The values of x satisfying sin^-1 x + sin^-1 (1- x) = cos^-1 x are

The area bounded by the curve y = f(x), the x-axis and the ordinates x = 1 and x = b is (b - 1) cos (3b + 4) sq unit. Then f(x) is given by

If the function f(x)=2x³-9ax²+12a²x+1 has a local maximum at x=x₁ and a local minimum at x=x₂ such that x₂=x₁² then a is equal to

If P(x) is a polynomial such that P(x² + 1) = {P(x)}² + 1 and P(0) = 0, then P' (0) is equal to

If \(y={ 2 }^{ \frac { 1 }{ \log _{ x }{ 4 } } }\) , then x is equal to

If the roots of the equation ax2+bx+c=0 are of the form ∝/(∝-1) and (∝+1)/∝ then the value of (a+b+c)² is

Let p : 7 is not greater than 4.
q : Paris is in France.
be two statements. Then ~ (p v q) is the statement

In \(\Delta ABC,\) if 2a²b² + 2b²c²=a⁴ +b⁴+c⁴, then \(\angle B\) is equal to 

The value of \(tan(cos^{-1}\frac{4}{5}+tan^{-1}\frac{2}{3})=\)

sin x + cos x = 1+ sin x cos x, if

If the foci of the ellipse \({x^2\over 9}+{y^2\over 16}=1\) are (0, √7) and (0, -√7) then the foci of the ellipse \({x^2\over 9+t^2}+{y^2\over 16+t^2}=1, t\in R \), are

The tangents drawn from the origin to the circle x² + y² - 2px - 2qy + q² = 0 are perpendicular, if

The angle between the lines \(\sqrt{3}\)X + Y = 1 and X + \(\sqrt{3}\)Y = 1 is equal to

The area enclosed by the curve y=\(\sqrt { x } \) and x = -\(\sqrt { y } \) , the circle x² + y² = 2 above the x-axis, is

The sum to infinity of the series, \(1+2\left( 1-\frac { 1 }{ n } \right) +3{ \left( 1-\frac { 1 }{ n } \right) }^{ 2 }+....\) is

The value of \(^{ 50 }{ C }_{ 4 }+\overset { 6 }{ \underset { r=1 }{ \Sigma } } ^{ 56-r }{ C }_{ 3 }\) is

If A ={1,2,3} and B ={3,6,8}, then \(\left( A\cap B \right) \times A\) is equal to 

Let a, b, c be three cube roots of unity, the value of
\(\left| \begin{matrix} { e }^{ a } & { e }^{ 2a } & { e }^{ 3a }-1 \\ { e }^{ b } & { e }^{ 2b } & { e }^{ 3b }-1 \\ { e }^{ c } & { e }^{ 2c } & { e }^{ 3c }-1 \end{matrix} \right| \) is

Lanthanoids and actinoids differ from each other because

Which of the following represents correct order of first ionisation energy?

The density (in g mL^-1) of a 3.60 M sulphuric acid solution that is 29% H₂SO₄ (98 g mol^-1) by mass will be 

Which of the following reactions is not correct according to the law of conservation of mass?

Which of the folllowing is not an organic water pollutants?

Which one of the following fertilisers contains the maximum percentage of nitrogen?

Glycogen is a branched chain polymer of \(\alpha -D\)-glucose units in which chain is formed by C1-C4 glycosidic linkage whereas branching occurs by the formation of C1-C6 glycosidic linkage. Structure of glycogen is similar to

Commercial name of PMMA is 

The process of concentrating Ag and Au ores is based on their solubility in

Which one of the following is known as Hinsberg reagent?

Hydroboration - oxidation of propene yield 

The chief constituent of coal tar is

What is the product formed when acetylene reacts with hypochlorous acid.

 A solution of a potassium ferrocyanide would contain......ions 

Solid K₂Cr₂O₇ is used to detect the presence of which one of the following acid radicals in analytical chemistry?

Asthma patients use a mixture of ........ for respiration.

Which one of the following  refers to Na₂SO₄.10H₂O?

Which difficulty is encountered in pennis method for the preparation of fluorine and is removed in whyflaw-Grat's method?

Hydrogen exists in 

Cupellation process is used in the metallurgy of 

Hydrogen peroxide in aq.  soln.  decomposes  on warming according to the equation \({ 2H }_{ 2 }{ O }_{ 2 }\quad (aq)\rightarrow \quad { 2H }_{ 2 }O\quad (l)\quad +{ O }_{ 2 }\quad (g)\)  If one mole of gas occupies 24 cm³ and 100 cm³ of  an  XM solution of H₂O₂  produces 3 dm³ of O₂ , then X is

If 22 g of a substance when vaporised  isCH₂, 1 mole of the compound has mass 14 g.Its moleculer formula is

The end product of (4n+2) disintegration series is

Langmuir adsorption is best represented mathematically as

A reaction is of first order with respect to reactant A, has a rate constant 6 min^-1. If one starts with [A]= 0.5 mol L^-1, the concentraction of [A] would reach to 0.05 mol L^-1 in

Relative lowering of vapour pressure is a colligative property because

The oxidation number of sulphur in Na₂S₄O₆ is

How many coulombs of electricity are required to reduce 1 mol of MnO^-₄  to Mn^2+?

What is the pH of the resulting solution when equal volumes of 0.1 M NaOH and 0.01 M HCl are mixed?

The equilbrium constant at  a certain temperature for the reactions \(H_{2}+1/2 \ S_{2}\rightarrow \ H_{2}S\) and \(H_{2}+Br_{2}\rightarrow2HBr\) are \(K_{1}\) and \(K_{2}\) respectively.   The value of K for the reaction \(Br_{2}+H_{2}S\rightarrow2HBr+1/2 \ S_{2}\) would be

Carbon and carbon monoxide burn in oxygen to give carbon dioxide according to the equations :

\(C(s)+{ O }_{ 2 }(g)\rightarrow { CO }_{ 2 }(g);\Delta H=-394kJ\quad \quad ...(i)\)

\(2CO(g)+{ O }_{ 2 }(g)\rightarrow { 2CO }_{ 2 }(g);\Delta H=-569kJ\quad \quad ...(ii)\)

The heat of formation of carbon monoxide would be

Frenkel defect is produced due to 

Which of the following molecule will be stabilised by loosing one electron from its HOMO? 

The maximum number of electrons in a subshell for which l=3 is:

The speed of sound through oxygen at T K is v ms^-1 As the temperature becomes 2T and oxygen gas dissociated into atomic oxygen, the speed of sound:

For a simple pendulum, when a graph is plotted betwen displacement d, KE=\(1\over 2\)mv² and PE=mgh, taking d along X-axis and \(1\over 2\)mv² and mgh along Y-axis, the graph comes as

The I-V characteristic of an LED is

Let m_p be the mass of a proton m_n the mass of a neutron, M₁ the mass of a \(^{20}_{10}Ne\) nucleus and M₂ the mass of a \(^{40}_{20}Ca\)

Light coming from a discharge tube filled with hydrogen falls on the cathode of the photoelectric cell. The work function of the surface of cathode is 4 eV. Which of the following values of the anode voltage with respect to the cathode will likely to make the photo current zero?

The value of magnetic susceptibility for superconductors is 

Two SHM x₁=A sin ωt and x₂=A cos ωt are superimposed on a particle having mass m. Total mechanical energy of particle is

If average velocity becomes 4 times then what will be the effect on rms velocity at that temperature?

At what temperature (in ^o C), the fahrenheit and celsius scale gives same reading? 

A body of density \(\rho is\) dropped from height h into a liquid having density \(\sigma \) \((\sigma >\rho )\)  . If the body just touches the base of the container, then the depth of the container would be proportional to (Neglect viscous forces)

A wire suspended vertically from one of its ends is stretched by attaching a weight of 200 N to the lower end. The weight stretches the wire by 1 mm. Then, the elastic energy stored in the wire is 

Two infinite long wires at a distance of 1 m carry current of 1 mA in the same direction. Which one of the following is CORRECT?

The medium wave broadcast signals may get propagated via ionosphere. This happens

When we apply reverse bias to a junction diode it

The ionisation energy of 10 times ionised sodium atom is

The work function of sodium is 2.3 eV.  Sodium will  NOT show photoelectric effect with which of the following colours ?

Two waves \({ y }_{ 1 }={ A }_{ 1 }\sin { (\omega t-{ \beta }_{ 1 }) } \)and\({ y }_{ 2 }={ A }_{ 2 }\sin { (\omega t-{ \beta }_{ 2 }) } \)superimpose to form a resultant wave whose amplitude is

Length of a Galilean telescope in normal adjustment, in terms of the focal lengths of the objectives (f₀) and that of the eye piece (f_e) is

If an a.c.signal is applied to a non-ideal inductor, then

Two tangent galvanometers having coils of the same radius are connected in series. A current flowing in them produces deflections of \(60º\) and \(45º\) respectively. The ratio of the number of turns in the coils is

Two resistance filaments of same length are connected first in series and then in parallel. Find the ratio of power dissipated in both cases assuming that equal current flows in the main circuit.

Electrical resistance can be

A. wire-bound resistance

B. carbon resistance

C. markedly with distance from the centre

D. decreases with distance from the centre

If the amount of heat energy received per unit area from the sun is measured on the earth, mars and jupiter, it will be:

The volume expansion coefficient is:

Two sources of sound of frequencies \({ \nu }_{ 1 }\quad and\quad { \nu }_{ 2 }\) superpose each other to give rise to the formation of beats. In beats,

A mass M is attached to a spring whose upper end is fixed. The mass and stiffness k of the spring are m and k respectively. The natural frequency of the spring-mass system is

Two capillary tubes of radii 0.2 cm and 0.4 cm are dipped in the same liquid.The ratio of heights through which liquid will rise in the tubes is

Match the statement given in column I with their formula given in column II and choose the correct option from the choices given below.

A circular disc of moment of inertia I_t is rotating in a horizontal plane, about its symmetry axis, with a constant angular speed \(\omega_i\). Another disc of moment of inertia I_b is dropped coaxially onto the rotating disc. Initially the second disc has zero angular speed. Eventually both the discs rotate with a constant angular speed \(\omega_f\). The energy lost by the initially rotating disc to friction is:

A tank of 2 x 2 x 3 is to be filled with water from a well of average depth 10 m. The work done will be

A pendulum is suspended from the roof of the compartment of a train. The string makes a constant angle of 30° with the vertical, towards the rear of the train. What is the nature of the)IlOtion of the train?

A particle moving along the circular path with a speed v and its speed increases by g in one second. If the radius of the circular path be r, then the net acceleration of the particle is

Two stones are thrown up simultaneously from the edge of a cliff 240 m high with initial speed of 10 m/s and 40 m/s respectively. Which of the following graph best represents the time variation of relative position of the second stone with respect to the first?
(Assume stones do not rebound after hitting the ground and neglect air resistance, take g = 10 m/s² )
(The figures are schematic and not drawn to scale)

The direction of  \(\vec { A } \) is vertically upward and direction of \(\vec { B } \)  is in north direction. The direction of \(\vec { A } \times \vec { B } \) will be:

The owner of a milk store finds that be can sell 980 L of milk each week  at Rs 14 per liter and 1220 L of milk each week at Rs 16 per litre. Assuming a linear relationship between selling price and demand. How many litres could he sell weekly at rs 17 per litre?

If \(\frac{1}{2}\) < | x |< 1, then which of the following are real?

The value of c for which the area of the figure bounded by the curve y = 8x2- x⁵, the straight lines x = 1 and x = c an d the x-axis is equal to16/3 is

Let f(x)=\(\int _{ 0 }^{ x }{ \frac { cost }{ t } } dt\) (x>0); then f(x) has 

Let x^{cos y} + y ^{cos x} = 5. Then

\(\log _{ p }{ \log _{ p }{ \underbrace { \sqrt [ p ]{ \sqrt [ p ]{ \sqrt [ p ]{ ....\sqrt [ p ]{ p } } } } }_{ n\quad times } , } } \) p>0 and p\(\ne\)1, is equal to

The number of negative integral solutions of  \({ x }^{ 2 }.{ 2 }^{ x+1 }+{ 2 }^{ |x-3|+2 }={ x }^{ 2 }.{ 2 }^{ |x-3|+4 }+2^{ x-1 }\) is

Write the contrapositive of the following statement:
If you are born in India, then you are a citizen of India

Statement I: The value of \(tan\{cos^{-1}\frac{4}{5}+tan^{-1}\frac{2}{3}\}\) is \(\frac{17}{6}\)
Statement II: \(tan^{-1}x+tan^{-1}y=tan^{-1}(\frac{x-y}{1+xy})\)

The solution of the equation \(sin^{10}x+cos^{10}x={29\over 16}cos^42x\)is 

If the tangent and normal to a rectangular hyperbola cut off intercepts X₁ and X₂ on one axis and y₁ and y₂ on the other axis, then

If the derivative of f(x) w.r.t  x is \(\frac { (1/2)-sin^{ 2 }x }{ f(x) } \), then f(x) is periodic function with period

If the derivative of f(x) w.r.t  x is \(\frac { (1/2)-sin^{ 2 }x }{ f(x) } \), then f(x) is periodic function with period

Find the maximum value of the following functions. \(f(x)=sin3x+4, x\in \left(-{\pi\over2},{\pi\over2}\right)\)

If x = \(acos^{ 4 }\theta ,y=asin^{ 4 }\theta ,\)then \(\frac { dy }{ dx } at\theta =\frac { 3\pi }{ 4 } is\)

For these values of a and b, (gof)x ∀ x ∈ ( - 1, 1) is

\(\lim _{ x\rightarrow \infty }{ \sqrt { x } } \left( \sqrt { x+1 } -\sqrt { x } \right) \)  equals

If domain of f is D₁ and domain of g is D₂, then domain of f+g is

A ball is projected with a velocity of 9.8 m/sec at a plane where g = 9.8 m/\({ sec }^{ 2 }\). The maximum range of the projectile on the horizontal ground, is

P and Q are two like parallel forces.If P is moved parallel to itself through a distance x,then resultant of these forces moves through a distance

The tangent at any point P on y = f(x) meets x-axis and y-axis at A and B respectively. If PA : PB = 2: 1, then the equation of the curve, is (where c is arbitrary constant)

\(\begin{matrix} lim \\ n\rightarrow \infty \end{matrix}\frac { \pi }{ n } \left( sin\frac { \pi }{ n } +sin\frac { 2\pi }{ n } +......sin(n-1)\frac { \pi }{ n } \right) \) equals

The derivative with respect to x of the function \(\tan { ^{ -1 }{ \left( \frac { \sqrt { 1+{ x }^{ 2 } } +\sqrt { 1-{ x }^{ 2 } } }{ \sqrt { 1+{ x }^{ 2 } } -\sqrt { 1-{ x }^{ 2 } } } \right) } } \) is

Observe the following columns

Find the standard deviation for the following data:

The distance between the planes 2x+y+2z=8 and 4x+2y+4z+5=0, is

If e and e' are the eccentricities of the hyperbolas \({x^{2}\over a^{2}}-{y^{2}\over b^{2}}=1\) and \({y^{2}\over b^{2}}-{x^{2}\over a^{2}}=1\) ,then value of (e)^-2+(e')^-2,is

The sides of an euilateral triangle, a square and a regular hexagon circumscribed in a circle are in

Let a₁, a₂, a₃, .... be in a AP with common difference not a multiple of 3. Then maximum number of consecutive terms so that all are primes is 

The positive integer just greater than S = (1+0.0001)¹⁰⁰⁰⁰ is

P(n):2.7ⁿ+3.5ⁿ-5 is divisible by 

The number of permutations of the letters a, b, c, d such that b does not follow a, and c does not follow band d does not follow c, is

A real root of the equation log₄ \(\left\{ { log }_{ 2 }(\sqrt { x+8 } -\sqrt { x } ) \right\} =0\) is

If the matrix product AB=0 then

If z1=\(\sqrt3+i\sqrt3\)and z2=\(\sqrt3+i\) , then find the quadrant in which\(\left( \frac { { z }_{ 1 } }{ { z }_{ 2 } } \right) \) lies

If the relation R:A\(\rightarrow \) B, where A={1,2,3} and B={1,3,5} is defined by R={(x,y):x\(x\epsilon A,y\epsilon B\) }, then 

The titration of Mohr's salt vs KMnO₄ is an example of redox titration. In this titration, KMnO₄ oxidises only ferrous salt to the ferric salt ( no effect on other ions) but we cannot use ferrous sulphate in place of Mohr's salt because

Luminal, a barbiturate drug, is used as 

Large number of oxidation states are exhibited by the antinoids than those by the lanthanoids, the main reason being

Most abundant element in earth's crust  ( in terms of number of atoms per 100 atoms) is

Which of the following group elements belongs to chalcogens?

Which one of the following reactions is not a redox reaction?

When 401 J of heat is supplied to system and work done by system is  8 J then \(\triangle U\) of system during this process is

The structure of \(Na_2O\) crystal is

Lithium forms body central cubic structure.  The length of the side of its unit cell is 351 pm.  Atomic radius of lithium will be

In the following reaction, 
     \({ N }_{ 2 }(g)+3{ H }_{ 2 }(g)\rightarrow 2{ NH }_{ 3 }(g)\) 
Calculate the moles of NH₃ obtained when 2 moles of N₂ react with 3 moles of H₂ .

Calculate the number of moles left after removing 10²¹ molecules from 200 mg of CO₂

Photochemical smog occurs in warm, dry and sunny climate. Amongst the following one is not the components of photochemical smog, identify it.

Each polypeptide in a protein has amino acids linked each other in a specific sequence. This sequence of amino acid is said to be

The length of the chain of the polymer can be controlled during polymerisation by the addition of a reagent called

On evaporation of an aqueous solution of ammonium cyanate we get urea. This is a reaction known as

Which is the strongest acid in the following?

A sample of petrol contains 30% n-heptane and 70% iso-octane.The octane number of the sample is 

Acetylene on oxidation with chromic acid gives

Which of the following compounds displays geometrical isomerism?

The oxidation state of Pt [(en)H₂O)₄(NO₂)Cl]²⁺ is 

The common oxidation state of the elements of transition series is

Number of \(p\pi-d\pi\)bonds present in XeO₄ are

Cassiterite is an alloy contains

Which one the following statement about KClO₃  is wrong?

As the automic weight of halogen increases from flouring to astatine the halogens

Zone refining process is used for the

The reactant which is  completely consumed during the  reaction  is  called 

The purity of an organic compound is determind by 

A radioactive isotope decays according to the given reaction under STP condition as follows \(_{ Z }^{ A }{ X\quad \longrightarrow _{ Z-2 }^{ A-4 }{ X+_{ 2 }^{ 4 }\quad } }\)He.Its half-time is 10 days The volume of helium collected in 20 days if one mole of  \(_{ Z }^{ A }{ X\quad }\)is kept in a sealed container ,is

Butter is a colloid formed when 

t_1/4 can be taken as the time taken for the concentration of a reactant to drop to 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

A liquid mixture which boils without change in composition is called a/an

Which substance is serving as a reducing agent in the following reaction?\(14{ H }^{ + }+{ Cr }_{ 2 }{ O }_{ 7 }^{ 2- }+3\quad Ni\longrightarrow 2{ Cr }^{ 3+ }+7{ H }_{ 2 }O+3{ Ni }^{ 2+ }\)

From an electric wire \(1.27\times 10^{18} \) electrons are transferred per minute.The current in A that would be flowing per second is

A 0.2 molar solution of formic acid is 3.2% ionised. Its ionisation constant is

At constant temperature, the equilibrium constant (\(K_{p}\)) for the decomposition \(N_{2}O_{4}\rightleftharpoons 2NO_{2}\) is expressed by \(K_{p}={4x^{2}-P\over1-x^{2}}\), where p=pressure, x=extent of decomposition.  Which of the following statement is true ?

Since the enthalpy of elements in their natural state is taken to be zero the heat of formation (\(\Delta H_f\))  of compounds

It is true that

The forces present in the crystals of naphthalene are

Which one of the following statements is most appropriate?

In the experiment of measuring speed of sound by resonance tube, it is observed that for tuning fork of frequency v = 480 Hz, length of air column in cm, \(l_1=30\ cm, l_2=70\ cm\)  then \(v_1\) is equal to

When pressure increased by 1 atmosphere and temperature increases by 1⁰ C, the velocity of sound:

When an n-p-n transistor is used as an amplifier,

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

In a photoelectric experiment, the potential difference V that must be maintained between the illuminated surface and the collector so as just to prevent any electron from reaching the collector is determined for different frequencies f of the incident illumination. The graph obtained is shown in figure.

The maximum kinetic energy of the electrons emitted at frequency f₁ is

The magnetic property inherent in all materials is 

A capacitor with capacity 100 \(\mu F\) is charged with a battery of 24 V. It is then connected to a battery of 12 V with positive plate is joined to positive terminal of battery. Heat developed in the process of flow of charge after connection is

A body doing SHM with amplitude 1 cm and frequency 60 Hz. The maximum acceleration will be

A real gas behaves like an ideal gas if its:

In absence of gravity, which of the following will not be there for a fluid? 

If a stretched wire under tension suddenly breaks, then the temperature of wire would

A circular loop of radius R carrying a current I is placed in a un9iform magnetic field with its plane perpendicular to B. The force on the loop is

In an amplitude modulated wave for audio frequency of 500 cycle/s, the appropriated carrier frequency will be 

In a halfwave rectifier, the r.m.s value of the a.c. component of the wave is

In the process of fission, the binding energy per nucleon

In Thomson's experiment, if the ratio E/B is less than velocity \(\vartheta \) of electron, then electron will more

If \({ \mu }_{ 0 }\) and \({ \epsilon }_{ 0 }\) represent the permeability and permittivity of vacuum and \(\mu\) and \(\epsilon\) represent the permeability and permittivity of the medium, then the refractive index of the medium is :

Two waves \({ y }_{ 1 }={ A }_{ 1 }\sin { (\omega t-{ \beta }_{ 1 }) } \)and\({ y }_{ 2 }={ A }_{ 2 }\sin { (\omega t-{ \beta }_{ 2 }) } \)superimpose to form a resultant wave whose amplitude is

If the luminous intensity of a unidirectional bulb is 100 candela, then total luminous flux emitted from the bulb is

An inductive load has

The neutral point in a magnetic field is a point at which

The e.m.f. in a thermocouple one junction of which is kept at 0⁰C, is given by E=at+bt². The peltier coefficient is

On dipping a conducting rod in boiling water, its conducitivity

Choose the INCORRECT statement

A thin aluminium rod (diameter = 1 mm) is very long and is under a uniform tension of 1000 N. Its Young's modulus (Y) is 7.0 X 10¹⁰ Pa and its density \((\rho )\) is 2.70 g cm^-3. The velocity of transverse waves on the rod is

A light spring AC of stiffness 2k is cut at B into two halves AB and BC. The point A is connected to a upper rigid support whereas point C is connected to a lower rigid support. The common point B is connected to a certain mass m. The new system will vibrate with frequency, as compared to spring AC with the same mass m,

The kinetic energy E per unit volume of a gas is related to its pressure P by the expression

A satellite with kinetic energy E is revolving round the earth in a circular orbit. The minimum additional kinetic energy required for it to escape into outer space is

Angle of friction has value

The trajectory of a projectile in vertical plane is \(y=ax-bx^2,\)  where a and b are constants and x and y are horizontal and vertical distance of the projection respectively,  The maximum height attained by the particle and the angle of projection from the horizontal are

A 100 m long train at 15 m/s overtakes a man running on the platform in the same direction in 10 s. How long the train will take to cross the man if he was running in the opposite direction?

The dimensional formula for Y is

Equation of line is 3x - 4y + 10 = 0, find its x and y -intercept respectively

Let f : A\(\rightarrow \)B be a function defined by y = f(x) such that f is both one-one (Injective) and onto (surjective)(ie, bijective), then there exists a unique function g: B\(\rightarrow \)A such that \(f\left( x \right) =y\Leftrightarrow g\left( y \right) =x,\ \forall x\epsilon A\ y\epsilon B\), then g is said to be inverse of f. Thus, g = f^-1: B\(\rightarrow \)A = \(\left[ \left\{ f\left( x \right) ,x \right\} :\left\{ x,\ f(x) \right\} \epsilon { f }^{ -1 } \right] \).If no branch of an inverse trigonometric function is mentioned, then it means the principal value branch of that functon. On the basis of above information, answer the following question: The value of cos(tan^-1tan 4) is

The area bounded by the graph y = |[x - 3]|, the x-axis and the lines x = -2 and x = 3 is ([.] denotes the greatest integer function)

Let f(x) be a differential function for all x, if f(1) = - 2 and f' (x) ≥2 for all x in [1, 6], then minimum value of f(6) is equal to

If \(\frac { { d }^{ 2 }y }{ { dy }^{ 2 } } \left( \frac { dy }{ dx } \right) ^{ 3 }+\frac { { d }^{ 2 }y }{ { dy }^{ 2 } } =k\) then k is equal to

If log₁₀3=0.477, the number of digit in 3⁴⁰ is

If one root of the equation \({ x }^{ 2 }-\lambda x+12=0\) is even prime while \({ x }^{ 2 }+\lambda x+\mu =0\) has equal roots, then μ is

Statement-I: The negation of the statement "I become a teacher, then I will open a school" is I will become a teacher and I will not open a school.
Statement-II: A statement which is formed by changing the truth value of a given statement by using word like 'no' or 'not' is called negation of the given statement.

If the angle of elevation of the top of a hill from each of the vertices A,B and C of a horizontal triangle is \(\alpha \) . Then, the height of the hill is 

The number of triplets(x,y,z) satisfies the equation sin^-1x+sin^-1 y+sin^-1=\(\frac{3\pi}{2}\) is

The general solution of the trigonometrical equation sin x + cos x = 1 for n = 0, ± 1, ± 2 ... is given by

The length of the latus rectum of the ellipse 3x² + y² = 12 is

If the distances from the origin of the centres of three circles \({ x }^{ 2 }+{ y }^{ 2 }+2{ \lambda }_{ i }x-{ c }^{ 2 }\) = 0 (i = 1, 2, 3) are in GP, then the lengths of the tangents drawn to them from any point on the circle x² + y² = c² are in

A straight line L with negative slope passes through the point (8, 2) and cuts the positive coordinates axes at points P and Q. As L varies, the absolute minimum value of OP + OQ is (O is origin)

The value of the integral  \(\int _{ a }^{ b }{ \frac { |x| }{ x } dx } ,a is

\(\int { \sqrt { \left( x-3 \right) } \{ \sin { ^{ -1 } } (ln\quad x)+{ cos }^{ -1 } } (ln\quad x)\} \)dx equals

If \(f(x)={x\over sin}\) and \(g(x)={x\over tanx}, 0<x\le 1,\) then in the interval

Let f(x)=sinx, g(x)=x² and h(x)=log_ex, If F(x)=(hogof) (x), then F"(x) is equal to

The number of points at which the function \(f(x)={1\over x-[x]}\)([.] denotes the greatest integer function) is not continuous, is

Evaluate of the following limis.
\(\lim _{ x\rightarrow 0 }{ \frac { tanx }{ x } } \) is

If f(x)=\(3\sin { \sqrt { \left( \frac { { \pi }^{ 2 } }{ 16 } -{ x }^{ 2 } \right) } } \), then its range is

If you want to kick a football to the maximum distance, then the angle at which it should be kicked is

If the mean deviation of numbers 1,1+d,1+2d,...,1+100d from their mean is 255, then d is equal to

If (1 + x) \(\frac { dy }{ dx } \) - xy = 1, y(0) = -1, then y(1) = 

\(\int { \frac { \sqrt { cos2x } }{ sinx } dx, } \) equals

The values of a and b such that\(\underset { x->0 }{ lim } \frac { x(1+a\cos { x)-b\sin { x } } }{ { x }^{ 3 } } =1\)are

A box contains 100 tickets numbered as 1, 2, .......,100. Two tickets are chosen at random. It is given that the maximum number on the two chosen tickets is not more than 10. The minimum number on them is 5 with probability.

The resultant R of two forces P and Q act at right angles to P. Then the angle between these forces, is

A variable plane is at a constant distance p from the origin, meets the axes in A,B and C. Through A,B, and C planes are drawn parallel to the coordinates planes. Then locus of their point of intersection is given by

The volume of the tetrahedron whose vertices are with position vectors \(\hat { i } -6\hat { j } +10\hat { k } ,-\hat { i } -3\hat { j } +7\hat { k } ,5\hat { i } -\hat { j } +\lambda \hat { k } \) and \(7\hat { i } -4\hat { j } +7\hat { k } \) is 11 cubic unit if \(\lambda \) equals

Three normals to the parabola y² =x are drawn through a point  (c,0);then

Area of the parallelogram formed by the lines y=mx, y=mx+1, y=nx, y=nx+1, is

The sides a,b,c of the triangle ABC are in A.P., then cot \(A\over 2\) ,cot \(B\over 2\) ,cot \(C\over 2\) ,are in

If S₁,S₂,S₃ are the sum of n,2n,3n terms respectively of an A.P., then S₃/(S₂-S₁)=

10ⁿ +3(4ⁿ⁺²) +5(n∈N) is divisible by 

If n is a natural number, then \(\left( \frac { n+1 }{ 2 } \right) ^{ n }\ge n!\) is true, then

There are six men A,B,C,D,E and F. The number of ways of forming a committee of four so as to always include C and exclude D, is

The expression \({ x }^{ 2 }+2bx+c\) has the positive value, if

The equations x+ky+3z=0, 3x+ky-2z=0, 2x+3y-4z=0 have a non-trivial solution, if value of k is

If  \(|a_i|<1,\lambda_i\ge0\) for i=1,2,3,...,n and \(\lambda_1+\lambda_2+\lambda_3+...+\lambda_1=1\)then the value of \(|\lambda_1a_1+\lambda_2a_2+...+\lambda_na_n|\) is

A town has total population 25000, out of which 13000 read 'The Hindustan Times' and 10500 read 'The Indian Express' and 2500 read both papers. The percentage of population who read nither of these newspapers is