JEE Main 2005 · 08 May 2005 · Offline

Q1.A projectile can have the same range $R$ for two angles of projection. If $t_1$ and $t_2$ be the times of flights in the two cases, then the product of the two time of flights is proportional to

Q2.An annular ring with inner and outer radii $R_1$ and $R_2$ 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, $F_1/F_2$ is

Q3.A smooth block is released at rest on a $45^{\circ }$ incline and then slides a distance d. The time taken to slide is $n$ times as much to slide on rough incline than on a smooth incline. The coefficient of friction is

Q4.The upper half of an inclined plane with inclination $\varphi$ 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

Q5.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?

Q6.Out of the following pair, which one does NOT have identical dimensions is

Q7.The relation between time $t$ and distance $x$ is $t=a{x}^2+bx$ where $a$ and $b$ are constants.

The acceleration is

Q8.A particle is moving eastwards with a velocity of $5\mathrm{m}/\mathrm{s}$ in 10 seconds the velocity changes to $5\mathrm{m}/\mathrm{s}$ northwards. The average acceleration in this time is

Q9.A parachutist after bailing out falls 50 m without friction. When parachute opens, it decelerates at $2\mathrm{m}/{\mathrm{s}}^2$. He reaches the ground with a speed of $3\mathrm{m}/\mathrm{s}$. At what height, did he bail out?

Q10.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

Q11.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 …

Q12.A body A of mass M while falling vertically downwards under gravity breaks into two parts; a body B of mass $1/3\mathrm{M}$ and a body C of mass $2/3\mathrm{M}$. The centre of mass of bodies B and C taken together shifts compared to that of body A towards

Q13.The moment of inertia of a uniform semicircular disc of mass M and radius r about a line perpendicular to the plane of the disc through the centre is

Q14.A particle of mass 0.3 kg is subjected to a force $\mathrm{F}=-\mathrm{kx}$ with $\mathrm{k}=15\mathrm{N}/\mathrm{m}$. What will be its initial acceleration if it is released from a point 20 cm away from the origin?

Q15.The block of mass M moving on the frictionless horizontal surface collides with a spring of spring constant K and compresses it by length L. The maximum momentum of the block after collision is

Q16.A mass '$m$' moves with a velocity $v$ and collides inelastically with another identical mass. After collision the $1^{\text{st }}$ mass moves with velocity $\mathrm{v}/\sqrt{}3$ in a direction perpendicular to the initial direction of motion. Find the speed of the …

Q17.A 20 cm 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

Q18.If S is stress and Y is Young's modulus of material of a wire, the energy stored in the wire per unit volume is

Q19.Average density of the earth

Q20.A body of mass $m$ is accelerated uniformly from rest to a speed $v$ in a time $T$. The instantaneous power delivered to the body as a function time is given by

Q21.Consider a car moving on a straight road with a speed of $100\mathrm{m}/\mathrm{s}$. The distance at which car can be stopped is $\left[{\mu }_{\mathrm{k}}=0.5\right]$

Q22.A 'T' shaped object with dimensions shown in the figure, is lying on a smooth floor. A force $F$ is applied at the point P parallel to AB, such that the object has only the translational motion without rotation. Find the location of P with respect to C

Q23.The change in the value of $g$ 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 ' $h$ ' are much smaller than the radius of earth, then which one of the following is correct?

Q24.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 …

Q25.A gaseous mixture consists of 16 g of helium and 16 g of oxygen. The ratio $\frac{{\mathrm{C}}_{\mathrm{P}}}{{\mathrm{C}}_{\mathrm{v}}}$ of the mixture is

Q26.The intensity of gamma radiation from a given source is I. On passing through 36 mm of lead, it is reduced to $\frac{1}{8}$. The thickness of lead which will reduce the intensity to $\frac{1}{2}$ will be

Q27.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

Q28.A photocell is illuminated by a small bright source placed 1 m away. When the same source of light is placed $\frac{1}{2}\mathrm{m}$ away, the number of electrons emitted by photo cathode would

Q29.Starting with a sample of pure ${}^{66}\mathrm{Cu}$, 7/8 of it decays into Zn in 15 minutes. The corresponding half-life is

Q30.If radius of ${}_{13}^{27}\mathrm{Al}$ nucleus is estimated to be 3.6 Fermi then the radius ${}_{52}^{125}\mathrm{Te}$ nucleus be nearly

Q31.The temperature-entropy diagram of a reversible engine cycle is given in the figure. Its efficiency is

Q32.The figure shows a system of two concentric spheres of radii $r_1$ and $r_2$ and kept at temperatures $T_1$ and $T_2$ respectively. The radial rate of flow of heat in a substance between the two concentric sphere is proportional to

Q33.A system goes from A to B via two processes I and II as shown in the figure. If $\Delta {\mathrm{U}}_1$ and $\Delta {\mathrm{U}}_2$ are the changes in internal energies in the processes I and II respectively, the

Q34.The function ${\sin }^2(\omega t)$ represents

Q35.A Young's double slit experiment uses a monochromatic source. The shape of the interference fringes formed on a screen is

Q36.Two simple harmonic motions are represented by the equation $y_1=0.1\sin \left(100\pi t+\frac{\pi }{3}\right)$ 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

Q37.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

Q38.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]

Q39.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?

Q40.If the kinetic energy of a free electron doubles, its de-Broglie wavelength changes by the factor

Q41.In a common base amplifier, the phase difference between the input signal voltage and output voltage is

Q42.In a full wave rectifier circuit operating from 50 Hz mains frequency, the fundamental frequency in the ripple would be

Q43.A nuclear transformation is denoted by $\mathrm{X}(\mathrm{n},\alpha {)}_3^7\mathrm{Li}$. Which of the following is the nucleus of element $X$?

Q44.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 b …

Q45.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 ${\mathrm{z}}_1$ and ${\mathrm{z}}_2$ respectively t …

Q46.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

Q47.Two sources of equal emf are connected to an external resistance $R$. The internal resistance of the two sources are $R_1$ and $R_2\left(R_2>R_1\right)$. If the potential difference across the source having internal resistance $R_2$ is zero, then

Q48.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 differen …

Q49.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 cir …

Q50.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

Q51.Two thin, long parallel wires separated by a distance '$d$' carry a current of '$i$' $A$ in the same direction. They will

Q52.When an unpolarized light of intensity $I_0$ is incident on a polarizing sheet, the intensity of the light which does not get transmitted is

Q53.A charged ball $B$ hangs from a silk thread $S$ which makes an angle $\theta$ with a large charged conducting sheet $P$, as shown in the figure. The surface charge density $\sigma$ of the sheet is proportional to

Q54.Two point charges $+8q$ and $-2q$ 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

Q55.Two thin wire 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

Q56.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

Q57.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 second are heard. If the frequency of fork 1 is 200 Hz, then what was th …

Q58.If a simple harmonic motion is represented by $\frac{d^{2}x}{d{t}^2}+\alpha x=0$, its time period is

Q59.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

Q60.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?

Q61.If ${\mathrm{I}}_0$ 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?

Q62.Two concentric coils each of radius equal to $2\pi \mathrm{cm}$ 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 $/{\mathrm{m}}^2$ at the centre of the coils will be ( …

Q63.A coil of inductance 300 mH and resistance $2\Omega$ is connected to a source of voltage 2 V. The current reaches half of its steady state value in

Q64.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

Q65.An energy source will supply a constant current into the load of its internal resistance is

Q66.A circuit has a resistance of $12\Omega$ and an impedance of $15\Omega$. The power factor of the circuit will be

Q67.The phase difference between the alternating current and emf is $\pi /2$. Which of the following cannot be the constituent of the circuit?

Q68.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

Q69.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

Q70.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

Q71.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?

Q72.A magnetic needle is kept in a non-uniform magnetic field. It experiences

Q1.Which of the following oxides is amphoteric in character?

Q2.Which one of the following species is diamagnetic in nature?

Q3.If $\alpha$ is the degree of dissociation of ${\mathrm{Na}}_2{\mathrm{SO}}_4$, the vant Hoff's factor (i) used for calculating the molecular mass is

Q4.The oxidation state of Cr in ${\left[\mathrm{Cr}{\left({\mathrm{NH}}_3\right)}_4{\mathrm{Cl}}_2\right]}^{+}$ is

Q5.Hydrogen bomb is based on the principle of

Q6.An ionic compound has a unit cell consisting of $A$ ions at the corners of a cube and $B$ ions on the centres of the faces of the cube. The empirical formula for this compound would be

Q7.For a spontaneous reaction the $\Delta \mathrm{G}$, equilibrium constant ( K ) and ${\mathrm{E}}_{\text{cell }}^{\circ }$ will be respectively

Q8.Which of the following is a polyamide?

Q9.Which one of the following types of drugs reduces fever?

Q10.Due to the presence of an unpaired electron, free radicals are:

Q11.Lattice energy of an ionic compounds depends upon

Q12.The highest electrical conductivity of the following aqueous solutions is of

Q13.Aluminium oxide may be electrolysed at $1000^{\circ }\mathrm{C}$ to furnish aluminium metal (Atomic mass $=27\mathrm{amu};1$ Faraday $=96,500$ Coulombs). The cathode reaction is ${\mathrm{Al}}^{3+}+3{\mathrm{e}}^{-}⟶{\mathrm{Al}}^{\circ }$
To prepare 5.12 kg of alumi …

Q14.Consider an endothermic reaction, $X⟶Y$ with the activation energies $E_b$ and $E_f$ for the backward and forward reactions, respectively. In general

Q15.Consider the reaction: ${\mathrm{N}}_2+3{\mathrm{H}}_2⟶2{\mathrm{NH}}_3$ carried out at constant temperature and pressure. If $\Delta \mathrm{H}$ and $\Delta \mathrm{U}$ are the enthalpy and internal energy changes for the reaction, which of the following expres …

Q16.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?

Q17.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

Q18.The solubility product of a salt having general formula ${\mathrm{MX}}_2$, in water is: $4\times 10^{-12}$. The concentration of ${\mathrm{M}}^{2+}$ ions in the aqueous solution of the salt is

Q19.Benzene and toluene form nearly ideal solutions. At $20^{\circ }$ C, the vapour pressure of benzene is 75 torr and that of toluene is 22 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

Q20.The exothermic formation of ${\mathrm{ClF}}_3$ is represented by the equation:
${\mathrm{Cl}}_{2(\mathrm{g})}+3{\mathrm{F}}_{2(\mathrm{g})}\rightleftharpoons 2{\mathrm{ClF}}_{3(\mathrm{g})};\Delta \mathrm{rH}=-329\mathrm{kJ}$
Which of the following will increase the quant …

Q21.Two solutions of a substance (non electrolyte) are mixed in the following manner. 480 ml of 1.5 M first solution +520 mL of 1.2 M second solution. What is the molarity of the final mixture?

Q22.For the reaction
$2{\mathrm{NO}}_{2(\mathrm{g})}\rightleftharpoons 2{\mathrm{NO}}_{(\mathrm{g})}+{\mathrm{O}}_{2(\mathrm{g})}$,
( ${\mathrm{K}}_{\mathrm{c}}=1.8\times 10^{-6}$ at $184^{\circ }$ C )
( $\mathrm{R}=0.0831$ kJ/(mol.K) )
When ${\mathrm{K}}_{\mathrm{p}}$ and …

Q23.Hydrogen ion concentration in mol/L in a solution of pH = 5.4 will be

Q24.A reaction involving two different reactants can never be

Q25.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

Q26.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) $n=1,I=0,m=0$
(b) $n=2,I=0,m=0$
(c) $n=2,I=1,m=1$
(d) $n=3,I=2,m=1$
(e) …

Q27.During the process of electrolytic refining of copper, some metals present as impurity settle as 'anode mud' These are

Q28.Calculate ${\Lambda }_{\mathrm{HOAc}}^{\infty }$ Using appropriate molar conductances of the electrolytes listed above at infinite dilution in ${\mathrm{H}}_2\mathrm{O}$ at $25^{\circ }\mathrm{C}$
| Electrolyte | KCl | ${\mathrm{KNO}}_3$ | HCl | NaOAc | NaCl ||-------------| …

Q29.A schematic plot of $\ln K_{\mathrm{eq}}$ versus inverse of temperature for a reaction is shown below

The reaction must be

Q30.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?

Q31.Based on lattice energy and other considerations which one of the following alkali metal chlorides is expected to have the highest melting point.

Q32.Heating mixture of ${\mathrm{Cu}}_2\mathrm{O}$ and ${\mathrm{Cu}}_2\mathrm{S}$ will give

Q33.The molecular shapes of ${\mathrm{SF}}_4,{\mathrm{CF}}_4$ and ${\mathrm{XeF}}_4$ are

Q34.The number and type of bonds between two carbon atoms in calcium carbide are

Q35.The oxidation state of chromium in the final product formed by the reaction between KI and acidified potassium dichromate solution is

Q36.The number of hydrogen atom(s) attached to phosphorus atom in hypophosphorous acid is

Q37.What is the conjugate base of ${\mathrm{OH}}^{-}$?

Q38.The correct order of the thermal stability of hydrogen halides $(H-X)$ is

Q39.Heating an aqueous solution of aluminium chloride to dryness will give

Q40.Calomel $\left({\mathrm{Hg}}_2{\mathrm{Cl}}_2\right)$ on reaction with ammonium hydroxide gives

Q41.In which of the following arrangements the order is NOT according to the property indicated against it?
(1) ${\mathrm{Al}}^{3+}<{\mathrm{Mg}}^{2+}<{\mathrm{Na}}^{+}<{\mathrm{F}}^{-}$ — Increasing ionic size
(2) $\mathrm{B}<\mathrm{C}<\mathrm{N}<\mathrm{O}$ — Increasing first ionization e …

Q42.In silicon dioxide

Q43.Of the following sets which one does NOT contain isoelectronic species?

Q44.The lanthanide contraction is responsible for the fact that

Q45.The IUPAC name of the coordination compound ${\mathrm{K}}_3\left[\mathrm{Fe}(\mathrm{CN}{)}_6\right]$ is

Q46.Which of the following compounds shows optical isomerism?

Q47.Which one of the following cyano complexes would exhibit the lowest value of paramagnetic behaviour?
(At. No. $\mathrm{Cr}=24,\mathrm{Mn}=25,\mathrm{Fe}=26,\mathrm{Co}=27$)

Q48.2 methylbutane on reacting with bromine in the presence of sunlight gives mainly

Q49.The photon of hard gamma radiation knocks a proton out of ${}_{12}^{24}\mathrm{Mg}$ nucleus to form

Q50.The best reagent to convert pent -3- en-2-ol into pent -3-en-2-one is

Q51.Tertiary alkyl halides are practically inert to substitution by ${\mathrm{S}}_{\mathrm{N}}^2$ mechanism because of

Q52.In both DNA and RNA, heterocyclic base and phosphate ester linkages are at-

Q53.Reaction of one molecule of HBr with one molecule of 1,3 -butadiene at $40^{\circ }\mathrm{C}$ gives predominantly

Q54.Among the following acids which has the lowest ${\mathrm{pK}}_{\mathrm{a}}$ value?

Q55.The decreasing order of nucleophilicity among the nucleophiles
(a) (see structure)
(b) ${\mathrm{CH}}_3{\mathrm{O}}^{-}$
(c) ${\mathrm{CN}}^{-}$
(d) (see structure)

Q56.Which one of the following methods is neither meant for the synthesis nor for separation of amines?

Q57.Which of the following is fully fluorinated polymer?

Q58.Of the five isomeric hexanes, the isomer which can give two monochlorinated compounds is

Q59.Alkyl halides react with dialkyl copper reagents to give

Q60.Acid catalyzed hydration of alkenes except ethene leads to the formation of

Q61.Amongst the following the most basic compound is

Q62.Which types of isomerism is shown by 2,3-dichlorobutane?

Q63.Elimination of bromine from 2-bromobutane results in the formation of-

Q64.Equimolar solutions in the same solvent have

Q65.Which of the following statements in relation to the hydrogen atom is correct?

Q66.The structure of diborane (${\mathrm{B}}_2{\mathrm{H}}_6$) contains

Q67.The value of the 'spin only' magnetic moment for one of the following configurations is 2.84 BM. The correct one is

Q68.Which of the following factors may be regarded as the main cause of lanthanide contraction?

Q69.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

Q70.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

Q71.An organic compound having molecular mass 60 is found to contain $\mathrm{C}=20\%,\mathrm{H}=6.67\%$ and $\mathrm{N}=46.67\%$ while rest is oxygen. On heating it gives ${\mathrm{NH}}_3$ alongwith a solid residue. The solid residue give violet colour with alkaline copper sulp …

Q72.$t_{1/4}$ 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

Q73.An amount of solid ${\mathrm{NH}}_4\mathrm{HS}$ is placed in a flask already containing ammonia gas at a certain temperature and 0.50 atm . Pressure. Ammonium hydrogen sulphide decomposes to yield ${\mathrm{NH}}_3$ and ${\mathrm{H}}_2\mathrm{S}$ gases in the flask. When the de …

Q74.The reaction $\mathrm{R}-\mathrm{CO}-\mathrm{X}+{\mathrm{Nu}}^{-}⟶\mathrm{R}-\mathrm{CO}-\mathrm{Nu}+{\mathrm{X}}^{-}$ is fastest when X is

Q1.If $A^2-A+I=0$, then the inverse of $A$ is

Q2.If the cube roots of unity are $1,\omega ,{\omega }^2$ then the roots of the equation
$(x-1{)}^3+8=0$, are

Q3.Let $\mathrm{R}=\{(3,3),(6,6),(9,9),(12,12),(6,12),(3,9),(3,12),(3,6)\}$ 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

Q4.Area of the greatest rectangle that can be inscribed in the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is

Q5.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:

Q6.${\lim }_{\mathrm{n}\to \infty }\left[\frac{1}{{\mathrm{n}}^2}{\sec }^2\frac{1}{{\mathrm{n}}^2}+\frac{2}{{\mathrm{n}}^2}{\sec }^2\frac{4}{{\mathrm{n}}^2}+\dots .+\frac{1}{{\mathrm{n}}^2}{\sec }^{2}1\right]$ equals

Q7.$ABC$ is a triangle. Forces $\vec{P},\vec{Q},\vec{R}$ acting along $IA$, $IB$ and $IC$ respectively are in equilibrium, where $I$ is the incentre of $△ABC$. Then $P:Q:R$ is

Q8.If in a frequently distribution, the mean and median are 21 and 22 respectively, then its mode is approximately

Q9.Let $P$ be the point $(1,0)$ and $Q$ a point on the locus $y^2=8x$. The locus of mid point of $PQ$ is

Q10.If $C$ is the mid point of $AB$ and $P$ is any point outside $AB$, then

Q11.If the coefficients of $r$th, $(r+1)$th and $(r+2)$th terms in the binomial expansion of $(1+y{)}^m$ are in A.P., then $m$ and $r$ satisfy the equation

Q12.In a triangle $\mathrm{PQR}$, $\angle \mathrm{R}=\frac{\pi }{2}$. If $\tan \left(\frac{\mathrm{P}}{2}\right)$ and $\tan \left(\frac{\mathrm{Q}}{2}\right)$ are the roots of $a{x}^2+bx+c=0,a\ne 0$ then

Q13.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

Q14.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

Q15.If roots of the equation $x^2-bx+c=0$ be two consecutive integers, then $b^2-4c$ equals

Q16.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

Q17.The value of ${}^{50}{C}_4+{\sum }_{r=1}^6{}^{56-r}{C}_3$ is

Q18.If $A=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]$ and $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, then which one of the following holds for all $n\ge 1$, by the principle of mathematical induction

Q19.If the coefficient of $x^7$ in ${\left[a{x}^2+\left(\frac{1}{bx}\right)\right]}^{11}$ equals the coefficient of $x^{-7}$ in ${\left[ax-\left(\frac{1}{b{x}^2}\right)\right]}^{11}$, then a and b satisfy the relation

Q20.Let $f:(-1,1)\to B$, be a function defined by $f(x)={\tan }^{-1}\frac{2x}{1-x^2}$, then $f$ is both one-one and onto when $B$ is the interval

Q21.If $z_1$ and $z_2$ are two non-zero complex numbers such that $\left|z_1+z_2\right|=\left|z_1\right|+\left|z_2\right|$ then $\arg z_1-\arg z_2$ is equal to

Q22.If $\omega =\frac{z}{z-\frac{1}{3}i}$ and $|\omega |=1$, then $z$ lies on

Q23.If $a^2+b^2+c^2=-2$ and $f(x)=\left|\begin{array}{ccc}1+a^{2}x& \left(1+b^2\right)x& \left(1+c^2\right)x\\ \left(1+a^2\right)x& 1+b^{2}x& \left(1+c^2\right)x\\ \left(1+a^2\right)x& \left(1+b^2\right)x& 1+c^{2}x\end{array}\right|$ then …

Q24.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

Q25.A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched?
| (2) $[2,\infty )$ | …

Q26.Let $\alpha$ and $\beta$ be the distinct roots of $a{x}^2+bx+c=0$, then ${\lim }_{x\to \alpha }\frac{1-\cos \left(a{x}^2+bx+c\right)}{(x-\alpha {)}^2}$ is equal to

Q27.Suppose $f(x)$ is differentiable at $x=1$ and ${\lim }_{h\to 0}\frac{1}{h}f(1+h)=5$, then $f^{\prime }(1)$ equals

Q28.Let $f$ be differentiable for all $x$. If $f(1)=-2$ and $f^{\prime }(x)\ge 2$ for $x\in [1,6]$, then

Q29.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

Q30.If $x$ is so small that $x^3$ and higher powers of $x$ may be neglected, then $\frac{(1+x{)}^{3/2}-{\left(1+\frac{1}{2}x\right)}^3}{(1-x{)}^{1/2}}$ may be approximated as

Q31.If $x={\sum }_{n=0}^{\infty }{a}^n,y={\sum }_{n=0}^{\infty }{b}^n,z={\sum }_{n=0}^{\infty }{c}^n$ where $a,b,c$ are in A.P. and $|a|<1,|b|<1,|c|<1$, then $x,y,z$ are in

Q32.In a triangle ABC, let $\angle \mathrm{C}=\frac{\pi }{2}$. If r is the inradius and R is the circumradius of the triangle $ABC$, then $2(r+R)$ equals

Q33.If ${\cos }^{-1}x-{\cos }^{-1}\frac{y}{2}=\alpha$, then $4x^2-4xy\cos \alpha +y^2$ is equal to

Q34.If in a triangle $ABC$, the altitudes from the vertices $A,B,C$ on opposite sides are in H.P., then $\sin A,\sin B,\sin C$ are in

Q35.If ${\mathrm{I}}_1={\int }_0^1{2}^{{\mathrm{x}}^2}\mathrm{dx},{\mathrm{I}}_2={\int }_0^1{2}^{{\mathrm{x}}^3}\mathrm{dx},{\mathrm{I}}_3={\int }_1^2{2}^{{\mathrm{x}}^2}\mathrm{dx}$ and ${\mathrm{I}}_4={\int }_1^2{2}^{{\mathrm{x}}^3}\mathrm{dx}$ then

Q36.The area enclosed between the curve $y={\log }_e(x+e)$ and the coordinate axes is

Q37.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 …

Q38.If $x\frac{dy}{dx}=y(\log y-\log x+1)$, then the solution of the equation is

Q39.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 $(a,b)\ne (0,0)$ is

Q40.A spherical iron ball 10 cm in radius is coated with a layer of ice of uniform thickness than melts at a rate of $50{\mathrm{cm}}^3/\min$. When the thickness of ice is 5 cm , then the rate at which the thickness of ice decreases, is

Q41.$\int {\left\{\frac{(\log x-1)}{(1+(\log x{)}^2}\right\}}^{2}dx$ is equal to

Q42.Let $f:R\to R$ be a differentiable function having $f(2)=6,f^{\prime }(2)=\left(\frac{1}{48}\right)$. Then ${\lim }_{x\to 2}{\int }_6^{f(x)}\frac{4t^3}{x-2}dt$ equals

Q43.Let $f(x)$ be a non-negative continuous function such that the area bounded by the curve $y=f(x)$, $x$-axis and the ordinates $x=\frac{\pi }{4}$ and $x=\beta >\frac{\pi }{4}$ is $\left(\beta \sin \beta +\frac{\pi }{4}\cos \beta +\sqrt{2}\beta \right)$. Then …

Q44.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 $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is

Q45.If the angle $\theta$ between the line $\frac{x+1}{1}=\frac{y-1}{2}=\frac{z-2}{2}$ and the plane $2x-y+\sqrt{\lambda }z+4=0$ is such that $\sin \theta =\frac{1}{3}$, the value of $\lambda$ is

Q46.The angle between the lines $2x=3y=-z$ and $6x=-y=-4z$ is

Q47.If the plane $2ax-3ay+4az+6=0$ passes through the midpoint of the line joining the centres of the spheres
$x^2+y^2+z^2+6x-8y-2z=13$ and
$x^2+y^2+z^2-10x+4y-2z=8$, then $a$ equals

Q48.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 $\vec{r}\cdot (\hat{i}+5\hat{j}+\hat{k})=5$ is

Q49.For any vector $\vec{a}$, the value of $(\vec{a}\times \hat{i}{)}^2+(\vec{a}\times \hat{j}{)}^2+(\vec{a}\times \hat{k}{)}^2$ is equal to

Q50.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

Q51.If a vertex of a triangle is $(1,1)$ and the mid-points of two sides through this vertex are $(-1,2)$ and $(3,2)$, then the centroid of the triangle is

Q52.If the circles $x^2+y^2+2ax+cy+a=0$ and $x^2+y^2-3ax+dy-1=0$ intersect in two distinct points P and Q then the line $5x+by-a=0$ passes through P and Q for

Q53.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

Q54.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

Q55.An ellipse has OB as semi minor axis, F and ${\mathrm{F}}^{\prime }$ its focii and the angle ${\mathrm{FBF}}^{\prime }$ is a right angle. Then the eccentricity of the ellipse is

Q56.Let $a,b$ and $c$ 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 $c$ is

Q57.If $\vec{a},\vec{b},\vec{c}$ are non-coplanar vectors and $\lambda$ is a real number then
$\left[\lambda (\vec{a}+\vec{b}){\lambda }^2\vec{b}\lambda \vec{c}\right]=[\vec{a}\vec{b}+\vec{c}\vec{b}]$ for

Q58.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 $[\vec{a},\vec{b},\vec{c}]$ depends on

Q59.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

Q60.A random variable $X$ has Poisson distribution with mean 2. Then $P(X>1.5)$ equals

Q61.Let $A$ and $B$ be two events such that $P(\overline{A\cup B})=\frac{1}{6},P(A\cap B)=\frac{1}{4}$ and $P(\bar{A})=\frac{1}{4}$, where $\bar{A}$ stands for complement of event $A$. Then events $A$ and $B$ are

Q62.A lizard, at an initial distance of 21 cm behind an insect, moves from rest with an acceleration of $2\mathrm{cm}/{\mathrm{s}}^2$ and pursues the insect which is crawling uniformly along a straight line at a speed of $20\mathrm{cm}/\mathrm{s}$. Then the lizard will ca …

Q63.Two points $A$ and $B$ move from rest along a straight line with constant acceleration $f$ and $f^{\prime }$ respectively. If $A$ takes $m$ sec. more than $B$ and describes '$n$' units more than $B$ in acquiring the same speed then

Q64.$A$ and $B$ are two like parallel forces. A couple of moment $H$ lies in the plane of $A$ and $B$ and is contained with them. The resultant of $A$ and $B$ after combining is displaced through a distance

Q65.The resultant R 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

Q66.The sum of the series $1+\frac{1}{4.2!}+\frac{1}{16.4!}+\frac{1}{64.6!}+\dots \dots .$ ad inf. is

Q67.The value of ${\int }_{-\pi }^{\pi }\frac{{\cos }^{2}x}{1+a^x}dx,a>0$, is

Q68.The plane $x+2y-z=4$ cuts the sphere $x^2+y^2+z^2-x+z-2=0$ in a circle of radius

Q69.If the pair of lines $a{x}^2+2(a+b)xy+b{y}^2=0$ 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

Q70.Let $x_1,x_2,\dots ,x_n$ be n observations such that $\sum x_i^2=400$ and $\sum x_i=80$. Then a possible value of $n$ among the following is

Q71.A particle is projected from a point O with velocity $u$ at an angle of $60^{\circ }$ with the horizontal. When it is moving in a direction at right angles to its direction at O, its velocity then is given by

Q72.If both the roots of the quadratic equation $x^2-2kx+k^2+k-5=0$ are less than 5, then k lies in the interval

Q73.If $a_1,a_2,a_3,\dots ,a_n,\dots$ are in G.P., then the determinant …

Q74.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

Q75.If the equation $a_n{x}^n+a_{n-1}{x}^{n-1}+\dots +a_{1}x=0$, $a_1\ne 0$, $n\ge 2$, has a positive root $x=\alpha$, then the equation $n{a}_n{x}^{n-1}+(n-1)a_{n-1}{x}^{n-2}+\dots +a_1=0$ has a positive root, which is