1 Crore+ students have signed up on EduRev. Have you? 
What is the moment of inertia of a solid sphere of density r and radius ρ about its diameter?
For solid sphere
A body moves with uniform acceleration, then which of the following graph is correct ?
An object is said to be moving with a uniform acceleration, if its velocity changes by equal amount in equal intervals of time. The velocitytime graph of uniformly accelerated motion is a straight line inclined to time axis.
Acceleration of an object in a uniformly accelerated motion in one dimension is equal to the slope of the velocitytime graph with time axis.
A projectile can have the same range R for two angles of projection. If t_{1} and t_{2} be the times of flight in two cases, then what is the product of two times of flight?
where R is the range.
Hence t_{1} t_{2} ∝ R
A horizontal overhead powerline is at height of 4m from the ground and carries a current of 100A from east to west. The magnetic field directly below it on the ground is (μ_{0} = 4π × 10^{–7} Tm A^{–1})
The magnetic field is
According to right hand palm rule, the magnetic field is directed towards south.
A man of mass 100 kg. is standing on a platform of mass 200 kg. which is kept on a smooth ice surface. If the man starts moving on the platform with a speed 30 m/sec relative to the platform then calculate with what velocity relative to the ice the platform will recoil?
If the unit of force and length be each increased by four times, then the unit of energy is increased by
Since unit of energy = (unit of force).(unit of length) so if we increase unit of length and force, each by four times, then unit of energy will increase by sixteen times.
Which of the following must be known in order to determine the power output of an automobile?
Power is defined as the rate of doing work.
For the automobile, the power output is the amount of work done (overcoming friction) divided by the length of time in which the work was done.
If the force is given by F = at + bt^{2} with t as time. The dimensions of a and b are
Dimension of at = Dimension of F
Dimension of bt^{2} = Dimension of F
A wheel of radius R rolls on the ground with a uniform velocity v. The relative acceleration of topmost point of the wheel with respect to the bottom most point is
As a_{CM }= 0 [v_{CM} = constant],
Tangential acceleration of each point
If the radius of the earth were to shrink by one per cent, its mass remaining the same, the value of g on the earth’s surface would
The Young’s modulus of a perfectly rigid body is
For a perfectly rigid body strain produced is zero for the given force applied, so Y = stress/strain = ∞
An ice block floats in a liquid whose density is less than water. A part of block is outside the liquid. When whole of ice has melted, the liquid level will
Ice is lighter than water. When ice melts, the volume occupied by water is less than that of ice. Due to which the level of water goes down.
A large drop of oil (density 0.8 g/cm^{3} and viscosity η_{0}) floats up through a column of another liquid (density 1.2 g/cm^{3} and viscosity η_{L}).
Assuming that the two liquids do not mix, the velocity with which the oil drop rises will depend on:
A solid body of constant heat capacity 1 J/°C is being heated by keeping it in contact with reservoirs in two ways:
(i) Sequen tially keepin g in con tact with 2 reservoirs such that each reservoir supplies same amount of heat.
(ii) Sequen tially keepin g in con tact with 8 reservoirs such that each reservoir supplies same amount of heat.
In both the cases body is brought from initial temperature 100°C to final temperature 200°C.
Entropy change of the body in the two cases respectively is :
The entropy change of the body in the two cases is same as entropy is a state function.
Which of the following process is possible according to the first law of thermodynamics?
For an isothermal expansion of a perfect gas, the value of ΔP/P is equal to
Differentiate PV = constant w.r.t V
A sample of ideal monoatomic gas is taken round the cycle ABCA as shown in the figure. The work done during the cycle is
ΔW = area under the p – V curve
The average translational kinetic energy of O_{2} (molar mass 32) molecules at a particular temperature is 0.048 eV. The translational kinetic energy of N_{2} (molar mass 28) molecules in eV at the same temperature is
For a gas if ratio of specific heats at constant pressure and volume is γ then value of degrees of freedom is
One end of a long metallic wire of length L tied to the ceiling. The other end is tied with a massless spring of spring constant K. A mass hangs freely from the free end of the spring. The area of cross section and the young’s modulus of the wire are A and Y respectively. If the mass slightly pulled down and released, it will oscillate with a time period T equal to :
The transverse displacement y(x, t) of a wave on a string is given by
This represents a
It is a function of type y = f (ωt + kx)
∴ y (x, t) represents wave travelling along –x direction.
A sound source is moving towards stationary listener with 1/10th of the speed of sound. The ratio of apparent to read frequency is
In a region of space having a uniform electric field E, a hemispherical bowl of radius r is placed.The electric flux ϕ through the bowl is
ϕ = E(ds) cos θ = E(2πr^{2}) cos 0° = 2πr^{2} E.
The electric field intensity just sufficient to balance the earth’s gravitational attraction on an electron will be: (given mass and charge of an electron respectively are 9.1 × 10^{–31} kg and 1.6 × 10^{–19} C.)
– eE = mg
Two capacitors C_{1} and C_{2} are charged to 120 V and 200 V respectively. It is found that by connecting them together the potential on each one can be made zero. Then
For potential to be made zero, after connection
⇒ 3C_{1} = 5C_{2}
Three voltmeters A, B and C having resistances R, 1.5 R and 3R, respectively, are connected as shown. When some potential difference is applied between X and Y, the voltmeter readings are V_{A}, V_{B} and V_{C} respectively. Then –
V_{A} = IR
∴ V_{A} = V_{B }= V_{C}
The range of the particle when launched at an angle of 15º with the horizontal is 1.5 km. What is the range of the projectile when launched at an angle of 45º to the horizontal.
If m is magnetic moment and B is the magnetic field, then the torque is given by
Magnetic moment of bar magnet is M. The work done to turn the magnet by 90° of magnet in direction of magnetic field B will be
Work done, W = MB (1 – cos θ)
θ = 90°
W = MB
The laws of electromagnetic induction have been used in the construction of a
The impedance of a circuit consists of 3 Ω resistance and 4Ω reactance. The power factor of the circuit is
Power = cos ϕ = 3/5 = 0.6
The r.m.s. value of potential difference V shown in the figure is
A ray of light is incident at the glasswater interface at an angle i, it emerges finally parallel to the surface of water, then the value of μ_{g} would be
A mica slit of thickness t and refractive index µ is introduced in the ray from the first source S_{1}. By how much distance of fringes pattern will be displaced?
In a Young’s double slit experiment the angular width of a fringe formed on a distant screen is 1°. The wavelength fo the light used is 6280 Å. What is the distance between the two coherent sources?
The angular fringe width is given by α = λ/d
where λ is wavelength and d is the distance between two coherent sources. Thus d = λ/α
A light having wavelength 300 nm fall on a metal surface. The work function of metal is 2.54 eV, what is stopping potential ?
If the total binding energies of nuclei are 2.22, 28.3, 492 and 1786 MeV respectively, identify the most stable nucleus of the following.
is most stable as it has maximum binding energy per nucleon.
An oscillator is nothing but an amplifier with
A positive feedback from output to input in an amplifier provides oscillations of constant amplitude.
In an experiment on photoelectric effect photons of wavelength 300 nm eject electrons from a metal of work function 2.25eV. A photon of energy equal to that of the most energetic electron corresponds to the following transition in the hydrogen atom:
A letter 'A' is constructed of a uniform wire with resistance 1.0 Ω per cm, The sides of the letter are 20 cm and the cross piece in the middle is 10 cm long. The apex angle is 60. The resistance between the ends of the legs is close to:
Solving we get
x = 10 Ω
Putting the value of x = 10 Ω in equation (i)
We get
Number of atoms of He in 100 amu of He (atomic wt. of He is 4) are :
100 amu of He = 100/4 atoms of He
= 25 atoms.
[1 a.m.u. = mass of one proton (approx.)]
If the radius of H is 0.53 Å, then what will be the radius of _{3}Li^{2+} ?
Which of the following does not have valence electron in 3dsubshell?
P (At no. 15) has electronic configuration 1s^{2}, 2s^{2} p^{6}, 3s^{2} p^{3}, hence no electron in dsubshell.
Orthonitrophenol has intramolecular
and paranitrophenol has intermolecular Hbonding.
Hence former is more volatile than latter.
In an ideal gas, the intermolecular forces of attraction are negligible and hence it cannot be liquefied.
In which of the following reactions, standard entropy change (ΔS°) is positive and standard Gibb’s energy change (ΔG°) decreases sharply with increasing temperature ?
Since, in the first reaction gaseous products are forming from solid carbon hence entropy will increase i.e. ΔS = +ve.
Since, ΔG° = ΔH° – TΔS hence the value of ΔG decrease on increasing temperature.
Bond enthalpies of H_{2}, X_{2} and HX are in the ratio 2 : 1 : 2. If enthalpy of formation of HX is –50 kJ mol^{–1}, the bond enthalpy of X_{2} is
Let the bond enthalpy of X – X bond be x.
ΔH_{f} (HX) = – 50
∴ x = 50 × 2 = 100 kJ mol^{–1}
The pOH value of a solution whose hydroxide ion concentration is 6.2 × 10^{–9} mol/litre is
–log (OH) = pOH; – log 6.2 × 10^{–9} = pOH;
∴ pOH = 8.21
Which of the following combinations would not result in the formation of a buffer solution?
Combination of NaOH and CH_{3}COOH is the mixture of alkali and acetic acid. Therefore this combination can not be buffer forming solution.
The reaction, SO_{2} + Cl_{2} → SO_{2}Cl_{2} is exothermic and reversible. A mixture of SO_{2} (g), Cl_{2} (g) and SO_{2}Cl_{2} (l) is at equilibrium in a closed container. Now a certain quantity of extra SO_{2} is introduced into the container, the volume remaining the same. Which of the following is/ are true?
By addition of SO_{2}, equilibrium will shift to RHS which is exothermic. Hence temp, will increase.
O.N. of Br_{2} changes from 0 to –1 and +5, hence it is reduced as well as oxidised.
The boiling point of water is exceptionally high because
The high boiling point of water is due to Hbonding.
Which of the following has correct increasing basic strength?
The basic character of oxides increases down the group.
The given two structures are optical isomers but as these are mirror image of each other, hence they represent enantiomers of each other.
Paper chromatography is a special case of partition chromatography where the special quality paper containing water trapped in it acts as a stationary phase and solvent as a mobile phase. Thus, both phases are liquids.
In which case the NO_{2} will attack at the meta position
mdirecting in nature.
Which alkene on ozonolysis gives CH_{3}CH_{2}CHO and
Formation of ozone in the upper atmosphere from oxygen takes place by the action of
In presence of U.V. rays O_{2} is converted into O_{3}.
CO_{2} goes to air, causes green house effect and gets dissolved in water. What will be the effect on soil fertility and pH of the water?
Here [H^{+}] increases hence, pH decreases due to which soil fertility will also decreases.
The van’t Hoff factor i for an electrolyte which undergoes dissociation and association in solvents are respectively
When an electrolyte dissociates van’t Hoff factor i is greater than 1 and when it associates the i is less than 1.
If the elevation in boiling point of a solution of 10 g of solute (mol. wt. = 100) in 100 g of water is ΔTb, the ebullioscopic constant of water is
The ionic conductance of Ba^{2+} and Cl^{–} respectively are 127 and 76Ω^{–1}cm^{2} at infinite dilution.The equivalent conductance of BaCl_{2} at infinite dilution will be
2N_{2}O_{5} ⇌ 4NO_{2} + O_{2}
If rate and rate constant for above reaction are 2.40 × 10^{–5} mol L^{–1} s^{–1} and 3 × 10^{–5} s^{–1} respectively, then calculate the concentration of N_{2}O_{5}.
The reaction is of first and for a first order reaction, rate, R = k [N_{2}O_{5}] 2.4 × 10^{–5} = 3 × 10^{–5} × [N_{2}O_{5}]
Which of the following gas molecules have maximum value of enthalpy of physisorption?
The more the liquifiable nature of a gas, the more is the enthalpy of adsorption. Water is more liquifiable.
Which of the following will be the most effective in the coagulation of Fe(OH)_{3} soil?
According to HardySchulze rule, coagulation power of ions is directly proportional to charge on ion.
∵ Fe(OH)_{3} is positively charged colloid.
∴ It will be coagulated by anion.
Because has highest charge among the given anions, therefore, Mg_{3}(PO_{4})_{2} is the most effective in the coagulation of Fe(OH)_{3} solution.
When chlorine water is exposed to sunlight, O_{2} is liberated. Hence,
Hydrogen has more affinity for chlorine.
An extremely hot copper wire reacts with steam to give
Among the following the lowest degree of paramagnetism per mole of the compound at 298 K will be shown by
Hence lowest paramagnetism is shown by CuSO_{4}.5H_{2}O
At 120140°C temperature and 1.5 atm pressure, sodium phenoxide reacts with CO_{2} to yield sodium salicylate which on further hydrolysis give to salicylic acid.
This reaction is known as Kolbe’s reaction.
Which of the following is process used for the preparation of acetone?
In Wacker process, when mixture of propene and air is passed through mixture of Pd and CuCl_{2} at high pressure acetone is formed.
The preparation of ethyl acetoacetate involves:
In Claisen con den sation inter molecular condensation of esters containing ahydrogen atom in presence of strong base produce βketo ester.
Which one of the following pairs is not correctly matched?
Like clemmensen reduction, WolfKishner reduction involves reduction of > C = O to > CH_{2} , of course by different reagent.
The helical structure of protein is stabilised by
Fibrous proteins have thread like molecules which lie side by side to form fibres. The various molecules are held together by hydrogen bonds.
Alizarin is an anthraquinone dye. It gives a bright red colour with aluminium and a blue colour with barium.
2,4dichlorophenoxyacetic acid is used as a herbicide.
0.45 g of acid molecular weight 90 is neutralised by 20 ml of 0.5N caustic potash. The basicity of acid is
Eq. of acid = Eq of base,
In the reaction of KMnO_{4} with an oxalate in acidic medium, MnO^{}_{4} is reduced to Mn^{2+} and is oxidised to CO_{2}. Hence, 50 mL of 0.02 M KMnO_{4} is equivalent to
Which of the following is soluble in yellow ammonium sulphide?
SnS +(NH_{4})_{2} S_{2} → (NH_{4})_{2} SnS_{3} soluble
Direction: In the following questions choose the word opposite in meaning to the given word.
Florid
The word Flor id (Adjective) means : rosy; gaudy; ornated; red; having too much decoration or detail.
The word Pale (Adjective) means : light in colour; not strong or bright; having skin that is almost white because of illness.
Hence, the words florid and pale are antonymous.
Direction: In the following questions choose the word opposite in meaning to the given word.
Verity
The word Verity (Noun) means : a belief or principle about life that is accepted as true; truth.
Hence, the words verity and falsehood are antonymous.
Direction: In the following questions choose the word opposite in meaning to the given word.
Perspicuity
The word Perspicuity (Noun) means : clarity.
The word vagueness (Noun) means : no clarity in a person’s mind.
Hence, the words perspicuity and Vagueness are antonymous.
Direction: In question out of the four alternative, choose the one which best expresses the meaning of the given word.
Disgrace
Disgrace means a state of shame.
Direction: In question out of the four alternative, choose the one which best expresses the meaning of the given word.
Striking
Striking means extraordinary, attractive.
Direction: In question out of the four alternative, choose the one which best expresses the meaning of the given word.
Fiasco
Fiasco means a complete failure.
Direction: In the following questions a part of the sentence is bold. Below are given alternatives to the Underline part at (a), (b) and (c) which may improve the sentence. Choose the correct alternative. In case no improvement is needed, your answer is (d).
Q. Power got with money is the most craved for today.
Direction: In the following questions a part of the sentence is bold. Below are given alternatives to the Underline part at (a), (b) and (c) which may improve the sentence. Choose the correct alternative. In case no improvement is needed, your answer is (d).
Q. You are asked to copy this letter word by word.
Word for word means : in exactly the same words or when translated exactly equivalent words.
Direction: Sentences are given with blanks to be filled in with an appropriate word(s). Four alternatives are suggested for each question. Choose the correct alternative out of the four:
Q. Let us quickly __________.
Huddle : come close in a gr oup
Direction: Sentences are given with blanks to be filled in with an appropriate word(s). Four alternatives are suggested for each question. Choose the correct alternative out of the four:
Q. Rajesh’s car wasn’t __________ Ramesh’s, so we were too exhausted by the time we reached home.
Right use of as  as comparison
Direction: In the following questions, the 1st and the last sentences of the passage are numbered 1 and 6. The rest of the passage is split into four parts and named P, Q, R and S. These four parts are not given in their proper order. Read the sentence and find out which of the four combinations is correct. Then find the correct answer.
1. The most vulnerable section of the society are the students.
P. Revolutionary and new fledged ideas have a great appeal to them.
Q. Agitations may be nonviolent methods of protest.
R. They cannot resist the charm of persuasion.
S. They are to be taught that without discipline they cannot get proper education.
6. However if th ese become violent , the antisocial elements get encouraged and they put all proper working out of gear.
Direction: In the following questions, the 1st and the last sentences of the passage are numbered 1 and 6. The rest of the passage is split into four parts and named P, Q, R and S. These four parts are not given in their proper order. Read the sentence and find out which of the four combinations is correct. Then find the correct answer.
1. Venice is a strange city.
P. There are about 400 odd bridges connecting the islands of Venice.
Q. There are no motor cars, no horses and no buses there.
R. Th ese small islands ar e close to on e another.
S. It is not one island but a hundred islands.
6. This is because Venice has no streets.
The World health Organisation is briefly called W.H.O. It is a specialised agency of the United Nations and was established in 1948.
International health workers can be seen working in all kinds of surroundings in deserts, jungles, mountains, coconut groves, and rice fields. They help the sick to attain health and the healthy to maintain their health.
This global health team assists the local health workers in stopping the spread of what are called communicable diseases, like cholera. These diseases can spread from one country to another and so can be a threat to world health.
W.H.O. assists different national health authorities not only in controlling diseases but also in preventing them altogether. Total prevention of diseases is possible in a number so ways. Everyone knows how people, particularly children, are vaccinated against one disease or another. Similarly, most people are familiar with the spraying of houses with poisonous substances which kill diseasecarrying insects.
Q. "It is a specialised agency of the United Nations and was established in 1948". Here specialised means :
The World health Organisation is briefly called W.H.O. It is a specialised agency of the United Nations and was established in 1948.
International health workers can be seen working in all kinds of surroundings in deserts, jungles, mountains, coconut groves, and rice fields. They help the sick to attain health and the healthy to maintain their health.
This global health team assists the local health workers in stopping the spread of what are called communicable diseases, like cholera. These diseases can spread from one country to another and so can be a threat to world health.
W.H.O. assists different national health authorities not only in controlling diseases but also in preventing them altogether. Total prevention of diseases is possible in a number so ways. Everyone knows how people, particularly children, are vaccinated against one disease or another. Similarly, most people are familiar with the spraying of houses with poisonous substances which kill diseasecarrying insects.
Q. "International health workers can be seen working in all kinds of surroundings: in deserts, jungles, mountains, coconout groves, and rice fields". Here International means
The World health Organisation is briefly called W.H.O. It is a specialised agency of the United Nations and was established in 1948.
International health workers can be seen working in all kinds of surroundings in deserts, jungles, mountains, coconut groves, and rice fields. They help the sick to attain health and the healthy to maintain their health.
This global health team assists the local health workers in stopping the spread of what are called communicable diseases, like cholera. These diseases can spread from one country to another and so can be a threat to world health.
W.H.O. assists different national health authorities not only in controlling diseases but also in preventing them altogether. Total prevention of diseases is possible in a number so ways. Everyone knows how people, particularly children, are vaccinated against one disease or another. Similarly, most people are familiar with the spraying of houses with poisonous substances which kill diseasecarrying insects.
Q. They help the sick to attain health and the healthy to maintain their health. here they stands for:
In a code language, if SUMMER is coded as SDNLVR, then the word WINTER will be coded as:
Direction: In question number, select the missing number from the given responses.
(1 × 2 × 3 × 5) + (1 + 2 + 3 + 5) = 41
(3 × 4 × 2 × 6) + (3 + 4 + 2 + 6) = 159
(9 × 8 × 3 × 4) + (9 + 8 + 3 + 4) = 888
Each day of the week is repeated after 7 days.
So, after 63 days, it will be Monday.
After 61 days, it will be Saturday.
Rahul and Nitesh are standing in a row of persons. Rahul is 12th from left side and Nitesh is 18th from the right side of the row. If they interchanged their positions Rahul becomes 25th from left. Find the new position of Nitesh from right side?
One of the numbers does not fit into the series. Find the wrong number.
52, 152, 414, 1312, 5348, 26840
The number should be 404. × 1 + 100, × 2 + 100, × 3 + 100……
In the following question and Δ stands for any of Mathematical signs at different places, which are given as choices under each question. Select the choice with the correct sequence of signs which when substituted makes the question as correct equation? 24 Δ 4 Δ 5 Δ 4
After putting sign 24 = 4 × 5 + 4
24 = 24
Hence, (b) is correct choice.
"All the members of the Tennis club are members of the badminton club too". No woman plays badminton?
Which answer figure is the exact mirror image of the given figure when the mirror held form the right at PQ?
Let A and B be two sets then (A ∪ B)‘∪ (A ‘∩ B) is equal to
From VennEuler ’s Diagram.
∴ (A ∪ B)' ∪(A'∩ B) = A'
Let x and y be two natural numbers such that xy = 12(x + y) and x ≤ y. Then the total number of pairs (x, y) is
xy – 12x – 12y = 0 ⇒ (x – 12) (y – 12) = 144
Now 144 can be factorised into two factors x and y where x ≤ y and the factors are (1, 144), (2, 72), (3, 48), (4, 36), (6, 24), (8, 18),
(9, 16), (12, 12).
Thus there are eight solutions.
If sin^{2}θ + sin^{2}ϕ = 1/2, cos^{2}θ +cos^{2}ϕ = 3/2, then cos^{2} (q – ϕ) is equal to
Using cosine formula
2 sin (θ + ϕ) cos (θ – ϕ) = 1/2 .....(i)
2 cos (θ + ϕ) cos (θ – ϕ) =3/2 .....(ii)
Squaring (1) and (2) and then adding
Let T(k) be the statement 1 + 3 + 5 + ... + (2k – 1) = k^{2} +10
Which of the following is correct?
When k = 1, LHS = 1 but RHS = 1 + 10 = 11
∴ T(1) is not true
Let T(k) is true. That is
1+ 3+ 5+ ..... + (2k  1) = k^{2} + 10
Now, 1+ 3 + 5 + ..... + (2k1) + (2k+1)
= k^{2} + 10 + 2k +1 = (k + 1)^{2} + 10
∴ T(k+1) is true.
That is T(k) is true ⇒ T(k+ 1) is true.
But T(n) is not true for all n ∈ N , as T(1) is not true.
If x = ω – ω^{2} –2, then the value of x^{4} + 3x^{3} + 2x^{2} – 11x – 6 is
We have, x = ω – ω^{2} –2 or x + 2 = ω – ω^{2}
Squaring, x^{2} + 4x + 4 = ω^{2} + ω^{4} – 2ω3
= ω^{2} + ω^{3}ω. –2ω^{3} = ω^{2} + ω – 2 [ω^{3} = 1]
= –1 – 2 = – 3 ⇒ x^{2} + 4x + 7 = 0
Dividing x^{4} + 3x^{3} + 2x^{2} –11x – 6 by x^{2} + 4x + 7, we get
x^{4} + 3x^{3} + 2x^{2} – 11x – 6 = (x^{2} + 4x + 7)(x^{2} – x – 1) + 1
= (0) (x^{2} – x – 1)+1 = 0 + 1 = 1
In how many ways can 5 prizes be distributed among 4 boys when every boy can take one or more prizes ?
First prize may be given to any one of the 4 boys, hence first prize can be distributed in 4 ways. similarly every one of second, third, fourth and fifth prizes can also be given in 4 ways.
∴ the number of ways of their distribution = 4 × 4 × 4 × 4 × 4 = 4^{5} = 1024
The number of positive integral solution of abc = 30 is
We have : 30 = 2 × 3 × 5. So, 2 can be assigned to either a or b or c i.e. 2 can be assigned in 3 ways. Similarly, each of 3 and 5 can be assigned in 3 ways. Thus, the number of solutions is 3 × 3 × 3 = 27.
The coefficient of x^{20} in the expansion of
= (1 + x^{2})^{30}. x^{10}
The coefficient of x^{20} in x^{10} (1 + x^{2})^{30}
= the coefficient of x^{10} in (1 + x^{2})^{30}
= ^{30}C_{5} = ^{30}C_{30–5} = ^{30}C_{25}
If x is positive then the sum to infinity of the series
The series is a G.P. with common ratio is less than 1 since x is
positive
The nearest point on the line 3x + 4y = 12 from the origin is
If ‘D’ be the foot of altitude, drawn from origin to the given line, then ‘D’ is the required point.
Let ∠OBA = θ
⇒ tan q = 4/3
⇒ ∠ DOA = θ we have OD = 12/5.
If D is (h, k) then h = OD cosθ, k = OD sinθ
⇒ h = 36/25, k = 48/25.
The length of the tangent drawn from any point on the circle x^{2} + y^{2} + 2fy + λ = 0 to the circle x^{2} + y^{2} + 2fy + μ = 0, where μ > λ > 0, is
Let the radius of the first circle be CT = r_{1}.
Also, let the radius of the second circle be CP = r_{2}.
In the triangle PCT, T is a right angle
Find the eccentricity of the conic represented by x^{2} – y^{2} – 4x + 4y + 16 = 0
We have x^{2} – y^{2} – 4x + 4y + 16 = 0
⇒ (x^{2} – 4x) – (y^{2} – 4y) = 16
⇒ (x^{2} – 4x + 4) – (y^{2} – 4y + 4) = – 16
⇒ (x – 2)^{2} – (y – 2)^{2} = – 16
This is rectangular hyperbola, whose eccentricity is always √2.
∴ Given limit
Let f (x + y) = f (x) . f (y) for all x, y where f (0) ≠ 0. If f (5) = 2 and f ' (0) = 3, then f ' (5) is equal to –
If sample A contains 100 observations 101, 102, .... 200 and sample B contains 100 obsections 151, 152, .......... 250, then ratio of variance v_{A}/v_{B} =
But both A and B have 100 observations, then both the sample A and B have same standard deviation and the same variance.
Hence, V_{A}/V_{B} = 1
The probability of simultaneous occurrence of atleast one of two events A and B is p. If the probability that exactly one of A, B occurs is q, then P(A’) + P(B’) is equal to
Since, P (exactly one of A, B occurs) = q.
∴ P (A∪B) P (A∩B) = q
⇒ p  P (A∩B)= q ⇒ P ( A∩B)= p  q
⇒ 1 P ( A'∪B') = pq ⇒ P (A'∪B') =1 p+q
⇒ P (A') + P(B')  P (A'∩B') =1 p+ q
⇒ P (A') + P( B') =(1 p+ q) + [1 P(A ∪B)]
= (1 – p + q) + (1 – p) = 2 – 2p + q
If f is an even function and g is an odd function, then the function fog is
We have, fog (–x) = f [g (–x)] = f [–g(x)
(∵ g is odd)
= f[g (x)] (∵ f is even)
= fog (x) ∀ x ∈ R.
∴ fog is an even function.
If k ≤ sin^{–1} x + cos^{–1} x + tan^{–1} x ≤ K, then –
Since domain of the function x ∈ [1,1]
The equations 2x + 3y + 4 = 0; 3x + 4y + 6 = 0 and 4x + 5y + 8 = 0 are
Consider first two equations :
2x + 3y = –4 and 3x + 4y = –6
∴ x = –2 and y = 0
Now this solution satisfies the third, so the equations are consistent with unique solution.
Applying C_{1} – C_{2} and C_{2} – C_{3}, we get
[by C_{1} – C_{2}, C_{3} –10C_{2}]
= 4 (180 – 180) = 0
If x = a sin θ and y = b cos θ, then d^{2}y/dx^{2} is
Given x = a sin θ and y = b cos θ
⇒ dx/dθ = a cos θ and dy/dθ = b sin θ
If f(x) = x^{α} log x and f(0) = 0, then the value of a for which Rolle’s theorem can be applied in [0, 1] is
For Rolle’s theorem in [a, b], f(a) = f(b),
In [0, 1] ⇒ f(0) = f(1) = 0
∵ the function has to be continuous in [0, 1]
If the function
is continuous at x = 2 and 4, then the values of a and b are
Since f (x) is continuous at x = 2
∴ 1 = 2a + b ..... (1)
Again f(x) is continuous at x = 4,
Solving (1) and (2), we get a = 3, b = – 5
is a decreasing function of x in R, then the set of possible values of a (independent of x) is
f'(x) < 0 for all x if a^{2} 1 ≤ 0 ⇒ 1 ≤ a ≤ 1
The diagonal of a square is changing at the rate of 0.5 cm/sec. Then the rate of change of area, when the area is 400 cm^{2}, is equal to
when area A is 400 cm^{2} then a = 20
If the normal to the curve y = f (x) at the point (3, 4) makes an angle 3π/4 with the positive xaxis, then f'(3) =
Slope of normal to y = f(x) at (3, 4) is 1/f'(3)
The area bounded by the curve y = sinx, xaxis and the ordinates x = 0 and x = π/2 is
The differential equation whose solution is Ax^{2} + By^{2} = 1 where A and B are arbitrary constants is of
Ax^{2} + By^{2} = 1 ......(1)
From (2) and (3)
Dividing both sides by –B, we get
Which is a DE of order 2 and degree 1
Unit vector perpendicular to both the given vectors is,
If a.b = a.c and a × b = a × c, then correct statement is
a. b. = a.c ⇒ a.(b – c) = 0
⇒ a = 0 or b – c = 0 or a ⊥ (b – c)
⇒ a = 0 or b = c or a ⊥ (b – c) ...(1)
Also a x b = a x c ⇒ a × (b – c) = 0
⇒ a = 0 or b – c = 0 or a  (b – c)
⇒ a = 0 or b = c or a  (b – c) ...(2) Observing to (1) and (2) we find that a = 0 or b = c
What is the value of n so that the angle between the lines having direction ratios (1, 1, 1) and (1, –1, n) is 60°?
If (l_{1}, m_{1}, n_{1}) and (l_{2}, m_{2}, n_{2}) are the direction ratios then angle between the lines is
The foot of the perpen dicular from the point (7, 14, 5) to the plane 2x + 4y – z = 2 are
We know that the length of the per pendicular from the point (x_{1} , y_{1}, z_{1}) to the plane ax + by + cz + d = 0 is
and the coor dinate (α,β,γ) of the foot of the ⊥ are given by
In the given ques,, x_{1} = 7, y_{1} = 14, z_{1} = 5, a = 2 b = 4, c = 1, d = 2
By putting these values in (1), we get
Hence, foot of ⊥ is (1, 2, 8)
Find the coordinates of the point where the line joining the points (2, –3, 1) and (3, – 4, – 5) cuts the plane 2x + y + z = 7.
The direction ratios of the line are 3 – 2, – 4 – (–3), – 5 –1 i.e. 1, –1, – 6
Hence equation of the line joining the given points is
Coordinates of any point on this line are (r + 2, – r – 3, – 6r + 1)
If this point lies on the given plane 2x + y + z = 7, then 2(r + 2) + (– r – 3) + (– 6r + 1) = 7
⇒ r = – 1
Coordinates of any point on this line are (– 1 + 2, – (– 1) – 3, – 6 (–1) + 1) i.e. (1, – 2, 7)
A boy is throwing stones at a target. The probability of hitting the target at any trial is 1/2. The probability of hitting the target 5th time at the 10th throw is :
The probability of hitting the target 5th time at the 10th throw = P(the probability of hitting the target 4 times in the first 9 throws) × P(the probability of hitting the target at the 10th throw) =
Two dice are thrown together 4 times. The probability that both dice will show same numbers twice is 
The probability of showing same number by both dice p = 6/36 = 1/6
In binomial distribution here n = 4, r = 2, p = 1/6 , q = 5/6
In a triangle ABC, if a = 2, B = 60° and C = 75°, then b equals
A = 180° – 60° – 75° = 180° – 135° = 45°
Prabh at wan ts to invest the total amount of ₹ 15,000 in saving certificates and national saving bonds. According to rules, he has to invest at least ₹ 2000 in saving certificates and ₹ 2500 in national saving bonds. The interest rate is 8% on saving certificate and 10% on national saving bonds per annum. He invest ₹ x in saving certificate and ₹ y in national saving bonds. Then the objective function for this problem is
The function is given by profit function
For the function
f '(1) = mf ' (0), where m is equal to
Given,
⇒ f ' (1) = 100 ...(ii)
Again, putting x = 0, we get
f ' (0) = 0 + 0 + ... + 0 + 1
⇒ f ' (0) = 1 ...(iii)
From eqs. (ii) and (iii), we get; f ' (1) = 100f ' (0) Hence, m = 100
As A^{2} = 0, A^{k} = 0 ∀ k ≥ 2.
Thus, (A + I)^{50} = I + 50A ⇒ (A + I)^{50} – 50A = I
∴ a = 1, b = 0, c = 0, d = 1
abc + abd + bcd + acd = 0
If the line x cos α + y sin α = p represents the common chord of the circles x^{2} + y^{2} = a^{2} and x^{2} + y^{2} + b^{2} (a > b), where A and B lie on the first circle and P and Q lie on the second circle, then AP is equal to
The given circles are concentric with centre at (0, 0) and the length of the perpendicular from (0, 0) on the given line is p. Let OL = p
Let a_{1}, a_{2}, a_{3}............ be terms on A.P. If
2 videos15 docs70 tests

Use Code STAYHOME200 and get INR 200 additional OFF

Use Coupon Code 
2 videos15 docs70 tests









