =8,\beta =6$

• question_answer8) The angular speed of the electric in the nth orbit of Bohr hydrogen atom is

A) directly proportional to n

B) inversely proportional to $\sqrt{n}$

C) inversely proportional to ${{n}^{2}}$

D) inversely proportional to ${{n}^{3}}$

• question_answer9) If the electron in hydrogen atom jumps from the third to second orbit, the wavelength of the emitted radiation in terms of Rydberg constant R is given by

A) $\lambda =\frac{36}{5R}$

B) $\lambda =\frac{5R}{36}$

C) $\lambda =\frac{5}{R}$

D) $\lambda =\frac{R}{6}$

• question_answer10) When a monochromatic point source of light is at a distance of $0.2\text{ }m$from a photocell, the cut-off voltage and the saturation current are respectively ${{\text{V}}_{0}}=0.6$volt and${{I}_{s}}=18.0\,mA$. If the same source is placed 0.6 m away from the photocell, then

A) stopping potential ${{V}_{0}}\text{=}0.2$volt and saturation current ${{I}_{s}}=18.0\text{ }mA$

B) stopping potential ${{V}_{0}}=0.6$volt and saturation current ${{I}_{s}}=18.0\text{ }mA$

C) stopping potential ${{V}_{0}}=0.6$ volt and saturation current ${{I}_{s}}=2.0\,mA$

D) stopping potential ${{V}_{0}}=2.0$ volt and saturation current ${{I}_{s}}=2.0\,mA$

• question_answer11) When light of wavelength $300~nm$(nanometer) falls on a photoelectric emitter, photoelectrons are just liberated. For another emitter, however, light of $600\text{ }nm$wavelength is sufficient for creating photoemission. What is the ratio of the work function of the two emitters?

A) $1:2$

B) $2:1$

C) $4:1$

D) $1:4$

• question_answer12) In Millikan's oil drop experiment, a charged drop of mass $1.8\times {{10}^{-14}}\text{ }kg$ is stationary between the plates. The distance between the plates $0.9\text{ }cm$and potential difference between the plates is$2000\text{ }V$. The number of electons on the oil drop is

A) $10$

B) $5$

C) $50$

D) $20$

• question_answer13) In the experiment for the determination of $\frac{e}{m}$of electrons by the Thomson method, electric and magnetic fields are

A) parallel and both are perpendicular to the motion of the electron

B) both mutually perpendicular and parallel to the motion of electron

C) both mutually perpendicular and also perpendicular to the motion of electron

D) both mutually perpendicular and have no relation with motion of the electron

• question_answer14) Blue colour of sky is due to

A) interference

B) scattering of light

C) dispersion of light

D) sun emits more of blue light

• question_answer15) In Young's double slit experiment distance between source is 1 mm and distance between the screen and source is 1 m. If the fringe width on the screen is 0.06 cm, then K is

A) $6000\text{ }\overset{\text{o}}{\mathop{\text{A}}}\,$

B) $4000\text{ }\overset{\text{o}}{\mathop{\text{A}}}\,$

C) $1200\text{ }\overset{\text{o}}{\mathop{\text{A}}}\,$

D) $2400\text{ }\overset{\text{o}}{\mathop{\text{A}}}\,$

• question_answer16) Angle of minimum deviation for a prism of refractive index $1.5$ is equal to the angle of the prism. The angle of the prism is (given $\cos \,{{41}^{o}}-24'-36''=0.75$)

A) ${{82}^{o}}-49'-12''$

B) ${{72}^{o}}-48'-30''$

C) ${{41}^{o}}-24'-36''$

D) ${{31}^{o}}-49'-30''$

• question_answer17) Two media having speeds of light $2\times {{10}^{8}}\text{ }m/s$ and $2.4\times {{10}^{8}}\text{ }m/s,$ are separated by a plane surface. What is the critical angle for a ray going from medium I to medium II ?

A) ${{\sin }^{-1}}\left( \frac{5}{6} \right)$

B) ${{\sin }^{-1}}\left( \frac{5}{12} \right)$

C) ${{\sin }^{-1}}\left( \frac{1}{\sqrt{2}} \right)$

D) ${{\sin }^{-1}}\left( \frac{1}{2} \right)$

• question_answer18) When sunlight is incident son a prism, it produces a spectrum due to

A) interference of light

B) diffraction of light

C) total internal reflection

D) variation in speeds of different colours of light in the prism

• question_answer19) The transverse nature of light is shown by

A) interference of light

B) diffraction of light

C) polarization

D) radiation spectrum of black body

• question_answer20) An electromagnetic radiation of wavelength $\lambda ,$ frequency v and propagating in air with velocity v is incident on a glass plate and is transmitted through. Which of the following statement is true for the wave inside the glass plate?

A) The velocity v remains unchanged but $\lambda$changes

B) The frequency v and wavelength remain unchanged but v changes

C) The wavelength $\lambda$ remains unchanged but velocity v changes

D) The frequency v remains unchanged but $\lambda$changes

• question_answer21) The oscillating electric and magnetic vectors of an electromagnetic wave in vacuum are oriented along

A) the same direction but differ in phase by ${{90}^{o}}$

B) the same direction and are in phase

C) mutually perpendicular directions and differ in phase by ${{90}^{o}}$

D) mutually perpendicular directions and are in phase

• question_answer22) In an AC circuit the instantaneous values of emf and current are $e=200\text{ }sin\text{ }300\,t\text{ }volt$ and $i=2\,\sin \left( 300t+\frac{\pi }{3} \right)amp$ the average power consumed in watts is

A) $200$

B) $100$

C) $50$

D) $400$

• question_answer23) A transformer is used to light a 140 W, $24\text{ }V$ lamp from $240\text{ }VAC$mains. The current in the mains cable of $0.7\text{ }A$. The efficiency of the transformer is

A) $48%$

B) $63.8%$

C) $83.3%~$

D) $90%$

• question_answer24) A solenoid has 2000 turns wound over a length of $0.30\text{ }m$. The area of its cross-section is $1.2\times {{10}^{-3}}{{m}^{2}}$. Around its central section a coil of 300 turns is wound. If an initial current of 2 A in the solenoid is reversed in $0.25\text{ }s,$ the emf induced in the coil is

A) $48\text{ }V$

B) $4.8\text{ }V$

C) $4.8\times {{10}^{-1}}\,V$

D) $4.8\times {{10}^{-2}}\,V$

• question_answer25) A copper ring is held horizontally and a bar magnet with its length along the axis of the ring is dropped through the ring. The acceleration of the falling magnet is

A) less than that due to gravity

B) equal to that due to gravity

C) more than that due to gravity

D) depends on the length of the magnet and diameter of the ring

• question_answer26) The dimensional formula of Planck's constant (h) is

A) $[M{{L}^{-2}}{{T}^{-3}}]$

B) $[M{{L}^{2}}{{T}^{-1}}]$

C) $[ML{{T}^{-2}}]$

D) $[M{{L}^{-2}}{{T}^{-1}}]$

• question_answer27) A particle starts from rest and experiences constant acceleration for 6 s. If it travels a distance d-^ in the first two seconds, a distance ${{d}_{2}}$ in the next two seconds and a distance ${{d}_{3}}$ in the last two seconds, then

A) ${{d}_{1}}:{{d}_{2}}:{{d}_{3}}=1:1:1$

B) ${{d}_{1}}:{{d}_{2}}:{{d}_{3}}=1:2:3$

C) ${{d}_{1}}:{{d}_{2}}:{{d}_{3}}=1:3:5$

D) ${{d}_{1}}:{{d}_{2}}:{{d}_{3}}=1:5:9$

• question_answer28) A particle originally at rest at the highest point of a smooth circle in a vertical plane, is gently pushed and starts sliding/along the circle. It will leave the circle at a' vertical distance h. below the highest point such that

A) $h=2R$

B) $h=\frac{R}{2}$

C) $h=R$

D) $h=\frac{R}{3}$

• question_answer29) An inclined plane has an inclination $\theta$ with horizontal. A body of mass m rests on it. If the coefficient of friction between the body and the plane is $\mu ,$ then the minimum force that needs to be applied parallel to the inclined plane is

A) $mg\,\,\sin \,\theta$

B) $\mu mg\,\,\cos \,\theta$

C) $\mu mg\,\,\cos \,\theta +mg\,\,\sin \,\theta$

D) $\mu mg\,\,\cos \,\theta -mg\,\,\sin \,\theta$

• question_answer30) A $60\text{ }kg$man stands on a spring scale in a lift. At some instant he finds that the scale reading has changed from $60\text{ }kg$ to $50\text{ }kg$ for a while and then comes again to $60\text{ }kg$ mark. What should he conclude?

A) The lift was in constant motion upwards

B) The lift was in constant motion downwards

C) The lift while in motion suddenly stopped

D) The lift while in motion upwards suddenly' stopped

• question_answer31) A diatomic molecule is formed by two atoms which may be treated as mass points ${{m}_{1}}$ and ${{m}_{2}},$ joined by a massless rod of length r. Then the moment of inertia of the molecule about an axis passing through the centre of mass and perpendicular to rod is

A) zero

B) $({{m}_{1}}+{{m}_{2}}){{r}^{2}}$

C) $\left( \frac{{{m}_{1}}+{{m}_{2}}}{{{m}_{1}}{{m}_{2}}} \right){{r}^{2}}$

D) $\left( \frac{{{m}_{1}}{{m}_{2}}}{{{m}_{1}}+{{m}_{2}}} \right){{r}^{2}}$

• question_answer32) If the polar ice caps of earth melt, how long it affect the length of day?

A) Length of day would remain unchanged

B) Length of day would increase

C) Length of day would decrease

D) None of the above

• question_answer33) A planet revolves in an elliptical orbit around the sun. The semi-major and semi-minor axes are a and b, their the period is given by

A) ${{T}^{2}}\propto {{b}^{3}}$

B) ${{T}^{2}}\propto {{\left( \frac{a+b}{2} \right)}^{3}}$

C) ${{T}^{2}}\propto {{a}^{3}}$

D) ${{T}^{2}}\propto {{\left( \frac{a-b}{2} \right)}^{3}}$

• question_answer34) In a capillary tube of which the lower end dips in a liquid, the liquid rises to a height of$10\text{ }cm$If a capillary of the same bore is taken, whose length is 5 cm and dipped in liquid, then

A) a fountain of liquid will be obtain

B) the liquid will not rise in the tube at all

C) the liquid will rise upto the top and slowly ooze out of it

D) the liquid will rise to the top and will stay there

• question_answer35) There relation between Y (Young's modulus), K (bulk modulus) and $\eta$ (shear modulus) is

A) $\frac{9}{Y}=\frac{1}{K}+\frac{3}{\eta }$

B) $\frac{1}{Y}=\frac{1}{3\lambda }+\frac{1}{9K}$

C) $\frac{9}{Y}=\frac{1}{\eta }+\frac{3}{K}$

D) $\frac{1}{\eta }=\frac{1}{K}+\frac{1}{Y}$

• question_answer36) Solar radiation emitted by sun correspond to that emitted by black body at a temperature of $6000\text{ }K$. Maximum intensity is emitted at wavelength of$4800\text{ }\overset{\text{o}}{\mathop{\text{A}}}\,$. If the sun was to cool down from $6000\text{ }K$to $3000\text{ }K,$ then the peak intensity of emitted radiation would occur at a wavelength

A) $4800\overset{\text{o}}{\mathop{\text{A}}}\,$

B) $9600\overset{\text{o}}{\mathop{\text{A}}}\,$

C) $2400\overset{\text{o}}{\mathop{\text{A}}}\,$

D) $19200\overset{\text{o}}{\mathop{\text{A}}}\,$

• question_answer37) An ideal gas heat engine operates in a Carnot cycle between ${{227}^{o}}C$and${{127}^{o}}C$. It absorbs $6\times {{10}^{4}}cal$ at the higher temperature. The amount of heat converted into work is

A) $4.8\times {{10}^{4}}\text{ }cal$

B) $1.2\times {{10}^{4}}\text{ }cal$

C) $3.5\times {{10}^{4}}\text{ }cal$

D) $1.6\times {{10}^{4}}\text{ }cal$

• question_answer38) A certain mass of gas at NTP is expanded to three times its volume under adiabatic conditions. The resulting temperature of gas ,' will be (7 for gas is $1.40$)

A) $273\times {{\left( \frac{1}{3} \right)}^{1.4}}$

B) $273\times {{(3)}^{0.4}}$

C) $273\times {{\left( \frac{1}{3} \right)}^{0.4}}$

D) $273\times {{(3)}^{1.4}}$

• question_answer39) If for a gas $\frac{R}{{{C}_{V}}}=0.67,$ this gas is made up of molecules which "are

A) monoatomic

B) diatomic

C) polyatomic

D) mixture of diatomic and polyatomic molecules

• question_answer40) The velocity of a small ball of mass M and density ${{d}_{1}}$ when dropped in a container filled with glycerine becomes constant after some time. If the density of glycerine is ${{d}_{2}},$ the viscous force acting on the ball is

A) $Mg\left( 1-\frac{{{d}_{2}}}{{{d}_{1}}} \right)$

B) $Mg\frac{{{d}_{1}}}{{{d}_{2}}}$

C) $Mg({{d}_{1}}-{{d}_{2}})$

D) $Mg{{d}_{1}}{{d}_{2}}$

• question_answer41) The length of an elastic spring is a meters when a force of $4\text{ }N$is applied, and b meters when the $\text{5 }N$ force is applied. Then the length of the spring when the $9N$ force is applied is

A) $a+b$

B) $~9b-9a$

C) $5b-4a$

D) $4a-5b$

• question_answer42) Moment of inertia of a uniform disc about a diameter is I. Its moment of inertia about an axis perpendicular to its plane and passing through a point on its rim will be

A) $5I$

B) $3I$

C) $4I$

D) $6I$

• question_answer43) The escape velocity of a body from the surface of earth is $11.2\text{ }km/s$. It is thrown up with a velocity 4 times this velocity of escape. The velocity of the body when it has escaped the gravitational pull of earth (neglecting presence of all other heavenly bodies) is

A) $4\times 11.2\text{ }km/s$

B) $\sqrt{15}\times 11.2\text{ }km/s$

C) zero

D) $3\times 11.2\text{ }km/s$

• question_answer44) If a planet of given density were made larger (keeping its density unchanged) its force of attraction for an object on its surface would increase because of increased mass of the planet but would decrease because of larger separation between the centre of the planet and its surface. Which effect would dominate?

A) Increase in mass

C) Both affect the attraction equally

D) None of the above

• question_answer45) Two balls each of mass m are placed on the vertices A and B of an equilateral triangle ABC of side$1\text{ }m$. A ball of mass $2m$ is placed at vertex C. The centre of mass of this system from vertex A (located at origin) is

A) $\left( \frac{1}{2}m,\,\frac{1}{2}m \right)$

B) $\left( \frac{1}{2}m,\sqrt{3}m \right)$

C) $\left( \frac{1}{2}m,\frac{\sqrt{3}}{4}m \right)$

D) $\left( \frac{\sqrt{3}}{4}m,\frac{\sqrt{3}}{4}m \right)$

• question_answer46) A particle moves along the x-axis from $x={{x}_{1}}$ to $x={{x}_{2}}$ under the action of a force given by$F=2\text{ }x$. Then the work done in the process is

A) zero

B) ${{x}_{2}}^{2}-{{x}_{1}}^{2}$

C) $2{{x}_{2}}({{x}_{2}}-{{x}_{1}})$

D) $2{{x}_{1}}({{x}_{1}}-{{x}_{2}})$

• question_answer47) A body of mass 5 kg strikes another body of mass $2.5\text{ }kg$initially at rest. The bodies after collision coalesce and begin to move as whole with a kinetic energy of$5\text{ }J$. The kinetic energy of the first body before collision is

A) $7.5\text{ }J$

B) $5\text{ }J$

C) $2.5\text{ }J$

D) $10\text{ }J$

• question_answer48) For traffic moving at $60\text{ }km/h$along a circular track of radius $0.1\text{ }km,$the correct angle of banking is

A) ${{\tan }^{-1}}\left( \frac{{{60}^{2}}}{0.1} \right)$

B) ${{\tan }^{-1}}\left( \frac{{{(50/3)}^{2}}}{100\times 9.8} \right)$

C) ${{\tan }^{-1}}{{(60\times 0.1\times 9.8)}^{1/2}}$

D) ${{\tan }^{-1}}\left( \frac{100\times 9.8}{{{(50/3)}^{2}}} \right)$

• question_answer49) Two projectiles are fired at different angles with the same magnitude of velocity, such that they have the same range. At what angles they might have been projected?

A) ${{25}^{o}}$ and ${{65}^{o}}$

B) ${{35}^{o}}$ and ${{75}^{o}}$

C) ${{10}^{o}}$ and ${{50}^{o}}$

D) None of these

• question_answer50) If two non-zero vectors obey the relation $|\vec{P}+\vec{Q}|=|\vec{P}-\vec{Q}|$.then the angle between the vectors $\vec{P},$ $\vec{Q}$ is

A) ${{0}^{o}}$

B) $\pi$

C) $\frac{\pi }{4}$

D) $\frac{\pi }{2}$

• question_answer51) A weightless spring which has a force constant k oscillates with a frequency r% when a mass m is suspended from it. The spring is cut into two equal halves and a mass 2m is attached to one of the halves. The frequency of oscillation will not become

A) $\sqrt{2n}$

B) $\frac{n}{\sqrt{2}}$

C) $2n$

D) $n$

• question_answer52) A particle moves so that its acceleration a is given by $a=-bx,$ where x is displacement from equilibrium position and b is a non-negative real constant. The time period of oscillation of the particle is

A) $2\pi \sqrt{b}$

B) $\frac{2\pi }{b}$

C) $\frac{2\pi }{\sqrt{b}}$

D) $2\frac{\pi }{\sqrt{b}}$

• question_answer53) A tuning fork of frequency $500\text{ }cycles/s$is sounded on a resonance tube. The first and second resonance is obtained at $17\text{ }cm$and$52\text{ }cm$. The velocity of sound in m/s is

A) $175$

B) $350$

C) $525$

D) $700$

• question_answer54) Which of the following characteristics does not change due to the damping of simple harmonic motion?

A) Angular frequency

B) Time period

C) Initial phase

D) Amplitude

• question_answer55) A hollow charged metal sphere has a radius r. If the potential difference between its surface and a point at a distance $3\text{ }r$from the centre is V, then electrical intensity at distance $3\text{ }r$from the centre is

A) $\frac{V}{2r}$

B) $\frac{V}{3r}$

C) $\frac{V}{4r}$

D) $\frac{V}{6r}$

• question_answer56) Current is flowing with a current density $J=480\,A/c{{m}^{2}}$in a copper wire. Assuming that each copper atom contributes one free electron and given that Avogadro number $=6.0\times {{10}^{23}}\text{ }atoms/mol$ Density of copper $=9.0\text{ }g/c{{m}^{3}}$ Atomic weight of copper $=64\text{ }g/mol$ Electronic charge $=1.6\times {{10}^{-19}}C$ The drift velocity of electrons is

A) $1\text{ }mm/s$

B) $\text{2 }mm/s$

C) $0.5\text{ }mm/s$

D) $0.36\text{ }mm/s$

• question_answer57) A tap supplies water at ${{20}^{o}}C$. A man takes $1\text{ }L$of water per minute at ${{35}^{o}}C$from a geyser connected to the tap. The power of geyser is

A) $1050\text{ }W$

B) $2100\text{ }W$

C) $1500\text{ }W$

D) $3000\text{ }W$

• question_answer58) A charge is fired through a magnetic field. The force acting on the charge is maximum when the angle between the direction of motion of charge and the magnetic field is

A) zero

B) $\frac{\pi }{4}$

C) $\pi$

D) $\frac{\pi }{2}$

• question_answer59) At a place the angle of dip is ${{30}^{o}}$. If the horizontal component of earth's magnetic field is H, then the total field intensity is

A) $\frac{H}{2}$

B) $\frac{2H}{\sqrt{3}}$

C) $H\sqrt{2}$

D) $H\sqrt{3}$

• question_answer60) A particle is vibrating in simple harmonic motion with an amplitude of$4\text{ }cm$. At what displacement from the equilibrium position is its energy half potential and half-kinetic?

A) $2\sqrt{2}cm$

B) $\sqrt{2}cm$

C) $2\,cm$

D) $1\,cm$

• question_answer61) A tuning fork of frequency $480\text{ }Hz$produces 10 beats/s, when sounded with a sonometer string. A slight increase in tension in the sonometer string produces fewer beats/s than before. What was the frequency of sonometer string?

A) $470\text{ }Hz$

B) $490\text{ }Hz$

C) $480\text{ }Hz$

D) $460\text{ }Hz$

• question_answer62) A hollow metal sphere of radius $5\text{ }cm$is charged such that potential at its surface is$10\text{ }V$. The potential at the centre of the sphere is

A) zero volt

B) $10\text{ }V$

C) same as at point $5\text{ }cm$away from the surface

D) same as at point $\text{10 }cm$ away from the surface

• question_answer63) An electric dipole consisting of two opposite charges of $2\times {{10}^{-6}}\text{ }C$each, separated by a distance of $3\text{ }cm$is placed in an electric field of $2\times {{10}^{5}}\text{ }N/C$. The maximum torque on the dipole is

A) $12\times {{10}^{-1}}\text{ }Nm$

B) $12\times {{10}^{-3}}\text{ }Nm$

C) $24\times {{10}^{-1}}Nm$

D) $24\times {{10}^{-3}}\text{ }Nm$

• question_answer64) A given piece of wire of length I and radius r is having a resistance R. This wire is stretched uniformly to a wire of radius $\frac{r}{2}$. What is the new resistance?

A) $4R$

B) $8R$

C) $16R$

D) $2R$

• question_answer65) Two .thin long parallel wires separated by a distance b are carrying current i ampere each. The magnitude of the force per unit length exerted by one wire on the other is

A) $\frac{{{\mu }_{0}}}{2\pi }\frac{{{i}^{2}}}{b}$

B) $\frac{{{\mu }_{0}}}{2\pi }\frac{{{i}^{2}}}{{{b}^{2}}}$

C) $\frac{{{\mu }_{0}}}{2\pi }\frac{i}{b}$

D) $\frac{{{\mu }_{0}}}{2\pi }\frac{i}{{{b}^{2}}}$

• question_answer66) To increase the range of a voltmeter, we need to connect a suitable

A) high resistance in parallel

B) high resistance in series

C) low resistance in series

D) low resistance in parallel

• question_answer67) Two wires made up of the same material are of equal lengths but their radii are in the ratio of$1:2$. On stretching each of these two strings by the same tension, the ratio between the fundamental frequencies is

A) $1:4$

B) 4 : 1

C) $2:1$

D) $1:2$

• question_answer68) An object producing a pitch of $400\text{ }Hz$flies past a stationary person. The object was moving in a straight line with a velocity 200 m/s. The velocity of sound is 300 m/s. What is the change in frequency noted by the person as the object flies past him?

A) $1440\text{ }Hz$

B) $240\text{ }Hz$

C) $1200\text{ }Hz$

D) $960\text{ }Hz$

• question_answer69) Four capacitors each of $1\mu F$ are connected as shown. The equivalent capacitance between P and Q is

A) $4\mu F$

B) $\frac{1}{4}\mu F$

C) $\frac{3}{4}\mu F$

D) $\frac{4}{3}\mu F$

• question_answer70) The magnitude of electric field at distance r from an infinitely thin rod having a linear charge density $\lambda ,$is (use Gauss's law)

A) $E=\frac{\lambda }{2\pi {{\varepsilon }_{0}}r}$

B) $E=\frac{2\lambda }{\pi {{\varepsilon }_{0}}r}$

C) $E=\frac{2\lambda }{4\pi {{\varepsilon }_{0}}r}$

D) $E=\frac{4\lambda }{\pi {{\varepsilon }_{0}}r}$

• question_answer71) A $200\text{ }W$and a $100\text{ }W$bulb, both. meant for operation at $220\text{ }V$are connected in series. When connected to a $220\text{ }V$ supply the power consumed by the, combination is

A) $33.3\text{ }W$

B) $66.7\text{ }W$

C) $300W$

D) $100W$

• question_answer72) The sensitivity of a moving coil galvanometer depends on

A) angle of deflection -

B) earth's magnetic field

C) torsional constant of the spring

D) None of the above

• question_answer73) Domain formation is the necessary feature of

A) diamagnetic substances

B) paramagnetic substances

C) ferromagnetic substances

D) All of the above

• question_answer74) In a charged capacitor the energy stored in

A) the positive charges

B) the negative charges

C) the field between the plates

D) None of the above

• question_answer75) For a given temperature difference which of the following pairs will generate maximum thermo-emf?

B) copper-iron

C) gold-silver

D) antimony-bismuth

• question_answer76) The electrons identified by quantum numbers:

 (I) $n=4,l=1$ (II) $n=4,l=0$ (III) $n=3,l=2$ (IV) $n=2,l=1$
can be placed in order of increasing energy from the lowest to highest as

A) $(IV)<(II)<(III)<(I)$

B) $(II)<(IV)<(I)<(III)$

C) $(I)<(III)<(II)<(IV)$

D) $(III)<(I)<(IV)<(II)$

• question_answer77) Ground state electronic configuration of nitrogen atom can be represented as

A)

B)

C)

D)

• question_answer78) The first emission line in the electronic spectrum of hydrogen in the Balmer series appears at$c{{m}^{-1}}$

A) $\frac{9R}{400}c{{m}^{-1}}$

B) $\frac{7R}{144}c{{m}^{-1}}$

C) $\frac{3R}{4}c{{m}^{-1}}$

D) $\frac{5R}{36}c{{m}^{-1}}$

• question_answer79) $\text{KMn}{{\text{O}}_{\text{4}}}$reacts with oxalic acid according to the equation $2MnO_{4}^{-}+5{{C}_{2}}O_{4}^{2-}+16{{H}^{+}}\xrightarrow{{}}2M{{n}^{2+}}$$+\,10C{{O}_{2}}+8{{H}_{2}}O$ Here, 20 mL of $\text{0}\text{.1}\,\text{M}\,\text{KMn}{{\text{O}}_{\text{4}}}$ is equivalent to

A) $\text{20 mL}$of $\text{0}\text{.5}\,\text{M }{{\text{H}}_{\text{2}}}{{\text{C}}_{\text{2}}}{{\text{O}}_{\text{4}}}$

B) $\text{50}\,\text{mL}$of $\text{0}\text{.1}\,\text{M}\,{{\text{H}}_{\text{2}}}{{\text{C}}_{\text{2}}}{{\text{O}}_{\text{4}}}$

C) $\text{ }\!\!~\!\!\text{ 50 mL}$of $\text{0}\text{.01}\,\text{M}\,{{\text{H}}_{2}}{{C}_{2}}{{O}_{4}}$

D) $\text{20}\,\text{mL}$of $\text{0}\text{.1 M}\,{{\text{H}}_{\text{2}}}{{\text{C}}_{\text{2}}}{{\text{O}}_{\text{4}}}$

• question_answer80) Half-life of a sample is 160 days. After 800 days, 1 g of the sample shall be reduced to

A) $\frac{1}{2}g$

B) $\frac{1}{5}g$

C) $\frac{1}{4}g$

D) $\frac{1}{32}g$

A) GN Lewis

B) J Williard Gibbs

C) WF Libby

D) W Nernst

• question_answer82) An aqueous solution of 6.3 g oxalic acid dihydrate is made up to 250 mL. The volume of N sodium hydroxide required to completely neutralise 10 mL of this solution is

A) 40 mL

B) 20 mL

C) 10 mL

D) 4 mL

• question_answer83) How will increase of pressure affect the equation? $C(s)+{{H}_{2}}O(g)CO(g)+{{H}_{2}}(g)$

A) Shift in the forward direction

B) Shift in the reverse direction

C) Increase in the yield of hydrogen

D) No effect

• question_answer84) For which of the following reactions,${{K}_{p}}={{K}_{c}}?$

A) ${{N}_{2}}+3{{H}_{2}}2N{{H}_{3}}$

B) ${{N}_{2}}+{{O}_{2}}2NO$

C) $PC{{l}_{5}}PC{{l}_{3}}+C{{l}_{2}}$

D) $2S{{O}_{3}}2S{{O}_{2}}+{{O}_{2}}$

• question_answer85) Which of the following salts is most soluble?

A) $B{{i}_{2}}{{S}_{3}}({{K}_{sp}}=1\times {{10}^{-17}})$

B) $MnS({{K}_{sp}}=7\times {{10}^{-16}})$

C) $CuS({{K}_{sp}}=8\times {{10}^{-37}})$

D) $A{{g}_{2}}S({{K}_{sp}}=6\times {{10}^{-51}})$

• question_answer86) Ostwald dilution law is applicable to

A) strong electrolytes only

B) weak electrolytes only

C) non-electrolytes only

D) strong as well as weak electrolytes

• question_answer87) In the reaction $2A+B\xrightarrow{{}}{{A}_{2}}B,$ if the concentration of A is doubled and that of B is halved, then the rate of reaction will

A) increase by 4 times

B) decrease by 2 times

C) increase by 2 times

D) remains the same

• question_answer88) The Arrhenius equation expressing the effect of temperature on the rate constant of a reaction is

A) $k={{e}^{-E/RT}}$

B) $k=\ln \frac{E}{RT}$

C) $k=\frac{A.E}{RT}$

D) $k=A.{{e}^{-{{E}_{a}}/RT}}$

• question_answer89) If the half-time for a particular reaction is found to be constant and independent of the initial concentration of the reactants, then the reaction is of

A) first order

B) zero order

C) second order

D) none of these

• question_answer90) Normality of $\text{2}\,\text{M}{{\text{H}}_{\text{2}}}\text{S}{{\text{O}}_{\text{4}}}$ is

A) $\text{2N}$

B) $\text{4}\,\text{N}$

C) $\frac{N}{2}$

D) $\frac{N}{4}$

• question_answer91) Which of the following is a non-colligative property?

A) Elevation in boiling point

B) Osmotic pressure

C) Optical activity

D) Depression in freezing point

• question_answer92) The freezing point of equimolal aqueous solution will be highest for

A) ${{C}_{6}}{{H}_{5}}N{{H}_{3}}Cl$

B) $La{{(N{{O}_{3}})}_{3}}$

C) glucose

D) $Ca{{(N{{O}_{3}})}_{2}}$

• question_answer93) The van't Hoff factor for $\text{0}\text{.1}\,\text{M}\,\text{Ba(N}{{\text{O}}_{\text{3}}}{{\text{)}}_{\text{2}}}$ solution is 2.74. The degree of dissociation is

A) 91.3%

B) 87%

C) 100%

D) 74%

• question_answer94) The enthalpy of vaporisation of liquid water using the data ${{H}_{2}}(g)+\frac{1}{2}{{O}_{2}}(g)\xrightarrow{{}}{{H}_{2}}O(l)$ $\Delta H=-285.77\,kJ/mol$ ${{H}_{2}}(g)+\frac{1}{2}{{O}_{2}}(g)\xrightarrow{{}}{{H}_{2}}O(g);$ $\Delta H=-241.84\,kJ/mol$is

A) $+\,43.93\,kJ/mol$

B) $-43.93\,kJ/mol$

C) $527.61\,kJ/mol$

D) $-527.61\,kJ/mol$

• question_answer95) The enthalpy change for the transition of liquid water to steam is $40.8\,kJ\,mo{{l}^{-1}}$at $373\,K.$ Calculate the entropy of vaporisation of water.

A) $109.4\,J{{K}^{-1}}\,mo{{l}^{-1}}$

B) $-109.4\,J{{K}^{-1}}\,mo{{l}^{-1}}$

C) $218.8\,J{{K}^{-1}}\,mo{{l}^{-1}}$

D) $-218.8\,J{{K}^{-1}}\,mo{{l}^{-1}}$

• question_answer96) The occurrence of the reaction is impossible if

A) $\Delta H$ is$+\,ve;\Delta S$ is also $+\,ve$

B) $\Delta H$is$~-ve;\text{ }\Delta S$ is also $-ve$

C) $\Delta H$is $-ve;\Delta \,S$is $+\,ve$

D) $\Delta H$ is $+\,ve;\Delta S$is $-ve$

• question_answer97) The bond energy of an$O-H$ bond is 109 kcal/mol. When a mole of water is formed, then

A) 109 kcal is released

B) 218 kcal is absorbed

C) 109 kcal is absorbed

D) 218 kcal is released

• question_answer98) Of the following reaction, only one is a redox reaction. Identify this reaction.

A) $Ca{{(OH)}_{2}}+2HCl\xrightarrow{{}}CaC{{l}_{2}}+2{{H}_{2}}O$

B) $2{{S}_{2}}O_{7}^{2-}+2{{H}_{2}}O\xrightarrow{{}}2SO_{4}^{2-}+4{{H}^{+}}$

C) $BaC{{l}_{2}}+MgS{{O}_{4}}\xrightarrow{{}}BaS{{O}_{4}}+MgC{{l}_{2}}$

D) $C{{u}_{2}}S+2FeO\xrightarrow{{}}2Cu+2Fe+S{{O}_{2}}$

• question_answer99) Oxidation number of nitrogen is highest in

A) ${{N}_{3}}H$

B) ${{N}_{2}}{{O}_{4}}$

C) $N{{H}_{2}}OH$

D) $N{{H}_{3}}$

• question_answer100) Correct order of first ionization potential among the following elements Be, B, C, N, O is

A) $B<Be<C<O<N$

B) $B<Be<C<N<O$

C) $~Be<B<C<N<\text{O}$

D) $~Be<B<C<O<N$

• question_answer101) The unit of equivalent conductivity is

A) ohm cm

B) $~oh{{m}^{-1}}c{{m}^{2}}\text{ }g\text{ }equi{{v}^{-1}}$

C) $oh{{m}^{-2}}c{{m}^{-2}}\text{ }g\text{ }equiv$

D) $oh{{m}^{-2}}c{{m}^{-2}}g\,equiv$

• question_answer102) In which of the following crystals of ionic compounds would you expect maximum distance between the centres of cations and anions?

A) $\text{ }\!\!~\!\!\text{ LiF}$

B) $\text{CsF}$

C) $CsI$

D) $LiI$

• question_answer103) 50 mL of hydrogen diffuses through small hole from a vessel in 20 min time. Time taken for 40 mL of oxygen to diffuse out under similar conditions will be

A) 12 min

B) 32 min

C) 8 min

D) 64 min

• question_answer104) A gas will approach ideal behaviour at

A) low temperature and high pressure

B) low temperature and low pressure

C) high temperature and low pressure

D) high temperature and high pressure

• question_answer105) A colloidal system in which gas bubbles are dispersed in a liquid is known as

A) foam

B) aerosol

C) sol

D) emulsion

• question_answer106) In which of the following, Tyndall effect is not observed?

A) Smoke

B) Emulsions

C) Sugar solution

D) Gold sol

• question_answer107) Which is not correct regarding the adsorption of a gas on surface of a solid?

A) Enthalpy and entropy change is negative

B) Adsorption is more for some specific substance

C) On increasing temperature, adsorption increases progressively

D) It is a reversible reaction

• question_answer108) The ionic radii of isoelectronic species${{N}^{3-}},{{O}^{2-}}$and ${{F}^{-}}$are in the order

A) 1.36, 1.40, 1.71

B) 1.36, 1.71, 1.40

C) 1.71, 1.40, 1.36

D) 1.71, 1.36, 1.40

• question_answer109) Which of the following element is most electropositive?

A) Al

B) Mg

C) P

D) S

• question_answer110) The compound in which carbon atom uses only $s{{p}^{3}}-$hybrid orbitals for bond formation is

A) $HCOOH$

B) $N{{H}_{2}}CON{{H}_{2}}$

C) ${{(C{{H}_{3}})}_{3}}COH$

D) $C{{H}_{3}}CHO$

• question_answer111) The only molecule having dipole moment is

A) 2,2-dimethylpropane

B) trans-2-pentene

C) trans-3-hexene

D) 2,2,3,3-tetramethylbutane

• question_answer112) ${{N}_{2}}$and${{O}_{2}}$are converted into$N_{2}^{+}$and $O_{2}^{+}$respectively. Which of the following is .not correct?

A) In ${{N}_{2}}^{+},$the N?N bond weakens

B) In $O_{2}^{+},O-O$ bond order increases

C) ln ${{O}_{2}}^{+},$ paramagnetism decreases

D) ${{\text{N}}_{\text{2}}}^{\text{+}}$becomes diamagnetic

• question_answer113) The hybridisation of carbon atoms in C?C single bond of$HC\equiv C-CH=C{{H}_{2}}$is

A) $s{{p}^{3}}-s{{p}^{3}}$

B) $s{{p}^{2}}-s{{p}^{2}}$

C) $sp-s{{p}^{2}}$

D) $s{{p}^{3}}-sp$

• question_answer114) Chemical A is used for water softening to remove temporary hardness. A reacts with sodium carbonate to generate caustic soda. When$\text{C}{{\text{O}}_{\text{2}}}$ is bubbled through A, it turns cloudy. What is A?

A) $\text{CaC}{{\text{O}}_{3}}$

B) $CaO$

C) $Ca{{(OH)}_{2}}$

D) $Ca{{(HC{{O}_{3}})}_{3}}$

• question_answer115) Which of the following pentafluorides can't be formed?

A) $P{{F}_{5}}$

B) $As{{F}_{5}}$

C) $Sb{{F}_{5}}$

D) $Bi{{F}_{5}}$

• question_answer116) The correct order of increasing oxidising power is

A) $~{{F}_{2}}<C{{l}_{2}}<{{I}_{2}}>B{{r}_{2}}$

B) $~{{F}_{2}}<B{{r}_{2}}<C{{l}_{2}}<{{I}_{2}}$

C) $~C{{l}_{2}}<B{{r}_{2}}<{{F}_{2}}<{{I}_{2}}$

D) ${{I}_{2}}<B{{r}_{2}}<C{{l}_{2}}<{{F}_{2}}$

• question_answer117) Amongst $TiF_{6}^{2-},CoF_{6}^{3-},C{{u}_{2}}C{{l}_{2}}$and $NiCl_{4}^{2-}$at. no.$Ti=22,Co=27,Cu=29,Ni=28$). The. colourless species are

A) $CoF_{6}^{3-}$and $NiCl_{4}^{2-}$

B) $TiF_{6}^{2-}$and $CoF_{6}^{3-}$

C) $C{{u}_{2}}C{{l}_{2}}$ and $NiCl_{4}^{2-}$

D) $TiF_{6}^{2-}$ and $C{{u}_{2}}C{{l}_{2}}$

• question_answer118) The number of S?S bonds in sulphur trioxide is

A) three

B) two

C) one

D) zero

• question_answer119) Choose the correct statement.

A) Transition elements have low melting point

B) Transition elements do not have catalytic activity

C) Transition elements exhibit variable oxidation states

D) Transition elements show inert pair effect

• question_answer120) Two of the constituents of German silver are

A) $Ag+Cu$

B) $Ag+Zn$

C) $Cu+Zn$

D) $Cu+Sn$

• question_answer121) Aufbau principle does not Jgrve the correct arrangement of filling Up T of atomic orbitals in

A) Cu and Zn

B) Co and Zn

C) Mn and Cr

D) Cu and Cr

A) $[Co{{(en)}_{3}}]C{{l}_{3}}$

B) $[Co{{(en)}_{3}}]C{{l}_{3}}$

C) $[Co{{(en)}_{2}}N{{O}_{2}}Cl]Br$

D) $[Co{{(N{{H}_{3}})}_{5}}Cl]B{{r}_{2}}$

• question_answer123) IUPAC name of$[Pt{{(N{{H}_{3}})}_{3}}Br(N{{O}_{2}})Cl]Cl$

A) triamminechlorobromonitro platinum (IV) chloride

B) triamminebromonitrochloro platinum (IV) chloride

C) triamminebromochloronitro platinum (IV) chloride

D) triamminenitrochlorobromo platinum (IV)chloride

• question_answer124) According to postulates ofWemefs theory for coordination compounds, which of the following is true?

A) Primary valencies are ionisable

B) Secondary valencies are ionisable

C) Only primary valencies are non-ionizable

D) Primary and secondary valencies are non- ionisable

• question_answer125) Ferrocene is an example of

A) sand-wiched complex

B) pi-bonded complex

C) a complex in which all the five carbon atoms of cyclopentadiene anion are bonded to the metal

D) all of the above

• question_answer126) Which form of iron is least ductile?

A) Hard steel

B) Cast iron

C) Mild steel

D) Wrought steel

• question_answer127) Which of the following is known as Lunar caustic when in the fused state?

A) Silver nitrate

B) Silver sulphate

C) Silver chloride

D) Sodium sulphate

• question_answer128) One of the following metals is obtained by leaching its ore with dilute cyanide solution. Identify it.

A) Titanium

C) Silver

D) Zinc

• question_answer129) Which of the following species could be expected to exhibit aromatic character? Select the correct answer from the following

A) I and IV

B) II and IV

C) I and III

D) II and III

• question_answer130) The correct IUPAC name of the following compound is $C{{H}_{3}}C{{H}_{2}}\underset{C{{H}_{3}}}{\mathop{\underset{|}{\mathop{CH}}\,}}\,-C=\underset{{{C}_{2}}{{H}_{5}}}{\mathop{\underset{|}{\mathop{C}}\,}}\,-\underset{{{C}_{2}}{{H}_{5}}}{\mathop{\underset{|}{\mathop{C}}\,}}\,HC{{H}_{2}}C{{H}_{2}}C{{H}_{2}}C{{H}_{3}}$

A) 5, 6-dimethyl-8-methyl dec-6-ene

B) 6-butyl-5-ethyl-3-methyl oct-4-ene

C) 5, 6-diethyl-3-methyl dec-4-ene

D) 2, 4, 5-triethyl non-3-ene

• question_answer131) The kind of delocalisation involving sigma bond is called

A) inductive effect

B) hyperconjugation effect

C) electromeric effect

D) mesomeric effect

• question_answer132) Which of the following compounds react with HBr obeying Markownikoff?s rule?

A) $C{{H}_{2}}=C{{H}_{2}}$

B)

C)

D)

• question_answer133) How many cyclic isomers of ${{\text{C}}_{\text{5}}}{{\text{H}}_{\text{10}}}$are possible?

A) Four

B) Three

C) Two

D) Six

• question_answer134) The (R) and (S) enantiomers of an optically active compound differ in

A) their reactivity

B) their optical rotation of plane polarized light

C) their melting point

D) their solubility in achiral reagents

• question_answer135) Which of the following organic compounds exhibit acidic character?

A) ${{H}_{3}}C-C\equiv CH$

B) ${{H}_{3}}C-C\equiv C-C{{H}_{3}}$

C) ${{H}_{2}}C=C{{H}_{2}}$

D) ${{H}_{3}}C-C{{H}_{3}}$

• question_answer136) Methane is produced by the hydrolysis of

A) $A{{l}_{4}}{{C}_{3}}$

B) $Ca{{C}_{2}}$

C) Dry ice

D) $n-{{C}_{3}}{{H}_{7}}MgBr$

• question_answer137) In the reaction of phenol with chloroform and aqueous solution of$\text{NaOH}$at $\text{70}{{\,}^{o}}C,$ the electrophile attacking the ring is

A) $CHC{{l}_{3}}$

B) $CHC{{l}_{2}}$

C) $_{.}^{.}CC{{l}_{2}}$

D) $COC{{l}_{2}}$

• question_answer138) Which of the following compounds can exist in optically active form?

A) 1-butanol

B) 2-butanol

C) 3-pentanol

D) 4-heptanol

• question_answer139) Chloroform is slowly oxidised by air in the presence of light and air to form

A) formyl chloride

B) trichloro methanol

C) phosgene

D) formaldehyde

• question_answer140) Chlorination of toluene in the presence of light and heat followed by treatment with aqueous NaOH solution gives

A) o-cresol

B) p-cresol

C) benzoic acid

D) 2, 4-dihydroxytoluene

• question_answer141) Formation of cyanohydrin from the reaction of acetone with HCN is called

C) electrophilic substitution

D) nucleophilic substitution

• question_answer142) In the following reaction, $C{{H}_{3}}COCl\frac{BaS{{O}_{4}}}{Pd/{{H}_{2}}}X$ Identify X out of the following

A) acetaidehyde

B) propionaldehyde

C) acetone

D) acetic anhydride

• question_answer143) Reaction of t-butyl bromide with sodium methoxide produces

A) isobutane

B) isobutylene

C) sodium-t-butoxide

D) t-butylmethylether

• question_answer144) The acid which does not contain $-\text{COOH}$ group is

A) ethanoic acid

B) picric acid

C) lactic acid

D) palmitic acid

• question_answer145) The decreasing order of basic character of the three amines and ammonia is

A) $N{{H}_{3}}>C{{H}_{3}}N{{H}_{2}}>{{C}_{2}}{{H}_{5}}N{{H}_{2}}>{{C}_{6}}{{H}_{5}}N{{H}_{2}}$

B) ${{C}_{2}}{{H}_{5}}N{{H}_{2}}>C{{H}_{3}}N{{H}_{2}}>N{{H}_{3}}>{{C}_{6}}{{H}_{5}}N{{H}_{2}}$

C) ${{C}_{6}}{{H}_{5}}N{{H}_{2}}>{{C}_{2}}{{H}_{5}}N{{H}_{2}}>C{{H}_{3}}N{{H}_{2}}>N{{H}_{3}}$

D) $C{{H}_{3}}N{{H}_{2}}>{{C}_{2}}{{H}_{5}}N{{H}_{2}}>{{C}_{6}}{{H}_{5}}N{{H}_{2}}>N{{H}_{3}}$

• question_answer146) Aniline is treated with a mixture of sodium nitrite and hypophosphorus acid, the product formed is

A) aniline diazonium hypophosphate

B) benzene

C) anilinium hypophosphite

D) aniline diazonium hypophosphite

• question_answer147) Nitrosoamines $({{R}_{2}}N-N=O)$ are soluble in water. On heating them with cone ${{\text{H}}_{\text{2}}}\text{S}{{\text{O}}_{\text{4}}}\text{,}$they give secondary amines. The reaction is called

A) Perkin's reaction

B) Fries reaction

C) Libermann nitroso reaction

D) Etard reaction

• question_answer148) The compound obtained by heating a mixture of${{\text{1}}^{\text{o}}}$amine and chloroform with ethanolic potassium hydroxide is

A) an alkyl isocyanide

B) an alkyl isothiocyanate

C) an amide

D) an amide and nitro compound

• question_answer149) Which of the following is not an amino acid?

A) Glycine

B) Alanine

C) Histidine

D) Benzidine

• question_answer150) Which of the following is a fat soluble vitamin?

A) Vitamin A

B) Riboflavin

C) Pyridoxin

D) Thiamine

• question_answer151) $\int_{0}^{1}{{{\cot }^{-1}}(1-x+{{x}^{2}})dx}$ is equal to

A) $\pi -\log 2$

B) $\pi +\log 2$

C) $\frac{\pi }{2}+\log 2$

D) $\frac{\pi }{2}-\log 2$

• question_answer152) $\int_{0}^{1}{\log \,\left\{ \sin \left( \frac{\pi \,\,x}{2} \right) \right\}}\,\,dx$ is equal to

A) $-\frac{\pi }{2}\log \,2$

B) $-\,\log \,\,\,2$

C) $-\frac{\pi }{2}\,\log \,2$

D) $\frac{\pi }{2}\,\log \,2$

• question_answer153) The value of $2\cos \frac{\pi }{13}.\cos \frac{9\pi }{13}+\cos \frac{3\pi }{13}+\cos \frac{5\pi }{13}$ is

A) $-1$

B) $0$

C) $1$

D) None of these

• question_answer154) In a triangle ABC, $a=5,$ $b=4$ and $\cos (A+B)=\frac{31}{32}$ In this triangle, c is equal to

A) $\sqrt{6}$

B) $36$

C) $6$

D) None of these

• question_answer155) The locus of the centre of a circle of radius 2 which rolls on the outside of the circle ${{x}^{2}}+{{y}^{2}}+3x-6y-9=0$is

A) ${{x}^{2}}+{{y}^{2}}+3x-6y+5=0$

B) ${{x}^{2}}+{{y}^{2}}+3x-6y-31=0$

C) ${{x}^{2}}+{{y}^{2}}+3x-6y+\frac{29}{4}=0$

D) None of these

• question_answer156) $\int{x|x|\,dx}$ is

A) $\frac{{{x}^{2}}}{3}$

B) $-\frac{{{x}^{2}}}{3}$

C) $\frac{{{x}^{2}}|x|}{3}$

D) None of these

• question_answer157) The solution set of the equation $\sqrt{3{{x}^{2}}-7x-30}-\sqrt{2{{x}^{2}}-7x-5}=x-5$ is

A) $\{6\}$

B) $\left\{ 6,-\frac{5}{2} \right\}$

C) $\{5,6\}$

D) $\left\{ 5,\frac{7}{2} \right\}$

• question_answer158) If one root of the equation ${{x}^{2}}+px+q=0$is $2+\sqrt{3},$ then the value of p and q are

A) $-4,1$

B) $4,-1$

C) $2,\,\,\,\sqrt{3}$

D) $-2,-\,\,\,\sqrt{3}$

• question_answer159) In an AP the sum of any two terms, such that the distance of one of them from the beginning is same as that of the other from the end, is

A) first term

B) sum of first and last terms

C) last term

D) half of the sum of the series

• question_answer160) If $x,\,2x+2,\,\,3x+3,.....$ are in GP, then the fourth term is

A) $27.5$

B) $4x+5$

C) $-13.5$

D) $4x+4$

• question_answer161) The third term of a GP is 4. The product of its first five terms is

A) ${{4}^{5/2}}$

B) ${{4}^{5}}$

C) ${{4}^{3}}$

D) None of these

• question_answer162) The number of terms in the expansion of ${{[{{(a+4b)}^{3}}{{(-4b+a)}^{3}}]}^{2}}$are

A) $6$

B) $7$

C) $8$

D) $18$

• question_answer163) The coefficient of ${{x}^{4}}$ in ${{\left( \frac{x}{2}-\frac{3}{{{x}^{2}}} \right)}^{10}}$is

A) $\frac{405}{256}$

B) $\frac{504}{259}$

C) $\frac{450}{263}$

D) None of these

• question_answer164) Let A and B be two events such that $P(A)=0.3,$$P(A\cup B)=0.8,$if A and B are independent events, then $P(B)$ is equal to

A) $5/7$

B) $5/13$

C) $1/3$

D) $1/2$

• question_answer165) If for some real number k $\underset{x\to 0}{\mathop{\lim }}\,kx\,\,\text{cosec (x) =}\underset{x\to 0}{\mathop{\text{lim}}}\,\,\,x\,\,\text{cosec}\,\,(kx),$ then the possible values of k are

A) $1,-1$

B) $0,\,\,\,1$

C) $1,\,\,\,2$

D) $0,\,\,\,\pi$

• question_answer166) If ${{\log }_{2}}[{{\log }_{3}}\{lo{{g}_{4}}({{\log }_{5}}x)\}]=0,$then the value of x is

A) ${{5}^{24}}$

B) $1$

C) ${{2}^{25}}$

D) ${{5}^{64}}$

• question_answer167) The tangent to the curve $y=2{{x}^{2}}-x+1$is parallel to the line $y=3x+9$at the point

A) $(3,\,\,9)$

B) $(2,\,\,-1)$

C) $(2,\,1)$

D) $(1,\,2)$

• question_answer168) The system of equations $x+3y+2z=0$ $3x+y+z=0$ and $2x-2y-z=0$

A) possesses a trivial solution only

B) possesses a non-zero unique solution

C) does not have a common non-zero solution

D) has infinitely many solutions

• question_answer169) If $\omega$ is a cube root of unity, then a root of the equation $\left| \begin{matrix} x+1 & \omega & {{\omega }^{2}} \\ \omega & x+{{\omega }^{2}} & 1 \\ {{\omega }^{2}} & 1 & x+\omega \\ \end{matrix} \right|=0$ is

A) $x=0$

B) $x=1$

C) $x=\omega$

D) $x={{\omega }^{2}}$

• question_answer170) If A is a square matrix, then AA' is a (A' is the transpose of the matrix A)

A) unit matrix

B) null matrix

C) symmetric matrix

D) skew-symmetric matrix

• question_answer171) If A is a $3\times 3$ non-singular matrix, then $\det \,\,({{A}^{-1}}\,adj\,A)$ is equal to

A) $det\text{ }A$

B) $1$

C) ${{(det\text{ }A)}^{2}}$

D) ${{(det\text{ }A)}^{-1}}$

• question_answer172) In case of a linear programming problem, feasible region is always

A) a convex set

B) a concave set

C) a bounded convex set

D) a bounded concave set

• question_answer173) Which of the following is an odd function?

A) $|x|+1$

B) $\sin x+\cos x$

C) ${{x}^{2}}\,\,\sec x+x\,{{\tan }^{2}}x$

D) ${{x}^{2}}\cot x+4{{x}^{4}}\,\text{cosec x+}{{\text{x}}^{5}}$

• question_answer174) The incentre of the triangle with vertices $(1,\sqrt{3}),\,(0,0)$and $(2,0)$is

A) $(1,\,\sqrt{3}/2)$

B) $(2/3,\,\,\,1\,\sqrt{3})$

C) $(2/3,\,\,\sqrt{3}/2)$

D) $(1,\,\,1/\sqrt{3})$

• question_answer175) If r is Karl Pearson's coefficient of correlation between two sets of variates, then

A) $r<1$

B) $r>1$

C) $r<-1$

D) $|r|\,\le \,1$

• question_answer176) Three dice are thrown simultaneously, then probability of throwing a total greater than 4 is

A) $1/54$

B) $53/54$

C) $5/108$

D) None of these

• question_answer177) An approximate evaluation of $\int_{a}^{b}{f(x)\,\,dx}$ using trapezoidal rule requires the division of $[a,b]$ into

A) an even number of sub-intervals

B) an odd number of sub-intervals

C) any number of sub-intervals

D) None of the above

• question_answer178) In a triangle ABC, $C={{90}^{o}}$. If r is the in radius and R is the circum radius of the triangle, then $2\,(r+R)$is equal to

A) $a+c$

B) $a+b+c$

C) $b+c$

D) $a+b$

• question_answer179) The relation ${{\tan }^{-1}}\left( \frac{1+x}{1-x} \right)=\frac{\pi }{4}+{{\tan }^{-1}}x$ holds true for all

A) $x\,\in R$

B) $x\,\in (-\infty ,\,1)$

C) $x\,\in (-1,\,\,\infty )$

D) $x\,\in (-\,\infty ,-1)$

• question_answer180) $\int_{0}^{\pi /2}{\frac{1}{1+{{\tan }^{3}}x}}dx$is equal to

A) $\pi /2$

B) $\pi /4$

C) $\pi$

D) None of these

• question_answer181) A letter is taken out at random from the word 'ASSISTANT and another is taken-out from the word 'STATISTICS'. The chance that the two selected letters are identical, is

A) $19/45$

B) $19/90$

C) $89/90$

D) $1/90$

• question_answer182) The probability that the 6th day of a randomly chosen month of a year is a Sunday, is

A) $1/12$

B) $1/17$

C) $1/84$

D) None of these

• question_answer183) $\underset{x\to \infty }{\mathop{\lim }}\,\left( \sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x} \right)$ is equal to

A) $1/2$

B) $0$

C) $1$

D) None of these

• question_answer184) $\underset{x\to \infty }{\mathop{\lim }}\,{{\left( \frac{{{x}^{2}}+5x+3}{{{x}^{2}}+x+2} \right)}^{x}}$ is equal to

A) $e$

B) ${{e}^{2}}$

C) ${{e}^{3}}$

D) ${{e}^{4}}$

• question_answer185) If ${{b}^{2}}-4ac=0$and $a>0,$then domain of the function $f(x)=\log \{(a{{x}^{2}}+bx+c)\,\,(x+1)\}$ is

A) $R-\left( -\frac{b}{2a} \right)$

B) $R-(-\infty ,\,\,-1)$

C) $(-1,\infty )-\left\{ -\frac{b}{2a} \right\}$

D) $R-\left( \left\{ -\frac{b}{2a} \right\}\cap (-\infty ,-1) \right)$

• question_answer186) $\int{\frac{1}{{{({{x}^{4}}+1)}^{5/4}}}}$ is equal to

A) $-\frac{4}{{{({{x}^{4}}+1)}^{1/4}}}+c$

B) $\frac{1}{{{({{x}^{4}}+1)}^{1/4}}}+c$

C) $\frac{x}{{{({{x}^{4}}+1)}^{1/4}}}+c$

D) None of the above

• question_answer187) Let $f(x)=\left\{ \begin{matrix} |x|,\,\,for\,\,0<|x|\le 2 \\ 1,\,\,\,\,\,for\,\,\,\,x=0 \\ \end{matrix} \right.$, then at $x=0,$ $f$ has

A) a local maximum

B) a local minimum

C) no local extremum

D) no local maximum

• question_answer188) There are n different books and m copies of each in a college library. The number of ways in which a student can make a selection of one or more books is

A) ${{(m+1)}^{n}}$

B) $\frac{(mn)!}{{{(m!)}^{n}}}$

C) $^{mn}{{C}_{n}}{{\times }^{n}}{{C}_{1}}$

D) ${{(m+n)}^{n}}-1$

• question_answer189) Five digited numbers with distinct digits arc formed by using the digits,$5,4,3,2,1,0$. The number of those numbers which are multiples of 3, is

A) $720$

B) $240$

C) $216$

D) $120$

• question_answer190) If $A+B+C=\pi ,$then the value of $\left| \begin{matrix} \sin \,(A+B+C) & \sin B & \cos \,C \\ -\sin \,B & 0 & \tan \,A \\ \cos \,(A+B) & -\tan \,A & 0a \\ \end{matrix} \right|$ is

A) 0

B) 1

C) 2 sin B tan A cos C

D) None of the above

• question_answer191) If A and B are two matrices such that both $A+B$and AB are defined, then

A) A and B are of same order

B) A is of order $m\times m$and B is of order $n\times n$

C) both A and B are of same order $n\times n$

D) A is of order m x n and B is of order $n\times m$

• question_answer192) The number of dissimilar terms in expansion of ${{(a+b)}^{n}}$ is $n+1,$ therefore number of dissimilar terms of the expansion ${{(a+b+c)}^{12}}$ is

A) $13$

B) $39$

C) $78$

D) $91$

• question_answer193) If $|x|<1,$ then $1+n\left( \frac{2x}{1+x} \right)+\frac{n\,\,(n+1)}{2!}$${{\left( \frac{2x}{1+x} \right)}^{2}}+.......$ is equal to

A) ${{\left( \frac{2x}{1+x} \right)}^{n}}$

B) ${{\left( \frac{1+x}{2x} \right)}^{n}}$

C) ${{\left( \frac{1-x}{1+x} \right)}^{n}}$

D) ${{\left( \frac{1+x}{1-x} \right)}^{n}}$

• question_answer194) Eccentricity of the curve ${{x}^{2}}-{{y}^{2}}={{a}^{2}}$is equal to

A) $2$

B) $\sqrt{2}$

C) $4$

D) None of these

• question_answer195) If forces of $12,\text{ }5$ and $13$ unit weight balance at a point, two of them are inclined at

A) ${{30}^{o}}$

B) ${{45}^{o}}$

C) ${{60}^{o}}$

D) ${{90}^{o}}$

• question_answer196) If the resultant of two forces P and Q acting on a particle is at right angle to Q, then the angle between the forces is

A) ${{\cos }^{-1}}\left( \frac{P}{Q} \right)$

B) $\pi -{{\cos }^{-1}}\left( \frac{Q}{P} \right)$

C) $\pi -{{\cos }^{-1}}\left( \frac{P}{Q} \right)$

D) None of these

• question_answer197) To open a lock, a key is taken out from a collection of n keys at random. If the lock is not opened with this key, it is put back into the collection and another key is tried. The process is repeated again and again. If it is given that with only one key in the collection, the lock can be opened, then the probability that the lock will open in n trials, is

A) ${{\left( \frac{1}{n} \right)}^{n}}$

B) ${{\left( \frac{n-1}{n} \right)}^{n}}$

C) $1-{{\left( \frac{n-1}{n} \right)}^{n}}$

D) None of these

• question_answer198) The function $f(x)=x+\frac{1}{x}$ has

A) a local maxima at $x=1$and a local minima at $x=-1$

B) a local minima at $x=1$ and a local maxima at $x=-1$

C) absolute maxima at $x=1$ and absolute minima at $x=-1$

D) Absolute minima at $x=1$ and absolute maxima at $x=-1$

• question_answer199) If the sides of a triangle are $13,7,8,$ then the greatest angle of the triangle is

A) $\frac{\pi }{2}$

B) $\frac{3\pi }{2}$

C) $\frac{2\pi }{3}$

D) $\frac{\pi }{3}$

• question_answer200) $\underset{h\to \frac{x}{4}}{\mathop{\lim }}\,\frac{\tan \,x-1}{x-\frac{\pi }{4}}$is equal to

A) $1$

B) $1/2$

C) $2$

D) $0$

• question_answer201) $\underset{h\to 0}{\mathop{lim}}\,\frac{\sin \,\sqrt{x+h}-\sin \,\sqrt{x}}{h}$ is equal to

A) $\cos \,\,\sqrt{x}$

B) $1/(2\,\sin \sqrt{x})$

C) $(\cos \sqrt{x})/2\sqrt{x}$

D) $\sin \sqrt{x}$

• question_answer202) If $f(x)={{x}^{2}}+4x+1,$ then

A) $f(x)=f(-x),$ for all x

B) $f(x)\ne 1,$ for $x=0$

C) $f''(x)>0,$ for all x

D) $f(x)>1,$ for $x\le 1$

• question_answer203) If a, b, c are different real numbers and $a\,\,\hat{i}+b\,\,\hat{j}+c\,\,\hat{k},$$b\,\,\hat{i}+c\,\,\hat{j}+a\,\,\hat{k}$ and $c\,\,\hat{i}+a\,\,\hat{j}+b\,\,\hat{k}$are position vectors of three non-collinear points, then

A) centroid of $\Delta ABC$ is s$\frac{a+b+c}{3}\,(\hat{i}+\hat{j}+\hat{k})$

B) $(\hat{i}+\hat{j}+\hat{k})$ is not really inclined to three vectors

C) triangle ABC is a scalene triangle

D) perpendicular from the origin to the plane of the triangle does not meet it at the centroid

• question_answer204) Given four lines with equations $x+2y-3=0,$ $2x+3y-4=0,$ $3x+4y-5=0,$ $4x+5y-6=0.$ These lines are

A) concurrent

C) the sides of a parallelogram

D) the sides of a square

• question_answer205) The equations of lines through the point $(1,\,1)$ and making angles of ${{45}^{o}}$with the line $x+y=0$are

A) $x-1=0,\,\,x-y=0$

B) $x-1=0,\,\,y-1=0$

C) $x-y=0,\,\,y-1=0$

D) $x+y-2=0,\,\,\,y-1=0$

• question_answer206) If a function F is such that $f(0)=2,\,\,F(1)=3,$ $F(n+2)=2F(n)-F(n+1)$for $n\ne 0,$ then $F(5)$ is equal to

A) $-7$

B) $-3$

C) $7$

D) $13$

• question_answer207) If $f(x)=\frac{1}{\sqrt{-x}},$ then domain of fof is

A) $(0,\,\,\infty )$

B) $(-\,\infty ,\,\,0)$

C) $\{0\}$

D) $\{\,\,\}$

• question_answer208) If x and y are odd integers, then ${{x}^{2}}+{{y}^{2}}$is

A) an odd integer

B) an even integer divisible by 4

C) an even integer not divisible by 4

D) None of the above

• question_answer209) The minimum value of $sin{{\,}^{4}}x+co{{s}^{4}}x$is

A) $1$

B) $0$

C) $1/2$

D) None of these

• question_answer210) The locus of the point of intersection of the lines $x\,cot\,\theta +y\,cosec\,\theta \text{=}2$and $x\,cosec\,\theta +y\,cot\,\theta =6$is

A) a straight line

B) circle

C) a hyperbola

D) an ellipse

• question_answer211) If $2/3,\,k,\,5/8$are in AP, find the value of k.

A) $15$

B) $21$

C) $12$

D) $31/48$

• question_answer212) If $x=\sqrt{3018+\sqrt{36+\sqrt{169}}},$ then the value of x is

A) $55$

B) $44$

C) $63$

D) $42$

• question_answer213) The number of diagonals that can be drawn by joining the vertices of an octagon, is

A) $28$

B) $48$

C) $20$

D) $15$

• question_answer214) If the fourth term in the expansion of ${{\left( ax+\frac{1}{x} \right)}^{n}}$ is $\frac{5}{2},$ then find the value of a and n.

A) $a=1/2$and $n=6$

B) $a=1/3$ and $n=5$

C) $a=2$and $n=3$

D) $a=1/4$and $n=1$

• question_answer215) If ${{e}^{x}}=y+\sqrt{1+{{y}^{2}}},$ then the value of y is

A) ${{e}^{x}}-{{e}^{-x}}$

B) $\frac{1}{2}({{e}^{x}}-{{e}^{-x}})$

C) ${{e}^{x}}+{{e}^{-x}}$

D) None of these

• question_answer216) In the group $G=\{1,3,7,9\}$under multiplication modulo 10, the inverse of 7 is

A) $7$

B) $3$

C) $9$

D) $1$

• question_answer217) If A and B are two mutually exclusive events then

A) $P(A)<P(\bar{B})$

B) $P(A)>P(\bar{B})$

C) $P(A)<P(B)$

D) None of these

• question_answer218) The equation of the directrix of the parabola ${{y}^{2}}+4y+4x+2=0$ is

A) $-1$

B) $x=1$

C) $x=-3/2$

D) $x=3/2$

• question_answer219) Let ${{T}_{n}}$ denote the number of triangles which can be formed using the vertices of a regular polygon of n sides. If ${{T}_{n+1}}-{{T}_{n}}=21,$ then n equals

A) $5$

B) $7$

C) $6$

D) $4$

• question_answer220) In a triangle ABC, $2ca\,\sin \left( \frac{A-B+C}{2} \right)$ is equal to

A) ${{a}^{2}}+{{b}^{2}}+{{c}^{2}}$

B) ${{c}^{2}}+{{a}^{2}}-{{b}^{2}}$

C) ${{b}^{2}}-{{c}^{2}}-{{a}^{2}}$

D) ${{c}^{2}}-{{a}^{2}}-{{b}^{2}}$

• question_answer221) A fair die is tossed eight times. The probability that a third six is observed on the eight throw is

A) ${{a}^{2}}+{{b}^{2}}+{{c}^{2}}$

B) $\frac{^{7}{{C}_{2}}\times {{5}^{5}}}{{{6}^{8}}}$

C) $\frac{^{7}{{C}_{2}}\times {{5}^{5}}}{{{6}^{6}}}$

D) None of these

• question_answer222) The greatest distance of the point from the circle ${{x}^{2}}+{{y}^{2}}-4x-2y-20=0$is

A) $10$

B) $15$

C) $5$

D) None of these

• question_answer223) The vectors $\overrightarrow{AB}=3\hat{i}+4\hat{k}$ and $\overrightarrow{AC}=5\hat{i}-2\hat{j}+4\hat{k}$ are the sides of a triangle ABC. The length of the median through A is

A) $\sqrt{18}$

B) $\sqrt{72}$

C) $\sqrt{33}$

D) $\sqrt{288}$

• question_answer224) A particle acted on by constant forces $4\hat{i}+\hat{j}-3\hat{k}$ and $3\hat{i}+\hat{j}-\hat{k}$ is displaced from the point $\hat{i}+2\hat{j}+3\hat{k}$ to the point $5\hat{i}+4\hat{j}+\hat{k}$. The total work done by the forces is

A) 20 unit

B) 30 unit

C) 40 unit

D) 50 unit

• question_answer225) If $f(x)$ is a differentiable function, then $\underset{x\to a}{\mathop{\lim }}\,$ $\frac{a\,\,f(x)-x\,f(a)}{x-a}$ is equal to

A) $a\,\,f'(a)-f(a)$

B) $a\,\,f(a)-f'(a)$

C) $a\,\,f'(a)+f(a)$

D) $\,f'(a)+a\,f(a)$


text:
View Single Post
 2005-11-09, 04:15 #1 lsoule     Nov 2004 California 6A816 Posts Choose your own K and work on finding a top-5000 prime! Below is a list of K values that can be reserved. Each K has at least 90 primes in the range n=0-50k. I have checked the top-5000 list (as of Jan 2006) and none of these K values have primes reported so they should be 'clean'. Code:  K # primes Notes n=0-50k 165054615 102 (Reserved by Larry, tested to n=340k, 1 prime reported) 8331405 101 (Reserved by Larry, testing at n=900k) 1501946985 100 (Available) 22932195 99 (Reserved by Larry, 17 primes in n=210-770k) 2028893625 98 (Reserved by CedricVonck) 687218805 98 (Reserved by CedricVonck) 1222853775 98 (Available, tested to 348822, 7 primes reported) 1346178405 98 (Reserved by Larry, 1 prime in n=200-255k) 1194281385 98 1748348745 98 857996205 98 (Reserved by Larry) 62984955 97 (Reserved by Larry) 1310150985 97 686701125 97 (Tested to n=173307, 19 more primes in 50k-173k) 1693514745 97 (Reserved by Larry) 8713308285 96 317372055 96 (Available, 4 primes reported in n=200-410k) 19217385 96 (Reserved by nuggetprime) 146921775 96 445419975 96 (Reserved by Kosmaj, 7 primes reported in n=200-302k) 228461805 96 (Reserved by kuratkull) 317372055 96 949473525 96 3289888875 95 1169473305 95 776668035 95 14549535 95 (Reserved by Larry, 3 primes in n=210-290k) 1062034545 95 2037910875 95 (Reserved by Larry, 3 primes in n=200-272k) 80555475 95 (Reserved by Larry, 3 primes in n=200-285k) 24107655 95 (Available, tested 230-262k with 0 primes) 174956925 95 175977945 95 74784765 95 (Available, tested 230-259k with 0 primes) 9158021565 94 2241069285 94 3988569585 94 555567045 94 (Reserved by Edorah, tested to 429k, 10 primes reported) 39418665 94 (Reserved by NiceMedTxMd, 5 primes in n=200-400k) 107337555 94 828512685 94 228986175 94 152751885 93 2904473715 93 (Reserved by Larry, 7 primes in n=200-370k) 264766425 93 (Tested to 250k, 2 primes in n=200-250k) 244133175 93 28397655 93 (Reserved by levitate) 51010245 93 281804985 93 28397655 93 99311355 93 (Reserved by Kar Bon) 96623835 93 1256369595 93 234686595 93 (Available, tested by CedricVonck to n=145k, 9 more primes from n=50k-145k) 78290355 93 2481187995 92 8594430075 92 3870706125 92 2641908225 92 1311455145 92 100542585 92 272936235 92 249671565 92 3250995 92 (Tested to n=114832) 1386560175 92 (Reserved by Templus) 105330225 92 384158775 92 44702775 92 199140045 92 240924255 92 226005 91 (Available, tested to n=260k, sieved to 450bn, 1 prime) 8614926705 91 1533677145 91 4259877765 91 759237765 91 (Reserved by Larry, 4 primes in n=200-275k) 178140105 91 226231005 91 360949875 91 (Available, tested to n=147627) 35900025 91 8670014925 90 272605245 90 418791945 90 682980375 90 27114615 90 20934375 90 8629569795 89 8858158725 89 719053335 89 11340615 89 19474455 89 4471935 88 (Reserved by NiceMedTxMd) 850612425 88 (Tested to 275965k, 2 primes reported) 5169615 82 (Available. Tested to 430k by amphoria, 8 primes in n=200-430k) 7753875 70 up to n=10k (Reserved by NiceMedTxMd, 4 primes reported) 3545685 62 (Reserved by gd_barnes) 692835 60 (Reserved by Larry, 16 primes in n=200-655k) 465 ?? (Reserved by VBCurtis) 290499495 ?? (Reserved by gd_barnes) 968911515 ?? (Reserved by gd_barnes) 775784295 ?? (Reserved by gd_barnes) 3428677395 ?? (Reserved by gd_barnes) There are many considerations to picking a K: LLR runs faster on smaller values of K, higher weight K's will have more candidates to test, and luck! You may want to check the most recent top-5000 list and run the K in the range 50-100K or more before embarking on a quest for a top-5000 prime. To reserve a K, just leave a reply on this forum. Last fiddled with by lsoule on 2007-10-15 at 20:40

format: plain


text:
Search for a tool
Birthday Probabilities

Tool to calculate the birthday paradox problem in probabilities. How many people are necessary to have 50% chance that 2 of them share the same birthday.

Results

Birthday Probabilities -

Tag(s) : Statistics, Fun/Miscellaneous

Share
dCode and more

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!

Thanks to your feedback and relevant comments, dCode has developed the best 'Birthday Probabilities' tool, so feel free to write! Thank you!

# Birthday Probabilities

### Probability to share a same birthday

 Probability That at least N people that exactly N people That not any one
 To be born a same day (at any date) a given date (ie. May 1st)

### Format

 Probability Format Numeric (between 0 and 1) Percentage (in %) Fraction A/B (exact or approximate) Odds (About ≈) X in N Chance

### What is the birthday paradox? (Definition)

The paradox of birthdays is a mathematical problem put forward by Von Mises, who looks for the value N in the problem: In a group of N people there is 50% chance that at least 2 people in the group share the same birthday (day + month). The answer is N = 23, which is quite counterintuitive, hence the paradox.

### What are the hypothesis made to calculate the birthday probabilities?

During the calculation of the birthdate paradox, it is supposed that births are equally distributed over the days of a year (it is not exactly true in reality). In the following, a year has 365 days (leap years are ignored).

### What is the probability for a person to be born a given day of the year?

Probability is 1/365 = 0.0027 = 0.27%, indeed, one chance out of 365 to be bord a precise day.

### What is the probability for a person to be born a different day of mine?

The probability is 364/365 = 0.9973 = 99.73%, indeed, he must be born a distinct day of mine, so there is 364 possibilities out of 365.

### What is the probability for a person to be born the same day as me?

This calculation is the same as What is the probability for a person to be born a given day of the year? This given day is my birthday.

Example: 1/365 = 0.0027 = 0.27%

The calculation could also be formulated by noting that the probability for a person to be born the same day as me is the opposite as the probability for a person to be born a different day than mine.

Example: 1 - 364/365 = 1 - 0.9973 = 0,0027 = 0.27%

It is also the probability that a mother has a second child born on the same day (not the same year) as the first.

### What is the probability for a mother and its child to be born the same day?

This calculation is the same as What is the probability for a person (the child) to be born a given day of the year (the mother birthday)?.

Example: 1 chance in 365 or P = 0.0027 or 0.27%

The probability that a mother and her 2 children were born on the same day

Example: 1 chance in 365 * 365 = 133225 or P = 0.0000075 or 0.00075%

The probability that a mother and her N children were born on the same day is 1 in 365 ^ N

$$P(X) = 1/365^N$$

### What are the odds for 2 people in a group of N to be born the same day?

For N = 2, it is the same as probability for a person to be born the same day as me or the opposite as the probability for a person to be born a different day than mine.

Example: P(N=2) = 1 - (364/365) = 0,0027 = 0.27%

For other people, they have to be born also a distinct day than me, but also a distinct day of the other one (then calculate the opposite):

Example: $$P(N=3) = 1 - (364/365) * (363/365) = 0,0082 = 0.82\% \\ P(N=4) = 1 - (364/365) * (363/365) * (362/365) = 0,0164 = 1.64\% \\ P(N=5) = 1 - (364/365) * (363/365) * (362/365) * (361/365) = 0,0271 = 2.71\%$$

### What is the probability that among 23 people, 2 share the same birth day?

About 0.5, 50% (1 chance out of 2)

### How many people are needed in a group to be sure that 2 share the same birthday?

366 are needed (367 if leap years are taken into account). Indeed, every day of the year plus one are needed to be sure that at least a couple of two people share the same birthday.

### How many people are necessary to have the probability > 50% of 2 people to be born the same given day?

For a given date, 253 people are needed to get a probability of 50% that 2 people to be born a same precise day.

### What is the probability of 2 people among N to be born a given day?

Probabilities are multiplied

Example: P(N=2) = 1 - (364/365) = 0,0027 = 0.27%

Example: P(N=3) = 1 - (364/365)^2 = 0,0055 = 0.55%

Example: P(N=n) = 1 - (364/365)^(n-1)

### What is the probability that N people not to be born a given day of the year?

The probability is

Example: P = (364/365)^N

### How many people are born the same day as myself?

With the hypothesis of a world population of 8 billion people, there are 8000000000 / 365.25 or about 22 million people born on a given day / month.

Limiting to USA (350 millions inhabitants), there are 350000000 / 365.25 or nearly 1 millions US people born on a given day / month date.

## Source code

dCode retains ownership of the online 'Birthday Probabilities' tool source code. Except explicit open source licence (indicated CC / Creative Commons / free), any 'Birthday Probabilities' algorithm, applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any 'Birthday Probabilities' function (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and no data download, script, copy-paste, or API access for 'Birthday Probabilities' will be for free, same for offline use on PC, tablet, iPhone or Android ! dCode is free and online.

## Need Help ?

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!


text:
Share

# In a Sphere the Rate of Change of Surface Area is (A) 8π Times the Rate of Change of Diameter (B) 2π Times the Rate of Change of Diameter (C) 2π Times the Rate of Change of Radius - Mathematics

#### Question

In a sphere the rate of change of surface area is

##### Options
• 8π times the rate of change of diameter

• 2π times the rate of change of diameter

• 2π times the rate of change of radius

• 8π times the rate of change of radius

#### Solution

8π times the rate of change of radius

$\text { Let r be the radius and S be the surface area of the sphere at any time t .Then },$

$S = 4\pi r^2$

$\Rightarrow \frac{dS}{dt} = 8\pi r\frac{dr}{dt}$

$\therefore \text { The rate of change of surface area is } 8\pi \text { times the rate of change of the radius.}$

Is there an error in this question or solution?


text:
# Multiple rotated text boxes with absolute page positions

As part of a layout for a home made recipe book, I'm trying to lay out a series of text boxes at absolute positions on a page. I'm using eso-pic to put a background image on the page which has openings for the text I'm trying to lay out such as ingredients and directions.

I'm currently using the eso-pic AddToShipoutPictureFg to handle each block of text which resides in a rotated minipage. If I understand correctly, the arguments to AddToShipoutPicture will only get expanded at the end of the page. This is creating problems when I'm using a changing definition such as a counter inside the AddToShipoutPicture macro.

Here is an example showing the method I'm using for multiple rotated text boxes.

\documentclass[twoside,openright,letterpaper,11pt]{memoir}

\usepackage{rotating}
\usepackage{eso-pic}

\newcounter{mycounter}
\setcounter{mycounter}{1}

\begin{document}

% First chunk of angled boxed text.
\setlength{\unitlength}{1in}
\put(2,6){% Or in my document \put(Xcoord,Ycoord)
\begin{rotate}{45}
\begin{minipage}[t]{1in}
This is some text.  I expect
the counter to be 1, counter is \themycounter.
\end{minipage}
\end{rotate}
}
}

\stepcounter{mycounter}
% Or in my document:
% \renewcommand{\Xcoord}{calculated X position}
% \renewcommand{\Ycoord}{calculated Y position}

% Second chunk of angled boxed text.
\setlength{\unitlength}{1in}
\put(2,8){% Or in my document \put(Xcoord,Ycoord)
\begin{rotate}{-30}
\begin{minipage}[t]{1in}
This is some more text.  I expect
the counter to be 2, counter is \themycounter.
\end{minipage}
\end{rotate}
}
}

\null

\end{document}

This is the result, showing that the counter's value is always at the last value set:

How can the expansion of the counter's value be forced at the time AddToShipoutPictureFG appears in the source text so that the output at the red arrow shows 1 rather than 2?

NOTE 1: I'm a newbie using pdflatex. If there's an easier way to get multiple rotated, absolute positioned text boxes on the page please advise. I tried using the rotating package with textpos environment but they don't appear to play nice together - rotation occurs about a point in the text block that I can't determine.

NOTE 2: I've used a counter for the purposes of this example to illustrate a changing definition between AddToShipoutPictureFG calls. What I have in my document in place of \stepcounter{mycounter} is a macro which redefines an X, Y coordinate pair for the following text box which is used inside the next AddToShipoutPictureFG call. Of course the X, Y definitions at page end are the last X,Y set and so the text boxes always appear on top of each other rather than at the X,Y coordinate set before each AddToShipoutPictureFG call.

-

\stepcounter{mycounter}

immediately after the \themycounter so it is incremented after it is used. As is you are incrementing it on the main flow but using it twice in the output routine


text:
1. ## Find the anti-derivative

$\int \frac{2}{3}(1-x^2)^{3/2}\; dx$

Hmm... I don't think you can simply use the u-substitution technique here, as there is no $kx, k \in \mathbb{R}$ term on the outside.

Any idea?

2. Originally Posted by scorpion007
$\int \frac{2}{3}(1-x^2)^{3/2}\; dx$

Hmm... I don't think you can simply use the u-substitution technique here, as their is no $kx, k \in \mathbb{R}$ term on the outside.

Any idea?
Let $x = \sin \theta$. Then use some double angle formulas.


text:
If an error of 1 ° is made in measuring the angle of a sector of radius 30 cm, then the approximate error in its area, is

If an error of 1 ° is made in measuring the angle of a sector of radius 30 cm, then the approximate error in its area, is

1. A

2. B

3. C

$2.5\pi {\mathrm{cm}}^{2}$

4. D

none of these

Register to Get Free Mock Test and Study Material

+91

Verify OTP Code (required)

I agree to the terms and conditions and privacy policy.

Solution:

Let $A$ be the area and 0 (in radians) be the sector angle.  Then,

Let $\mathrm{\Delta }\theta$ be an error in $\theta$ and $\mathrm{\Delta A}$ be the corresponding error in $A.$

Then,

Register to Get Free Mock Test and Study Material

+91

Verify OTP Code (required)

I agree to the terms and conditions and privacy policy.


text:
?

Free Version
Easy

# Higher Order Matrix: Determinant of a $3\times3$

ALGEBR-KJJELV

Evaluate the determinant of the $3 \times 3$ matrix as shown below:

$$\left| \begin{array}{ccc} 2 & -2 & 0\\\ -3 & 1 & 2\\\ 1 & -3 & -1 \end{array} \right|$$

A

$1$

B

$12$

C

$4$

D

$5$


text:
# Some basic questions from beginner

This topic is 1761 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

## Recommended Posts

First of all, hello everyone.

I have already made some super-basic 2D games like TicTacToe, snake and pong. I think I am starting to understand how simple games work. Collision detection in pong is simple, because there is only one ball and 2 objects that it can hit. But when I think about some simple 2D shooter game i feel confused.. Let's say I have my character, some enemies and some platforms, on which I can jump.. How am I supposed to check the collision  of every object ( me, all bullets flying, all enemies and all platforms ). When I have 1 enemy and many bullets I can store the bullets in the array and then use loop to check the collision between the enemy and every element of the array. When I have many enemies and many bullets I can use double loop.. this is the way I see it, I have no idea if it is correct. But what if I have many elements of different types? Enemies, trees, fences, cars etc.. This topic really confuses me.. I hope everything is clear.

Second question, programming side:

In my poorly written code I am mostly using numbers instead of const variables..

// this is my code
enemy array[10];

// this is the experienced programmers code
const int MAX_ENEMIES=10;
enemy array[MAX_ENEMIES];

what is the point of doing it?

##### Share on other sites

You only need to check for collision with surrounding entities. If entityA is only 50 units wide, and entityB is only 100 units wide, then if their distance between each other is more than 100, you don't need to check anything else.

Also, you can partition up your loaded area into sections (think of each section as a 'bucket' of entities), if myEntity is in section7, it only needs to check for collisions against other entities in section7, and maybe a few of the sections next to section7 (incase myEntity is on the edge of both sections).

Further, look into std::vector for your entities and bullets. Twenty thousand bullets or more won't be a problem, but if you have them as a local variable of some function, they very well could be. std::vector stores its memory internally as non-local (dynamic) memory, "on the heap", whereas local variables are stored "on the stack", and the stack has limited memory.

Furtherly also, treat your bullets as a point (x,y) instead of a rect (x,y,width,height), and it'll be easier to check collision for.

Rect to rect collision is simple, but if you are hurting for performance, point to rect collision is even simpler. And when you multiply that collision by twenty thousand bullets, once per frame, you might save quite a bit.

Alsoly further, who says you need to check every bullet every frame?

If it's really a problem, and you have literally over half a million bullets, you could always update half one frame, and half the next frame, or divide the work up in some other method.

Finally, don't worry about performance until your program starts to slow down. Pre-mature optimization slows down your development time, even if it speeds up your program's execution time.

When speed really is noticeably a problem, then use a profiler, and it'll tell you exact what parts of your code are running too slow. People are often surprised at what the real slowdown in their code is - humans are fairly bad at guessing where the slow parts are, but a computer will tell you exactly where they are.

Your focus shouldn't be on fast code, but on clean, easy to read, easy to expand code, in my opinion.

##### Share on other sites

Thanks a lot for the fast response, it seems clear now. I know about the Vector but I never had to use it. I made this topic because when something bothers me I can't stop thinking about it :D I am coding the Pong game while thinking about hundreds of other things.

Now I see that games are much more complicated than I thought..

Again, thanks a lot :)

##### Share on other sites

You had another question in there I missed:

But what if I have many elements of different types? Enemies, trees, fences, cars etc.. This topic really confuses me.. I hope everything is clear.

What is a tree? What is an enemy? What is a fence? What is a car?

I see:

• A rectangle (includes position and size)
• Possibly a velocity (for continuing movement)
• An image
• Logic

All four (tree, enemy, fence, car) have the exact same 'pieces'. The only thing that changes is what image, and what logic.

Trees and fences have a velocity of zero.

So create your 'Entity' as one single type of entity, and initialize them with different values:

Entity = {rectangle{pos,size}, image, velocity, typeOfEntity};

Entity myCar(car_size, "CarImage.png", TYPE_CAR);
Entity myTree(tree_size, "TreeImage.png", TYPE_TREE);
Entity myFence(fence_size, "FenceImage.png", TYPE_FENCE);
Entity myEnemy(enemy_size, "EnemyImage.png", TYPE_ENEMY);

...add all of them to your single array (std::vector) of entities...

Figure out what's the same between the objects, and add it to your Entity class. Anything that's not the same, separate it out.

If you're using C++, you can use polymorphism here for handling the logic, but that might be a bit advanced.

Instead, how about having an 'UpdateEntity()' function, which checks what TYPE of an entity it is, and then calls separate more-specializes functions like "UpdateCar()" and "UpdateEnemy()"?

There are many other methods or techniques (polymorphism, function pointers, components), but I don't want to throw too much at you at one time, so I'm trying to use language features that I'm guessing you already know about. If you have any questions, keep on asking.

I'm not sure whether you're using 'C', C++, or some other language, but the concepts are the same for most languages, even if the syntax looks different.

##### Share on other sites

Yep, I am using C++, but I still don't know much about OO part of this language. In my programs I use just simple classes, constructors, destructors and friend classes. I am reading a C++ book, so eventually I will learn the more advanced techniques.

I have one more question, this time about the game loop..

I know it should look like this:

while(gameRunning)
{
handle_events();
logics();
rendering();
}

handling events is just taking the player input?

rendering.. this is obvious

but the logic part bothers me.. I see it as a main part of the loop, where most things happen. Calculating collision, velocities, tons of other stuff and at the end changing the coordinates of the objects. But finally I come to the point, where I have no idea where to move the objects and where to calculate stuff.


text:
# Torque Converter

## Formula

Torque = Distance from the center * Force

Torque is defined as a force that tends to rotate or turn things. It can be thought of as rotational force.

A hydraulic device for the smooth transmission of power in which an engine-driven impeller transmits its momentum to a fluid held in a sealed container, which in turn drives a rotor is called as torque converter. It is also called as hydraulic coupling.

The torque units contain a distance and a force. To calculate the torque, you multiply the force by the distance from the center. Torque unit in SI system is newton meter (nm). Other force units, pound force foot (lbs ft), ounce inch, kilogram force meter are widely used.

Example:
Convert the 25 Newton Meters to various units.

Solution:
Newton Meters = 25
Milli Newton Meters = 25000
Kilogram Meters = 2.55
Kilogram Centi Meters = 254.99999999999997
Gram Centi Meters = 254925
Ounce Inches = 3540
Pound Inches = 221.27500000000003
Pound Feet = 18.45


text:
# Question - Solutions of Quadratic Equations by Completing the Square

Account
Register

Share

Books Shortlist
ConceptSolutions of Quadratic Equations by Completing the Square

#### Question

Solve the following quadratic equation for x: 4x^2 + 4bx – (a^2 – b^2) = 0

#### Solution

You need to to view the solution
Is there an error in this question or solution?

S


text:
# stoch

0th

Percentile

##### Stochastic Oscillator / Stochastic Momentum Index

The stochastic oscillator is a momentum indicator that relates the location of each day's close relative to the high/low range over the past n periods. Developed by George C. Lane in the late 1950s. The SMI relates the close to the midpoint of the high/low range. Developed by William Blau in 1993.

Keywords
ts
##### Usage
stoch(HLC, nFastK = 14, nFastD = 3, nSlowD = 3, maType, bounded = TRUE,
smooth = 1, ...)SMI(HLC, n = 13, nFast = 2, nSlow = 25, nSig = 9, maType,
bounded = TRUE, ...)
##### Arguments
HLC

Object that is coercible to xts or matrix and contains High-Low-Close prices. If only a univariate series is given, it will be used. See details.

nFastK

Number of periods for fast %K (i.e. the number of past periods to use).

nFastD

Number of periods for fast %D (i.e. the number smoothing periods to apply to fast %K).

nSlowD

Number of periods for slow %D (i.e. the number smoothing periods to apply to fast %D).

maType

Either:

1. A function or a string naming the function to be called.

2. A list with the first component like (1) above, and additional parameters specified as named components. See Examples.

bounded

Logical, should current period's values be used in the calculation?

smooth

Number of internal smoothing periods to be applied before calculating FastK. See Details.

n

Number of periods to use.

nFast

Number of periods for initial smoothing.

nSlow

Number of periods for double smoothing.

nSig

Number of periods for signal line.

Other arguments to be passed to the maType function in case (1) above.

##### Details

If a High-Low-Close series is provided, the indicator is calculated using the high/low values. If a vector is provided, the calculation only uses that series. This allows stochastics to be calculated for: (1) series that have no HLC definition (e.g. foreign exchange), and (2) stochastic indicators (e.g. stochastic RSI - see examples).

The smooth argument is the number of periods of internal smoothing to apply to the differences in the high-low-close range before calculating Fast K. Thanks to Stanley Neo for the suggestion.

##### Value

A object of the same class as HLC or a matrix (if try.xts fails) containing the columns:

fastK

Stochastic Fast %K

fastD

Stochastic Fast %D

slowD

Stochastic Slow %D

SMI

Stochastic Momentum Index

signal

Stochastic Momentum Index signal line

##### Note

The calculation for William's %R is similar to that of stochastics' fast %K.

The value for fast %K will be 0.5 whenever the highest high and lowest low are the same over the last n periods.

The stochastic oscillator and SMI calculate relative value of the close versus the high/low range and the midpoint of the high/low range, respectively.

The stochastic oscillator and the stochastic momentum index are interpreted similarly. Readings below 20 (above 80) are considered oversold (overbought). However, readings below 20 (above 80) are not necessarily bearish (bullish). Lane believed some of the best sell (buy) signals occurred when the oscillator moved from overbought (oversold) back below 80 (above 20).

For the stochastic oscillator, buy (sell) signals can also be given when %K crosses above (below) %D. Crossover signals are quite frequent however, which may result in whipsaws.

##### References

The following site(s) were used to code/document these indicators: Stochastic Oscillator: http://www.fmlabs.com/reference/StochasticOscillator.htm http://www.equis.com/Customer/Resources/TAAZ?c=3&p=106 http://linnsoft.com/tour/techind/stoc.htm http://stockcharts.com/school/doku.php?id=chart_school:technical_indicators:stochastic_oscillator_fast_slow_and_full SMI: http://www.fmlabs.com/reference/default.htm?url=SMI.htm

##### See Also

See EMA, SMA, etc. for moving average options; and note Warning section. See WPR to compare it's results to fast %K.

• SMI
• %D
• %K
• stoch
• stochastic
• stochastics
##### Examples
library(TTR) # NOT RUN { data(ttrc) stochOSC <- stoch(ttrc[,c("High","Low","Close")]) stochWPR <- WPR(ttrc[,c("High","Low","Close")]) plot(tail(stochOSC[,"fastK"], 100), type="l", main="Fast %K and Williams %R", ylab="", ylim=range(cbind(stochOSC, stochWPR), na.rm=TRUE) ) lines(tail(stochWPR, 100), col="blue") lines(tail(1-stochWPR, 100), col="red", lty="dashed") stoch2MA <- stoch( ttrc[,c("High","Low","Close")], maType=list(list(SMA), list(EMA, wilder=TRUE), list(SMA)) ) SMI3MA <- SMI(ttrc[,c("High","Low","Close")], maType=list(list(SMA), list(EMA, wilder=TRUE), list(SMA)) ) stochRSI <- stoch( RSI(ttrc[,"Close"]) ) # }
Documentation reproduced from package TTR, version 0.23-1, License: GPL-2

### Community examples

Looks like there are no examples yet.


text:
# Moments - Crane

1. Nov 10, 2007

### steven10137

1. The problem statement, all variables and given/known data
A crane with jib AB, 7m long. Centre of gravity 3m from A, inclined at 30 degrees to the vertical.
The horizontal tie is attached at C, 5m from A.
If the mass of the jib is 200kg and the load at B is 600kg, what is the tension in the tie.

http://img139.imageshack.us/img139/2818/cranezu9.jpg [Broken]

2. Relevant equations
$$\sum {M_{CW} } = \sum {M_{ACW} }$$

3. The attempt at a solution
OK, well if we take the pivot about the point A, we have;
$$\begin{array}{l} \sum {M_{CW} } = \sum {M_{ACW} } \\ \left( 3 \right)\left( {200 \times 9.8} \right) + \left( 7 \right)\left( {600 \times 9.8} \right) = \left( 5 \right)\left( {T\sin 30} \right) \\ \Rightarrow T = \frac{{5880 + 41160}}{{2.5}} = 18816N \\ \end{array}$$

This tension seems very high for the system.

Could someone please explain the principle behind the Tension and angle components of the crane, because I don't think I fully understand it.

Steven.

Last edited by a moderator: May 3, 2017
2. Nov 10, 2007

### PhanthomJay

You are not evaluating the moments correctly. The moment of a force about a point is equal to the force times the perpendicular distance from the line of action of the force to the point.

Last edited by a moderator: May 3, 2017
3. Nov 10, 2007

### Vidatu

Draw a free body diagram. That should help you see what you need to do to solve this. Its actually just a one step problem.

4. Nov 10, 2007

### steven10137

"The moment of a force about a point is equal to the force times the perpendicular distance from the line of action of the force to the point."

So for taking the moments about the point A:
The centre of mass for the jib has moment; (3sin30)(200x9.8)

Following this procedure;
(3sin30)(200x9.8) + (7sin30)(600x9.8) = 5T
therefore T = 4704N

5. Nov 10, 2007

### PhanthomJay

Almost there. But the perpendicular distance from the tension force tothe point is (5)sin60, right?

6. Nov 10, 2007

### aq1q

ya but thats not perpendicular.. thats horizontal, its COS not sin

7. Nov 10, 2007

### aq1q

ya exactly cos30 or sin60.. either works

8. Nov 10, 2007

### steven10137

OK so this perpencidular distance we are looking for the is the distance 'x' in the diagram yea?

http://img165.imageshack.us/img165/5121/craneforcepp2.jpg [Broken]

cos30 = x / 5
the distance x is the 5cos30

Using this;
(3sin30)(200x9.8) + (7sin30)(600x9.8) = 5cos30T
therefore T = 5.43x10^3 N

Last edited by a moderator: May 3, 2017
9. Nov 10, 2007

### Vidatu

The perpendicular distance, for each of the forces, will be the shortest distance from the green dot to each dotted line.

Its the distance to the line of action of the force, which is infinite.

10. Nov 10, 2007

### PhanthomJay

Correct.

Last edited by a moderator: May 3, 2017
11. Nov 10, 2007

### steven10137

excellent :)
thanks for your help Vidatu and PhanthomJay


text:
# Atiyah–Hirzebruch spectral sequence

In mathematics, the Atiyah–Hirzebruch spectral sequence is a spectral sequence for calculating generalized cohomology, introduced by Michael Atiyah and Friedrich Hirzebruch (1961) in the special case of topological K-theory. For a CW complex ${\displaystyle X}$ and a generalized cohomology theory ${\displaystyle E^{\bullet }}$, it relates the generalized cohomology groups

${\displaystyle E^{i}(X)}$

with 'ordinary' cohomology groups ${\displaystyle H^{j}}$ with coefficients in the generalized cohomology of a point. More precisely, the ${\displaystyle E_{2}}$ term of the spectral sequence is ${\displaystyle H^{p}(X;E^{q}(pt))}$, and the spectral sequence converges conditionally to ${\displaystyle E^{p+q}(X)}$.

Atiyah and Hirzebruch pointed out a generalization of their spectral sequence that also generalizes the Serre spectral sequence, and reduces to it in the case where ${\displaystyle E=H_{\text{Sing}}}$. It can be derived from an exact couple that gives the ${\displaystyle E_{1}}$ page of the Serre spectral sequence, except with the ordinary cohomology groups replaced with ${\displaystyle E}$. In detail, assume ${\displaystyle X}$ to be the total space of a Serre fibration with fibre ${\displaystyle F}$ and base space ${\displaystyle B}$. The filtration of ${\displaystyle B}$ by its ${\displaystyle n}$-skeletons ${\displaystyle X_{n}}$ gives rise to a filtration of ${\displaystyle X}$. There is a corresponding spectral sequence with ${\displaystyle E_{2}}$ term

${\displaystyle H^{p}(B;E^{q}(F))}$

and abutting to the associated graded ring of the filtered ring

${\displaystyle E_{2}^{p,q}=E^{p+q}(X)}$.

This is the Atiyah–Hirzebruch spectral sequence in the case where the fibre ${\displaystyle F}$ is a point.

## Examples

### Topological K-theory

For example, the complex topological ${\displaystyle K}$-theory of a point is

${\displaystyle KU(*)=\mathbb {Z} [x,x^{-1}]}$ where ${\displaystyle x}$ is in degree ${\displaystyle 2}$

This implies that the terms on the ${\displaystyle E_{2}}$-page of a finite CW-complex ${\displaystyle X}$ looks like

${\displaystyle E_{2}^{p,q}(X)=H^{p}(X;KU^{q}(pt))}$

Since the ${\displaystyle K}$-theory of a point is

${\displaystyle K^{q}(pt)={\begin{cases}\mathbb {Z} &{\text{if q is even}}\\0&{\text{otherwise}}\end{cases}}}$

we can always guarantee that

${\displaystyle E_{2}^{p,2k+1}(X)=0}$

This implies that the spectral sequence collapses on ${\displaystyle E_{2}}$ for many spaces. This can be checked on every ${\displaystyle \mathbb {CP} ^{n}}$, algebraic curves, or spaces with non-zero cohomology in even degrees. Therefore, it collapses for all (complex) even dimensional smooth complete intersections in ${\displaystyle \mathbb {CP} ^{n}}$.

#### Euclidean Space

One easy example for computing topological K-theory is on ${\displaystyle \mathbb {R} ^{n}}$ since the only non-zero terms on the ${\displaystyle E_{2}}$-page are the terms

${\displaystyle H^{0}(\mathbb {R} ^{n};K^{2k}({\text{pt}}))}$

This is useful because it makes computing the K-theory for vector bundles over a topological space more tractable.

#### Cotangent Bundle on a Circle

For example, consider the cotangent bundle ${\displaystyle S^{1}}$. This is a fiber bundle with fiber ${\displaystyle \mathbb {R} }$ so the ${\displaystyle E_{2}}$-page reads as

${\displaystyle {\begin{array}{c|cc}\vdots &\vdots &\vdots \\2&H^{0}(S^{1};\mathbb {Q} )&H^{1}(S^{1};\mathbb {Q} )\\1&0&0\\0&H^{0}(S^{1};\mathbb {Q} )&H^{1}(S^{1};\mathbb {Q} )\\-1&0&0\\-2&H^{0}(S^{1};\mathbb {Q} )&H^{1}(S^{1};\mathbb {Q} )\\\vdots &\vdots &\vdots \\\hline &0&1\end{array}}}$

#### Differentials

The odd-dimensional differentials of the AHSS for complex topological K-theory can be readily computed. For ${\displaystyle d_{3}}$ it is the Steenrod square ${\displaystyle Sq^{3}}$ where we take it as the composition

${\displaystyle \beta \circ Sq^{2}\circ r}$

where ${\displaystyle r}$ is reduction mod ${\displaystyle 2}$ and ${\displaystyle \beta }$ is the Bockstein homomorphism (connecting morphism) from the short exact sequence

${\displaystyle 0\to \mathbb {Z} \to \mathbb {Z} \to \mathbb {Z} /2\to 0}$

#### Complete Intersection 3-fold

Consider a smooth complete intersection 3-fold ${\displaystyle X}$ (such as a complete intersection Calabi-Yau 3-fold). If we look at the ${\displaystyle E_{2}}$-page of the spectral sequence

${\displaystyle {\begin{array}{c|ccccc}\vdots &\vdots &\vdots &\vdots &\vdots &\vdots &\vdots &\vdots \\2&H^{0}(X;\mathbb {Z} )&0&H^{2}(X;\mathbb {Z} )&H^{3}(X;\mathbb {Z} )&H^{4}(X;\mathbb {Z} )&0&H^{6}(X;\mathbb {Z} )\\1&0&0&0&0&0&0&0\\0&H^{0}(X;\mathbb {Z} )&0&H^{2}(X;\mathbb {Z} )&H^{3}(X;\mathbb {Z} )&H^{4}(X;\mathbb {Z} )&0&H^{6}(X;\mathbb {Z} )\\-1&0&0&0&0&0&0&0\\-2&H^{0}(X;\mathbb {Z} )&0&H^{2}(X;\mathbb {Z} )&H^{3}(X;\mathbb {Z} )&H^{4}(X;\mathbb {Z} )&0&H^{6}(X;\mathbb {Z} )\\\vdots &\vdots &\vdots &\vdots &\vdots &\vdots &\vdots &\vdots \\\hline &0&1&2&3&4&5&6\end{array}}}$

we can see immediately that the only potentially non-trivial differentials are

{\displaystyle {\begin{aligned}d_{3}:E_{3}^{0,2k}\to E_{3}^{3,2k-2}\\d_{3}:E_{3}^{3,2k}\to E_{3}^{6,2k-2}\end{aligned}}}

It turns out that these differentials vanish in both cases, hence ${\displaystyle E_{2}=E_{\infty }}$. In the first case, since ${\displaystyle Sq^{k}:H^{i}(X;\mathbb {Z} /2)\to H^{k+i}(X;\mathbb {Z} /2)}$ is trivial for ${\displaystyle k>i}$ we have the first set of differentials are zero. The second set are trivial because ${\displaystyle Sq^{2}}$ sends ${\displaystyle H^{3}(X;\mathbb {Z} /2)\to H^{5}(X)=0}$ the identification ${\displaystyle Sq^{3}=\beta \circ Sq^{2}\circ r}$ shows the differential is trivial.

### Twisted K-Theory

The Atiyah–Hirzebruch spectral sequence can be used to compute twisted K-theory groups as well. In short, twisted K-theory is the group completion of the isomorphism classes of vector bundles defined by gluing data ${\displaystyle (U_{ij},g_{ij})}$ where

${\displaystyle g_{ij}g_{jk}g_{ki}=\lambda _{ijk}}$

for some cohomology class ${\displaystyle \lambda \in H^{3}(X,\mathbb {Z} )}$. Then, the spectral sequence reads as

${\displaystyle E_{2}^{p,q}=H^{p}(X;KU^{q}(*))\Rightarrow KU_{\lambda }^{p+q}(X)}$

but with different differentials. For example,

${\displaystyle E_{3}^{p,q}=E_{2}^{p,q}={\begin{array}{c|cccc}\vdots &\vdots &\vdots &\vdots &\vdots \\2&H^{0}(S^{3};\mathbb {Z} )&0&0&H^{3}(S^{3};\mathbb {Z} )\\1&0&0&0&0\\0&H^{0}(S^{3};\mathbb {Z} )&0&0&H^{3}(S^{3};\mathbb {Z} )\\-1&0&0&0&0\\-2&H^{0}(S^{3};\mathbb {Z} )&0&0&H^{3}(S^{3};\mathbb {Z} )\\\vdots &\vdots &\vdots &\vdots &\vdots \\\hline &0&1&2&3\end{array}}}$

On the ${\displaystyle E_{3}}$-page the differential is

${\displaystyle d_{3}=Sq^{3}+\lambda }$

Higher odd-dimensional differentials ${\displaystyle d_{2k+1}}$ are given by Massey products for twisted K-theory tensored by ${\displaystyle \mathbb {R} }$. So

{\displaystyle {\begin{aligned}d_{5}&=\{\lambda ,\lambda ,-\}\\d_{7}&=\{\lambda ,\lambda ,\lambda ,-\}\end{aligned}}}

Note that if the underlying space is formal, meaning its rational homotopy type is determined by its rational cohomology, hence has vanishing Massey products, then the odd-dimensional differentials are zero. Pierre Deligne, Phillip Griffiths, John Morgan, and Dennis Sullivan proved this for all compact Kähler manifolds, hence ${\displaystyle E_{\infty }=E_{4}}$ in this case. In particular, this includes all smooth projective varieties.

#### Twisted K-theory of 3-Sphere

The twisted K-theory for ${\displaystyle S^{3}}$ can be readily computed. First of all, since ${\displaystyle Sq^{3}=\beta \circ Sq^{2}\circ r}$ and ${\displaystyle H^{2}(S^{3})=0}$, we have that the differential on the ${\displaystyle E_{3}}$-page is just cupping with the class given by ${\displaystyle \lambda }$. This gives the computation

${\displaystyle KU_{\lambda }^{k}={\begin{cases}\mathbb {Z} &k{\text{ is even}}\\\mathbb {Z} /\lambda &k{\text{ is odd}}\end{cases}}}$

### Rational Bordism

Recall that the rational bordism group ${\displaystyle \Omega _{*}^{\text{SO}}\otimes \mathbb {Q} }$ is isomorphic to the ring

${\displaystyle \mathbb {Q} [[\mathbb {CP} ^{0}],[\mathbb {CP} ^{2}],[\mathbb {CP} ^{4}],[\mathbb {CP} ^{6}],\ldots ]}$

generated by the bordism classes of the (complex) even dimensional projective spaces ${\displaystyle [\mathbb {CP} ^{2k}]}$ in degree ${\displaystyle 4k}$. This gives a computationally tractable spectral sequence for computing the rational bordism groups.

### Complex Cobordism

Recall that ${\displaystyle MU^{*}(pt)=\mathbb {Z} [x_{1},x_{2},\ldots ]}$ where ${\displaystyle x_{i}\in \pi _{2i}(MU)}$. Then, we can use this to compute the complex cobordism of a space ${\displaystyle X}$ via the spectral sequence. We have the ${\displaystyle E_{2}}$-page given by

${\displaystyle E_{2}^{p,q}=H^{p}(X;MU^{q}(pt))}$

## References

• Davis, James; Kirk, Paul, Lecture Notes in Algebraic Topology (PDF)
• Atiyah, Michael Francis; Hirzebruch, Friedrich (1961), "Vector bundles and homogeneous spaces", Proc. Sympos. Pure Math., Vol. III, Providence, R.I.: American Mathematical Society, pp. 7–38, MR 0139181
• Atiyah, Michael, Twisted K-Theory and cohomology, arXiv:math/0510674, Bibcode:2005math.....10674A