# Solved papers for VIT Engineering VIT Engineering Solved Paper-2008

### done VIT Engineering Solved Paper-2008

• question_answer1) Two beams of light will not give rise to an interference pattern, if

A) they are coherent

B) they have the same wavelength

C) they are linearly polarized perpendicular to each other

D) they are not monochromatic

• question_answer2) A slit of width a is illuminated with a monochromatic light of wavelength $\lambda$ from a distant source and the diffraction pattern is observed on a screen placed at a distance D from the slit. To increase the width of the central maximum one should

A) decrease D

B) decrease $\alpha$

C) decrease $\lambda$

D) The width cannot be changed

• question_answer3) A thin film of soap solution (n = 1.4) lies on the top of a glass plate (n = 1.5). When visible light is incident almost normal to the plate, two adjacent reflection maxima are observed at two wavelengths 400 and 630 nm. The minimum thickness of the soap solution is

A) 420 nm

B) 450 nm

C) 630 nm

D) 1260 nm

• question_answer4) If the speed of a wave doubles as it passes from shallow water into deeper water, its wavelength will be

A) unchanged

B) halved

C) doubled

• question_answer5) A light whose frequency is equal to 6 x 1014 Hz is incident on a metal whose work function is $2eV\left[ h=6.63\,\times \,{{10}^{-34}}\,Js,1eV=1.6\,\times \,{{10}^{-19}}J \right]$ The maximum energy of the electrons emitted will be

A) 2.49 eV

B) 4.49 eV

C) 0.49 eV

D) 5.49 eV

• question_answer6) An electron microscope is used to probe the atomic arrangements to a resolution of 5 A. What should be the electric potential to which the electrons need to be accelerated?

A) 2.5V

B) 5V

C) 2.5 kV

D) 5kV

• question_answer7) Which phenomenon best supports the theory that matter has a wave nature?

A) Electron momentum

B) Electron diffraction

C) Photon momentum

D) Photon diffraction

• question_answer8) The radioactivity of a certain material drops to $\frac{1}{16}$of the initial value in 2 h. The half-life of this radio nuclide is

A) 10 min

B) 20 min

C) 30 min

D) 40 min

• question_answer9) An observer A sees an asteroid with a radioactive element moving by at a speed $=0.3c$ and measures the radioactivity decay time to be${{T}_{A}}$. Another observer B is moving with the asteroid and measures its decay time as${{T}_{B}}$. Then ${{T}_{A}}$ and ${{T}_{B}}$ are related as

A) ${{T}_{B}}<\text{ }{{T}_{A}}$

B) ${{T}_{A}}={{T}_{B}}$

C) ${{T}_{B~}}>{{T}_{A}}$

D) Either [A] or [C] depending on whether the asteroid is approaching or moving away from A

• question_answer10) $^{234}U$has 92 protons and 234 nucleons total in its nucleus. It decays by emitting an alpha particle. After the decay it becomes

A) $^{232}U$

B) $^{232}Pa$

C) $^{230}Th$

D) $^{230}Ra$

• question_answer11) ${{K}_{\alpha }}\,and\,{{K}_{\beta \,}}X-rays$ are emitted when there is a transition of electron between the levels

A) $n=2$ to $n=1$ and $n=3$ to $n=1$ respectively

B) $n=2$ to $n=1$ and $n=3$ to $n=2$respectively

C) $n=3$ to $n=2$ and $n=4$ to $n=2$respectively

D) $n=3$ to $n=2$ and $n=4$ to $n=3$respectively

• question_answer12) A certain radioactive material $_{Z}{{X}^{A}}$starts emitting $\alpha$ and $\beta$ panicles successively such that the end product is $_{Z-3}{{Y}^{A-8}}.$. The number of $\alpha$and $\beta$ particles emitted are

A) 4 and 3 respectively

B) 2 and 1 respectively

C) 3 and 4 respectively

D) 3 and 8 respectively

• question_answer13) In the circuit shown above, an input of 1 V is fed into the inverting input of an ideal Op-amp A. The output signal ${{\text{V}}_{\text{out}}}$ will be

A) + 10 V

B) -10 V

C) 0 V

D) infinity

• question_answer14) When a solid with a band gap has a donor level just below its empty energy band, the solid is

A) an insulator

B) a conductor

C) p-type semiconductor

D) n-type semiconductor

• question_answer15) A $p-n$ junction has acceptor impurity concentration of ${{10}^{17}}$$c{{m}^{-3}}$in the $P$ side and donor impurity concentration of ${{10}^{16}}c{{m}^{-3}}$in the $N$ side. What is the contact potential at the junction? ($kT=$thermal energy, intrinsic carrier concentration ${{n}_{i}}1.4\times {{10}^{10}}c{{m}^{-3}}$)

A) $\text{(}kT/e)\text{ }In\text{ }(4\times {{10}^{12}})$

B) $\text{(}kT/e)\text{ }In\text{ }(2.5\times {{10}^{23}})$

C) $(kT/e)\text{ }ln\text{ }({{10}^{23}})$

D) $(kT/e)\text{ }ln\text{ }({{10}^{9}})$

• question_answer16) A Zener diode has a contact potential of 1 V in the absence of biasing. It undergoes Zener breakdown for an electric field of ${{10}^{6}}\,\,V/m$ at the depletion region of $p-n$ junction. If the width of the depletion region is 2.5 $\mu$m, what should be the reverse biased potential for the Zener breakdown to occur?

A) 3.5 V

B) 2.5 V

C) 1.5 V

D) 0.5 V

• question_answer17) In Colpitt oscillator the feedback network consists of

A) two inductors and a capacitor

B) two capacitors and an inductor

C) three pairs of RC circuit

D) three pairs of RL circuit

• question_answer18) The reverse saturation of$p-n$diode

A) depends on doping concentrations

B) depends on diffusion lengths of carriers

C) depends on the doping concentrations and diffusion lengths

D) depends on the doping concentrations, diffusion length and device temperature

• question_answer19) A radio station has two channels. One is AM at $\text{1020 kHz}$and the other FM at 89.5 MHz. For good results you will use

A) longer antenna for the AM channel and shorter for the FM

B) shorter antenna for the AM channel and longer for the FM

C) Same length antenna will work for both

D) Information given is not enough to say which one to use for which

• question_answer20) The communication using optical fibres is based on the principle of

A) total internal reflection

B) Brewster angle

C) polarization

D) resonance

• question_answer21) In nature, the electric charge of any system is always equal to

A) half integral multiple of the least amount of charge

B) zero

C) square of the least amount of charge

D) integral multiple of the least amount of charge.

• question_answer22) The energy stored in the capacitor as shown in Fig. [a] is $\text{4}\text{.5}\times \text{1}{{\text{0}}^{\text{-6}}}\text{J}$. If the battery is replaced by another capacitor of 900 pF as shown in Fig. [b], then the total energy of system is

A) $4.5\times {{10}^{-6}}J$

B) $\text{2}\text{.25}\times \text{1}{{\text{0}}^{\text{-6}}}\text{J}$

C) zero

D) $\text{9}\times \text{1}{{\text{0}}^{\text{-6}}}\text{J}$

• question_answer23) Equal amounts of a metal are converted into cylindrical wires of different lengths L and cross-sectional area A. The wire with the maximum resistance is the one, which has

A) length $=\text{ }L$ and area $=\text{ }A$

B) length = $\frac{L}{2}$ and area $=\text{ }2A$

C) length $=\text{ }2L$ and area = $\frac{A}{2}$

D) All have the same resistance, as the amount of the metal is the same

• question_answer24) If the force exerted by an electric dipole on a charge $q$ at a distance of 1 m is $F$, the force at a point 2 m away in the same direction will be

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

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

C) $\frac{F}{6}$

D) $\frac{F}{8}$

• question_answer25) A solid sphere of radius${{R}_{1}}$ and volume charge density $\rho =\frac{{{\rho }_{0}}}{r}$is enclosed by a hollow sphere of radius ${{R}_{2}}$with negative surface charge density $\alpha$, such that the total charge in the system is zero, ${{\rho }_{0}}$is a positive constant and $r$ is the distance from the centre of the sphere. The ratio $\frac{{{R}_{2}}}{{{R}_{1}}}$is

A) $\frac{\sigma }{{{\rho }_{0}}}$

B) $\sqrt{2\sigma /{{\rho }_{0}}}$

C) $\sqrt{{{\rho }_{0}}/(2\sigma )}$

D) $\frac{{{\rho }_{0}}}{\sigma }$

• question_answer26) A solid spherical conductor of radius R has a spherical cavity of radius $a\left( a<R \right)$ at its centre. A charge $+Q$ is kept at the centre. The charge at the inner surface, outer surface and at a position $r\left( a<r<R \right)$ are respectively

A) $+Q$,$-Q$, $0$

B) $Q$, $+Q$, $0$

C) $0$,$-Q$, $0$

D) $+Q$, $0$, $0$

• question_answer27) A cylindrical capacitor has charge Q and length L. If both the charge and length of the capacitors are doubled, by keeping other parameters fixed, the energy stored in the capacitor

A) remains same

B) increases two times

C) decreases two times

D) increases four times

• question_answer28) Three resistances of 4$\Omega$ each are connected as shown in figure. If the point D divides the resistance into two equal halves, the resistance between point A and D will be

A) $12\Omega$

B) $6\Omega$

C) $3\Omega$

D) $\frac{1}{3}\Omega$

• question_answer29) The resistance of a metal increases with increasing temperature because

A) the collisions of the conducting electrons with the electrons increase

B) the collisions of the conducting electrons with the lattice consisting of the ions of the metal increase

C) the number of conduction electrons decrease

D) the number of conduction electrons increase

• question_answer30) In the absence of applied potential, the electric current flowing through a metallic wire is zero because

A) the electrons remain stationary

B) the electrons are drifted in random direction with a speed of the order of ${{10}^{-2}}cm/s$

C) the electrons move in random direction with a speed of the order close to that of velocity of light

D) electrons and ions move in opposite direction

• question_answer31) A meter bridge is used to determine the resistance of an unknown wire by measuring the balance point length l. If the wire is replaced by another wire of same material but with double the length and half the thickness, the balancing point is expected to be

A) $\frac{1}{8l}$

B) $\frac{1}{4l}$

C) $8l$

D) $16l$

• question_answer32) Identify the incorrect statement regarding a superconducting wire

A) transport current flows through its surface

B) transport current flows through the entire area of cross-section of the wire

C) it exhibits zero electrical resistivity and expels applied magnetic field

D) it is used to produce large magnetic field

• question_answer33) A sample of $\text{HCl}$ gas is placed in an electric field of $3\times {{10}^{4}}\text{N}{{\text{C}}^{\text{-1}}}$. The dipole moment of each$\text{HCl}$molecule is $6\times {{10}^{-30}}\text{Cm}$. The maximum torque that can act on a molecule is

A) $2\times {{10}^{-34}}{{\text{C}}^{\text{2}}}\text{m}{{\text{N}}^{\text{-1}}}$

B) $2\times {{10}^{-34}}\text{Nm}$

C) $18\times {{10}^{-34}}\text{Nm}$

D) $0.5\times {{10}^{34}}{{\text{C}}^{\text{-2}}}\text{N}{{\text{m}}^{\text{-1}}}$

• question_answer34) When a metallic plate swings between the poles of a magnet

A) no effect on the plate

B) eddy currents are set up inside the plate and the direction of the current is alone the motion of the plate

C) eddy currents are set up inside the plate and the direction of the current oppose the motion of the plate

D) eddy currents are set up inside the plate

• question_answer35) When an electrical appliance is switched on, it responds almost immediately, because

A) the electrons in the connecting wires move with the speed of light

B) the electrical signal is carried by electromagnetic waves moving with the speed of light

C) the electrons move with the speed which is close to but less than speed of light

D) the electrons are stagnant

• question_answer36) Two identical incandescent light bulbs are connected as shown in the figure. When the circuit is an AC voltage source of frequency $f$, which of the following observations will be correct?

A) Both bulbs will glow alternatively

B) Both bulbs will glow with same brightness provided frequency $f=\frac{1}{2\pi }\sqrt{1/LC}$

C) Bulb ${{b}_{1}}$will light up initially and goes off, bulbs ${{b}_{2}}$will be constantly

D) Bulb ${{b}_{1}}$ will blink and bulb ${{b}_{2}}$ will be ON constantly

• question_answer37) A transformer rated at 10 kW is used to connect a 5 kV transmission line to a 240 V circuit. The ratio of turns in the windings of the transformer is

A) 5

B) 20.8

C) 104

D) 40

• question_answer38) Three solenoid coils of same dimension, same number of turns and same number of layers of winding are taken. Coil 1 with inductance ${{L}_{1}}$ was wound using a Mn wire of resistance 11$\Omega /m;$Coil 3 with inductance ${{L}_{3}}$ was wound using the similar wire but the direction of winding was reversed in each layer; Coil 3 with inductance ${{L}_{3}}$ was wound using a superconducting wire. The self-inductance of the coils ${{L}_{1}}$, ${{L}_{2}}$, ${{L}_{3}}$ are

A) ${{L}_{1}}={{L}_{2}}={{L}_{3}}$

B) ${{L}_{1}}={{L}_{2}};{{L}_{3}}=0$

C) ${{L}_{1}}={{L}^{3}};{{L}^{2}}=0~$

D) ${{L}_{1}}>{{L}_{2}}>{{L}_{3}}$

• question_answer39) Light travels with a speed of $2\times {{10}^{8}}\,m/s$ in crown glass of refractive index 1.5. What is the speed of light in dense flint glass of refractive index 1.8?

A) $1.33\times {{10}^{8}}m/s$

B) $1.67\times {{10}^{8}}m/s$

C) $2.0\times {{10}^{8}}m/s$

D) $3.0\times {{10}^{8}}m/s$

• question_answer40) A parallel beam of fast moving electrons is incident normally on a narrow slit. A screen is placed at a large distance from the slit. If the speed of the electrons is increased, which of the following statement is correct?

A) Diffraction pattern is not observed on the screen in the case of electrons

B) The angular width of the central maximum of the diffraction pattern will increase

C) The angular width of the central maximum will decrease

D) The angular width of the central maximum will remains the same

• question_answer41) $C{{H}_{3}}C{{H}_{3}}+HN{{O}_{3}}\xrightarrow{675K}$?

A) $C{{H}_{3}}C{{H}_{2}}N{{O}_{2}}$

B) $C{{H}_{3}}C{{H}_{2}}N{{O}_{2}}+C{{H}_{3}}N{{O}_{2}}$

C) $2C{{H}_{3}}N{{O}_{2}}$

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

• question_answer42) When acetamide is hydrolysed by boiling with acid, the product obtained is

A) acetic acid

B) ethyl amine

C) ethanol

D) acetamide

• question_answer43) Which will not go for diazotisation?

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

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

C)

D)

• question_answer44) Secondary nitroaikanes can be converted into ketones by using Y. Identify Y from the following

A) aqueous $HCl$

B) aqueous $NaOH$

C) $KMn{{O}_{4}}$

D) $CO$

• question_answer45) Alkyi cyanides undergo Stephen reduction to produce

A) aldehyde

B) secondary amine

C) primary amine

D) amide

• question_answer46) The continuous phase contains the dispersed phase throughout, example is

A) water in milk

B) fat in milk

C) water droplets in mist

D) Oil in water

• question_answer47) The number of hydrogen atoms present in 25.6 g of $({{C}_{12}}{{H}_{22}}{{O}_{11}})$which has a molar mass of 342.3 g is

A) $342.3g$

B) $9.91\times {{10}^{23}}$

C) $11\times {{10}^{23}}$

D) $44\times {{10}^{23}}H$atoms

• question_answer48) Milk changes after digestion into

A) cellulose

B) fructose

C) glucose

D) lactose

• question_answer49) Which of the following set consists only of essential amino acids?

A) Alanihe, tyrosine, cystme

B) Leucine, lysme,tryptophane

C) Alanine, glutamine, lycine

D) leucine, proline, glycine

• question_answer50) Which of the following is a ketohexose?

A) Glucose

B) Sucrose

C) Fructose

D) Ribose:-

• question_answer51) The oxidation number .of oxygen in $K{{O}_{3}},N{{a}_{2}}{{O}_{2}}$is

A) $3,2$

B) $1,0$

C) $0,1$

D) $-0.33,-1$

• question_answer52) Reaction of $PC{{l}_{3}}$ and PhMgBr would give

A) bromobenzene

B) chlorobenzene

C) triphenylphosphine

D) dichlorobenzene

• question_answer53) Which of the following is not a characteristic of transition elements?

A) Variable oxidation states

B) Formation of coloured compounds

C) Formation of interstitial compounds

• question_answer54) $Cl-P-Cl$ bond angles in PCIs molecule are

A) ${{120}^{o}}$ and ${{90}^{o}}$

B) ${{60}^{o}}$ and ${{90}^{o}}$

C) ${{60}^{o}}$ and ${{120}^{o}}$

D) ${{120}^{o}}$ and ${{30}^{o}}$

• question_answer55) The magnetic moment of a salt containing $Z{{n}^{2+}}$ ion is

A) $0$

B) $1.87$

C) $5.92$

D) $2$

• question_answer56) The number of formula, units of calcium fluoride, $Ca{{F}_{2}}$ present in 146.4 g of $Ca{{F}_{2}}$ (the molar mass of $Ca{{F}_{2}}$ is 78.08 g/mol) is

A) $1.129\times {{10}^{24}}Ca{{F}_{2}}$

B) $1.146\times {{10}^{24}}Ca{{F}_{2}}$

C) $7.808\times {{10}^{24}}Ca{{F}_{2}}$

D) $1.877\times {{10}^{24}}Ca{{F}_{2}}$

• question_answer57) The IUPAC name of the given compound $[Co{{(N{{H}_{3}})}_{5}}Cl]C{{l}_{2}}$is

A) penta amino cobalt chloride chlorate

B) cobalt penta ammine chloro chloride

C) pentamine chloro cobalt (III) chloride

D) penta, amino cobalt (III) chlorate

• question_answer58) When $SC{{N}^{-}}$ is added to an aqueous solution containing $Fe{{(N{{O}_{3}})}_{3}},$the complex ion produced is

A) ${{[Fe{{(O{{H}_{2}})}_{2}}(SCN)]}^{2+}}$

B) ${{[Fe{{(O{{H}_{2}})}_{5}}(SCN)]}^{2+}}$

C) ${{[Fe{{(O{{H}_{2}})}_{8}}(SCN)]}^{2+}}$

D) ${{[Fe(O{{H}_{2}})(SCN)]}^{6+}}$

A) copper nitrate

B) gold chloride

C) silver nitrate

D) copper sulphate

• question_answer60) Schottky defects occurs mainly in electrovalent compounds where

A) positive ions and negative ions are of different size

B) positive ions and negative ions are of same size

C) positive ions are small and negative ions are big

D) positive ions are big and negative ions are Small

• question_answer61) The number of unpaired electrons calculated in ${{[Co{{(N{{H}_{3}})}_{6}}]}^{3+}}$and ${{[Co({{F}_{6}})]}^{3-}}$ are

A) 4 and 4

B) 0 and 2

C) 2 and 4

D) 0 and 4

• question_answer62) The standard free energy change of a reaction is $\Delta {{G}^{o}}=-115kJ$ at 298 K. Calculate the equilibrium constant ${{K}_{P}}$ in $\log {{K}_{P}}$ $(R=8.314\,\,J{{K}^{-1}}mo{{l}^{-1}}).$

A) $20.16$

B) $2.303$

C) $2.016$

D) $13.83$

• question_answer63) If an endothermic reaction occurs spontaneously at constant temperature (T) and pressure (p), then which of the following is true?

A) $\Delta G>0$

B) $\Delta H<0$

C) $\Delta S>0$

D) $\Delta S<0$

• question_answer64) If a plot of ${{\log }_{10}}C$ versus t gives a straight line for a given reaction, then the reaction is

A) zero order

B) first order

C) second order

D) third order

• question_answer65) A spontaneous process is one in which the system suffers

A) no energy change

B) a lowering of free energy

C) a lowering of entropy

D) an increase in internal energy

• question_answer66) The half-life period of a first order reaction is 1 min 40 s. Calculate its rate constant.

A) $6.93\times {{10}^{-3}}{{\min }^{-1}}$

B) $6.93\times {{10}^{-3}}{{s}^{-1}}$

C) $6.93\times {{10}^{-3}}s$

D) $6.93\times {{10}^{3}}s$

• question_answer67) The molar conductivities of $KCl,Nacl$ and $KN{{O}_{3}}$ are 152, 128 and $111\,\,S\,\,\,c{{m}^{2}}\,\,mo{{l}^{-1}}$ respectively. What is the molar conductivity of $NaN{{O}_{3}}$?

A) $101\,\,S\,c{{m}^{2}}mo{{l}^{-1}}$

B) $87S\,c{{m}^{2}}\,mo{{l}^{-1}}$

C) $-101S\,c{{m}^{2}}\,mo{{l}^{-1}}$

D) $-391\,\,S\,c{{m}^{2}}\,mo{{l}^{-1}}$

• question_answer68) The electrochemical cell stops working after sometime because

A) electrode potential of both the electrodes becomes zero

B) electrode potent : of both the electrodes becomes equal

C) one of the electrodes is eaten away

D) the cell reaction gets reversed

• question_answer69) The amount of electricity required to produce one mole of copper from copper sulphate solution will be

A) 1 F

B) 2.33 F

C) 2 F

D) 1.33 F

• question_answer70) Dipping iron article into a strongly alkaline solution of sodium phosphate

A) does not affect the article

B) forms $F{{e}_{2}}{{O}_{3}}.x{{H}_{2}}O$ on the surface

C) forms iron phosphate film

D) forms ferric hydroxide

• question_answer71) Hydro oration oxidation of 4-methyl octene would give

A) 4-methyl octanol

B) 2-methyl decane

C) 4-methyl heptanol

D) 4-mediyL2-octanone

• question_answer72) When ethyl alcohol is heated with cone. ${{H}_{2}}S{{O}_{4}}$, the product obtained is

A) $C{{H}_{3}}COO{{C}_{2}}{{H}_{5}}$

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

C) ${{C}_{2}}{{H}_{6}}$

D) ${{C}_{2}}{{H}_{4}}$

• question_answer73) Anisole is the product obtained from phenol by the reaction known as

A) coupling

B) etherification

C) oxidation

D) esterification

• question_answer74) Ethylene glycol gives oxalic acid on oxidation with:

A) acidified ${{K}_{2}}C{{r}_{2}}{{O}_{7}}$

B) acidified $KMn{{O}_{4}}$

C) alkaline $KMn{{O}_{4}}$

D) periodic acid

• question_answer75) Diamond is hard because

A) all the four valence electrons are bonded to each carbon atoms by covalent bonds

B) it is a giant molecule

C) it is made up of carbon atoms

D) it cannot be burnt

• question_answer76) A Wittig reaction with an aldehyde gives

A) ketone compound -

B) a long chain fatty acid

C) olefin compound

D) epoxide

• question_answer77) Cannizaro reaction is given by

A) $HCHO$

B)

C)

D) $C{{H}_{3}}C{{H}_{2}}OH$

A) ${{H}_{2}}O$

B) $HCHO$

C) $CO$

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

• question_answer79) Maleic acid and fumaric acid are

A) position isomers

B) geometric isomers

C) enantiomers

D) functional isomers

• question_answer80) The gas evolved on heating alkali formate with soda-lime is

A) $CO$

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

C) hydrogen

D) water vapour

• question_answer81) If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ be three unit vectors such that $\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})=\frac{1}{2}\overrightarrow{b},$$\overrightarrow{b}$ and $\overrightarrow{c}$ being non-parallel. If ${{\theta }_{1}}$ is the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ and ${{\theta }_{2}}$ is the angle between $\overrightarrow{a}$ and $\overrightarrow{c}$, then

A) ${{\theta }_{1}}=\frac{\pi }{6},{{\theta }_{2}}=\frac{\pi }{3}$

B) ${{\theta }_{1}}=\frac{\pi }{3},{{\theta }_{2}}=\frac{\pi }{6}$

C) ${{\theta }_{1}}=\frac{\pi }{2},{{\theta }_{2}}=\frac{\pi }{3}$

D) ${{\theta }_{1}}=\frac{\pi }{3},{{\theta }_{2}}=\frac{\pi }{2}$

• question_answer82) The${{\vec{r}}^{2}}-\vec{r}\cdot \vec{c}+h=0,$ $|\vec{c}|>\sqrt{h},$ represents

A) circle

B) ellipse

C) cone

D) sphere

• question_answer83) The simplified expression of $\sin \,(ta{{n}^{-1}}x),$ for any real number x is given by

A) $\frac{1}{\sqrt{1+{{x}^{2}}}}$

B) $\frac{x}{\sqrt{1+{{x}^{2}}}}$

C) $-\frac{1}{\sqrt{1+{{x}^{2}}}}$

D) $-\frac{x}{\sqrt{1+{{x}^{2}}}}$

• question_answer84) If $\left| \frac{z-25}{z-1} \right|=5,$find the value of$|z|$.

A) 3

B) 4

C) 5

D) 6

• question_answer85) Argument of the complex number$\left( \frac{-1-3i}{2+i} \right)$is

A) $45{}^\circ$

B) $135{}^\circ$

C) $225{}^\circ$

D) $240{}^\circ$

• question_answer86) In a triangle ABC, the sides b and c are the roots of the equation ${{x}^{2}}-61x+820=0$ and$A={{\tan }^{-1}}\left( \frac{4}{3} \right)$ then ${{a}^{2}}$ is equal to

A) 1098

B) 1096

C) 1097

D) 1095

• question_answer87) The shortest distance between the straight lines through the points ${{A}_{1}}=(6,\,2,\,2)$ and ${{A}_{2}}=(-4,\,0,\,-1),$in the directions of $(1,\,-2,\,\,2)$ and$(3,\,-2,-\,2)$is

A) 6

B) 8

C) 12

D) 9

• question_answer88) The centre and radius of the sphere${{x}^{2}}+{{y}^{2}}+{{z}^{2}}+3x-4z+1=0$are

A) $\left( -\frac{3}{2},\,0,\,-2 \right);\frac{\sqrt{21}}{2}$

B) $\left( \frac{3}{2},\,0,\,2 \right);\sqrt{21}$

C) $\left( -\frac{3}{2},\,0,\,2 \right);\frac{\sqrt{21}}{2}$

D) $\left( -\frac{3}{2},\,2,\,0 \right);\frac{21}{2}$

• question_answer89) Let A and B are two fixed points in a plane, then locus of another point C on the same plane such that CA + CB = constant, (> AB) is

A) circle

B) ellipse

C) parabola

D) hyperbola

• question_answer90) The directrix of the parabola${{y}^{2}}+4x+3=0$is

A) $x-\frac{4}{3}=0$

B) $x+\frac{1}{4}=0$

C) $x-\frac{3}{4}=0$

D) $x-\frac{1}{4}=0$

• question_answer91) If $g(x)$ is a polynomial satisfying $g(x)g(y)=g(x)+g(y)+g(xy)-2$ for all real $x$ and $y$ and $g(2)=5,$ then $\underset{x\to 3}{\mathop{\lim }}\,g(x)$ is

A) 9

B) 10

C) 25

D) 20

• question_answer92) The value of $f(0)$ so that $\frac{(-{{e}^{x}}+{{2}^{x}})}{x}$may be continuous at$x=0$is

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

B) 0

C) 4

D) $-1+\log 2$

• question_answer93) Let [ ] denotes the greatest integer function and$f(x)=[{{\tan }^{2}}x]$. Then,

A) $\underset{x\to 0}{\mathop{\lim }}\,f(x)$ does not exist

B) $f(x)$is continuous at $x=0$

C) $f(x)$ is not differentiable at $x=0$

D) $f(x)=1$

• question_answer94) A spherical balloon is expanding. If the radius is increasing at the rate of 2 cm/min, the rate at which the volume increases (in cubic centimetres per minute) when the radius is 5 cm, is

A) $10\,\pi$

B) $100\,\pi$

C) $200\,\pi$

D) $50\,\pi$

• question_answer95) The length of the parabola ${{y}^{2}}=12x$ cut off by the latusrectum is

A) $6\,[\sqrt{2}+\log (1+\sqrt{2})]$

B) $3\,[\sqrt{2}+\log (1+\sqrt{2})]$

C) $6\,[\sqrt{2}-\log (1+\sqrt{2})]$

D) $3\,[\sqrt{2}-\log (1+\sqrt{2})]$

• question_answer96) If $I=\int{\frac{{{x}^{5}}}{\sqrt{1+{{x}^{3}}}}}\,dx,$ then $I$ is equal to

A) $\frac{2}{9}{{(1+{{x}^{3}})}^{\frac{5}{2}}}+\frac{2}{3}{{(1+{{x}^{3}})}^{\frac{3}{2}}}+c$

B) $\log |\sqrt{x}+\sqrt{1+{{x}^{3}}}|+\,c$

C) $\log |\sqrt{x}-\sqrt{1+{{x}^{3}}}|+\,c$

D) $\frac{2}{9}{{(1+{{x}^{3}})}^{\frac{3}{2}}}-\frac{2}{3}{{(1+{{x}^{3}})}^{\frac{1}{2}}}+c$

• question_answer97) Area enclosed by the curve$\pi \,[4{{(x-\sqrt{2})}^{2}}+{{y}^{2}}]=8$ is

A) $\pi$ sq unit

B) $2$ sq unit

C) $3\pi$ sq unit

D) $4$sq unit

• question_answer98) The value of $\int_{0}^{a}{\sqrt{\frac{a-x}{x}}}dx$ is

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

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

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

D) $\frac{\pi a}{4}$

• question_answer99) Let y be the number of people in a village at rime$t$. Assume that the rate of change of the population is proportional to the number of people in the village at any time and further assume that the population never increases in time. Then, the population of the village at any fixed time$t$is given by

A) $y={{e}^{kt}}+c,$ for some constant $c\le 0$ and $k\ge 0$

B) $y=c{{e}^{kt}},$ for some constants $c\ge 0$ and $k\le 0$

C) $y={{e}^{ct}}+k,$ for some constants $c\le 0$ and $k\ge 0$

D) $y=k{{e}^{ct}},$ for some constants $c\ge 0$ and $k\le 0$

• question_answer100) The differential equation of all straight lines touching the circle${{x}^{2}}+{{y}^{2}}={{a}^{2}}$is

A) ${{\left( y-\frac{dy}{dx} \right)}^{2}}={{a}^{2}}\left[ 1+{{\left( \frac{dy}{dx} \right)}^{2}} \right]$

B) ${{\left( y-x\frac{dy}{dx} \right)}^{2}}={{a}^{2}}\left[ 1+{{\left( \frac{dy}{dx} \right)}^{2}} \right]$

C) $\left( y-x\frac{dy}{dx} \right)={{a}^{2}}\left[ 1+\frac{dy}{dx} \right]$

D) $\left( y-\frac{dy}{dx} \right)={{a}^{2}}\left[ 1-\frac{dy}{dx} \right]$

• question_answer101) The differential equation $\left| \frac{dy}{dx} \right|+|y|+3=0$ admits

A) infinite number of solutions

B) no solutions

C) a unique solution

D) many solutions

• question_answer102) Solution of the differential equation $xdy-ydx-\sqrt{{{x}^{2}}+{{y}^{2}}}dx=0$is

A) $y-\sqrt{{{x}^{2}}+{{y}^{2}}}=c\,{{x}^{2}}$

B) $y+\sqrt{{{x}^{2}}+{{y}^{2}}}=c\,{{x}^{2}}$

C) $y+\sqrt{{{x}^{2}}+{{y}^{2}}}=c\,{{y}^{2}}$

D) $x-\sqrt{{{x}^{2}}+{{y}^{2}}}=c\,{{y}^{2}}$

• question_answer103) Let p, q, r and s be statements and suppose that $p\to q\to r\to p.$If $\tilde{\ }s\to r,$then

A) $s\to \,\tilde{\ }q$

B) $\tilde{\ }q\to s$

C) $\tilde{\ }s\to \,\tilde{\ }q$

D) $q\to \,\tilde{\ }s$

• question_answer104) In how many number of ways can 10 students be divided into three teams, one containing four students and the other three?

A) 400

B) 700

C) 1050

D) 2100

• question_answer105) If R be a relation denned as $aRb$ iff $|a-b|\,>0,$ then the relation is

A) reflexive

B) symmetric

C) transitive

D) symmetric and transitive

• question_answer106) Let S be a finite set containing n elements. Then the total number of commutative binary operation on S is

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

B) ${{n}^{\left[ \frac{n(n-1)}{2} \right]}}$

C) ${{n}^{({{n}^{2}})}}$

D) ${{2}^{({{n}^{2}})}}$

• question_answer107) A manufacturer of cotter pins knows that 5% of his product is defective. He sells pins in boxes of 100 and guarantees that not more than one pin will be defective in a box. In order to find the probability that a box will fail to meet the guaranteed quality, the probability distribution one has to employ is

A) binomial

B) poisson

C) normal

D) exponential

• question_answer108) The probability that a certain kind of component will survive a given shock test is$\frac{3}{4}.$The probability that exactly 2 of the next 4 components tested survive is

A) $\frac{9}{41}$

B) $\frac{25}{128}$

C) $\frac{1}{5}$

D) $\frac{27}{128}$

• question_answer109) Mean and standard deviation from the following observations of marks of 5 students of a tutorial group (marks out of 25) 8 12 13 15 22 are

A) 14, 4.604

B) 15, 4.604

C) 14, 5.604

D) None of these

• question_answer110) A random variable X follows binomial distribution with mean $\alpha$ and variance$\beta$. Then

A) $0<\alpha <\beta$

B) $0<\beta <\alpha$

C) $\alpha <0<\beta$

D) $\beta <0<\alpha$

• question_answer111) The system of equations $x+y+z=0$ $2x+3y+z=0$ and $x+2y=0$ has

A) a unique solution; $x=0,y=0,z=0$

B) infinite solutions

C) no solution

D) finite number of non-zero solutions

• question_answer112) ${{\left[ \begin{matrix} 0 & a \\ b & 0 \\ \end{matrix} \right]}^{4}}=I,$then

A) $a=1=2b$

B) $a=b$

C) $a={{b}^{2}}$

D) $ab=1$

• question_answer113) If $D=$ diag $({{d}_{1}},{{d}_{2}},\,......\,{{d}_{n}}),$where ${{d}_{i}}\ne 0,$for$i=1,\,2,\,.....\,n,$then ${{D}^{-1}}$is equal to

A) ${{D}^{T}}$

B) $D$

C) $\text{adj}\,(D)$

D) $\text{diag}\,(d_{1}^{-1},\,d_{2}^{-1},\,....\,d_{n}^{-1})$

• question_answer114) If x, y, z are different from zero and$\Delta =\left| \begin{matrix} a & b-y & c-z \\ a-x & b & c-z \\ a-x & b-y & c \\ \end{matrix} \right|=0,$ then the value of the expression $\frac{a}{x}+\frac{b}{y}+\frac{c}{z}$ is

A) 0

B) -1

C) 1

D) 2

• question_answer115) Probability of getting positive integral roots of the equation ${{x}^{2}}-n=0$ for the integer n, $1\le n\le 40$ is

A) $\frac{1}{5}$

B) $\frac{1}{10}$

C) $\frac{3}{20}$

D) $\frac{1}{20}$

• question_answer116) The number of real roots of the equation ${{x}^{4}}+\sqrt{{{x}^{4}}+20}=22$is

A) 4

B) 2

C) 0

D) 1

• question_answer117) Let $\alpha ,\beta$ be the roots of the equation ${{x}^{2}}-ax+b=0$ and ${{A}_{n}}={{\alpha }^{n}}+{{\beta }^{n}}.$ Then, ${{A}_{n+1}}-a{{A}_{n}}+b{{A}_{n-1}}$ is equal to

A) $-a$

B) $b$

C) 0

D) $a-b$

• question_answer118) If the sides of a right angle triangle form an AP, the sin of the acute angles are

A) $\left( \frac{3}{5},\,\frac{4}{5} \right)$

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

C) $\left( \sqrt{\frac{\sqrt{5}-1}{2}},\sqrt{\frac{\sqrt{5}-1}{2}} \right)$

D) $\left( \sqrt{\frac{\sqrt{3}-1}{2}},\sqrt{\frac{\sqrt{3}-1}{2}} \right)$

• question_answer119) The plane through the point $\left( -1,\,\,-1,\,\,-1 \right)$ and containing the line of intersection of the planes$\vec{r}\cdot (\hat{i}+3\hat{j}-\hat{k})=0$and $\vec{r}\cdot (\hat{j}+2\hat{k})=0$ is

A) $\vec{r}\cdot (\hat{i}+2\hat{j}-3\hat{k})=0$

B) $\vec{r}\cdot (\hat{i}+4\hat{j}+\hat{k})=0$

C) $\vec{r}\cdot (\hat{i}+5\hat{j}-5\hat{k})=0$

D) $\vec{r}\cdot (\hat{i}+\hat{j}+3\hat{k})=0$

• question_answer120) $\vec{a}=\hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=2\hat{i}+4\hat{j}+3\hat{k}$ are one of the sides and medians respectively, of a triangle through the same vertex, then area of the triangle is

A) $\frac{1}{2}\sqrt{83}$

B) $\sqrt{83}$

C) $\frac{1}{2}\sqrt{85}$

D) $\sqrt{86}$