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All questions are of objective type (multiple choice questions); each question with choice of four answers..
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Online AIEEE 2005 Solved Paper

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Question

Which of the following oxides is amphoteric in character?

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Question

Which one of the following species is diamagnetic in nature?

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Question

If $\alpha$ is the degree of dissociation of Na_2SO_4

, the vant Hoff’s factor (i) used for calculating the molecular mass is

Possible Answers

Selected	Possible Answer
	$1 + \alpha$
	$1 + 2 \alpha$
	$1 - \alpha$
	$1 - 2 \alpha$

Question

The oxidation state of

Possible Answers

Selected	Possible Answer
	+2
	+3
	0
	+1

Question

Hydrogen bomb is based on the principle of

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Selected	Possible Answer
	Nuclear fusion
	Artificial radioactivity
	Natural radioactivity
	Nuclear fission

Question

An ionic compound has a unit cell consisting of

ions at the corners of a cube and

ions on the centres of the faces of the cube. The empirical formula for this compound would be

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Question

For a spontaneous reaction the $\Delta G$ , equilibrium constant (K)

and $E_{cell}^o$ will be respectively

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Question

Which of the following is a polyamide?

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	Teflon
	Nylon – 66
	Terylene
	Bakelite

Question

Which one of the following types of drugs reduces fever?

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Selected	Possible Answer
	Antipyretic
	Antibiotic
	Tranquiliser
	Analgesic

Question

Due to the presence of an unpaired electron, free radicals are:

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	Chemically inactive
	Chemically reactive
	Anions
	Cations

Question

Lattice energy of an ionic compounds depends upon

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	Charge on the ion only
	Charge on the ion and size of the ion
	Packing of ions only
	Size of the ion only

Question

The highest electrical conductivity of the following aqueous solutions is of

Possible Answers

Selected	Possible Answer
	0.1 M acetic acid
	0.1 M fluoroacetic acid
	0.1 M difluoroacetic acid
	0.1 M chloroacetic acid

Question

Aluminium oxide may be electrolysed at $1000^{\circ}C$ to furnish aluminium metal (Atomic mass = 27 amu; 1 Faraday = 96,500 Coulombs). The cathode reaction is
$Al^{3+} + 3e^- \rightarrow Al^{\circ}$
To prepare 5.12 kg of aluminium metal by this method would require

Possible Answers

Selected	Possible Answer
	$4.49 \times 10^4 \; C \; \text{of electricity}$
	$5.49 \times 10^7 \; C \; \text{of electricity}$
	$1.83 \times 10^7 \; C \; \text{of electricity}$
	$4.49 \times 10^1 \; C \; \text{of electricity}$

Question

Consider an endothermic reaction, $X \longrightarrow Y$ with the activation energies E_b

and

for the backward and forward reactions, respectively. In general

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Selected	Possible Answer
	There is no definite relation between and

Question

Consider the reaction: $N_2 +3H_2 \longrightarrow 2NH_3$ carried out at constant temperature and pressure. If $\Delta H$ and $\Delta U$ are the enthalpy and internal energy changes for the reaction, which of the following expressions is true?

Possible Answers

Selected	Possible Answer
	$\Delta H < \triangle U$
	$\Delta H=0$
	$\Delta H > \triangle U$
	$\Delta H= \triangle U$

Question

Which one of the following statements is NOT true about the effect of an increase in temperature on the distribution of molecular speeds in a gas?

Possible Answers

Selected	Possible Answer
	The area under the distribution curve remains the same as under the lower temperature
	The fraction of the molecules with the most probable speed increases
	The distribution becomes broader
	The most probable speed increases

Question

The volume of a colloidal particle, V_C

as compared to the volume of a solute particle in a true solution V_S

, could be

Possible Answers

Selected	Possible Answer
	$\cfrac{V_C}{V_S} \simeq 1$
	$\cfrac{V_C}{V_S} \simeq 10^{23}$
	$\cfrac{V_C}{V_S} \simeq 10^{-3}$
	$\cfrac{V_C}{V_S} \simeq 10^{-3}$

Question

The solubility product of a salt having general formula MX_2

, in water is: $4 \times 10^{-12}$ . The concentration of $M^{2+}$ ions in the aqueous solution of the salt is

Possible Answers

Selected	Possible Answer
	$2.0 \times 10{-6} M$
	$1.0 \times 10{-4} M$
	$4.0 \times 10{-10} M$
	$1.6 \times 10{-4} M$

Question

Benzene and toluene form nearly ideal solutions. At $20^{\circ} \; C$ , the vapour pressure of benzene is

torr and that of toluene is

torr. The partial vapour pressure of benzene at $20^{\circ} \; C$ for a solution containing 78 g

of benzene and 46 g

of toluene in torr is

Possible Answers

Selected	Possible Answer
	25
	53.5
	37.5
	50

Question

The exothermic formation of ClF_3

is represented by the equation:
$Cl_{2(g)} + 3F_{2(g)} \rightleftharpoons 2ClF_{3(g)}; \triangle rH = -329 kJ$
Which of the following will increase the quantity of ClF_3

in an equilibrium mixture of $Cl_2, F_2 \; \text{and} \; ClF_3$ ?

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Selected	Possible Answer
	Adding
	Increasing the volume of the container
	Increasing the temperature
	Removing

Question

Two solutions of a substance (non electrolyte) are mixed in the following manner. 480 ml

first solution + 520 mL

second solution. What is the molarity of the final mixture?

Possible Answers

Selected	Possible Answer
	2.70 M
	1.50 M
	1.344 M
	1.20 M

Question

For the reaction
$2NO_{2(g)} \rightleftharpoons 2NO_{(g)} + O_{2(g)}$ ,
$(K_c =1.8 \times 10^{-6} \; \text{at} \; 184^{\circ} C)$
$(R = 0.0831 \; kJ / (mol.K)$
When K_p

and

are compared at $184^{\circ}C$ , it is found that

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Selected	Possible Answer
	Whether is greater than, less than or equal to depends upon the total gas pressure
	$K_p \text{is greater than} K_c$

	$K_p \text{is less than} K_c$

Question

Hydrogen ion concentration in mol / L in a solution of pH = 5.4

will be

Possible Answers

Selected	Possible Answer
	$3.68 \times 10^{-6}$
	$3.98 \times 10^{-6}$
	$3.98 \times 10^8$
	$3.88 \times 10^8$

Question

A reaction involving two different reactants can never be

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Selected	Possible Answer
	second order reaction
	Bimolecular reaction
	Unimolecular reaction
	First order reaction

Question

If we consider that $\frac{1}{6}$ , in place of $\frac{1}{12}$ ; mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

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Selected	Possible Answer
	Increase two fold
	Remain unchanged
	Decrease twice
	Be a function of the molecular mass of the substance

Question

In a multi – electron atom, which of the following orbitals described by the three quantum numbers will have the same energy in the absence of magnetic acid and electric fields?

(a) n = 1, l = 0, m = 0
(b) n = 2, l = 0, m = 0
(c) n = 2, l = 1, m = 1
(d) n = 3, l = 2, m = 1
(e) n = 3, l = 2, m = 0

Possible Answers

Selected	Possible Answer
	(a) and (b)
	(c) and (d)
	(b) and©
	(d) and (e)

Question

During the process of electrolytic refining of copper, some metals present as impurity settle as ‘anode mud’ These are

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Selected	Possible Answer
	Sn and Ag
	Ag and Au
	Pb and Zn
	Fe and Ni

Question

$Electrolyte$
$\land^{\infty}(S \quad cm^2 \; mol^{-1})$	149.9	145.0	426.2	91.0	126.5

Calculate $\land^{\infty}_{HOAc}$ Using appropriate molar conductances of the electrolytes listed above at infinite dilution in $H_2O \; \text{at} \; 25^{\circ} C$

Possible Answers

Selected	Possible Answer
	552.7
	517.2
	390.7
	217.5

Question

A schematic plot of In $K_{eq}$ versus inverse of temperature for a reaction is shown below !{width:250px}

$1.5 \times 10^{-3} \quad \frac{1}{T}(K^{-1}) \quad 2.0 \times 10^{-3}$ The reaction must be

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Selected	Possible Answer
	highly spontaneous at ordinary
	endothermic
	exothermic
	one with negligible enthalpy change temperature

Question

The disperse phase in colloidal iron (III) hydroxide and colloidal gold is positively and negatively charged, respectively, which of the following statements is NOT correct?

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Selected	Possible Answer
	magnesium chloride solution coagulates, the gold sol more readily than the iron (III) hydroxide sol.
	mixing the sols has no effect
	sodium sulphate solution causes coagulation in both sols
	coagulation in both sols can be brought about by electrophoresis

Question

Based on lattice energy and other considerations which one of the following alkali metal chlorides is expected to have the highest melting point.

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Question

Heating mixture of $Cu_2O \;\text{and}\; Cu_2S$ will give

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Question

The molecular shapes of $SF_4, CF_4 \; \text{and} \; XeF_4$ are

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	different with 0, 1 and 2 lone pair of electrons on the central atoms, respectively
	the same with 1, 1 and 1 lone pair of electrons on the central atoms, respectively
	different with 1, 0 and 2 lone pairs of electron on the central atoms respectively
	the same with 2,0 and 1 lone pairs of electrons on the central atom, respectively

Question

The number and type of bonds between two carbon atoms in calcium carbide are

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Selected	Possible Answer
	Two sigma, one pi
	One sigma, two pi
	One sigma, one pi
	Two sigma, two pi

Question

The oxidation state of chromium in the final product formed by the reaction between

and acidified potassium dichromate solution is

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Selected	Possible Answer
	+2
	+4
	+3
	+6

Question

The number of hydrogen atom(s) attached to phosphorus atom in hypophosphorous acid is

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Selected	Possible Answer
	one
	three
	zero
	two

Question

What is the conjugate base of OH^-

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Selected	Possible Answer
	$O^{-2}$

Question

The correct order of the thermal stability of hydrogen halides (H - X)

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Selected	Possible Answer
	HI > HCl < HF < HBr
	HI > HBr > HCl > HF
	HCl < HF > HBr < HI
	HF > HCl > HBr > HI

Question

Heating an aqueous solution of aluminium chloride to dryness will give

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Question

Calomel

on reaction with ammonium hydroxide gives

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Question

In which of the following arrangements the order is NOT according to the property indicated against it?

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Selected	Possible Answer
	Increasing metallic radius
	$Al^{3+} < Mg^{2+} < Na^+ < F^-$ Increasing ionic size
	Increasing electron gain enthalpy (with negative sign)
	Increasing first ionization enthalpy

Question

In silicon dioxide

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Selected	Possible Answer
	Each silicon atom is surrounded by two oxygen atoms and each oxygen atom is bonded to two silicon atoms
	Each silicon atom is surrounded by four oxygen atoms and each oxygen atom is bonded to two silicon atoms
	Silicon atoms is bonded to two oxygen atoms
	there are double bonds between silicon and oxygen atoms

Question

Of the following sets which one does NOT contain isoelectronic species?

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Selected	Possible Answer
	$PO^{-3}_4,SO^{-2}_4,ClO^-_4$
	$CN^-, N_2, C^{-2}_2$
	$SO^{-2}_3,CO^{-2}_3,NO^-_3$
	$SO^{-3}_3,CO^{-2}_3,NO^-_3$

Question

The lanthanide contraction is responsible for the fact that

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Selected	Possible Answer
	and have about the same radius state
	and have about the same radius
	and have similar oxidation
	and have the same oxidation

Question

The IUPAC name of the coordination compound K_3[Fe(CN)_6]

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Selected	Possible Answer
	Potassium hexacyanoferrate (III)
	Potassium hexacyanoferrate (II)
	tripotassium hexcyanoiron (II)
	Potassium hexacyanoiron (II)

Question

Which of the following compounds shows optical isomerism?

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Selected	Possible Answer
	$[Cr(C_2)O_4)_3]^{-3}$
	$[Cu(NH_3)_4]^{+2}$
	$[ZnCl_4]^{-2}$
	$[Co(CN)_6]^{-3}$

Question

Which one of the following cyano complexes would exhibit the lowest value of paramagnetic behaviour? (At. No. Cr = 24, Mn = 25, Fe = 26, Co = 27)

Possible Answers

Selected	Possible Answer
	$[Co(CN)_6]^{-3}$
	$[Mn(CN)_6]^{-3}$
	$[Fe(CN)_6]^{-3}$
	$[Cr(CN)_6]^{-3}$

Question

2 methylbutane on reacting with bromine in the presence of sunlight gives mainly

Possible Answers

Selected	Possible Answer
	2 – bromo -2 – methylbutane
	2 – bromo -3 – methylbutane
	1 – bromo -2 – methylbutane
	1 – bromo -3 – methylbutane

Question

The photon of hard gamma radiation knocks a proton out of $^{24}_{12}Mg$ nucleus to form

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Selected	Possible Answer
	the isobar of $^{23}_{11}Na$
	the nuclide $^{23}_{11}Na$
	the isobar of parent nucleus
	the isotope of parent nucleus

Question

The best reagent to convert pent -3- en-2-ol

into pent

one is

Possible Answers

Selected	Possible Answer
	Chromic anhydride in glacial acetic acid
	Acidic permanganate
	Pyridinium chloro – chromate
	Acidic dichromate

Question

Tertiary alkyl halides are practically inert to substitution by S_N^2

mechanism because of

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Selected	Possible Answer
	instability
	insolubility
	steric hindrance
	inductive effect

Question

In both DNA and RNA, heterocyclic base and phosphate ester linkages are at-

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Selected	Possible Answer
	$C^{I}_5 \; \text{and} \; C^{I}_2$ respectively of the sugar molecule
	$C^{I}_5 \; \text{and} \; C^{I}_1$ respectively of the sugar molecule
	$C^{I}_2 \; \text{and} \; C^{I}_5$ respectively of the sugar molecule
	$C^{I}_1 \; \text{and} \; C^{I}_5$ respectively of the sugar molecule

Question

Reaction of one molecule of HBr

with one molecule of 1,3-butadiene at $40^{\circ}C$ gives predominantly

Possible Answers

Selected	Possible Answer
	1-bromo-2-butene under thermodymically controlled conditions
	3-bromobutene under kinetically controlled conditions
	3-bromobutene under thermodynamically controlled conditions
	1-bromo-2-butene under kinetically controlled conditions

Question

Among the following acids which has the lowest pK_a

value?

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Question

The decreasing order of nucleophilicity among the nucleophiles (a) !{width:100px}

(b)

(c)

(d)!{width:170px}

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Selected	Possible Answer
	(a), (b), (c), (d)
	(c), (b), (a), (d)
	(b), (c), (a), (d)
	(d), (c), (b), (a)

Question

Which one of the following methods is neither meant for the synthesis nor for separation of amines?

Possible Answers

Selected	Possible Answer
	Curtius reaction
	Hinsberg method
	Hofmann method
	Wurtz reaction

Question

Which of the following is fully fluorinated polymer?

Possible Answers

Selected	Possible Answer
	Thiokol
	Neoprene
	Teflon
	PVC

Question

Of the five isomeric hexanes, the isomer which can give two monochlorinated compounds is

Possible Answers

Selected	Possible Answer
	2, 3-dimethylbutane
	2,2-dimethylbutane
	n-hexane
	2-methylpentane

Question

Alkyl halides react with dialkyl copper reagents to give

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Selected	Possible Answer
	alkenes
	alkenyl halides
	alkyl copper halides
	None of these

Question

Acid catalyzed hydration of alkenes except ethene leads to the formation of

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	secondary or tertiary alcohol
	mixture of secondary and tertiary alcohols
	mixture of primary and secondary alcohols
	primary alcohol

Question

Amongst the following the most basic compound is

Possible Answers

Selected	Possible Answer
	aniline
	acetanilide
	p-nitroaniline
	benzylamine

Question

Which types of isomerism is shown by 2,3-dichlorobutane?

Possible Answers

Selected	Possible Answer
	Optical
	Geometric
	Diastereo
	Structural

Question

The reaction !{width:280px}

is fastest when X is

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Selected	Possible Answer

Question

Elimination of bromine from 2-bromobutane results in the formation of-

Possible Answers

Selected	Possible Answer
	equimolar mixture of 1 and 2-butene
	predominantly 2-butene
	predominantly 1-butene
	predominantly 2-butyne

Question

Equimolar solutions in the same solvent have

Possible Answers

Selected	Possible Answer
	Same freezing point but different boiling point
	Same boiling and same freezing points
	Same boiling point but different freezing point
	Different boiling and different freezing points

Question

Which of the following statements in relation to the hydrogen atom is correct?

Possible Answers

Selected	Possible Answer
	3s, 3p and 3d orbitals all have the same energy
	3s and 3p orbitals are of lower energy than 3d orbital
	3p orbital is lower in energy than 3d orbital
	3s orbital is lower in energy than 3p orbital

Question

The structure of diborane (B_2H_6)

contains

Possible Answers

Selected	Possible Answer
	two 2c-2e bonds and two 3c-3e bonds
	four 2c-2e bonds and two 3c-2e bonds
	four 2c-2e bonds and four 3c-2e bonds
	two 2c-2e bonds and four 3c-2e bonds

Question

The value of the ‘spin only’ magnetic moment for one of the following configurations is 2.84 BM. The correct one is

Possible Answers

Selected	Possible Answer
	(in strong ligand field)
	(in strong ligand filed)
	(in weak ligand field)
	(in weak as well as in strong fields)

Question

Which of the following factors may be regarded as the main cause of lanthanide contraction?

Possible Answers

Selected	Possible Answer
	Effective shielding of one of electrons by another in the subshell
	Greater shielding of 5d electrons by electrons
	Poorer shielding of 5d electrons by electrons
	Poor shielding of one of electron by another in the subshell

Question

Reaction of cyclohexanone with dimethylamine in the presence of catalytic amount of an acid forms a compound if water during the reaction is continuously removed. The compound formed is generally known as

Possible Answers

Selected	Possible Answer
	an amine
	a Schiff’s base
	an enamine
	an imine

Question

p-cresol reacts with chloroform in alkaline medium to give the compound A which adds hydrogen cyanide to form, the compound B. The latter on acidic hydrolysis gives chiral carboxylic acid. The structure of the carboxylic acid is

Possible Answers

Selected	Possible Answer
	!{width:180px}
	!{width:180px}
	!{width:180px}
	!{width:180px}

Question

An organic compound having molecular mass 60 is found to contain C = 20%, H = 6.67% and N = 46.67% while rest is oxygen. On heating it gives NH_3

alongwith a solid residue. The solid residue give violet colour with alkaline copper sulphate solution. The compound is

Possible Answers

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Question

If the bond dissociation energies of $XY, X_2 \; \text{and} \; Y_2$ (all diatomic molecules) are in the ratio of $1:1:0.5 \;\text{and} \; \Delta_f H$ for the formation of $XY \; \text{is}\; -200 \; kJ \; mole^{-1}$ . The bond dissociation energy of X_2

will be

Possible Answers

Selected	Possible Answer
	$800 \; \text{kJ \; mol}^{-1}$
	$400 \; \text{kJ \; mol}^{-1}$
	$100 \; \text{kJ \; mol}^{-1}$
	$200 \; \text{kJ \; mol}^{-1}$

Question

$t_{1/4}$ can be taken as the time taken for the concentration of a reactant to drop to $\frac{3}{4}$ of its initial value. If the rate constant for a first order reaction is K, the $t_{1/4}$ can be written as

Possible Answers

Selected	Possible Answer
	0.10 / K
	0.75 / K
	0.29 / K
	0.69 / K

Question

An amount of solid NH_4HS

is placed in a flask already containing ammonia gas at a certain temperature and 0.50 atm. Pressure. Ammonium hydrogen sulphide decomposes to yield NH_3

and

gases in the flask. When the decomposition reaction reaches equilibrium, the total pressure in the flask rises to 0.84 atm. The equilibrium constant for NH_4HS

decomposition at this temperature is

Possible Answers

Selected	Possible Answer
	0.17
	0.30
	0.18
	0.11

Question

A projectile can have the same range R for two angles of projection. If t_1

and

be the times of flights in the two cases, then the product of the two time of flights is proportional to

Possible Answers

Selected	Possible Answer

Question

An annular ring with inner and outer radii R_1

and

is rolling without slipping with a uniform angular speed. The ratio of the forces experienced by the two particles situated on the inner and outer parts of the ring, F_1/F_2

Possible Answers

Selected	Possible Answer
	$\frac{R1}{R2}$
	$\bigg(\frac{R1}{R2}\bigg)^2$

	$\frac{R2}{R1}$

Question

A smooth block is released at rest on a $45^{\circ}$ incline and then slides a distance

. The time taken to slide is

times as much to slide on rough incline than on a smooth incline. The coefficient of friction is

Possible Answers

Selected	Possible Answer
	$\mu_K=\sqrt{1-\cfrac{1}{n^2}}$
	$\mu_S=\sqrt{1-\cfrac{1}{n^2}}$
	$\mu_S=1-\cfrac{1}{n^2}$
	$\mu_K=1-\cfrac{1}{n^2}$

Question

The upper half of an inclined plane with inclination $\phi$ is perfectly smooth while the lower half is rough. A body starting from rest at the top will again come to rest at the bottom if the coefficient of friction for the lower half is given by

Possible Answers

Selected	Possible Answer
	$2\sin \phi$
	$\tan \phi$
	$2\tan \phi$
	$2\cos \phi$

Question

A bullet fired into a fixed target loses half of its velocity after penetrating 3 cm

. How much further it will penetrate before coming to rest assuming that it faces constant resistance to motion?

Possible Answers

Selected	Possible Answer
	2.0 cm
	1.5 cm
	1.0 cm
	3.0 cm

Question

Out of the following pair, which one does NOT have identical dimensions is

Possible Answers

Selected	Possible Answer
	work and torque
	impulse and momentum
	angular momentum and Planck’s constant
	moment of inertia and moment of a force

Question

The relation between time

and distance $x \text{is} t=ax^2+bx$ where

and

are constants. The acceleration is

Possible Answers

Selected	Possible Answer

Question

A car starting from rest accelerates at the rate

through a distance

, then continues at constant speed for time

and then decelerates at the rate f/2

to come to rest. If the total distance traversed is 15 S

, then

Possible Answers

Selected	Possible Answer

Question

A particle is moving eastwards with a velocity of 5 m/s

in $10 \text{seconds}$ the velocity changes to 5 m/s

northwards. The average acceleration in this time is

Possible Answers

Selected	Possible Answer
	$\cfrac{1}{2} \; m/s^2 \; \text{towards north}$
	$\cfrac{1}{\sqrt{2}} \; m/s^2 \; \text{towards north-east}$
	$\cfrac{1}{\sqrt{2}} \; m/s^2 \; \text{towards north-west}$
	$\text{Zero}$

Question

A parachutist after bailing out falls 50 m

without friction. When parachute opens, it decelerates at 2 m/s^2

. He reaches the ground with a speed of 3 m/s

. At what height, did he bail out?

Possible Answers

Selected	Possible Answer
	91 m
	182 m
	293 m
	111 m

Question

A block is kept on a frictionless inclined surface with angle of inclination $\alpha$ . The incline is given an acceleration a to keep the block stationary. Then a is equal to

!{width:180px}

Possible Answers

Selected	Possible Answer
	$g \; cosec \; \alpha$
	$g\tan \alpha$
	$g/\tan \alpha$

Question

A spherical ball of mass $20 \; kg$ is stationary at the top of a hill of height $100 \; m$ . It rolls down a smooth surface to the ground, then climbs up another hill of height $30 \; m$ and finally rolls down to a horizontal base at a height of $20 \; m$ above the ground. The velocity attained by the ball is

Possible Answers

Selected	Possible Answer
	$10 \; m/s$
	$20 \; m/s$
	$40 \; m/s$
	$10 \sqrt{30} \; m/s$

Question

A body

of mass

while falling vertically downwards under gravity breaks into two parts; a body

of mass

and a body

of mass

. The centre of mass of bodies

and

taken together shifts compared to that of body

towards

Possible Answers

Selected	Possible Answer
	body C
	does not shift
	depends on height of breaking
	body B

Question

The moment of inertia of a uniform semicircular disc of mass

and radius

about a line perpendicular to the plane of the disc through the centre is

Possible Answers

Selected	Possible Answer
	$\cfrac{2}{5}Mr^2$
	$\cfrac{1}{2}Mr^2$

	$\cfrac{1}{4}Mr^2$

Question

A particle of mass 0.3 kg

is subjected to a force $F=-kx \; \text{with} \; k=15 N/m$ . What will be its initial acceleration if it is released from a point 20 cm

away from the origin?

Possible Answers

Selected	Possible Answer

Question

The block of mass

moving on the frictionless horizontal surface collides with a spring of spring constant

and compresses it by length

. The maximum momentum of the block after collision is

!{width:180px}

Possible Answers

Selected	Possible Answer

	$\cfrac{KL^2}{2M}$
	$\sqrt{MK}L$
	$\cfrac{ML^2}{K}$

Question

A mass

moves with a velocity

and collides inelastically with another identical mass. After collision the $1^{st}$ mass moves with velocity $v/ \sqrt{3}$ in a direction perpendicular to the initial direction of motion. Find the speed of the $2^{nd}$ mass after collision

!{width:180px}

Possible Answers

Selected	Possible Answer
	$2v/\sqrt{3}$

	$v/\sqrt{3}$
	$\sqrt{3}v$

Question

long capillary tube is dipped in water. The water rises up to 8 cm

. If the entire arrangement is put in a freely falling elevator the length of water column in the capillary tube will be

Possible Answers

Selected	Possible Answer
	8 cm
	10 cm
	4 cm
	20 cm

Question

is stress and

is Young’s modulus of material of a wire, the energy stored in the wire per unit volume is

Possible Answers

Selected	Possible Answer

Question

Average density of the earth

Possible Answers

Selected	Possible Answer
	is directly proportional to g
	is inversely proportional to g
	is a complex function of g
	does not depend on g

Question

A body of mass

is accelerated uniformly from rest to a speed

in a time

. The instantaneous power delivered to the body as a function time is given by

Possible Answers

Selected	Possible Answer
	$\cfrac{1 mv^2}{2 T^2}.t^2$
	$\cfrac{mv^2}{T^2}.t^2$
	$\cfrac{1 mv^2}{2 T^2}.t$
	$\cfrac{mv^2}{T^2}.t$

Question

Consider a car moving on a straight road with a speed of 100 m/s

. The distance at which car can be stopped is $[\mu _k = 0.5]$

Possible Answers

Selected	Possible Answer
	800 m
	100 m
	1000 m
	400 m

Question

Which of the following is incorrect regarding the first law of thermodynamics?

Possible Answers

Selected	Possible Answer
	It introduces the concept of the entropy
	None
	It is a restatement of the principle of conservation of energy
	It is not applicable to any cyclic process
	It introduces the concept of the internal energy

Question

shaped object with dimensions shown in the figure, is lying on a smooth floor. A force

is applied at the point

parallel to

, such that the object has only the translational motion without rotation. Find the location of

with respect to

Possible Answers

Selected	Possible Answer
	$\cfrac{4}{3}l$
	$\cfrac{2}{3}l$

	$\cfrac{3}{2}l$

Question

The change in the value of

at a height 'h'

above the surface of the earth is the same as at a depth 'd'

below the surface of earth. When both 'd'

and

are much smaller than the radius of earth, then which one of the following is correct?

Possible Answers

Selected	Possible Answer

	$d = \cfrac{3h}{2}$

	$d = \cfrac{h}{2}$

Question

A particle of mass 10 g is kept on the surface of a uniform sphere of mass 100 kg and radius 10 cm. Find the work to be done against the gravitational force between them to take the particle far away from the sphere. (you may take $G = 6 . 67* 10^{-11} Nm^2 / kg^2)$

Possible Answers

Selected	Possible Answer
	$6.67 *10^{-10}J$
	$3.33 *10^{-10} J$
	$13.34 * 10^{-10} J$
	$6.67 *10^{-9}J$

Question

A gaseous mixture consists of 16 g of helium and 16 g of oxygen. The ratio $\cfrac{C_p}{C_v}.$ of the mixture is

Possible Answers

Selected	Possible Answer
	1.54
	1.62
	1.59
	1.4

Question

The intensity of gamma radiation from a given source is I. On passing through 36 mm of lead, it is reduced to $\cfrac{1}{8}$ . The thickness of lead which will reduce the intensity to $\cfrac{1}{2}$ will be.

Possible Answers

Selected	Possible Answer
	6 mm
	12 mm
	18 mm
	9 mm

Question

The electrical conductivity of a semiconductor increases when electromagnetic radiation of wavelength shorter than 2480 nm is incident on it. The band gap in (eV) for the semiconductor is

Possible Answers

Selected	Possible Answer
	0.5 eV
	0.7 eV
	1.1 eV
	2.5 eV

Question

A photocell is illuminated by a small bright source placed 1 m away. When the same source of light is placed $\cfrac{1}{2}$ m away, the number of electrons emitted by photo cathode would

Possible Answers

Selected	Possible Answer
	increase by a factor of 4
	decrease by a factor of 4
	increase by a factor of 2
	decrease by a factor of 2

Question

Starting with a sample of pure 66^Cu

, 7/8 of it decays into Zn in 15 minutes. The corresponding half-life is

Possible Answers

Selected	Possible Answer
	15 minutes
	5 minutes
	10 minutes
	$7 \cfrac{1}{2}$ minutes

Question

If radius of $^{27}_{13}\; Al$ nucleus is estimated to be 3.6 Fermi then the radius $^{125}_{52}\; Te$ nucleus be nearly

Possible Answers

Selected	Possible Answer
	4 fermi
	8 fermi
	6 fermi
	5 fermi

Question

The temperature-entropy diagram of a reversible engine cycle is given in the figure. Its efficiency is

Possible Answers

Selected	Possible Answer
	1/3
	2/3
	1/2
	1/4

Question

The figure shows a system of two concentric spheres of radii r_1

and

and kept at temperatures T_1

and

respectively. The radial rate of flow of heat in a substance between the two concentric sphere is proportional to

Possible Answers

Selected	Possible Answer
	$\cfrac{r_2-r_1}{r_1r_2}$
	$In\; \Big(\cfrac{r_2}{r_1}\Big)$
	$\cfrac{r_1r_2}{r_2-r_1}$

Question

A system goes from A to B via two processes I and II as shown in the figure. If $\Delta U_1$ and $\Delta U_2$ are the changes in internal energies in the processes I and II respectively, the

Possible Answers

Selected	Possible Answer
	$\Delta U_2 < \Delta U_1$
	relation between $\Delta U_1$ and $\Delta U_2$ can not be determined
	$\Delta U_1 = \Delta U_2$
	$\Delta U_2 > \Delta U_1$

Question

The function $sin^2(\omega t)$ represents

Possible Answers

Selected	Possible Answer
	a simple harmonic motion with a period $2 \pi /\omega$
	a periodic, but not simple harmonic motion with a period $2 \pi /\omega$
	a periodic, but not simple harmonic motion with a period $\pi /\omega$
	a simple harmonic motion with a period $\pi /\omega$ .

Question

A Young’s double slit experiment uses a monochromatic source. The shape of the interference fringes formed on a screen is

Possible Answers

Selected	Possible Answer
	straight line
	circle
	parabola
	hyperbola

Question

Two simple harmonic motions are represented by the equation $y_1 = 0.1 sin \Big(100 \pi t + \cfrac{\pi}{3}\Big)$ and $y_2 = 0.1 cos \pi t$ The phase difference of the velocity of particle 1 w.r.t. the velocity of the particle 2 is

Possible Answers

Selected	Possible Answer
	$-\pi /6$
	$\pi /6$
	$- \pi /6$
	$\pi /3$

Question

A fish looking up through the water sees the outside world contained in a circular horizon. If the refractive index of water is 4/3 and the fish is 12 cm below the surface, the radius of this circle in cm is

Possible Answers

Selected	Possible Answer
	$36 \sqrt{5}$
	$36/ \sqrt{7}$
	$36 \sqrt{7}$
	$4 \sqrt{5}$

Question

Two point white dots are 1 mm apart on a black paper. They are viewed by eye of pupil diameter 3 mm. Approximately, what is the maximum distance at which these dots can be resolved by the eye? [ Take wavelength of light = 500 nm ]

Possible Answers

Selected	Possible Answer
	3m
	6 m
	5 m
	1m

Question

A thin glass (refractive index 1.5) lens has optical power of - 5D

in air. Its optical power in a liquid medium with refractive index 1.6 will be

Possible Answers

Selected	Possible Answer
	- 25 D
	- 1 D
	1 D
	25 D
	None

Question

The diagram shows the energy levels for an electron in a certain atom. Which transition shown represents the emission of a photon with the most energy?

Possible Answers

Selected	Possible Answer
	IV
	III
	II
	I

Question

If the kinetic energy of a free electron doubles. Its deBroglie wavelength changes by the factor

Possible Answers

Selected	Possible Answer
	1/2
	2
	$\cfrac{1}{\sqrt{2}}$
	$\sqrt{2}$

Question

In a common base amplifier, the phase difference between the input signal voltage and output voltage is

Possible Answers

Selected	Possible Answer
	$\cfrac{\pi}{2}$
	$\pi$
	$\cfrac{\pi}{4}$
	0

Question

In a full wave rectifier circuit operating from 50 Hz mains frequency, the fundamental frequency in the ripple would be

Possible Answers

Selected	Possible Answer
	25 Hz
	100 Hz
	50 Hz
	70.7 Hz

Question

A nuclear transformation is denoted by $X(n, \alpha )\; ^7_3Li$ . Which of the following is the nucleus of element X ?

Possible Answers

Selected	Possible Answer
	$^{12}C_6$

	$^{11}_4 Be$
	$^{10}_5 B$

Question

A moving coil galvanometer has 150 equal divisions. Its current sensitivity is 10 divisions per milliampere and voltage sensitivity is 2 divisions per millivolt. In order that each division reads 1 volt, the resistance in ohms needed to be connected in series with the coil will be

Possible Answers

Selected	Possible Answer
	99995


	9995

Question

Two voltameters one of copper and another of silver, are joined in parallel. When a total charge q flows through the voltameters, equal amount of metals are deposited. If the electrochemical equivalents of copper and silver are z_1

and

respectively the charge which flows through the silver voltameter is

Possible Answers

Selected	Possible Answer
	$q \frac{z_1}{z_2}$
	$\cfrac{q}{1+\frac{z_2}{z_1}}$
	$\cfrac{q}{1+\frac{z_1}{z_2}}$
	$q \frac{z_2}{z_1}$

Question

In the circuit, the galvanometer G shows zero deflection. If the batteries A and B have negligible internal resistance, the value of the resistor R will be

Possible Answers

Selected	Possible Answer
	$500 \Omega$
	$200 \Omega$
	$1000 \Omega$
	$100 \Omega$

Question

Two sources of equal emf are connected to an external resistance R. The internal resistance of the two sources are R_1

and

. If the potential difference across the source having internal resistance R_2

is zero, then

Possible Answers

Selected	Possible Answer

Question

A fully charged capacitor has a capacitance ‘C’ it is discharged through a small coil of resistance wire embedded in a thermally insulated block of specific heat capacity ‘s’ and mass ‘m’. If the temperature of the block is raised by $\Delta T$ The potential difference V across the capacitance is

Possible Answers

Selected	Possible Answer
	$\cfrac{ms \Delta T}{C}$
	$\cfrac{\sqrt{2ms \Delta T}}{C}$
	$\cfrac{\sqrt{2mC \Delta T}}{s}$
	$\cfrac{mC \Delta T}{s}$

Question

One conducting U tube can slide inside another as shown in figure, maintaining electrical contacts between the tubes. The magnetic field B is perpendicular to the plane of the figure. if each tube moves towards the other at a constant speed V, then the emf induced in the circuit in terms of B, $\ell$ and V where $\ell$ is the width of each tube will be

Possible Answers

Selected	Possible Answer
	$2 B \ell V$
	$- B \ell V$
	$B \ell V$
	zero

Question

A heater coil is cut into two equal parts and only one part is now used in the heater. The heat generated will now be

Possible Answers

Selected	Possible Answer
	doubled
	halved
	four times
	one fourth

Question

Two thin, long parallel wires separated by a distance ‘d’ carry a current of ‘i’ A in the same direction. They will

Possible Answers

Selected	Possible Answer
	attract each other with a force of $\mu _0i^2/(2 \pi d^2)$
	repel each other with a force of $\mu _0i^2/(2 \pi d^2)$
	repel each other with a force of $\mu _0i^2/(2 \pi d)$
	attract each other with a force of $\mu _0i^2/(2 \pi d)$

Question

When an unpolarized light of intensity l_0

is incident on a polarizing sheet, the intensity of the light which does not get transmitted is

Possible Answers

Selected	Possible Answer

	$\cfrac{1}{2} l_0$
	zero
	$\cfrac{1}{4} l_0$

Question

A charged ball B hangs from a silk thread S which makes an angle $\theta$ with a large charged conducting sheet P, as show in the figure. The surface charge density $\sigma$ of the sheet is proportional to

Possible Answers

Selected	Possible Answer
	$cot \theta$
	$sin \theta$
	$cos \theta$
	$\tan \theta$

Question

Two point charges + 8q

and

are located at x = 0 and x = L respectively. The location of a point on the x axis at which the net electric field due to these two point charges is zero is

Possible Answers

Selected	Possible Answer
	4L
	L/4
	8L
	2L

Question

Two thin wires rings each having a radius R are placed at a distance d apart with their axes coinciding. The charges on the two rings are +q and -q. The potential difference between the centres of the two rings is

Possible Answers

Selected	Possible Answer
	zero
	$QR/4 \pi \epsilon _0 d^2$
	$\cfrac{Q}{2\pi \epsilon _0} \Big[\cfrac{1}{2}- \cfrac{1}{\sqrt{R^2+d^2}}\Big]$
	$\cfrac{Q}{4\pi \epsilon _0} \Big[\cfrac{1}{2}- \cfrac{1}{\sqrt{R^2+d^2}}\Big]$

Question

A parallel plate capacitor is made by stacking n equally spaced plates connected alternatively. If the capacitance between any two adjacent plates is C then the resultant capacitance is

Possible Answers

Selected	Possible Answer
	C
	nC
	(n – 1)C
	(n + 1)C

Question

When two tuning forks (fork 1 and fork 2) are sounded simultaneously, 4 beats per second are heard. Now, some tape is attached on the prong of the fork 2. When the tuning forks are sounded again, 6 beats per seconds are heard. If the frequency of fork 1 is 200 Hz, then what was the original frequency of fork 2?

Possible Answers

Selected	Possible Answer
	200 Hz
	196 Hz
	204 Hz
	202 Hz

Question

If a simple harmonic motion is represented by $\cfrac{d^2x}{dt^2}+\alpha x = 0,$ its time period is

Possible Answers

Selected	Possible Answer
	$\cfrac{2 \pi}{\sqrt{\alpha}}$
	$2 \pi \alpha$
	$\cfrac{2 \pi}{\alpha}$
	$2 \pi \sqrt{\alpha}$

Question

The bob of a simple pendulum is a spherical hollow ball filled with water. A plugged hole near the bottom of the oscillation bob gets suddenly unplugged. During observation, till water is coming out, the time period of oscillation would

Possible Answers

Selected	Possible Answer
	remain unchanged.
	first increase and then decrease to the original value.
	first decreased then increase to the original value.
	increase towards a saturation value.

Question

An observer moves towards a stationary source of sound, with a velocity one fifth of the velocity of sound. What is the percentage increase in the apparent frequency?

Possible Answers

Selected	Possible Answer
	20%
	zero
	5%
	0.5%

Question

is the intensity of the principal maximum in the single slit diffraction pattern, then what will be its intensity when the slit width is doubled?

Possible Answers

Selected	Possible Answer

Question

Two concentric coils each of radius equal to $2 \pi$ cm are placed at right angles to each other. 3 Ampere and 4 ampere are the currents flowing in each coil respectively. The magnetic induction in Weber/m^2

at the centre of the coils will be $( \mu _0 = 4 \pi * 10^{-7}\; Wb/A-m)$

Possible Answers

Selected	Possible Answer
	$12 * 10^{-5}$
	$5 * 10^{-5}$
	$7 * 10^{-5}$
	$10^{-5}$

Question

A coil of inductance 300 mH and resistance $2 \Omega$ is connected to a source of voltage 2V. The current reaches half of its steady state value in

Possible Answers

Selected	Possible Answer
	0.1 s
	0.3 s
	0.05 s
	0.15 s

Question

The self inductance of the motor of an electric fan is 10 H. In order to impart maximum power at 50 Hz, it should be connected to a capacitance of

Possible Answers

Selected	Possible Answer
	$2 \mu F$
	$4 \mu F$
	$8 \mu F$
	$1 \mu F$

Question

An energy source will supply a constant current into the load of its internal resistance is

Possible Answers

Selected	Possible Answer
	non-zero but less than the resistance of the load.
	equal to the resistance of the load.
	zero
	very large as compared to the load resistance.

Question

A circuit has a resistance of $12 \Omega$ and an impedance of $15 \Omega$ . The power factor of the circuit will be

Possible Answers

Selected	Possible Answer
	0.8
	1.25
	0.4
	0.125

Question

The phase difference between the alternating current and emf is $\pi /2$ . Which of the following cannot be the constituent of the circuit?

Possible Answers

Selected	Possible Answer
	C alone
	L alone
	R.L
	L. C

Question

A uniform electric field and a uniform magnetic field are acting along the same direction in a certain region. If an electron is projected along the direction of the fields with a certain velocity then

Possible Answers

Selected	Possible Answer
	its velocity will decrease.
	it will turn towards left of direction of motion
	its velocity will increase.
	it will turn towards right of direction of motion.

Question

A charged particle of mass m and charge q travels on a circular path of radius r that is perpendicular to a magnetic field B. The time taken by the particle to complete one revolution is

Possible Answers

Selected	Possible Answer
	$\cfrac{2 \pi m}{qB}$
	$\cfrac{2 \pi q^B}{m}$
	$\cfrac{2 \pi mq}{B}$
	$\cfrac{2 \pi qB}{m}$

Question

In a potentiometer experiment the balancing with a cell is at length 240 cm. On shunting the cell with a resistance of $2 \Omega$ the balancing length becomes 120 cm. The internal resistance of the cell is

Possible Answers

Selected	Possible Answer
	$0.5 \Omega$
	$2 \Omega$
	$1 \Omega$
	$4 \Omega$

Question

The resistance of hot tungsten filament is about 10 times the cold resistance. What will be the resistance of 100 W and 200 V lamp when not in use?

Possible Answers

Selected	Possible Answer
	$40 \Omega$
	$200 \Omega$
	$400 \Omega$
	$20 \Omega$

Question

A magnetic needle is kept in a non-uniform magnetic field. It experiences

Possible Answers

Selected	Possible Answer
	neither a force nor a torque
	a force but not a torque.
	a torque but not a force
	a force and a torque.

Question

, then the inverse of

Possible Answers

Selected	Possible Answer
	A + I
	A
	A – I
	I – A

Question

If the cube roots of unity are $1, \omega, \omega ^2$ then the roots of the equation (x - 1)^3 + 8 = 0,

are

Possible Answers

Selected	Possible Answer
	$-1 , - 1 + 2\omega, - 1 - 2\omega ^2$
	$-1 , 1 - 2\omega, 1 - 2\omega ^2$
	-1 , -1, – 1
	$-1 , 1 + 2\omega, 1 + 2\omega ^2$

Question

Let $R = \{(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)\}$ be a relation on the set A = {3, 6, 9, 12}

be a relation on the set A = {3, 6, 9, 12}

. The relation is

Possible Answers

Selected	Possible Answer
	reflexive and transitive only
	reflexive and symmetric only
	an equivalence relation
	reflexive only

Question

Area of the greatest rectangle that can be inscribed in the ellipse $\cfrac{x^2}{a^2}+\cfrac{y^2}{b^2}= 1.$ is

Possible Answers

Selected	Possible Answer
	$\cfrac{a}{b}$


	$\sqrt{ab}$

Question

The differential equation representing the family of curves $y^2 = 2c(x +\sqrt{c }),$ where c > 0

, is a parameter, is of order and degree as follows:

Possible Answers

Selected	Possible Answer
	order 1, degree 1
	order 1, degree 2
	order 2, degree 2
	order 1, degree 3

Question

$\underset{n \rightarrow \infty}{lim}$ $\Big[\cfrac{1}{n^2}\; sec^2 \; \cfrac{1}{n^2} + \cfrac{2}{n^2} \; sec^2 \; \cfrac{4}{n^2}+..$ $+ \cfrac{1}{n^2} \; sec^2\; 1 \Big] \; equals$

Possible Answers

Selected	Possible Answer
	$\cfrac{1}{2}\; tan 1$
	$\cfrac{1}{2}\; cosec 1$
	tan1
	$\cfrac{1}{2}\; sec 1$

Question

ABC is a triangle. Forces $\vec P, \vec Q, \vec R$ acting along IA, IB

and

respectively are in equilibrium, where I is the incentre of $\Delta ABC.$ Then P : Q : R

Possible Answers

Selected	Possible Answer
	$sin \cfrac{A}{2} : sin \cfrac{B}{2}: sin \cfrac{C}{2}$

	$cos \cfrac{A}{2} : cos \cfrac{B}{2}: cos \cfrac{C}{2}$

Question

If in a frequently distribution, the mean and median are 21 and 22 respectively, then its mode is approximately

Possible Answers

Selected	Possible Answer
	22.0
	24.0
	20.5
	25.5

Question

Let

be the point (1, 0)

and

a point on the locus y^2 = 8x

. The locus of mid point of

Possible Answers

Selected	Possible Answer

Question

is the mid point of

and

is any point outside

, then

Possible Answers

Selected	Possible Answer
	$\overrightarrow{ P A}+\overrightarrow{P B}+ \overrightarrow{P C}=0$
	$\overrightarrow{ P A}+\overrightarrow{P B} =2 \overrightarrow{P C}$
	$\overrightarrow{ P A}+\overrightarrow{P B}+2 \overrightarrow{P C}=0$
	$\overrightarrow{ P A}+\overrightarrow{P B} = \overrightarrow{P C}$

Question

If the coefficients of rth, (r+ 1)th

and

terms in the binomial expansion of (1 + y)^m

are in A.P., then m and r satisfy the equation

Possible Answers

Selected	Possible Answer

Question

$In\; a\; triangle\; PQR\;, \angle R= \cfrac{\pi}{2}.$ $If\; tan \Big(\cfrac{P}{2} \Big)\; and\;$ $tan \Big(\cfrac{Q}{2} \Big)\; are\; the\; roots\; of$ $ax^2 + bx + c = 0, a \neq 0 \; then$

Possible Answers

Selected	Possible Answer

Question

The system of equations $\alpha x + y + z = \alpha - 1,$ $x + \alpha y + z = \alpha - 1,$ $x + y + \alpha z = \alpha - 1$ $has\; no\; solution, if\; \alpha \; is$

Possible Answers

Selected	Possible Answer
	-2
	not -2
	1
	either – 2 or 1

Question

The value of $\alpha$ for which the sum of the squares of the roots of the equation x^2 - (a -2)x- a - 1 = 0

assume the least value is

Possible Answers

Selected	Possible Answer
	0
	3
	1
	2

Question

If roots of the equation x^2 - bx + c = 0

be two consectutive integers, then b^2 - 4c

equals

Possible Answers

Selected	Possible Answer
	1
	2
	3
	- 2

Question

If the letters of word SACHIN are arranged in all possible ways and these words are written out as in dictionary, then the word SACHIN appears at serial number

Possible Answers

Selected	Possible Answer
	603
	600
	601
	602

Question

The value of $^{50}C_4 + \overset{6}{\underset{r-1}{\sum}} ^{56-rC_3}\; is$

Possible Answers

Selected	Possible Answer
	$^{55}C_4$
	$^{56}C_4$
	$^{55}C_3$
	$^{56}C_3$

Question

then which one of the following holds for all $n \geq 1$ by the principle of mathematical indunction

Possible Answers

Selected	Possible Answer

	$A^n = 2^{n-1}A- (n - 1)I$

	$A^n = 2^{n-1}A + (n - 1)I$

Question

If the coefficient of x^7

in $\Big[ax^2 +\Big(\cfrac{1}{bx} \Big)\Big]^{11}$ equals the coefficient of $x^{-7}$ in $\Big[ax^2 -\Big(\cfrac{1}{bx} \Big)\Big]^{11}$ then

and

satisfy the relation

Possible Answers

Selected	Possible Answer
	$\cfrac{a}{b}=1$

Question

$Let\; f : (-1, 1) \rightarrow B$ , be a function defined by $f(x) = \tan ^{-1} \cfrac{2x}{1-x^2}$ then

is both one-one and onto when

is the interval

Possible Answers

Selected	Possible Answer
	$\Big[0, \cfrac{\pi}{2}\Big)$
	$\Big(0, \cfrac{\pi}{2}\Big)$
	$\Big(-\cfrac{\pi}{2}, \cfrac{\pi}{2}\Big)$
	$\Big[-\cfrac{\pi}{2}, \cfrac{\pi}{2}\Big]$

Question

and

are two non-zero complex numbers such that |z_1 + z_2| = |z_1| + |z_2|

then

is equal to

Possible Answers

Selected	Possible Answer
	$- \cfrac{\pi}{2}$
	$\cfrac{\pi}{2}$
	0
	$- \pi$

Question

$If \; \omega = \cfrac{z}{z-\frac{1}{3}i}\;$ and $|\omega| = 1,$ then

lies on

Possible Answers

Selected	Possible Answer
	a circle
	an ellipse
	a straight line
	a parabola.

Question

$a^2 + b^2 + c^2 = -2\; and\; f(x) =$

then

is a polynomial of degree

Possible Answers

Selected	Possible Answer
	3
	2
	0
	1

Question

The normal to the curve $x = a(cos \theta + \theta \; \sin \theta), y = a( \sin \theta - \theta \cos \theta)$ at any point $\theta$ is such that

Possible Answers

Selected	Possible Answer
	it is at a constant distance from the origin
	it makes angle $\cfrac{\pi}{2} + \theta$ with the x-axis
	it passes through the origin
	it passes through $\Big(a \cfrac{\pi}{2} , -a \Big)$

Question

A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched?

Interval	Function
$(1)\; (- \infty, \infty)$
$(2)\; [2, \infty)$
$(3)\; \Big(- \infty, \cfrac{1}{3} \Big)$
$(4)\; (- \infty, -4]$

Possible Answers

Selected	Possible Answer
	2
	4
	3
	1

Question

$Let\; \alpha$ and $\beta$ be the distinct roots of ax^2 + bx + c = 0,

then $\overset{\lim}{x \rightarrow \alpha} \cfrac{1-cos(ax^2+bx+c)}{(x-\alpha)^2}$ equal to

Possible Answers

Selected	Possible Answer
	$\cfrac{1}{2}(\alpha-\beta)^2$
	$\cfrac{a^2}{2}(\alpha-\beta)^2$
	$- \cfrac{a^2}{2}(\alpha-\beta)^2$
	0

Question

Suppose

is differentiable x = 1

and $\overset{\lim}{h\rightarrow 0} \cfrac{1}{h} f(1+h)=5,$ then f'(1)

equals

Possible Answers

Selected	Possible Answer
	4
	6
	3
	5

Question

Let f be differentiable for all

. If $f(1) = -2\; and\; f'(x) \geq 2\; for\; x \in [1, 6] ,$ then

Possible Answers

Selected	Possible Answer


	$f(6) \le 8$

Question

If f is a real-valued differentiable function satisfying $|f(x) - f(y)| \le (x - y)^2, x, y \in R$ and f(0) = 0

, then f(1) equals

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Selected	Possible Answer
	-1
	0
	1
	2

Question

is so small that x^3

and higher powers of

may be neglected, then $\cfrac{(1+x)^{2/3} - \Big( 1+ \frac{1}{2} x\Big)^3}{(1-x)^{1/2}}$ may be approximated as

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Selected	Possible Answer
	$-\cfrac{3}{8} x^2$
	$\cfrac{x}{2}-\cfrac{3}{8} x^2$
	$3x +\cfrac{3}{8} x^2$
	$1-\cfrac{3}{8} x^2$

Question

$If \;x = \overset{\infty}{\underset{n=0}{\Sigma}} a^n,$ $y = \overset{\infty}{\underset{n=0}{\Sigma}} b^n,$ $z= \overset{\infty}{\underset{n=0}{\Sigma}} c^n$ where a, b, c

are in A.P. and |a| < 1, |b|<1, |c|< 1

, then

are in

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Selected	Possible Answer
	G.P.
	Arithmetic − Geometric Progression
	A.P
	H.P.

Question

In a triangle $\text{ABC}$ , let $\angle C = \cfrac{\pi}{2}$ If

is the inradius and

is the circumradius of the the triangle $\text{ABC}$ , then 2 (r + R)

equals

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Question

$If\; cos^{-1}\; x - cos^{-1}\; \cfrac{y}{2}= \alpha\; then\; 4x^2 - 4xy\; cos\; \alpha + y^2$ is equal to

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Selected	Possible Answer
	4
	$4 sin^2 \alpha$
	$-4 sin^2 \alpha$
	$2 sin 2\alpha$

Question

If in a triangle $\text{ABC}$ , the altitudes from the vertices $\text{A, B, C}$ on opposite sides are in H.P., then $\sin A, \sin B, \sin C$ are in

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Selected	Possible Answer
	G.P.
	A.P.
	Arithmetic − Geometric Progression
	H.P.

Question

$If\; I_1 = \overset{1}{\underset{0}{\int}} 2^{x2} dx, l_2=\overset{1}{\underset{0}{\int}} 2^{x3} dx,$ $l_3=\overset{2}{\underset{1}{\int}}2^{x2} dx, and \; l_4=\overset{2}{\underset{1}{\int}} 2^{x3} \; dx\; then$

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Question

The area enclosed between the curve y = log_e (x + e)

and the coordinate axes is

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Selected	Possible Answer
	3
	2
	4
	1

Question

The parabolas $y^2 = 4x\; and\; x^2 = 4y$ divide the square region bounded by the lines x = 4, y = 4

and the coordinate axes. If S_1, S_2, S_3

are respectively the areas of these parts numbered from top to bottom; then S_1 : S_2 : S_3

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Selected	Possible Answer
	1 : 2 : 3
	2 : 1 : 2
	1 : 2 : 1
	1 : 1 : 1

Question

$If x \cfrac{dy}{dx}=y ((log y - log x + 1),$ then the solution of the equation is

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Selected	Possible Answer
	$y\; log \Big(\cfrac{x}{y}=cx \Big)$
	$log \Big(\cfrac{x}{y}=cy \Big)$
	$x\; log \Big(\cfrac{y}{x}=cy \Big)$
	$log \Big(\cfrac{y}{x}=cx \Big)$

Question

The line parallel to the x-axis and passing through the intersection of the lines $ax + 2by + 3b = 0 \;and\; bx - 2ay - 3a = 0, where (a, b) \neq (0, 0) \; is$

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Selected	Possible Answer
	above the x−axis at a distance of $\cfrac{3}{2}$ from it
	below the x−axis at a distance of $\cfrac{2}{3}$ from it
	below the x−axis at a distance of $\cfrac{3}{2}$ from it
	above the x−axis at a distance of $\cfrac{2}{3}$ from it

Question

A spherical iron ball 10 cm

in radius is coated with a layer of ice of uniform thickness than melts at a rate of $50\; cm^3/min.$ When the thickness of ice is 5 cm

, then the rate at which the thickness of ice decreases, is

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Selected	Possible Answer
	$\cfrac{5}{6 \pi}\; cm/min$
	$\cfrac{1}{36 \pi}\; cm/min$
	$\cfrac{1}{54 \pi}\; cm/min$
	$\cfrac{1}{18 \pi}\; cm/min$

Question

$Q. 191. \quad \int \Bigg\{ \cfrac{(log\; x -1)}{(1+log\; x)^2}\Bigg\}^2 \quad dx \; \text{is equal to}$

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Selected	Possible Answer
	$\cfrac{x}{(log x)^2+1}+c$
	$\cfrac{x}{( x^2+1)}+c$
	$\cfrac{xe^e}{(1+ x^2)}+c$
	$\cfrac{log x}{(log x)^2+1}+c$

Question

$Let\; f : R \rightarrow R$ be a differentiable function having $f (2) = 6, f' (2) =\Big(\cfrac{1}{48} \Big).$ then $\overset{lim}{x\rightarrow2} \overset{f(x)}{\underset{6}{\int}} \cfrac{4t^3}{x-2}\; dt\; equals$

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Selected	Possible Answer
	18
	12
	24
	36

Question

Let

be a non−negative continuous function such that the area bounded by the curve y = f (x)

, x−axis and the ordinates $x = \cfrac{\pi}{4}\; and$ $x = \beta > \cfrac{\pi}{4}\; is$ $\Bigg( \beta sin \beta + \cfrac{\pi}{4} cos \beta + \sqrt{2 \beta} \Bigg). then f \Big( \cfrac{\pi}{2}\Big) is$

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Selected	Possible Answer
	$\Bigg( \cfrac{\pi}{4}-\sqrt{2}-1 \Bigg)$
	$\Bigg( \cfrac{\pi}{4}+\sqrt{2}-1 \Bigg)$
	$\Bigg(1- \cfrac{\pi}{4}-\sqrt{2} \Bigg)$
	$\Bigg(1- \cfrac{\pi}{4}+\sqrt{2} \Bigg)$

Question

The locus of a point $P (\alpha, \beta)$ moving under the condition that the line $y = \alpha x + \beta$ is a tangent to the hyperbola $\cfrac{x^2}{a^2} - \cfrac{y^2}{b^2} = 1$ is

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Selected	Possible Answer
	an ellipse
	a parabola
	a hyperbola
	a circle

Question

If the angle $\theta$ between the line $\cfrac{x+1}{1} = \cfrac{y-1}{2} = \cfrac{z-2}{2}$ and the plane $2x - y + \sqrt{\lambda} z + 4 = 0$ is such that $\sin \theta = \cfrac{1}{3}$ the value of $\lambda$ is

Possible Answers

Selected	Possible Answer
	$\frac{-4}{3}$
	$\frac{5}{3}$
	$\frac{3}{4}$
	$\frac{-3}{5}$

Question

The angle between the lines 2x = 3y = - z

and

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Selected	Possible Answer
	$0^{\circ}$
	$30^{\circ}$
	$90^{\circ}$
	$45^{\circ}$

Question

If the plane 2ax - 3ay + 4az + 6 = 0

passes through the midpoint of the line joining the centers of the spheres x^2 + y^2 + z^2 + 6x - 8y - 2z = 13

and

, then a equals

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Question

The distance between the line $\vec{r} =2 \hat{i}-2\hat{j}+3\hat{k}+ \lambda (\hat{i} - \hat{j} + 4 \hat{k})$ and the plane $\vec{r}.(\hat{i} +5\hat{j} +\hat{k})=5$ is

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Selected	Possible Answer
	$\cfrac{10}{3\sqrt{3}}$
	$\cfrac{3}{10}$
	$\cfrac{10}{9}$
	$\cfrac{10}{3}$

Question

For any vectora $\hat{a}$ , the value of $(\vec{a} \times \hat{i})^2+(\vec{a} \times \hat{j})^2+(a \times \hat{k})^2$ is equal to

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Selected	Possible Answer
	$4\vec{a}^2$
	$2\vec{a}^2$
	$\vec{a}^2$
	$3\vec{a}^2$

Question

If non-zero numbers a, b, c

are in H.P., then the straight line $\frac{x}{a} + \frac{y}{b} + \frac{1}{c} = 0$ always passes through a fixed point. That point is

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Selected	Possible Answer

	$\bigg(1,- \frac{1}{2}$

Question

If a vertex of a triangle is (1, 1)

and the mid-points of two sides through this vertex are (-1, 2)

and

, then the centroid of the triangle is

Possible Answers

Selected	Possible Answer
	$\bigg(1, \cfrac{7}{3}\bigg)$
	$\bigg(\cfrac{1}{3}, \cfrac{7}{3}\bigg)$
	$\bigg( -1, \cfrac{7}{3}\bigg)$
	$\bigg(\cfrac{-1}{3}, \cfrac{7}{3}\bigg)$

Question

If the circles x^2 + y^2 + 2ax + cy + a = 0

and

intersect in two distinct points

and

then the line 5x + by - a = 0

passes through

and

for

Possible Answers

Selected	Possible Answer
	exactly two values of a
	no value of a
	infinitely many values of a
	exactly one value of a

Question

A circle touches the x-axis and also touches the circle with centre at (0, 3) and radius 2. The locus of the centre of the circle is

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Selected	Possible Answer
	an ellipse
	a circle
	a parabola
	a hyperbola

Question

If a circle passes through the point (a, b)

and cuts the circle x^2 + y^2 = p^2

orthogonally,then the equation of the locus of its centre is

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Question

An ellipse has

as semi minor axis,

and

its focii and the angle FBF'

is a right angle. Then the eccentricity of the ellipse is

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Selected	Possible Answer
	$\cfrac{1}{\sqrt{2}}$
	$\cfrac{1}{4}$
	$\cfrac{1}{2}$
	$\cfrac{1}{\sqrt{3}}$

Question

Let

and

be distinct non-negative numbers. If the vectors $a\hat{i} +a\hat{j} +c\hat{k}, \hat{i} +\hat{k}$ and $c\hat{i} +c\hat{j} +b\hat{k}$ lie in a plane, then

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Selected	Possible Answer
	the Geometric Mean of a and b
	the Arithmetic Mean of a and b
	equal to zero
	the Harmonic Mean of a and b

Question

If $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar vectors and $\lambda$ is a real number then $\bigg[ \lambda \big( \vec{a} + \vec{b} \big) \quad \lambda^2 \vec{b} \quad \lambda^2 \vec{c} \bigg]$ $= \bigg[\vec{a} \quad \vec{b}+\vec{c}\quad \vec{b}\bigg]$

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Selected	Possible Answer
	exactly three values of $\lambda$
	no value of $\lambda$
	exactly one value of $\lambda$
	exactly two values of $\lambda$

Question

Let $\vec{a}=\hat{i} - \hat{k}, \vec{b}=x \hat{i} + \hat{j}+(1-x)\hat{k}$ and $\vec{c}=y\hat{i}+x\hat{j}+(1+x-y)\hat{k}$ . Then $\big[\vec{a}, \; \vec{b}, \; \vec{c}\big]$

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Selected	Possible Answer
	only y
	neither x nor y
	both x and y
	only x

Question

Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three apply for the same house is

Possible Answers

Selected	Possible Answer
	$\cfrac{1}{9}$
	$\cfrac{2}{9}$
	$\cfrac{7}{9}$
	$\cfrac{8}{9}$

Question

A random variable

has Poisson distribution with mean 2. Then P(X > 1.5)

equals

Possible Answers

Selected	Possible Answer
	0
	$1-\cfrac{3}{e^2}$
	$\cfrac{2}{e^2}$
	$\cfrac{3}{e^2}$

Question

Let

and

be two events such that $P \overline {(A \cup B)} =\cfrac{1}{6}, p(A \cap B) = \cfrac{1}{4}\; and\; P(\bar{A}) =\cfrac{1}{4}$ where $\bar{A}$ stands for complement of event

. Then events

and

are

Possible Answers

Selected	Possible Answer
	independent but not equally likely
	equally likely but not independent
	equally likely and mutually exclusive
	mutually exclusive and independent

Question

Q. 212. A lizard, at an initial distance of 21 cm behind an insect, moves from rest with an acceleration of 2 cm/s^2 and pursues the insect which is crawling uniformly along a straight line at a speed of 20 cm/s . Then the lizard will catch the insect after

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Question

Q. 213. Two points and move from rest along a straight line with constant acceleration and respectively. If takes m sec. more than and describes 'n' units more than in acquiring the same speed then

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Selected	Possible Answer

	$(f' -f)n = \cfrac{1}{2} ff' m^2$
	$\cfrac{1}{2} (f + f' )m =ff' n^2$

Question

Q. 214. and are two like parallel forces. couple of moment lies in the plane of and and is contained with them. The resultant of and after combining is displaced through a distance

Possible Answers

Selected	Possible Answer
	$\cfrac{2H}{A-B}$
	$\cfrac{H}{2(A+B)}$
	$\cfrac{H}{A-B}$
	$\cfrac{H}{A+B}$

Question

Q. 215. The resultant of two forces acting on a particle is at right angles to one of them and its magnitude is one third of the other force. The ratio of larger force to smaller one is

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Selected	Possible Answer
	$3: 2\sqrt{2}$
	$3: \sqrt{2}$

Question

Q. 216. The sum of the series $1+ \cfrac{1}{4.2 !}+\cfrac{1}{16.4 !}+\cfrac{1}{64.6 !}+ \cdot{.....}$ ad inf. is

Possible Answers

Selected	Possible Answer
	$\cfrac{e+1}{\sqrt{e}}$
	$\cfrac{e-1}{2 \sqrt{e}}$
	$\cfrac{e+1}{2 \sqrt{e}}$
	$\cfrac{e-1}{\sqrt{e}}$

Question

The value of $\overset{x}{\underset{- \pi}{\int}} \cfrac{cos^2\; x}{1+a^x}\; dx, a>0, is$

Possible Answers

Selected	Possible Answer
	$2 \pi$
	$a \pi$
	$\cfrac{\pi}{2}$
	$\cfrac{\pi}{a}$

Question

Q. 218. The plane x + 2y - z = 4 cuts the sphere x^2 + y^2 + z^2 - x + z - 2 = 0 in a circle of radius

Possible Answers

Selected	Possible Answer
	1
	$\sqrt{2}$
	3
	2

Question

Q. 219. If the pair of lines ax^2 + 2(a + b)xy + by^2 = 0 lie along diameters of a circle and divide the circle into four sectors such that the area of one of the sectors is thrice the area of another sector then

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Question

$Let\; x_1, x_2, ....,x_n$ be

observations such that $\Sigma\; x^2_i = 400\; and\; \Sigma\; x_i =80.$ Then

possible value of

among the following is

Possible Answers

Selected	Possible Answer
	9
	18
	12
	15

Question

A particle is projected from a point

with velocity

at an angle of 60^o

with the horizontal. When it is moving in a direction at right angles to its direction at

, its velocity then is given by

Possible Answers

Selected	Possible Answer
	$\cfrac{u}{2}$
	$\cfrac{2u}{3}$
	$\cfrac{u}{\sqrt{3}}$
	$\cfrac{u}{3}$

Question

If both the roots of the quadratic equation x^2 - 2kx + k^2 + k - 5 = 0

are less than 5, then

lies in the interval

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Selected	Possible Answer


	$(-\infty, 4)$
	$(6, \infty)$

Question

are in G.P., then the determinant

is equal to

Possible Answers

Selected	Possible Answer
	1
	2
	4
	0

Question

A real valued function f(x)

satisfies the functional equation f(x - y) = f(x) f(y) - f(a - x) f(a + y)

where a is a given constant and f(0) = 1, f(2a - x)

is equal to

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Selected	Possible Answer

Question

If the equation $a_n x^n + a_{n-1} x^{n-1} + ....... + a_1 x= 0, a_1 \neq 0, n \geq 2,$ has a positive root $x =\alpha,$ then the equation $na_n x^{n-1} + (n-1) a_{n-1} x^{n-2} + ..... + a_1 = 0$ has a positive root, which is

Possible Answers

Selected	Possible Answer
	equal to $\alpha$
	smaller than $\alpha$
	greater than or equal to $\alpha$
	greater than $\alpha$

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