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Which one of the following graphs represents the variation of electric field with distance r from the centre of a charged spherical conductor of radius R?
The charged sphere is a conductor.
Therefore the field inside is zero and outside it is proportional to 1/r^{2}.
are the electric and magnetic field vectors of e.m. waves then the direction of propagation of e.m. wave is along the direction of
The direction of propagation of electromagnetic wave is perpendicular to the variation of electric field as well as to the magnetic field
The young's modulus of a wire of length L and radius r is Y N/m^{2}. If the length and radius are reduced to L/2 and r/2, then its young's modulus will be
Young's modulus of wire does not vary with dimention of wire. It is a constant quantity.
Twelve resistors each of resistance 16 Ω are connected in the circuit as shown. The net resistance between A and B is
Redraw the given circuit,
where, R = 16 Ω
R_{net} = 4 Ω
The time period of a satellite of earth is 5 hours.If the separation between the earth and the satellite is increased to 4 times the previous value, the new time period will become
Accor ding to Kepler ’s law of plan etary motion, T^{2} ∝ R^{3 }
Two trains are moving towards each other with speeds of 20 m/s and 15 m/s relative to the ground. The first train sounds a whistle of frequency 600 Hz. The frequency of the whistle heard by a passenger in the second train before the train meets, is (the speed of sound in air is 340 m/s)
You are asked to design a shaving mirror assuming that a person keeps it 10 cm from his face and views the magnified image of the face at the closest comfortable distance of 25 cm. The radius of curvature of the mirror would then be :
Concave morror is used as a shaving mirror.
Therefore radius of curvature,
R = 2f = – 60 cm
A block is kept on a frictionless inclined surface with angle of inclination ‘α’. The incline is given an acceleration ‘a’ to keep the block stationary.
Then ‘a’ is equal to
From free body diagram, For block to remain stationary,
⇒ a = g tan α
With the increase in temperature, the angle of contact
On increasing the temperature, angle of contact decreases.
Forward biasing is that in which applied voltage
Forward bias opposes the potential barrier and if the applied voltage is more than knee voltage it cancels the potential barrier.
In multiplication or division the final result should return as many significant figures as there are in the original number with the least significant figures. (Rounding off to three significant digits)
The ratio of the specific heats C_{p}/C_{v} = γ in terms of degrees of freedom (n) is given by
Let ‘n’ be the degree of freedom
A stone is thrown with a velocity u making an angle θ with the horizontal. The horizontal distance covered by its fall to ground is maximum when the angle θ is equal to
Since range on horizontal plane is
so it is maximum when, sin 2θ = 1
θ = π/4
A ball of mass 150 g, moving with an acceleration 20 m/s^{2}, is hit by a force, which acts on it for 0.1 sec. The impulsive force is
Force = Mass × acceleration
Impulsive force = F.Δt = 3 × 0.1 = 0.3 N
A man drags a block through 10 m on rough surface (µ = 0.5). A force of √3 kN acting at 30° to the horizontal. The work done by applied force is
Given, d = 10 m
θ = 30°
μ = 0.5
W = F_{s}dcosθ
Where,
F_{s} = μF
F_{s} = 0.5 × √3 kN
F_{s} = 0.866 kN
F_{s} = 866 N
So, W = 866 × 10 × cos 30°
A force of acts on a body for 4 second, produces a displacement of The power used is
The Earth is assumed to be a sphere of radius R. A platform is arranged at a height R from the surface of the Earth. The escape velocity of a body from this platform is fv, where v is its escape velocity from the surface of the Earth. The value of f is
Kepler ’s second law regarding constancy of areal velocity of a planet is a consequence of the law of conservation of
dA/dt = L/2m = Constant
Water is flowing through a horizonal tube having crosssectional areas of its two ends being A and A' such that the ratio A/A' is 5. If the pressure difference of water between the two ends is 3 × 10^{5} N m^{–2}, the velocity of water with which it enters the tube will be (neglect gravity effects)
According to Bernoulli’s theorem
According to the condition,
From equation of continuity,
From equation (i)
A thermodynamic system is taken from state A to B along ACB and is brought back to A along BDA as shown in the PV diagram. The net work done during the complete cycle is given by the area
Work done = Area under curve ACBDA
A boat crosses a river from port A to port B, which are just on the opposite side. The speed of the water is V_{w} and that of boat is V_{B} relative to still water. Assume V_{w} = 2V_{w}. What is the time taken by the boat, if it has to cross the river directly on the AB line [D = width of the river]
From figure, V_{B} sin θ = V_{W}
Time taken to cross the river.
Two springs, of force constants k_{1} and k_{2} are connected to a mass m as shown. The frequency of oscillation of the mass is f. If both k_{1} and k_{2} are made four times their original values, the frequency of oscillation becomes
The two springs are in parallel.
∴ Effective spring constant, k = k_{1} + k_{2} Now, frequency of oscillation is given by
When both k_{1} and k_{2} are made four times their original values, the new frequency is given by
When a potential difference V is applied across a conductor at a temperature T, the drift velocity of electrons is proportional to
Drift velocity,
The amplitude of a damped oscillator becomes (1/3)^{rd} in 2 secon ds. If its amplitude after 6 seconds is 1/n times the original amplitude, the value of n is
Amplitude of a damped oscillator
A = A_{0}e^{–bt/2m}
Case 1 :
The angular speed of the electron in the n^{th} orbit of Bohr hydrogen atom is
Angular speed of electron in the nth orbit of Bohr Hatom is inversely proportional to n3
In the given figure, the charge on 3 µF capacitor is
C = equivalent capacitance
Charge on each capacitor in series circuit will be same.
∴ q = CV = (1 × 10^{6}) × 10 = 10μC
∴ Charge across 3µF capacitor will be 10µC.
Two bodies A and B are placed in an evacuated vessel maintained at a temperature of 27ºC. The temperature of A is 327ºC and that of B is 227ºC.
The ratio of heat loss from A and B is about
If a rigid body is rotating about an axis with a constant velocity, then
The fundamental frequency of an open organ pipe is 300 Hz. The first overtone of this pipe has same frequency as first overtone of a closed organ pipe. If speed of sound is 330 m/s, then the length of closed organ pipe is
For open pipe, n_{1} = v/2ℓ , where n_{1} is the fundamental frequency of open pipe. length of open pipe is,
Ist overtone of open pipe,
Ist overtone of closed pipe
where, ℓ’ = length of closed pipe
As freq. of 1st overtone of open pipe = freq. of 1st overtone of closed pipe
In Young's experiment, the distance between the slits is reduced to half and the distance between the slit and screen is doubled, then the fringe width
If a rolling body’s angular momentum changes by 20 Sl units in 3 seconds, by a constant torque.Then find the torque on the body
As we know, τ is change in angular momentum.
Charge Q is distributed to two different metallic spheres having radii x and 2x such that both spheres have equal surface charge density, then charge on large sphere is
Let q and q' be the charges on spheres of radii x and 2x respectively.
Given, q + q' = Q …(i)
Surface charge densities are
From eq. (i), q' = Q – q or, 4q = Q – q
or, Q = 5q …(ii)
In an LR circuit f = 50 Hz, L = 2 H, E = 5 volts, R = 1 Ω then energy stored in inductor is
L = 2 H, E = 5 volts, R = 1Ω
A straight wire of length 0.5 metre and carrying a current of 1.2 ampere is placed in uniform magnetic field of induction 2 tesla. The magnetic field is perpendicular to the length of the wire.The force on the wire is
F = Biℓ = 2 ×1.2 × 0.5 = 1.2 N
A man drives a car from station B towards station A at speed 60 km/h. A car leaves station A for station B every 10 min. The distance between A and B is 60 km. The car travels at the speed of 60 km/h. A man drives a car from B towards A at speed of 60 km/h. If he starts at the moment when first car leaves the station B, then how many cars would be meet on the route ?
Distance between two cars leaving from the station A is,
Man meets the first car after time,
He will meet the next car after time,
In the remaining half an hour, the number of cars he will meet again is,
∴ Total number of cars would be meet on route will be 7.
In rotatory motion, linear velocities of all the particles of the body are
From v = r ω, linear velocities (v) for particles at different distances (r) from the axis of rotation are different.
If x, v and a denote the displacement, the velocity and the acceleration of a particle executing simple harmonic motion of time period T, then, which of the following does not change with time?
For an SHM, the acceleration a = ω^{2}x where, ω is a constant = 2π/T
The period of oscilation T is a constant.
∴ aT/x is a constant.
A conducting wire frame is placed in a magnetic field which is directed into the paper. The magnetic field is increasing at a constant rate.
The directions of induced current in wires AB and CD are
As the inward magnetic field increases, its flux also increases into the page and so induced current in bigger loop will be anticlockwise. i.e., from D to C in bigger loop and then from B to A in smaller loop.
Find the acceleration of block A and B. Assume pulley is massless.
Since A moves twice the distance moved by B.
If acceleration of B is ‘a’, then acceleration of A is ‘2a’.
T' – (T + T) = 0 (since pulley is massless)
⇒ T' = 2T ....(i)
For 5 kg block
5g – T' = 5a
for 2 kg block
⇒ 5g – 2T = 5a ....(ii) [T’ = 2T]
T = 2 × (2a) = 4a ....(iii)
From equations (ii) and (iii),
5g – (2 × 4a) = 5a
5g – 8a = 5a
5g = 13a
a = 5g/13
The nuclei of which one of the following pairs of nuclei are isotones?
Isotones means equal number of neutrons
i.e., (A–Z) = 74 – 34 = 71 – 31 = 40
Plots showing the variation of the rate constant (k) with temperature (T) are given below. The plot that follows Arrhenius equation is
As per Arrhenius equation (k = Ae ^{ Ea / RT}) , the rate constant increases exponentially with temperature.
3.6 g of oxygen is adsorbed on 1.2 g of metal powder. What volume of oxygen adsorbed per gram of the adsorbent at 1 atm and 273 K?
Mass of O_{2} absorbed per gram of adsorbent = 3.6/1.2 = 3
No. of moles of O_{2} absorbed per gram of adsorbent = 3/32
Volume of O_{2} absorbed per gram of adsorbent
PV = nRT
In the purification of impure nickel by Mond's process, metal is purified by :
When chlorine water is added to an aqueous solution of sodium iodide in the presence of chloroform, a violet colouration is obtained. On adding more of chlorine water and vigorous shaking, the violet colour disappears. This shows the conversion of ...... into ......
3Cl_{2} + 2NaI → 2NaCl + I_{2}
I_{2} gives violet colouration in CHCl_{3}.
In the clathrates of xenon with water, the nature of bonding between xenon and water molecule is
Clathrate formation in volves dipole– induced dipole interaction.
The electronic configurations of Eu(Atomic No. 63), Gd(Atomic No. 64) and Tb (Atomic No. 65) are
Eu (63) = [Xe] 4 f^{7} 6s^{2}
Gd (64) = [Xe] 4 f^{7} 5d^{1} 6s^{2}
Tb (65) = [Xe] 4 f^{9} 6s^{2}
Which of the following carbonyls will have the strongest C – O bond ?
As positive charge on the central metal atom increases, the less readily the metal can donate electron density into the π* orbitals of CO ligand (donation of electron density into π* orbitals of CO result in weakening of C – O bond). Hence, the C – O bond would be strongest in [Mn(CO)6]^{+}.
How many chiral compounds are possible on monochlorination of 2 methyl butane ?
Four monochloro derivatives are chiral.
Which of the following are intermediates in the reaction of excess of CH_{3}MgBr with C_{6}H_{5}COOC_{2}H_{5} to make 2phenyl  2propanol?
In compound (b), the lone pair of nitrogen is not involved in resonance therefore it is the strongest base.
Which of the following does not reduce Benedict’s solution?
Sucrose, being a nonreducing sugar, does not reduce Benedict’s solution. Remember that fructose has an αhydroxy ketonic group, which is also reducing group (different from ordinary ketonic group)
General formula of solid in zinc blende structure is:
Zn^{+2} present in alternate tetrahedral void
S^{2–} present in ccp = 4
∴ Zn_{4}S_{4} = ZnS i.e., AB type compound.
Glycine in alkaline solution exists as ______ and migrates to __________.
Glycine in alkaline solution exists as anion and migrates to anode.
Due to internal proton tranfer of H^{+} from the –COOH group to the –NH_{2}, the amino acid exists as an ion with both a negative charge and a positive charge, called a Zwitter ion
Adding an alkali to glycine
Now, during electrophoresis, glycine moves towards anode.
Product on reaction of ethanamide with phosphorus pentoxide is:
K_{a} of HX is 10^{–5}, then find concentration of H_{3}O^{+} when equal volumes of 0.25M HX and 0.05 M NaOH are mixed.
Net cell reaction of Pt  H_{2} (640 mm)  HCl  H_{2} (510 mm)  Pt.
Pt  H_{2} (640 mm)  HCl  H_{2} (510 mm)  Pt
E°cell = 0
(p_{1})H_{2}(g) → 2H^{+}+2e^{–}
2H^{+} + 2e^{–} → H_{2}(g)(p_{2})
Which of the following has zero net dipole moment?
XeF_{4} has zero net dipole moment
Which of the following element has the highest ionisation enthalpy?
Boron has the highest ionisation enthalpy amongst the following.
Ionisation enthalpy decreases down the group and increases across the period.
Out of the elements with atomic number 7, 8, 9, 13 which has the smallest size and highest ionization enthalpy?
Elemen t with atomic number 7 has the smallest size and highest ionization enthalpy Nitrogen – Atomic Number 7
N has a stable halffilled electronic configuration therefore it is difficult to remove electron and hence it has a high ionization enthalpy.
Which one is classified as a condensation polymer?
Except dacron all are additive polymers.
Terephthalic acid condenses with ethylene glycol to give dacron.
Which of the following compounds is not an antacid?
Phenelzine is an antidepressant, while others are antacids.
Mole fraction of the solute in a 1.00 molal aqueous solution is
1 molal solution means 1 mole of solute dissolved in 1000 g solvent.
∴ n_{solute} = 1 w_{solvent} = 1000 g
Most stable carbocation among the following is:
Stability of carbocation ∝ no. of α–H present on carbocation.
Only Lindlar ’s catalyst converts alkyne to alkene (cis addition) and alkenes with Baeyer’s reagent give cis glycols.
The stability of +1 oxidation state among Al, Ga, In and Tl increases in the sequence :
Lower oxidation state become more stable on moving down the group
Al < Ga < ln < Tl
Which of the following alkaline earth metal hydroxides is amphoteric in character?
Be(OH)_{2} is amphoteric while Ca(OH)_{2}, Sr(OH)_{2} and Ba(OH)_{2} are all basic.
Which reaction shows oxidising nature of H_{2}O_{2}?
H_{2}O_{2} + 2KI → I_{2}, O.S. of I^{–} (–1) changes to I_{2} (Zero) There is increase in oxidation number, hence oxidation.
aK_{2}Cr_{2}O_{7} + bKCl + cH_{2}SO_{4} → xCrO_{2}Cl_{2} + yKHSO_{4} + zH_{2}O
The above equation balances when
The balanced equation is
K_{2}Cr_{2}O_{7 }+ 4KCl + 6H_{2}SO_{4} → 2CrO_{2}Cl_{2 }+ 6KHSO_{4} + 3H_{2}O
For the reactions
A ⇌ B ; K_{c} = 2
B ⇌ C ; K_{c} = 4
C ⇌ D ; K_{c} = 6
K_{c} for the reaction A ⇌ D is
Multiply the three equations,
Which of the following will always lead to a nonspontaneous change?
ΔG = ΔH – TΔS = +ve for spontaneous change, ΔH < 0, ΔS > 0
for nonspontaneous change, ΔH > 0, ΔS < 0
The densities of two gasses are in the ratio of 1: 16. The ratio of their rates of diffusion is
In the reaction the change in hybridisation is from
Isoelectronic species have same no. of electrons.
∴ O^{2–}, F^{–}, Na^{+} , Mg^{+2} are isoelectronic
100 mL O_{2} and H_{2} kept at same temperature and pressure. What is true about their number of molecules
This is Avogadro’s hypothesis. According to this, equal volume of all gases contain equal no. of molecules under similar condition of temperature and pressure.
If m_{A} gram of a metal A displaces m_{B} gram of another metal B from its salt solution and if the equivalent mass are E_{A} and E_{B} respectively then equivalent mass of A can be expressed as:
Eq. of A = Eq. of B
Which one of the following set of quantum numbers is not possible for 4p electron?
For 4p electron n = 4, l = 1, m = –1, 0, + 1 and s = +½ or –½
Which of the following radial distribution graphs correspond to l = 2 for the H atom?
l = 2 represent d orbital for which
B_{2} is paramagnetic due to the presence of unpaired electron in π^{2}p_{x} = π^{2}p_{y} orbital.
M.O diagram for B_{2} molecule
Direction: In the following questions below, out of the four alternatives, choose the one which best expresses the meaning of the given word.
Garrulous
The word Garrulous (Adjective) means : talkative; talking a lot.
Direction: In the following questions below, out of the four alternatives, choose the one which best expresses the meaning of the given word.
Tinsel
The word Tinsel (Noun/Adjective) means : strips of shiny material like metal used as decorations.
Direction: In the following questions below, out of the four alternatives, choose the one which best expresses the meaning of the given word.
Labyrinth
The word Labyrinth (Noun) means : a place that has many confusing paths or passage.
The correct synonym will be 'meandering' which means, 'to have a lot of curves on a path'.
Direction: In the following questions, choose the word opposite in meaning to the given word.
Knack :
Knack means a clever way of doing something.
Direction: In the following questions, choose the word opposite in meaning to the given word.
Pernicious :
Pernicious means highly injurious or destructive.
Direction: In the following questions, choose the word opposite in meaning to the given word.
Opulence :
Opulence means wealthy.
Direction: Read the passage carefully and choose the best answer to each question out of the four alternatives and mark it by blackening the appropriate circle [·].
Like watering a plant, we grow our friendships [and all our relationships) by running them. Friendships need the same attention as other relationships. If they are to continue. These relationships can be delightfully nonjudgemental, supportive, understanding and fun.
Sometimes a friendship can bring out the positive side that you never show in any other relationship. This may be because the pressure of playing a 'role' (daughter, partner or child) is removed.
With a friend you are to be yourself and free to change.
Of course, you are free to do this in all other relationships as well, but in friendships you get to have lats of rehearsals and discussion about changes as you experience them. It is an unconditional experience where you receive as much as you give.
You can explain yourself to a friend openly without the fear of hurting a family member. How do friendships grow ? The answer is simple. By revealing yourself; being attentive: remembering what is most showing empathy; seeing the world through the eyes of your friend, you will understand the value of friendship. All this means learning to accept a person from a completely different family to your own or perhaps someone from a completely different cultural background. This is the way we learn tolerance. In turn we gain tolerance and acceptance for our own differences.
Q. In good friendships, we
In good friendships, we receive as much as we give.
Direction: Read the passage carefully and choose the best answer to each question out of the four alternatives and mark it by blackening the appropriate circle [·].
Like watering a plant, we grow our friendships [and all our relationships) by running them. Friendships need the same attention as other relationships. If they are to continue. These relationships can be delightfully nonjudgemental, supportive, understanding and fun.
Sometimes a friendship can bring out the positive side that you never show in any other relationship. This may be because the pressure of playing a 'role' (daughter, partner or child) is removed.
With a friend you are to be yourself and free to change.
Of course, you are free to do this in all other relationships as well, but in friendships you get to have lats of rehearsals and discussion about changes as you experience them. It is an unconditional experience where you receive as much as you give.
You can explain yourself to a friend openly without the fear of hurting a family member. How do friendships grow ? The answer is simple. By revealing yourself; being attentive: remembering what is most showing empathy; seeing the world through the eyes of your friend, you will understand the value of friendship. All this means learning to accept a person from a completely different family to your own or perhaps someone from a completely different cultural background. This is the way we learn tolerance. In turn we gain tolerance and acceptance for our own differences.
Q. Empathy means
Empathy means the ability to show and understand the feelings of others.
Direction: Read the passage carefully and choose the best answer to each question out of the four alternatives and mark it by blackening the appropriate circle [·].
Like watering a plant, we grow our friendships [and all our relationships) by running them. Friendships need the same attention as other relationships. If they are to continue. These relationships can be delightfully nonjudgemental, supportive, understanding and fun.
Sometimes a friendship can bring out the positive side that you never show in any other relationship. This may be because the pressure of playing a 'role' (daughter, partner or child) is removed.
With a friend you are to be yourself and free to change.
Of course, you are free to do this in all other relationships as well, but in friendships you get to have lats of rehearsals and discussion about changes as you experience them. It is an unconditional experience where you receive as much as you give.
You can explain yourself to a friend openly without the fear of hurting a family member. How do friendships grow ? The answer is simple. By revealing yourself; being attentive: remembering what is most showing empathy; seeing the world through the eyes of your friend, you will understand the value of friendship. All this means learning to accept a person from a completely different family to your own or perhaps someone from a completely different cultural background. This is the way we learn tolerance. In turn we gain tolerance and acceptance for our own differences.
Q. Through strong friendships, we gain
A strong friendship helps us gain acceptance and tolerance.
Direction: Read the passage carefully and choose the best answer to each question out of the four alternatives and mark it by blackening the appropriate circle [·].
Like watering a plant, we grow our friendships [and all our relationships) by running them. Friendships need the same attention as other relationships. If they are to continue. These relationships can be delightfully nonjudgemental, supportive, understanding and fun.
Sometimes a friendship can bring out the positive side that you never show in any other relationship. This may be because the pressure of playing a 'role' (daughter, partner or child) is removed.
With a friend you are to be yourself and free to change.
Of course, you are free to do this in all other relationships as well, but in friendships you get to have lats of rehearsals and discussion about changes as you experience them. It is an unconditional experience where you receive as much as you give.
You can explain yourself to a friend openly without the fear of hurting a family member. How do friendships grow ? The answer is simple. By revealing yourself; being attentive: remembering what is most showing empathy; seeing the world through the eyes of your friend, you will understand the value of friendship. All this means learning to accept a person from a completely different family to your own or perhaps someone from a completely different cultural background. This is the way we learn tolerance. In turn we gain tolerance and acceptance for our own differences.
Q. Friendships and relationships grow when they are
The very first line of the passage states that friendships and relationships grow when they are nurtured just like nurturing a plant.
Direction: In the following questions, sentences are given with blanks to be filled with an appropriate word(s). Four alternatives are suggested for each question. Choose the correct alternative out of the four as your answer.
Q. There are not solitary, freeliving creatures ; every form of life is ______ other forms.
Dependent on = needing som ebod y / something in order to survive or be successful; affected or decided by something.
Direction: In the following questions, sentences are given with blanks to be filled with an appropriate word(s). Four alternatives are suggested for each question. Choose the correct alternative out of the four as your answer.
Q. I'll take ______ now as I have another's appointment some where else.
Take your leave = to say good bye.
Direction: In the following questions, some parts of the sentences have errors and some are correct. Find out which part of a sentence has an error. The number of that part is the answer. If a sentence is free from error, then your answer is (d). i.e., No error.
When one hears of the incident (a)/about the plane crash (b)/ he feels very sorry. (c)/ No error (d)
Here, indefinite article i.e., 'about a plane crash' should be used. No particular incident is evident here.
Direction: In the following questions, some parts of the sentences have errors and some are correct. Find out which part of a sentence has an error. The number of that part is the answer. If a sentence is free from error, then your answer is (d). i.e., No error.
I went there (a)/ with a view to survey (b)/ the entire procedure. (c)/ No error (d)
‘With a View to’ should be followed by gerund i.e., surveying.
Direction: In the following questions, some parts of the sentences have errors and some are correct. Find out which part of a sentence has an error. The number of that part is the answer. If a sentence is free from error, then your answer is (d). i.e., No error.
It had laid (a)/ in the closet (b)/ for a week before we found it. (c)/ No error (d)
Here, time period is given. Hence, Past Perfect Continuous i.e., 'It had been lying' ....should be used.
Direction: In the following questions, which answer figure will complete the question figure?
Question Figures :
Direction: In the following questions, which answer figure will complete the question figure?
Question Figure:
A piece of paper is folded and cut/punched as shown below in the question figures. From the given answer figures, indicate how it will A appear when opened.
Question figures:
Select there lated word from the given alternatives:
Medicine : Patient : : Education : ?
Medicine is given to patient. Similar ly, Education is given to student.
Choose the correct alternative from the given ones that will complete the series.
A3E, F5J, K7O, _____?
Which one of the following numbers lacks the common property in the series?
81, 36, 25, 9, 5, 16
Except 5, all numbers are perfect square numbers.
In a certain code language, "TIRED" is written as "56" and "BRAIN" is written as "44". How is 'LAZY" written in that code language?
As,
TIRED = 20 + 9 + 18 + 5 + 4 = 56
BRAIN = 2 + 18 + 1 + 9 + 14 = 44
Similarly,
LAZY = 12 + 1 + 26 + 25 = 64.
Select the missing number from the given response.
8 × 8 × 88 = 5632
7 × 7 × 77 = 3773
Similarly, 6 × 6 × ? = 3132
Which one of the following diagrams best depicts the relationship among Human Society  Youth Club, Political Party and Youths?
Among her children, Ganga's favourites are Ram and Rekha. Rekha is the mother of Sharat, who is loved most by his uncle Mithun. The head of the family is Ram Lal, who is succeeded by his sons Gopal and Mohan. Gopal and Ganga have been married for 35 years and have 3 children.What is the relation between Mithun and Mohan?
Mohan is son of Ram Lal and uncle of Ram and Rekha. Mithun is uncle of Sharat who is son of Rekha. Rekha is niece of Mohan.
Therefore, Mithun is brother of Rekha's husband.
If x cos α + y sin α = P is a tangent to the ellipse
Given line is x cos α + y sin α = P ....(1)
Any tangent to the ellipse is
Comparing (1) and (2)
Eliminate θ, cos^{2}θ + sin^{2}θ
or a^{2} cos2 α + b^{2} sin2 α = P^{2}
If a_{1} , a_{2} , a_{3}........,a_{n} are in A.P. where a_{i} > 0 for all i, then
As a_{1}, a_{2} , a_{3} , .......,a_{n} , are in A.P. we get, a_{2}  a_{1} = a_{3}  a_{2} = ............. = a_{n}  a_{n1} = d (say)
[formula for n^{th} term]
In order to solve the differential equation the integrating factor is:
Given differential equation is :
x cos x dy/dx + y (x sin x + cos x) = 1 Dividing both the sides by x cos x,
which is of the form
Integrating factor
= e^{(log sec x + log x)} = e^{log (sec x . x)} = x sec x
Equation of the first line L_{1} is (x1)/2 = (y2)/3 = (z3)/4 and that of the second line
Clearly, these lines are not parallel (the ratios of D.R. are not equal).
Any point P on the first line is (1 + 2λ, 2 + 3λ, 3 + 4λ) and any point Q on the second line is (4 + 5μ, 1 + 2μ, μ). If these two points P and Q are identical then.
1 + 2λ = 4 + 5μ ...(1)
2 + 3λ = 1 + 2μ ...(2)
3 + 4λ = μ ...(3)
From (2) and (3), we get λ = μ = –1, which also satisfies (1). Thus the two lines L_{1} and L_{2} ; entersect and the coordinates of the point of intersection are (– 1, – 1, – 1).
The equation of the curve passing through the point and satisfying the differential equation
⇒ ydx  xdy = ay^{2} dx + ady
⇒ y(1  ay)dx = (x + a)dy
Integrating, we get
log(x + a)  log y + log(1  ay) = log C
Since the curve passes through
So, (x + a )(1  ay) = 4a^{2}y
The locus of the midpoint of a chord of the circle x^{2} + y^{2} = 4 , which subtends a right angle at the origin is
Equation of given circle is x^{2} +y^{2 }= 4 Its centre , O = (0, 0) and radius, r = 2 Draw OM ⊥ AB
Clearly M is the midpoint of AB which subtends a right angle at O.
In ΔAOB, OA = OB radius
∴ ∠A = ∠B = π/4
and in ΔOMA, sin A = OM/OA
⇒ OM = √2 ...(1)
Let M = (x, y) then OM ....(2)
From (1) and (2), x^{2} + y^{2 }= 2
This is the required equation of locus.
Squaring both sides, we get
x^{2}(1+ y) = y^{2}(1 + x)
⇒ x^{2} – y^{2} + x^{2}y – xy^{2} = 0
⇒ (x – y) (x + y + xy) = 0
⇒ y = x or y(1 + x) = – x ⇒ y = x or
If f (x) = 3x^{4} + 4x^{3} – 12x^{2} + 12, then f (x) is
Given : f(x) = 3x^{4} + 4x^{3} – 12x^{2} + 12
Differentiating with respect to x, we get
f'(x) = 12x^{3}+ 12x^{2}– 24x
For f (x) to be increasing
f '(x) > 0 ⇒ 12x^{3} + 12x^{2} – 24x > 0
⇒ 12x (x^{2} + x – 2) > 0
⇒ 12x (x – 1) (x + 2) > 0
⇒ x (x – 1) (x + 2) > 0
⇒ – 2 < x < 0 or x > 1
It means x ∈ ( 2, 0) ∪ (1,∞).
Hence f (x) is increasing in (– 2, 0) and (1,∞)
Consider
Then number of possible solutions are :
x, y ≥ 0 convert them into equation and solve them and draw the graph of these equations we get y = 1 and x = 3/2
The distance of a point (2, 5, –3) from the plane
Here, and d = 4.
Therefore, the distance of the point (2, 5, –3) from the given plane is
For the following feasible region, the linear constraints are
The general solution of differential equation (e^{x} + 1) ydy = (y + 1)e^{x} dx is
Since, (e^{x} + 1) ydy = (y + 1)e^{x} dx
After integrating on both sides, we have
Hence y = log [k (1 + y) (1 + e^{x})]
What is the slope of the normal at the point (at^{2}, 2at) of the parabola y^{2} = 4ax?
Equation of parabola is y^{2} = 4ax
(On differentiating w.r.t ‘x’)
∴ dy/dx = 2a/y , [slope of tangent]
So, slope of normal
From the definite integral property
we have
By adding (i) and (ii)
[∵ sin 2x = 2 sin x cos x]
= 6i [3i^{2} + 3] + 3i [4i + 20] + 1 [12 – 60i]
= 6i [–3 + 3] + 12i^{2} + 60i + 12 – 60i
= –12 + 12 = 0 = x + iy
∴ x = 0
If 2 cos^{2} x + 3 sin x – 3 = 0, 0 ≤ x ≤ 180°, then x =
2 cos^{2} x + 3 sin x – 3 = 0
2 – 2 sin^{2} x + 3 sin x – 3 = 0
⇒ (2 sin x – 1) (sin x – 1) = 0
⇒ sin x = 1/2 or sin x = 1
If the number of available constraints is 3 and the number of parameters to be optimized is 4, then
On differentiating w.r.t. x, we get
The maximum area of rectangle inscribed in a circle of diameter R is
The diagonal = R
Thus the area of rectangle
If A and B are two events, such that P(A∪B) = 3/4, P(A∩B) = 1/4 , P(A^{c}) = 2/3
where A^{c} stands for the complementary event of A, then P(B) is given by:
From the given problem: P(A ∪ B) = 3/4 , P(A ∩ B) = 1/4
Limit doesn't exist, so f (x) is not continuous at 0.
Put x = cos 2θ
The equation of chord of the circle x^{2} + y^{2} = 8x bisected at the point (4, 3) is
T = S_{1} ⇒ x(4) + y(3) – 4 (x + 4) = 16 + 9 – 32
⇒ 3y – 9 = 0 ⇒ y = 3
x and y are positive number. Let g and a be G. M. and AM of these numbers. Also let G be G. M. of x + 1 and y + 1. If G and g are roots of equation x^{2} – 5x + 6 = 0, then
The roots of equation are 2 and 3
∴ x = y = 2
The coefficient of x^{n} in the expansion of
A pair of tangents are drawn from the origin to the circle x^{2} + y^{2}+ 20 (x + y) + 20 = 0, then the equation of the pair of tangent are
Equation of pair of tangents is given by SS_{1} = T^{2},
or S = x^{2} + y^{2} + 20 (x + y) + 20, S, = 20,
T = 10 (x + y) + 20 = 0
∴ SS_{1} = T_{2}
⇒ 20 (x^{2} + y^{2} + 20 (x + y) + 20)
= 10^{2} (x + y + 2)^{2}
⇒ 4x^{2} + 4y^{2} + 10xy = 0
⇒ 2x^{2} + 2y^{2} + 5xy = 0
If the sum of a certain number of terms of the A.P. 25, 22, 19, ........ is 116. then the last term is
a = 25, d = 22 – 25 = –3.
Let n be the no. of terms
or 232 = n[50 – 3n + 3] = n[53 3n]
= –3n^{2} + 53n
⇒ 3n^{2} – 53 + 232 = 0 ⇒ (n – 8) (3n – 29) = 0
∴ Now, T_{8} = a + (8 – 1)d = 25 + 7 × (–3)
= 25 – 21
∴ Last term = 4
If 1, a and P are in A. P. and 1, g and P are in G. P., then
2a = 1 + P and g^{2} = P
⇒ g^{2} = 2a 1 ⇒ 1 2a +g^{2} = 0
If y = sin x + e^{x}, then d^{2}x/dy^{2} is equal to
y = sin x = e^{x}
The foci of the hyperbola 4x^{2} – 9y^{2} – 1 = 0 are
4x^{2} – 9y^{2} = 1
From the top of a cliff 50 m high, the angles of depression of the top and bottom of a tower are observed to be 30° and 45°. The height of tower is
Let height of the tower be h m and distance between tower and cliff be x m.
∴ CD = h, BD = x
In ΔABD, tan 45° = AB/BD or 1 = 50/x
x = 50 ... (i)
In ΔAEC
or 50 = 50√3 h√3
[From (i), x = 50]
or h√3 = 50√3  50
The coefficient of x^{2} term in the binomial expansion of
General term of the given binomial series is given by:
Put r = 4, we get
Thus coefficient of x^{2} = 70/243.
The value of λ, for which the circle x^{2} + y^{2} + 2λx + 6y + 1 = 0 intersects the circle x^{2} + y^{2} + 4x + 2y = 0 orthogonally, is
Two circles x^{2} + y^{2} + 2g_{1}x + 2f_{1}y + c_{1} = 0 and x^{2} + y^{2} + 2g_{2}x + 2f_{2}y + c_{2} = 0 cuts orthogonally if 2g_{1}g_{2} + 2f_{1} f_{2} = c_{1} + c_{2}
Given equations of two circles are
x^{2} + y^{2} + 2λ_{x} + 6y + 1 = 0 .... (i)
x^{2} + y^{2} +4x + 2y = 0 .... (ii)
On comparing (i) and (ii) with original equation, we get g_{1} = λ, f_{1} = 3, c_{1} = 1 and g_{2} = 2,f_{2} = 1, c_{2} = 0
So, from orthogonality condition, we have 4λ + 6 = 1 ⇒ 4λ = 5
∴ λ = 5/4
We know, scalar triple product
(By definition of scalar triple product)
If f (x) = (a  x^{n})^{1/n} , where a > 0 and n ∈ N , then fof (x) is equal to :
Given that f(x) = (a  x^{n})1/n
∴ fof (x ) = [a  {(a  x^{n})^{1/n}}^{n}]^{1/n}
= [a (a x^{n})]^{1/n}
= [x^{n}]^{1/n} = x
Sum of n terms of the series 8 + 88 + 888 + .... equals
The modulus of the complex number z such that  z + 3 – i  = 1 and arg(z) = π is equal to
Let z = x + iy
∴  z + 3 – i  =  (x + 3) + i(y – 1)  = 1
From equations (i) and (ii), we get x = –3, y = 0 ∴ z = –3
⇒  z  =  –3  = 3
Bag P contains 6 red and 4 blue balls and bag Q contains 5 red and 6 blue balls. A ball is transferred from bag P to bag Q and then a ball is drawn from bag Q. What is the probability that the ball drawn is blue?
Let E_{1}, E_{2} and A be the even ts defined as follows:
E_{1} = red ball is transferred from bag P to bag Q
E_{2} = blue ball is transferred from bag P to bag Q
A = the ball drawn from bag Q is blue
As the bag P contains 6 red and 4 blue balls,
Note that E_{1} and E_{2} are mutually exclusive and exhaustive events.
When E_{1} has occurred i.e., a red ball has already been transferred from bag P to Q, then bag Q will contain 6 red and 6 blue balls, So, P(AE_{1}) = 6/12 = 1/2
When E_{2} has occurred i.e., a blue ball has already been transferred from bag P to Q, then bag Q will contain 5 red and 7 blue balls, So, P(AE_{2}) = 7/12
By using law of total probability, we get P(A) = P(E_{1}) P(AE_{1}) + P(E_{2}) P(AE_{2})
The number of 4digit numbers that can be formed with the digits 1, 2, 3, 4 and 5 in which at least 2 digits are identical, is
Total number of 4digit numbers = 5 × 5 × 5 × 5 = 625
(as each place can be filled by anyone of the numbers 1, 2, 3, 4 and 5)
Numbers in which no two digits are identical = 5 × 4 × 3 × 2 = 120 (i.e. repetition not allowed) (as 1st place can be filled in 5 different ways, 2nd place can be filled in 4 different ways and so on)
Number of 4digits numbers in which at least 2 digits are identical
= 625 – 120 = 505
Consider the system of linear equations;
x_{1} + 2x_{2} + x_{3} = 3
2x_{1} + 3x_{2} + x_{3} = 3
3x_{1} + 5x_{2} + 2x_{3} = 1
The system has
⇒ Given system, does not have any solution.
⇒ No solution
What is the value of y so that the line through (3, y) and (2, 7) is parallel to the line through (– 1, 4) and (0, 6)?
Let A(3, y), B(2, 7), C(–1, 4) and D(0, 6) be the given points.
Since AB and CD are parallel.
∴ m_{1} = m_{2} ⇒ y = 9.
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