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QUESTION: 1

From the following combinations of physical constants (expressed through their usual symbols) the only combination, that would have the same value in different systems of units, is:

Solution:

A dimensionless and unitless term will have same value in all system of units. Option (2) is a dimensionless term.

Thus, this term would have same value in different systems of units.

QUESTION: 2

A person climbs up a stalled escalator in 60 s. If standing on the same but escalator running with constant velocity he takes 40 s. How much time is taken by the person to walk up the moving escalator ?

Solution:

The expression of time taken by man is,

The expression for time taken by escalator is,

QUESTION: 3

Three masses m, 2m and 3m are moving in x-y plane with speed 3u, 2u, and u respectively as shown in figure. The three masses collide at the same point P and stick together. The velocity of resulting mass will be:

Solution:

According to the law of conservation of momentum,

Left momentum Right momentum

QUESTION: 4

A 4 g bullet is fired horizontally with a speed of 300 m/s into 0.8 kg block of wood at rest on a table. If the coefficient of friction between the block and the table is 0.3, how far will the block slide approximately?

Solution:

According to the law of conservation of momentum,

mv = Mv'

0.004 x 300 = 0.8v'

v' = 1.5 m/s

From kinetics, write the expression for the velocity,

v^{2} = v'^{2} + 2as

0 = 1.5^{2} + 2 x 3 x s

s ≈ 0.379 m

QUESTION: 5

A spring of unstitched length l has a mass m with one end fixed to a rigid support. Assuming spring to be made of a uniform wire, the kinetic energy possessed by it if its free end is pulled with uniform velocity v is:

Solution:

The kinetic energy in elemental form is,

The velocity in elemental form is,

The mass in elemental form is,

The kinetic energy in integral form is,

QUESTION: 6

A particle is moving in a circular path of radius a, with a constant velocity v as shown in the figure. The center of circle is marked by 'C. The angular momentum from the origin O can be written as:

Solution:

Consider the diagram shown below,

The expression of the component of angular velocity through OC is,

OC = v × a ……(1)

By using the property of triangle,

The expression for the angular velocity for an arbitrary point N from the figure is,

ON = OC + CN ……(2)

The expression of the component of angular velocity through CN is,

Substitute the values into equation (2)

Angular momentum = va +vacos2θ

= va (1 + cos2θ)

QUESTION: 7

Two hypothetical planets of masses m_{1} and m_{2} are at rest when they are infinite distance apart. Because of the gravitational force they move towards each other along the line joining their centres. What is their speed when their separation is 'd' ? (Speed of m_{1} is v_{1} and that of m_{2} is v_{2}):

Solution:

Let reference point is at infinity. The total energy at infinity will be 0.

So, initial energy of the system is 0.

The expression for the final energy of the system will be,

From law of conservation of energy,

.....(1)

As the momentum is conserved for the system, the kinetic energy will be inversely proportional to the mass.

The expression of the kinetic energy for mass m_{1} is,

Substitute above expression in equation (1),

Similarly for mass m_{2} KE is,

Solve the above expression.

QUESTION: 8

Steel ruptures when a shear of 3.5 ×10^{8} Nm^{–2} is applied. The force needed to punch a 1 cm diameter hole in a steel sheet 0.3 cm thick is nearly :

Solution:

The expression for stress is,

The expression for area is,

Area 2πrt

The expression for the force is,

Force A x stress

QUESTION: 9

A cylindrical vessel of cross-section A contains water to a height h. There is a hole in the bottom of radius 'a'. The time in which it will be emptied is :

Solution:

If the liquid is at x height from orifice at time t then, the volume coming from the orifice is given by the expression,

V = vA …… (1)

The expression of the velocity of water is,

Write expression for rate of volume.

Compare equation (2) with the above expression.

Solve the above expression.

QUESTION: 10

Two soap bubbles coalesce to form a single bubble. If V is the subsequent change in volume of contained air and S the change in total surface area, T is the surface tension and P atmospheric pressure, which of the following relation is correct?

Solution:

Let the radius of the soap bubble is r and the radius of the larger bubble is R.

So, the surface tension is given by,

By solving the above expression,

Rearrange the above expression to form,

3PV + 4TS = 0

QUESTION: 11

Hot water cools from 60°C to 50°C in the first 10 minutes and to 42°C in the next 10 minutes. The temperature of the surroundings is :

Solution:

The rate of change temperature is given as,

Substitute the values.

....(1)

Similarly,

...(2)

Thus,

T = 10 °C

QUESTION: 12

A Carnot engine absorbs 1000 J of heat energy from a reservoir at 127°C and rejects 600 J of heat energy during each cycle. The efficiency of engine and temperature of sink will be :

Solution:

The values for heat in Carnot engine is,

The expression for efficiency is,

So,

QUESTION: 13

At room temperature a diatomic gas is found to have an r.m.s. speed of 1930 ms^{–1}. The gas is :

Solution:

The equation of rms velocity is,

From ideal gas equation,

Thus,

The molar mass of hydrogen is 2g.

QUESTION: 14

Which of the following expressions corresponds to simple harmonic motion along a straight line, where x is the displacement and a, b, c are positive constants?

Solution:

Only linear equations represent SHM along a straight line; and the degree of x as 1 is present only in option (4) so it is a linear equation. Thus option (4) is correct.

QUESTION: 15

A source of sound A emitting waves of frequency 1800 Hz is falling towards ground with a terminal speed v. The observer B on the ground directly beneath the source receives waves of frequency 2150 Hz. The source A receives waves, reflected from ground, of frequency nearly : (Speed of sound =343 m/s)

Solution:

By using Doppler effect,

The velocity from the above expression is,

v = 55.8 m/s

The reflected frequency is,

= 2500 Hz

QUESTION: 16

A spherically symmetric charge distribution is characterised by a charge density having the following variation:

Where r is the distance from the centre of the charge distribution and ρ_{0} is a constant. The electric field at an internal point (r <R ) is :

Solution:

The expression of charge for a symmetrically charged spherical body,

dq = ρ × 4πr^{2}dr

The equation of Charge density is,

So, the total charge is given by,

QUESTION: 17

The space between the plates of a parallel plate capacitor is filled with a 'dielectric' whose 'dielectric constant' varies with distance as per the relation:

K(x) = K_{o} + λx (λ = a constant)

The capacitance C, of this capacitor, would be related to its ‘vacuum’ capacitance C_{o} as per the relation :

Solution:

The equation of potential is,

Let the charge density is σ and the surface area is s.

The capacitance is given by,

Solve above expression by multiply the denominator and numerator by d in the right hand side.

QUESTION: 18

The circuit shown here has two batteries of 8.0 V and 16.0 V and three resistors 3 Ω, 9 Ω and 9 Ω and a capacitor 5.0 µF.

How much is the current I in the circuit in steady state?

Solution:

The current would not flow through the branch with capacitor in the steady state.

Therefore, net potential is 8 V and equivalent resistance is 12 Ω.

So,

I = 8/12

= 0.67 A

QUESTION: 19

A positive charge 'q' of mass 'm' is moving along the +x axis. We wish to apply a uniform magnetic field B for time ∆t so that the charge reverses its direction crossing the y axis at a distance d. Then :

Solution:

Write expression for the energy balance for the charge,

...(1)

The equation of the total time for charge particles is,

...(2)

The equation of the time for reversal of direction is,

Solve equation (1) and (2),

QUESTION: 20

Consider two thin identical conducting wires covered with very thin insulating material. One of the wires is bent into a loop and produces magnetic field B_{1}, at its centre when a current (I) passes through it. The second wire is bent into a coil with three identical loops adjacent to each other and produces magnetic field B_{2} at the centre of the loops when current 1/3 passes through it. The ratio B_{1} : B_{2} is :

Solution:

Consider the following diagram,

The equation of magnetic field for loop wire is,

The equation of magnetic field for coiled wire is,

Taking ratio,

=1/3

QUESTION: 21

A sinusoidal voltage V(t) = 100 sin (500t) is applied across a pure inductance of L = 0.02 H. The current through the coil is :

Solution:

The inductive reactance is,

X_{L} = ωL

Substitute the values.

X_{L} 500 x 0.2

= 10 Ω

The current lags behind voltage by π/2 because the circuit is

pure inductive. So,

QUESTION: 22

A lamp emits monochromatic green light uniformly in all directions. The lamp is 3% efficient in converting electrical power to electromagnetic waves and consumes 100 W of power. The amplitude of the electric field associated with the electromagnetic radiation at a distance of 5 m from the lamp will be nearly :

Solution:

The intensity of electric field is,

...(1)

Also,

From equation (1),

The electric field intensity will be half of the total intensity and thus,

QUESTION: 23

The refractive index of the material of a concave lens is µ. It is immersed in a medium of refractive index µ_{1}. A parallel beam of light is incident on the lens. The path of the emergent rays when µ_{1} > µ is :

Solution:

The light bends towards the normal as it passes through rarer to denser medium and as it goes through lens to medium.

Thus, option (1) is correct.

QUESTION: 24

Interference pattern is observed at 'P' due to superimposition of two rays coming out from a source 'S' as shown in the figure. The value ‘l’ for which maxima is obtained at 'P' is (R is perfect reflecting surface) :

Solution:

From the geometry of the given figure,

The expression of total path difference by light is,

For maxima, the path difference should be nλ,

QUESTION: 25

In an experiment of single slit diffraction pattern, first minimum for red light coincides with first maximum of some other wavelength. If wavelength of red light is 6600 Å , then wavelength of first maximum will be:

Solution:

Write the expression for the first minima of red.

...(1)

Write the expression for maxima for a wavelength λ_{2}.

...(2)

As the light coincides, So compare (1) and (2).

QUESTION: 26

A beam of light has two wavelengths 4972 Å and 6216 Å with a total intensity of 3.6 ×10^{–3} Wm^{−2} equally distributed among the two wavelengths. The beam falls normally on an area of 1 cm^{2} of a clean metallic surface of work function 2.3 eV. Assume that there is no loss of light by reflection and that each capable photon ejects one electron. The number of photo electrons liberated in 2s is approximately:

Solution:

Write the expression for the number of photons per second by a monochromatic light.

Substitute the values,

The number of photons ejected in 2s is,

QUESTION: 27

A piece of bone of an animal from a ruin is found to have ^{14}C activity of 12 disintegrations per minute per gm of its carbon content. The ^{14}C activity of a living animal is 16 disintegrations per minute per gm. How long ago nearly did the animal die ? (Given half life of ^{14}C is t_{1/2} = 5760 years):

Solution:

The radioactivity in t^{th} time,

Substitute the values.

Take log both sides, and use property of log.

QUESTION: 28

For LED's to emit light in visible region of electromagnetic light, it should have energy band gap in the range of:

Solution:

λ should lie between 4000 × 10^{−10} to 7600 ×10^{−10} m for emitting light. The equation of minimum energy is,

Substitute the values in above expression,

Similarly,

QUESTION: 29

For sky wave propagation, the radio waves must have a frequency range in between:

Solution:

The range of frequency for radio wave is 5 MHz to 25 MHz for sky wave propagation. Thus, option (2) is correct.

QUESTION: 30

In the experiment of calibration of voltmeter, a standard cell of e.m.f. 1.1 volt is balanced against 440 cm of potentiometer wire. The potential difference across the ends of resistance is found to balance against 220 cm of the wire. The corresponding reading of voltmeter is 0.5 volt. The error in the reading of voltmeter will be:

Solution:

The potential gradient is,

The voltage across 220 cm is,

The error in reading is,

0.5 − 0.55= −0.05 V

QUESTION: 31

If m and e are the mass and charge of the revolving electron in the orbit of radius r for hydrogen atom, the total energy of the revolving electron will be:

Solution:

The total energy of a revolving electron in the hydrogen atom is calculated by adding the potential energy and kinetic energy of the electron.

The value of Z for hydrogen atom is 1.

QUESTION: 32

The de-Broglie wavelength of a particle of mass 6.63 g moving with a velocity 100 ms^{–1} is:

Solution:

The momentum of the particle is,

p = m·v

The de Broglie wavelength of the given particle,

QUESTION: 33

What happens when an inert gas is added to an equilibrium keeping volume unchanged?

Solution:

The molar concentration of reactants and products does not change by the addition of inert gas at constant volume, and due to which the state of equilibrium remains unchanged. Thus, the addition of an inert gas at constant volume does not affect the equilibrium.

QUESTION: 34

The amount of BaS0_{4} formed upon mixing 100 mL of 20.8% BaCl_{2} solution with 50 mL of 9.8% H_{2}SO_{4 }solution will be:

(Ba = l37, Cl = 35.5, S=32, H = l and O = 16)

Solution:

The number of moles of BaCl_{2} is,

= 0.01mol

The molarity of the given BaCl_{2} solution is,

= 0.1M

The number of moles of H_{2}SO_{4} is,

= 0.01mol

The molarity of the given H_{2}SO_{4} solution is,

= 0.2 M

The number of moles of BaSO_{4} formed from BaCl_{2} and H_{2}SO_{4} solution is 0.1 mol because the number of moles of barium is limited. The mass of BaSO_{4} formed in the given system is calculated by multiplying its moles with the molar mass.

Mass of BaSO_{4} = 0.1 mol x (137 g/mol + 32 g/mol 4 x 16 g/mol)

= 23.3 g

QUESTION: 35

The rate coefficient (k) for a particular reaction is 1.3 ×10^{−4} M^{−1}s^{−1} at 100 C, and 1.3 ×10^{−3} M^{−1}s^{−1} at 150 C. What is the energy of activation (E_{A}) (in kJ) for this reaction? (R = molar gas constant = 8.314 JK^{−1}mol^{−1})

Solution:

The activation energy of the given reaction is,

QUESTION: 36

How many electrons would be required to deposit 6.35 g of copper at the cathode during the electrolysis of an aqueous solution of copper sulphate?

(Atomic mass of copper = 63.5 u, NA = Avogadro's constant) :

Solution:

The number of molecules of copper formed in the given electrolysis process is,

Since one copper molecule is formed by 2 electrons.

So, the number of electrons is double the number of molecules of copper.

Hence, the number of electrons required in the given system is calculated as,

QUESTION: 37

The entropy (Sº) of the following substances are :

CH_{4} (g) 186.2 J K^{–1} mol^{–1}

O_{2} (g) 205.0 J K^{–1} mol^{–1}

CO_{2} (g) 213.6 J K^{–1} mol^{–1}

H_{2}O (l) 69.9 J K^{–1} mol^{–1 }

The entropy change (∆Sº) for the reaction

CH_{4} (g) + 2O_{2} → CO_{2} (g) + 2H_{2}O (l) is :

Solution:

The standard entropy change of the given reaction is,

QUESTION: 38

The conjugate base of hydrazoic acid is:

Solution:

The expression of the dissociation of hydrazoic acid is,

Thus, the conjugate base of hydrazoic acid is

QUESTION: 39

In a monoclinic unit cell, the relation of sides and angles are respectively:

Solution:

In a monoclinic unit cell, the relation between sides is a ≠ b ≠c, and the relationship between angles is γ = β = 90° and α ≠ 90°.

QUESTION: 40

The standard enthalpy of formation (∆_{f}Hº_{298}) for methane, CH_{4} is– 74.9 kJ mol^{–1}. In order to calculate the average energy given out in the formation of a C–H bond from this it is necessary to know which one of the following?

Solution:

Write the expression of the balanced chemical equation for the formation of methane.

If the values of enthalpy of sublimation of carbon and bond dissociation energy of H_{2} are known, the average energy released in the given reaction can be calculated.

QUESTION: 41

Which of the following xenon-OXO compounds may not be obtained by hydrolysis of xenon fluorides ?

Solution:

The chemical reactions of xenon fluoride with water are,

The oxidation state of xenon in XeO_{4} is +8 .

QUESTION: 42

Excited hydrogen atom emits light in the ultraviolet region at 2.47 ×10^{15 }Hz . With this frequency, the energy of a single photon is:

(h = 6.63×10^{–34} Js)

Solution:

The energy of a photon is,

E=hν

QUESTION: 43

Which one of the following exhibits the largest number of oxidation states ?

Solution:

The number of oxidation states exhibited by an element depends upon the number of unpaired electrons present in the d orbital. Since, these are five unpaired electrons in the d orbital of manganese that is larger than that of titanium, vanadium, and chromium. Thus, manganese exhibits the largest number of oxidation states among the given compounds.

QUESTION: 44

Copper becomes green when exposed to moist air for a long period. This is due to:

Solution:

The copper metal in presence of air for a longer time reacts with carbon dioxide, water, and oxygen to form copper carbonate and copper hydroxide. The color of copper changes to green due to the formation of copper carbonate (green color).

The chemical equation involved in the reaction of copper with moist air is given as,

2Cu + H_{2}O + CO_{2} + O_{2}→ CuCO_{3} + Cu (OH)_{2}

QUESTION: 45

Among the following species the one which causes the highest CFSE, ∆o as a ligand is :

Solution:

The crystal field splitting energy of ligand depends upon the strength of ligands. The splitting of molecular orbitals occurs due to the higher strength of ligands, cause results in the high crystal field splitting energy. Since the strength of CO ligand is higher than that of the CN^{−}, NH_{3} and F^{−}. Therefore, CO causes highest crystal field splitting energy out of the given ligands.

QUESTION: 46

Similarity in chemical properties of the atoms of elements in a group of the Periodic table is most closely related to :

Solution:

According to the modern periodic law, the similarity in the chemical properties is shown by the elements having an equal number of valence electrons in their valence shell. Thus, the similarity in the chemical properties of elements in a group is related to the number of valence electrons as all elements present in a group have an equal number of valence electrons in their outermost shell.

QUESTION: 47

Which of the following arrangements represents the increasing order (smallest to largest) of ionic radii of the given species O^{2–}, S^{2–}, N^{3–}, P^{3–} ?:

Solution:

The atomic radii of O^{2−},N^{3−} and S^{2−} is smaller than that of P^{3−} due to the atomic radius of elements decreases along the period. The atomic radius of O^{2−} is smaller than that of N^{3−} because both the ions have an equal number of electrons but the number of protons is greater in O^{2−} as compared to N^{3−}. Thus, the increasing order of atomic radii of the given elements is O^{2−} < N^{3−} <S^{2−}< P^{3−}.

QUESTION: 48

Global warming is due to increase of:

Solution:

The increase in the amount of methane and carbon dioxide gas in the atmosphere causes global warming. The increase in the amount of these gases warms the earth’s atmosphere which results in the global warming.

QUESTION: 49

Hydrogen peroxide acts both as an oxidising and as a reducing agent depending upon the nature of the reacting species. In which of the following cases H_{2}O_{2} acts as a reducing agent in acid medium ?

Solution:

The reducing agent in the reaction with permanganate ion in the acidic medium is hydrogen peroxide as shown in the chemical equation,

QUESTION: 50

Which one of the following complexes will most likely absorb visible light?

(At nos. Sc = 21, Ti = 22, V = 23, Zn = 30)

Solution:

The electronic configuration of central metal ions in the given complexes is,

The central metal ion present in the complex has its valence electrons in the 4s orbital that can easily excite to the 3d orbital and can absorb visible light, but in case of other complexes, the central metal ion has fully filled electronic configuration. Thus, the complex is most likely to absorb visible light among the given complexes.

QUESTION: 51

on mercuration-demercuration produces the major product:

Solution:

The mechanism involved in the mercuration-demercuration of the given compound is,

The mercuration-demercuration reaction of the given compound involves the formation of cyclic carbocation intermediate. This cyclic carbocation intermediate cannot undergo rearrangement because it is a type of non-classical carbocation.

QUESTION: 52

In the Victor-Meyer's test, the colour given by 1°, 2° and 3 alcohols are respectively:

Solution:

In the Victor Meyer’s test, primary alcohols give blood red color solution of nitrolic acid in sodium hydroxide as shown below.

In the Victor Meyer's test, secondary alcohols give a blue color solution of pseudonitrol in sodium hydroxide as shown below.

At the end of Victor Meyer’s test, tertiary alcohols form a colorless solution as shown below.

QUESTION: 53

Conversion of benzene diazonium chloride to chloro benzene is an example of which of the following reactions?

Solution:

Sandmeyer reaction involves the reaction of benzene diazonium salts with the solution of copper halide, and hydrochloric acid leads to the formation of halogen substituted benzene. The conversion of benzene diazonium salt to chlorobenzene is shown below.

QUESTION: 54

In the presence of peroxide, HCl and HI do not give antiMarkownikoff s addition to alkenes because :

Solution:

The steps involved in the antimarkovnikov addition of hydrogen chloride and hydrogen iodide on alkenes in the presence of peroxide are endothermic in nature due to which the antimarkovnikov addition of hydrogen chloride and hydrogen iodide is not favorable.

QUESTION: 55

The major product obtained in the photo catalysed bromination of 2-methylbutane is:

Solution:

The photocatalyzed bromination of 2 − methylbutane proceeds via free radical mechanism. Since the stability of tertiary free radical is greater than that of primary and secondary radical, the major product obtained in the given reaction corresponds to that of tertiary free radical.

Thus, the major product formed in the given reaction is 2 − bromo − 2− methylbutane.

QUESTION: 56

Which of the following molecules has two sigma (σ) and two pi(π)bonds :

Solution:

The structure of HCN is shown below.

In the structure of HCN, there are two sigma bonds and two pi bonds.

QUESTION: 57

Which one of the following acids does not exhibit optical isomerism?

Solution:

The structure of maleic acid is shown below.

Maleic acid exhibits only geometrical isomerism because only cis and trans arrangement is possible in its structure due to the presence of two groups at each carbon atom. So, the compound which does not exhibit optical isomerism is maleic acid.

QUESTION: 58

Aminoglycosides are usually used as:

Solution:

Aminoglycosides are a class of antibiotics due to which they are very useful as antibiotics. Thus, the aminoglycosides are usually used as antibiotics.

QUESTION: 59

Which of the following will not show mutarotation?

Solution:

Sucrose does not show mutarotation because the bond present between glucose and fructose is between the alpha hydroxyl group of both glucose and fructose due to which it cannot exists in alpha and beta form. Thus, sucrose does not undergo mutarotation.

QUESTION: 60

Phthalic acid reacts with resorcinol in the presence of concentrated H_{2}SO_{4} to give :

Solution:

The formation of fluorescein takes place due to the reaction of phthalic acid with resorcinol as shown below.

QUESTION: 61

A relation on the set A = {x : |x| < 3, x∈ Z} , where Z is the set of integers is defined by R = {( x, y ) : y = x , x ≠ 1}. Then the number of elements in the power set of R is:

Solution:

For |x| <3 , the set A is,

A = {−2,− 1, 0, 1, 2}

For y = |x| and x ≠ −1, the set R is,

R = {( −2, 2) (0, 0) (1,1) (1, 2)}

The power set of R is,

2^{4} = 16

QUESTION: 62

Let z ≠ −i be any complex number such that is a purely imaginary number. Then is

Solution:

Let’s consider z = x + iy

The expression is,

This term cannot be purely imaginary. Thus, is any non-

zero real number.

QUESTION: 63

The sum of the roots of the equation, x^{2} + |2x − 3| − 4 = 0, is:

Solution:

Write the equation when

x^{2} + |2x-3| - 4 = 0

x^{2} + 2x - 7 = 0

Write the equation when

x^{2} + |2x-3| - 4 = 0

x^{2} - 2x + 3 - 4 = 0

x^{2} - 2x -1 = 0

Compute the roots of the quadratic equations when

Compute the roots of the quadratic equations when

Compute the sum of the roots as,

QUESTION: 64

then k is equal to :

Solution:

By gauss elimination, apply the first row transformation

R_{2} → R_{2} − R_{1} and R_{1} → R_{1} − R_{3}.

By gauss elimination, apply the second row transformation

R_{3} → R_{3} + R_{1} and R_{1} → R_{2} - 1/2R_{2}.

QUESTION: 65

If and be such that then:

Solution:

The multiplication of AB is,

Solve the above matrix for first equation.

y + 2x + x = 6

3x + y = 6

Solve the above matrix for second equation.

3y - x + 2 = 8

3y - x = 6

Solve the above equations.

3x + y = 3y - x

4x = 2y

y = 2x

QUESTION: 66

8-digit numbers are formed using the digits 1, 1, 2, 2, 2, 3, 4, 4. The number of such number in which the odd digits do not occupy odd places, is:

Solution:

The expression for the ways to select 3 odd places out of 4 odd places is,

= 120

QUESTION: 67

is expanded in the ascending powers of x and the coefficients of powers of x in two consecutive terms of the expansion are equal, then these terms are:

Solution:

The expression for the general term from Binomial theorem is,

Let’s consider the coefficients of T_{r +1} =T_{r+2}.

r = 7

Thus, the first term is given as,

= 8^{th}

And, the second term is given as,

=9^{th}

QUESTION: 68

Let G be the geometric mean of two positive numbers a and b, and M be the arithmetic mean of then a : b can be:

Solution:

Write expression for the geometric mean of numbers a and b.

The value of b is ar.

Thus, the expression becomes,

Calculate the arithmetic mean of numbers

The ratio of 1/M : G is,

Solve above equation,

4r^{2} + 4 + 8r = 25r

4r^{2} - 17r + 4 = 0

The roots of the above equation are,

Thus, the value of a:b is,

a = br

a/b r

= 1/4

QUESTION: 69

The least positive integer n such that is,

Solution:

Write the general binomial expression.

Solve the above expression.

Further solve the above expression,

n - 1 = 5

n = 6

QUESTION: 70

Let f, g : R → R be two functions defined by

Statement I : f is a continuous function at x = 0.

Statement II : g is a differentiable function at x = 0.

Solution:

Write the given function.

The value of the function f (x) at x → 0.

= f (0)

So, the f (x) is continuous function at x = 0.

The value of the function g (x) at x → 0.

g(0) = 0

The expression for the left hand limit of the function g (x) is,

The expression for the right hand limit of the function g (x) is,

Since, the left hand limit is equal to the right hand limit. Thus, the function g(x) at x = 0 is differentiable.

QUESTION: 71

If f(x) x^{2} - x + 5, x > 1/2, and g (x) is its inverse function, then g′ (7) is equals :

Solution:

Write the given function.

f(x) = x^{2} − x+ 5

Compute the value of x at which the value of f(x) is 7 as,

7 = x^{2} - x + 5

2 = x(x-1)

x = 2 and 3

The value of the function g (x) is,

g(f(x)) = x

Differentiate the above expression.

The value of the function g′ (7) is,

= 1/3

QUESTION: 72

Let f and g be two differentiable functions on R such that f'(x) > 0 and g'(x) < 0, for all x ∈ R. Then for all x:

Solution:

The term f′(x) > 0 shows that it is an increasing function.

The term g′(x) < 0 shows that it is an decreasing function.

In case (1),

x > x - 1

g(x) > g(x-1)

f(g(x)) < f(g(x-1))

In case (2),

In case (3),

In case (4),

QUESTION: 73

If 1 + x^{4} + x^{5} = for all x in R, then a_{2} is:

Solution:

Write the given equation.

Compare the coefficients of x^{5}.

a_{5} = 1

Compare the coefficients of x^{4}.

Compare the coefficients of x^{3}.

Compare the coefficients of x^{2}.

a_{2} = -4.

QUESTION: 74

The integral is equal to:

Solution:

Write the given integral.

Let, the term tan^{3} x + 1 is,

The integral is,

QUESTION: 75

If [ ] denote the greatest integer function, then the integral is equal to:

Solution:

Write the given integral.

QUESTION: 76

If for a continuous function for all t ≥− π, then is equal to:

Solution:

Write the given continuous function.

Write the expression of the function.

Thus, the value of f (x) is,

f(x) = −3x

= π

QUESTION: 77

The general solution of the differential equation,

Solution:

Write the given differential equation.

Compute the integration factor of the above equation as,

The expression for the general solution of the equation is,

QUESTION: 78

If a line intercepted between the coordinate axes is trisected at a point A (4, 3), which is nearer to x-axis, then its equation is:

Solution:

The below figure represents the diagram of the line,

Write the value of the abscissa at point B.

a/3 = 4

a = 12

Write the value of the ordinate at point C.

2b/3 = 3

b = 9/2

Write the equation of the line.

QUESTION: 79

If the three distinct lines x + 2ay + a = 0, x +3by +b = 0 and x + 4ay + a = 0 are concurrent, then the point (a, b) lies on a :

Solution:

Write the matrix form of the given equations.

By gauss elimination, apply the first row transformation

R_{2} → R_{2}− R_{1} and R_{3} → R_{3} −R_{1}.

Compute the determinant of the above matrix as,

a = b

The locus of (a, b) lies at x = 0 or y = x. Hence, the point (a, b) will lie on a straight line.

QUESTION: 80

For the two circles x^{2} + y^{2 }= 16 and x^{2} + y − 2y = 0, there is/are:

Solution:

The distance between their centers is 1 unit and sum of their radii is 3 from the two given equations of the circle. Thus, one of them lie completely inside the other. So, there will be no common tangent.

QUESTION: 81

Two tangents are drawn from a point (–2,–1) to curve y^{2} = 4x. If α is the angle between them, then tan α is equal to :

Solution:

Write the equation of the tangent.

Write the condition of tangency.

C = a/m

Compute the roots of the above equation as,

Compute the value of the tan α is,

=3

QUESTION: 82

The minimum area of a triangle formed by any tangent to the ellipse and the coordinate axes is:

Solution:

Let the point on an ellipse is M having coordinates (4 cosθ,9 sinθ ) as shown in figure.

Write the equation of the tangent.

Let the points P and Q intersect the tangent at M with coordinate axis.

Write the coordinates of point P.

Write the coordinates of point Q.

Compute the area of the triangle.

Compute the minimum area of the triangle as,

QUESTION: 83

A symmetrical form of the line of intersection of the planes x = ay + b and z = cy + d is :

Solution:

Write the first equation.

Write the second equation.

Equate the both equations.

QUESTION: 84

If the distance between planes, 4x – 2y– 4z + 1 = 0 and 4x – 2y– 4z + d = 0 is 7, then d is :

Solution:

The expression for the distance between the two lines is,

QUESTION: 85

If are three unit vectors in three-dimensional space, then the minimum value of is:

Solution:

Write the given expression.

The minimum value of given expression lies at

The minimum value of the given expression is,

= 6 - 3

= 3

QUESTION: 86

Let and M.D. be the mean and the mean deviation about of n observation x_{i}, i = 1, 2, ...., n. If each of the observation is increased by 5, then the new mean and the mean deviation about the new mean respectively, are:

Solution:

The expression for the mean of the n observation is,

The new mean after the increment 5 in each observation is given as,

The expression for the mean deviation of the n observation is,

The new mean deviation after the increment 5 in each observation is given as,

QUESTION: 87

A number x is chosen at random from the set {1, 2, 3, 4, ....., 100}. Define the event: A = the chosen number x satisfies Then P(A) is:

Solution:

Write the given expression.

Here, the range of x lies

x ∈ {10, 11, 12, 13K 29} ∪ {50, 51, 52 .....100} .

The total value of x is 71.

Thus, the value of P (A) is,

QUESTION: 88

Statement I : The equation has a solution for all

Statement II : For any x ∈ R, and

Solution:

Write the given equation.

Solve the above equation.

Since, the value of cos^{−1} x lies,

Thus, statement I is false.

For statement II, the given equation is not applicable for any x ∈ R. So, statement II is false.

QUESTION: 89

and A and B are respectively the maximum and the minimum values of f(θ), then (A, B) is equal to:

Solution:

The determinant of the given matrix is calculated as,

The value of sin 2θ + cos 2θ lies between

The maximum and minimum value of sin 2θ + cos 2θ is,

The maximum value of A is and the minimum value of B is

QUESTION: 90

Let p, q, r denote arbitrary statements. Then the logically equivalent of the statement p ⇒ (q∨r) is:

Solution:

Simplify the given statement as,

p ⇒ (q∨r)

~ pv(qvr)

(~pvq) v (~pvr)

(p⇒q)∨(p⇒r)

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