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What is the minimum energy required to launch a satellite of mass m from the surface of a planet of mass M and radius R in a circular orbit at an altitude of 2R?
As we know,
Gravitational potential energy
and orbital velocity
Therefore minimum required energy,
A mercury drop of radius 1 cm is sprayed into 10^{6} drops of equal size. The energy expressed in joule is (surface tension of Mercury is 460 × 10^{–3} N/m)
W = TΔA = 4pR^{2}T(n^{1/3} – 1)
= 4 × 3.14 × (10^{2})^{2} × 460 × 10^{–3} [(10^{6})^{1/3} –1]
= 0.057
Two planoconcave lenses (1 and 2) of glass of refractive index 1.5 have radii of curvature 25 cm and 20 cm. They are placed in contact with their curved surface towards each other and the space between them is filled with liquid of refractive index 4/3. Then the combination is
Now
A charged particle moves through a magnetic field perpendicular to its direction. Then
When a charged particle enters a magnetic field at a direction perpendicular to the direction of motion, the path of the motion is circular. In circular motion the direction of velocity changes at every point (the magnitude remains constant).
Therefore, the tangential momentum will change at every point. But kinetic energy will remain constant as it is given by 1/2mv^{2 }and v^{2} is the square of the magnitude of velocity which does not change.
After two hours, onesixteenth of the star ting amount of a certain radioactive isotope remained undecayed. The half life of the isotope is
Half life
A coil of inductance 300 mH and resistance 2Ω is connected to a source of voltage 2 V. The current reaches half of its steady state value in
The charging of inductance given by,
Taking log on both the sides,
Two concentric conducting thin spherical shells A, and B having radii r_{A} and r_{B} ((r_{B} > r_{A}) are charged to Q_{A} and –Q_{B} (Q_{B} > Q_{A}). The electric field along a line passing through the centre is
For, r < r_{A}, E = 0
These values are correctly represent in option (a).
A capillary tube of radius R is immersed in water and water rises in it to a height H. Mass of water in the capillary tube is M. If the radius of the tube is doubled, mass of water that will rise in the capillary tube will now be :
and
A sonometer wire resonates with a given tuning fork forming standing waves with five antinodes between the two bridges when a mass of 9 kg is suspended from the wire. When this mass is replaced by a mass M, the wire resonates with the same tuning fork forming three antinodes for the same positions of the bridges. The value of M is
From above equations, we get M = 25 kg.
When a metal surface is illuminated by light of wavelengths 400 nm and 250 nm, the maximum velocities of the photoelectrons ejected are v and 2v respectively. The work function of the metal is (h  Planck's constant, c = velocity of light in air)
We have,
E = W_{0} + K
On simplifying above equations, we get
W_{0 } = 2 hc x 10^{6} J
Two conductin g shells of radius a an d b are connected by conducting wire as shown in figure.
The capacity of system is :
V = 0, and so
When _{92}U^{235} under goes fission , 0.1% of its original mass is changed into energy. How much energy is released if 1 kg of _{92}U^{235} undergoes fission
Mass of uranium changed into energy
The energy released = mC^{2}
= 10^{–3} x (3 x 10^{8})^{2}
= 9 x 10^{13 }J.
One mole of an ideal gas is taken from state A to state B by three different processes,
(i) ACB (ii) ADB (iii) AEB as shown in the PV diagram. The heat absorbed by the gas is
The change in internal energy DU is same in all process.
Here W_{ACB} is positive and W_{AEB} is negative.
Hence Q_{ACB} > Q_{ADB} > Q_{AEB}.
In the formula X = 3 YZ^{2}, X and Z have dimensions of capacitance and magnetic induction respectively. The dimensions of Y in MKSA system are :
Two very long, straight, parallel wires carry steady currents I and I respectively. The distance between the wires is d. At a certain instant of time, a point charge q is at a point equidistant from the two wires, in the plane of the wires. Its instantaneous velocity v is perpendicular to this plane. The magnitude of the force due to the magnetic field acting on the charge at this instant is
Net magnetic field due to the wires will be downward as shown below in the figure.
Since angle between is 180°,
Two projectiles A and B thrown with speeds in the ratio 1: √2 acquired the same heights. If A is thrown at an angle of 45° with the horizontal, the angle of projection of B will be
For projectile A
For projectile B
Maximum height,
As we know, H_{A} = H_{B}
A meter bridge is set up as shown, to determine an unknown resistance ‘X’ using a standard 10 ohm resistor. The galvanometer shows null point when tappingkey is at 52 cm mark. The endcorrections are 1 cm and 2 cm respectively for the ends A and B. The determined value of ‘X’ is
At Null point
Here ℓ_{1} = 52 + End correction
= 52 + 1 = 53 cm
ℓ_{2} = 48 + End correction = 48 + 2 = 50 cm
A dis k of r ad ius a/4 having a uniformly distributed charge 6 C is placed in the x  y plane with its centre at (–a / 2, 0, 0). A rod of length a carrying a uniformly distributed charge 8C is placed on the xaxis from x = a /4 to x = 5a / 4.
Two point charges – 7 C and 3 C are placed at (a/4, – a/4, 0) and (– 3a/4, 3a/4, 0), respectively. Consider a cubical surface formed by six surfaces x = ± a/2, y = ± a/2, z = ± a/2. The electric flux through this cubical surface is
Total flux through the cubical surface,
A particle of mass m moving in the x direction with speed 2v is hit by another particle of mass 2m moving in the y direction with speed v. If the collision is perfectly inelastic, the percentage loss in the energy during the collision is close to
Initial momentum of the system
Final momentum of the system = 3mV
By the law of conservation of momentum
2√2mv = 3mV
Loss in energy
A coil is suspended in a uniform magnetic field, with the plane of the coil parallel to the magnetic lines of force. When a current is passed through the coil it starts oscillating; It is very difficult to stop. But if an aluminium plate is placed near to the coil, it stops. This is due to :
Because of the Lenz's law of conservation
of energy.
A steel wire of length ‘L’ at 40°C is suspended from the ceiling and then a mass ‘m’ is hung from its free end. The wire is cooled down from 40°C to 30°C to regain its original length ‘L’. The coefficient of linear thermal expansion of the steel is 10^{–5}/°C, Young’s modulus of steel is 10^{11} N/ m^{2} and radius of the wire is 1 mm. Assume that L >> diameter of the wire. Then the value of ‘m’ in kg is nearly
We know that
From (1) and (2)
On a hypotenuse of a right prism (30° – 60° – 90°) of refractive index 1.50, a drop of liquid is placed as shown in figure. Light is allowed to fall normally on the short face of the prism. In order that the ray of light may get totally reflected, the maximum value of refractive index is :
C_{max} = 60°
A tuning fork of frequency 392 Hz, resonates with 50 cm length of a string under tension (T). If length of the string is decreased by 2%, keeping the tension constant, the number of beats heard when the string and the tuning fork made to vibrate simultaneously is :
The frequency of tuning fork, f = 392 Hz.
Also
After decreasing the length by 2%, we have
From above equations,
f' = 400 Hz.
∴ Beats frequency=8 Hz.
Hydrogen (H), deuterium (D), sin gly ionized helium (He^{+}) and doubly ionized lithium (Li^{++}) all have one electron around the nucleus. Consider n = 2 to n = 1 transition. The wavelengths of emitted radiations are λ_{1}, λ_{2}, λ_{3} and λ_{4} respectively.Then approximately :
Z_{1} = 1, Z_{2} = 1, Z_{3} = 2 and Z_{4} = 3.
or λZ^{2} =constant
So
λ_{1}(1)^{2} = λ_{2} (1)^{2} = λ_{3}(2)^{2} = λa(3^{2} )
or λ_{1} = λ_{2} = 4λ_{3} = 9λ_{4.}
The following figure depict a circular motion. The radius of the circle, the period of revolution, the initial position and the sense of revolution are indicated on the figure.
The simple harmonic motion of the xprojection of the radius vector of the rotating particle P can be shown as :
There are two sources kept at distances 2λ. A large screen is perpendicular to line joining the sources. Number of maximas on the screen in this case is (λ = wavelength of light)
Δx_{max} = 0 and Δx_{max} = 2λ
Theortical maximas are = 2n + 1 = 2 × 2 + 1 = 5
But on the screen there will be three maximas.
In the circuit shown in figure the current through
The net resistance of the circuit is 9Ω as shown in the following figures.
The flow of current in the circuit is as follows.
The current divides into two equal parts if passes through two equal resistances in parallel.
Thus current through 4Ω resistor is 0.25 A.
A telescope has an objective lens of 10 cm diameter and is situated at a distance of one kilometer from two objects. The minimum distance between these two objects, which can be resolved by the telescope, when the mean wavelength of light is 5000 Å, is of the order of
x is of the order of 5 mm.
During vapourisation
I. change of state from liquid to vapour state occurs.
II. temperature remains constant.
III. both liquid and vapour states coexist in equilibrium.
IV. specific heat of substance increases.
Correct statements are
The change of state from liquid to vapour (for gas) is called vapourisation. It is observed that when liquid is heated, the temperature remains constant untill the entire amount of the liquid is converted into vapour.
The temperature at which the liquid and the vapour states of the substance coexists is called its boiling point.
A wire is connected to a battery between the point M and N as shown in the figure (1). The same wire is bent in the form of a square and then connected to the battery between the points M and N as shown in the figure (2). Which of the following quantities increases ?
When the wire is bent in the form of a square and connected between M and N as shown in fig. (2), the effective resistance between M and N decreases to one fourth of the value in fig. (1). The current increases four times the initial value according to the relation V = IR. Since H = I^{2}Rt, the decrease in the value of resistance is more than compensated by the increases in the value of current. Hence heat produced increases.
Percentage loss in energy during the collision ; 56%
A body moves in a circular orbit of radius R under the action of a central force. Potential due to the central force is given by V(r) = kr (k is a positive constant). Period of revolution of the body is proportional to :
U = mV = kmr.
Two equal heavy spheres, each of radius r, are in equilibrium within a smooth cup of radius 3r. The ratio of reaction between the cup and one sphere and that between the two sphere is
sin θ = 1/2
Thus, N_{1} sin θ =N_{2}
A long, hollow conducting cylinder is kept coaxially inside another long, hollow conducting cylinder of larger radius. Both the cylinders are initially electrically neutral
When charge is given to inner cylinder, an electric field will be in between the cylinders.
So there is potential difference between the cylinders.
A thin but rigid semicircular wire frame of radius r is hinged at O and can rotate in its own vertical plane. A smooth peg P starts from O and moves horizontally with constant speed v_{0}, lifting the frame upward as shown in figure.
Find the angular velocity w of the frame when its diameter makes an angle of 60° with the vertical :
⇒ x =2r sin θ
Given that A + B = R and A = B = R. What should be the angle between A and B ?
R^{2} = [A^{2} + B^{2} + 2AB cos θ]
R^{2} = R^{2} + R^{2} + 2R^{2} cos θ
 R^{2} = 2 R^{2} cos θ or cos θ = 1 / 2
or θ = 2 π/3
The basic magnetization curve for a ferromagnetic material is shown in figure. Then, the value of relative permeability is highest for the point
According to the given graph, slope of the graph is highest at point Q.
Five gas molecules chosen at random are found to have speeds of 500, 600, 700, 800 and 900 m/s:
and
Thus v_{rms} is greater than average speed by 14 m/s.
What is equivalent capacitance of circuit between points A and B?
The effective circuit is shown in figure.
The capacitance of upper series,
Now
A cyclic process ABCD is shown in the figure PV diagram. Which of the following curves represent the same process
Proce ss AB is is ob as i c and BC is isothermal, CD isochoric and DA isothermic compression.
In the circuit given below, V(t) is the sinusoidal voltage source, voltage drop V_{AB}(t) across the resistance R is
During the operation, either of D_{1} and D_{2} be in forward bias. Also R_{1} and R_{2} are different, so output across R will have different peaks.
Which of the following can be repeatedly soften on heating?
(i) Polystyrene
(ii) Melamine
(iii) Polyesters
(iv) Polyethylene
(v) Neoprene
Polystyrene and polyethylene belong to the category of thermoplastic polymers which are capable of repeatedly softening on heating and harden on cooling.
Which one of the following complexes is an outer orbital complex ?
Hybridisation :
Hence [Ni(NH_{3})_{6}]^{2+} is outer or bital complex.
For the reaction H_{2}(g) + Br_{2} (g) → 2HBr (g), the experimental data suggest, rate = k[H_{2}][Br_{2}]^{1/2}.
The molecularity and order of the reaction are respectively
The order of reaction is 3/2 and molecularity
is 2.
CaSO_{4}
More is more is the tendency to get itself reduced or more is oxidising power.
Which of the following relation represents correct relation between standard electrode potential and equilibrium constant?
Choose the correct statement(s)
ΔG = –2.303 RT log K
nFE° = –2.303 RT log K
Which of the following shows nitrogen with its increasing order of oxidation number?
Therefore increasing order of oxidation state of N is:
NH_{4}^{+} < N_{2}O < NO < NO_{2} < NO_{3}^{–}
Raoult’s law becomes a special case of Henry’s law when
Raoult’s law becomes special case of Henry’s law when KH become equal to p_{1}^{°}.
E° for the cell, Zn  Zn^{2+} (aq)   Cu^{2+} (aq)  Cu is 1.10 V at 25°C. The equilibrium constant for the cell reaction
is of the order of
∴ K_{c} = 1.9 x 10^{37}
Which of the following represents Gay Lussac's law ?
I. P/T = constant
II. P_{1}T_{2} = P_{2}T_{1}
III. P_{1}V_{1} = P_{2}V_{2}
P/T = constant (Gay Lussac's law)
PV = constant
P_{1}V_{1} = P_{2}V_{2 }[Boyle's law]
For the reaction
CO(g) + 1/2O_{2} (g) → CO_{2} (g)
Which one of the statement is correct at constant T and P ?
The energy of an electron in second Bohr orbit of hydrogen atom is :
For second orbit, n = 2
Z = At. no. = 1 (for hydrogen)
The right sequence of I.E_{1} of Li < B < Be < C.
Which of the following is not in volved in the formation of photochemical smog?
Photochemical smog does not involve SO_{2}.
Which of the following is not present in Portland cement?
There are four chief miner als present in a Portland cement tricalcium silicate (Ca_{3}SiO_{5}), dicalcium silicate (Ca_{2}SiO_{4}), tricalcium aluminate (Ca_{3}Al_{2}O_{6}) and calcium aluminoferrite (Ca_{4}Al_{n}Fe_{2n}O_{7}).
Which of the following can form buffer solution?
Ammonia is a weak base and a salt containing its conjugate acid, the ammonium cation, such as NH_{4}OH functions as a buffer solution when they are present together in a solution.
Which of the following complex shows sp^{3}d^{2} hybridization?
Among these ligands, ‘F’ is a weak field ligand, makes only high spin complexes which has sp^{3}d^{2} hybridization.
Glycosidic linkage is a type of covalent bond that joins either two carbohydrate (sugar) molecule or one carbohydrate to another group. All molecules show such type of linkages.
Which of the following represents Schotten Baumann reaction?
SchottenBaumann Conditions
The use of added base to drive the equilibrium in the formation of amides from amines and acid chlorides.
In the following structures, which two forms are staggered conformations of ethane ?
Note that in structures 1 and 2, every two adjacent hydrogen atoms are at maximum possible distance from each other (staggered conformation).
Which of the following shows correct order of bond length?
Bond length decreases with an increase in bond order. Therefore, the order of bond length in these species is O_{2}+ < O_{2}^{–} > O_{2} < O_{2}^{2} (bond order  O_{2}^{+} = 2.5, O_{2} =2, O_{2}^{–} =1.5, O_{2}^{2–} =1)
The number of radial nodes of 3s and 2p orbitals are respectively
For a given orbital with principal quan tum number (n) and azimuthal quantum number (l)
number of radial nodes = (n – l – 1) for 3s orbital: n = 3 and l = 0 therefore, number of radial nodes= 3 – 0 – 1 = 2
for 2p orbital: n = 2 and l = 1
therefore, number of radial nodes = 2 – 1 – 1 = 0
If a 25.0 mL sample of sulfuric acid is titrated with 50.0 mL of 0.025 M sodium hydroxide to a phenolphthalein endpoint, what is the molarity of the acid?
M_{1}V_{1} = M_{2}V_{2}
(0.025 M) (0.050 L) = (M_{2}) (0.025 L)
M_{2} = 0.05 M
but, there are 2 H’s per H_{2}SO_{4} so [H_{2}SO_{4}] = 0.025 M
Find which of the following compound can have mass ratios of C:H:O as 6:1:24
Given, mass ratio is C:H:O (6:1:24) so, molar ratio will be 6/12:1/1:24/16 = 1:2:3 therefore, HO(C=O)OH has molar ratio 1:2:3
The number of atoms per unit cell of bcc structure is
In bcc structure,
no. of atoms at corner = 1/8 × 8 = 1
no. of atom at body centre = 1
therefore, total no of atom per unit cell = 2.
PH_{5} does not exist because dor bital of ‘P’ interacts with sorbital of H. Bond formed is not stable and not energetically favorable. It depends on size and orientation of interaction.
Ion ic bonding is non directional, wher eas covalent bonding is directional. So, CO2 is directional.
For a given reaction, ΔH = 35.5 kJ mol^{1} and ΔS = 83.6 JK^{1} mol^{1}. The reaction is spontaneous at : (Assume that ΔH and ΔS do not vary with tempearature)
Given ΔH 35.5 kJ mol^{–1} ΔS = 83.6 JK^{–1} mol^{–1}
∴ ΔG = ΔH – TΔS
For a reaction to be spontaneous, ΔG = –ve
i.e., ΔH < TΔS
So, the given reaction will be spontaneous at T > 425 K
Specific conductance of 0.1 M HA is 3.75 x 10^{–4} ohm^{–1} cm^{–1}. If λ^{∞} (HA) = 250 ohm^{–1} cm^{2} mol^{–1}, the dissociation constant Ka of HA is :
The rate of reaction between two reactants A and B decreases by a factor of 4 if the concentration of reactant B is doubled. The order of this reaction with respect to reactant B is:
A compound of molecular formula of C_{7}H_{16} shows optical isomerism, compound will be
A compound is said to exhibit optical isomerism if it atleast contains one chiral carbon atom, which is an atom bonded to 4 different atoms or groups.
Which of the following does not contain Plane of symmetry?
Meso compounds ar e ch aracterized by an internal plane of symmetry that renders them achiral.
Control rods slowdown the motion of neutrons and help in controlling the rate of fission. Cadmium is efficient for this purpose.
Which reagent converts nitrobenzene to Nphenyl hydroxyamine?
Reducing reagent is needed, as shown in given reaction.
Which of the following can act as both Bronsted acid and Bronsted base?
Identify the structure of water in the gaseous phase.
Electrometallurgical process is used to extract
Because Na is very reactive and cannot be extracted by means of the reduction by C, CO etc. So it is extracted by electrolysis.
The correct statement about the compounds A, B, and C
Rotation of B through 180º within the plane of the paper gives D which is an enantiomer of A, hence A and B are enantiomers
Correct formula of the complex formed in the brown ring test for nitrates is
Which one of the following is an amine hormone ?
Thyroxine is an amine hormone.
choose the one which best expresses the meaning of the given word.
Loquacious
The word Loquacious (Adjective) means : talking a lot; talkative. Option (a) is the right synonym while others have different meanings.
Choose the word opposite in meaning to the given word.
Meticulous
The word Meticulous (Adjective) means : paying careful attention to every detail; fastidious; thorough.
Careless in option (c) is the correct antonomy.
Direction: Read the passage carefully and choose the best answer to each question out of the four alternatives.
PASSAGE
To write well you have to be able to write clearly and logically, and you cannot do this unless you can think clearly and logically. If you cannot do this yet you should train yourself to do it by taking particular problems and following them through, point by point, to a solution, without leaving anything out and without avoiding any difficulties that you meet.
At first you find clear, stepbystep thought very difficult. You may find that your mind is not able to concentrate. Several unconnected ideas may occur together. But practice will improve your ability to concentrate on a single idea and think about it clearly and logically. In order to increase your vocabulary and to improve your style, you should read widely and use a good dictionary to help you find the exact meanings and correct usages of words.
Always remember that regular and frequent practice is necessary if you want to learn to write well.
It is no good waiting until you have an inspiration before you write. Even with the most famous writers, inspiration is rare. Someone said that writing is ninetynine percent hard work and one percent inspiration, so the sooner you get into the habit of disciplining yourself to write, the better.
To write well, a person must train himself in
Direction: Read the passage carefully and choose the best answer to each question out of the four alternatives.
PASSAGE
To write well you have to be able to write clearly and logically, and you cannot do this unless you can think clearly and logically. If you cannot do this yet you should train yourself to do it by taking particular problems and following them through, point by point, to a solution, without leaving anything out and without avoiding any difficulties that you meet.
At first you find clear, stepbystep thought very difficult. You may find that your mind is not able to concentrate. Several unconnected ideas may occur together. But practice will improve your ability to concentrate on a single idea and think about it clearly and logically. In order to increase your vocabulary and to improve your style, you should read widely and use a good dictionary to help you find the exact meanings and correct usages of words.
Always remember that regular and frequent practice is necessary if you want to learn to write well.
It is no good waiting until you have an inspiration before you write. Even with the most famous writers, inspiration is rare. Someone said that writing is ninetynine percent hard work and one percent inspiration, so the sooner you get into the habit of disciplining yourself to write, the better.
Initially it is difficult to write because
Direction: Read the passage carefully and choose the best answer to each question out of the four alternatives.
PASSAGE
To write well you have to be able to write clearly and logically, and you cannot do this unless you can think clearly and logically. If you cannot do this yet you should train yourself to do it by taking particular problems and following them through, point by point, to a solution, without leaving anything out and without avoiding any difficulties that you meet.
At first you find clear, stepbystep thought very difficult. You may find that your mind is not able to concentrate. Several unconnected ideas may occur together. But practice will improve your ability to concentrate on a single idea and think about it clearly and logically. In order to increase your vocabulary and to improve your style, you should read widely and use a good dictionary to help you find the exact meanings and correct usages of words.
Always remember that regular and frequent practice is necessary if you want to learn to write well.
It is no good waiting until you have an inspiration before you write. Even with the most famous writers, inspiration is rare. Someone said that writing is ninetynine percent hard work and one percent inspiration, so the sooner you get into the habit of disciplining yourself to write, the better.
According to the passage, writing style can be improved by
Direction: Read the passage carefully and choose the best answer to each question out of the four alternatives.
PASSAGE
To write well you have to be able to write clearly and logically, and you cannot do this unless you can think clearly and logically. If you cannot do this yet you should train yourself to do it by taking particular problems and following them through, point by point, to a solution, without leaving anything out and without avoiding any difficulties that you meet.
At first you find clear, stepbystep thought very difficult. You may find that your mind is not able to concentrate. Several unconnected ideas may occur together. But practice will improve your ability to concentrate on a single idea and think about it clearly and logically. In order to increase your vocabulary and to improve your style, you should read widely and use a good dictionary to help you find the exact meanings and correct usages of words.
Always remember that regular and frequent practice is necessary if you want to learn to write well.
It is no good waiting until you have an inspiration before you write. Even with the most famous writers, inspiration is rare. Someone said that writing is ninetynine percent hard work and one percent inspiration, so the sooner you get into the habit of disciplining yourself to write, the better.
Famous writers have achieved success by
DIRECTIONS: In question below, 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.
China is a big country, in area it is bigger than any other country _________ Russia.
China is a big country. In area it is bigger than any other country except Russia. [except means other than, accept means consent, expect means to anticipate and access means entrance].
DIRECTIONS: In question below, 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.
The treasure was hidden ______ a big shore.
The treasure was hidden off the shore.
When something is hidden "off the shore," it just means that it's hidden somewhere near it.
DIRECTIONS: In question, some parts of the sentences have errors and some are correct.
Find out which part of a sentence has an error. If a sentence is free from error, mark (d) in your Answer.
My father gave me (a) / a pair of binocular (b) / on my birthday. (c) / No error. (d)
Delete 'pair of' before binocular because the word 'binocular' itself suggests a pair.
DIRECTIONS: In question, some parts of the sentences have errors and some are correct.
Find out which part of a sentence has an error. If a sentence is free from error, mark (d) in your Answer.
The teacher as well as his students, (a) / all left (b) / for the trip. (c) / No error. (d)
Delete 'all' before 'left'. Here the usage of 'all' is superfluous as 'the teacher as well as his students' itself signifies everyone.
Which answer figure complete the form in question figure ?
DIRECTIONS: In the following question which answer figure will complete the question figure?
Option (d) will complete the question figure.
Which answer figure includes all the components given in the question figure ?
Which of the answer figures include the separate components found in the question figure?
All the components of question figure are present in Answer Figure (c)
Which of the answer figures include the separate components found in the question figure?
Question Figure
Answer Figures
In each row, the second figure is obtained from the first figure by adding two mutually perpendicular line segments at the centre and the third figure is obtained from the first figure by adding four circles outside the main figure.
Which of the answer figures include the separate components found in the question figure?
Question Figure
Answer Figures
In each row, the second figure is obtained by rotating the first figure through 90° CW or 90° ACW and adding a circle to it. Also, the third figure is obtained by adding two circles to the first figure (without rotating the figure).
M is the son of P. Q is the grand daughter of O who is the husband of P. How is M related to O?
O is the husband of P. M is the son of P.
Therefore , M is the son of O.
Vinod introduces Vishal as the son of the only brother of his father's wife. How is Vinod related to Vishal?
Wife of Vinod’s father means the mother of Vinod.
Only brother of Vinod’s mother means maternal uncle of Vinod.
Therefore, Vinod is cousin of Vishal.
The pattern is as follows:
The pattern is as follows :
Therefore, the first term should be
In the following question, one statement is given followed by two assumptions I and II. You have to consider the statement to be true even if it seems to be at variance from commonly known facts. You have to decide which of the given assumptions, if any, follow from the given statement.
Statements : Politicians become rich by the votes of the people.
Assumptions :
I. People vote to make politicians rich.
II. Politicians become rich by their virtue.
The statemen t implies that politicians win elections by the votes of people. Therefore, neither of the assumptions is implicit in the statement.
Two statements are given followed by four conclusions, I, II, III and IV. You have to consider the statements to be true, even if they seem to be at variance from commonly known facts. You have to decide which of the given conclusions can definitely be drawn from the given statements. Indicate your answer.
Statements:
(A) No cow is a chair
(B) All chairs are tables.
Conclusions:
I. Some tables are chairs.
II. Some tables are cows
III. Some chairs are cows
IV. No table is a cow
DIRECTIONS: In question one/two statements are given, followed by two conclusions I and II. You have to consider the statements to be true, even if they seem to be at variance from commonly known facts. You have to decide which of the given conclusions, if any follow from the given statement.
Statements :
1. Temple is a place of worship.
2. Church is also a place of worship.
Conclusions :
I. Hindus and Christians use the same place for worship.
II. All churches are temples.
Temple and Church are places of worship. It does not imply that Hindus and Christians use the same place for worship. Church is different temple. Therefore, neither Conclusion I nor II follows.
DIRECTIONS: In question one/two statements are given, followed by two conclusions I and II. You have to consider the statements to be true, even if they seem to be at variance from commonly known facts. You have to decide which of the given conclusions, if any follow from the given statement.
Statement : The human organism grows and develops through stimulation and action.
Conclusions:
I. Inert human organism cannot grow and develop.
II. Human organisms do not react to stimulation and action.
Growth and development of human organism is a continuous process. Some changes take place in human body now and then.
Therefore, neither Conclusion I nor II follows.
Choose the set of figure which follows the given rule.
Rule: Closed figures gradually become open and open figures gradually become closed.
Let f and g be functions from R to R defined as
Then
We have
g (–3) = 0
⇒ f (g(–3)) = f (0) = 7 (0)2 +0 – 8 = – 8
∴ fog (–3) = –8 g (9) = 92 + 4 = 85
⇒ f (g(9)) = f (85) = 8(85) + 3= 683
∴ fog (9) = 683 f (0) = 7.02 + 0 – 8 = – 8
⇒ g (f (0)) = g (–8) =  –8  = 8
∴ gof (0) = 8 f (6) = 4(6) +5 = 29
⇒ g (f (6)) = g (29) = (29)2 + 4 = 845
∴ gof (6) = 845
How many different nine digit numbers can be formed from the number 223355888 by rearranging its digits so that the odd digits occupy even positions ?
X  X  X  X  X. The four digits 3, 3, 5,5 can be arranged at () places in
The five digits 2, 2, 8, 8, 8 can be arranged at (X) places in
Total no. of ar rangements = 6 x 10 = 60 ways
if where p, q, t and s are constants, then the value of s is equal to
The length of the semilatus rectum of an ellipse is one thrid of its major axis, its eccentricity would be
Let eq. of ellipse be
length of semilatus rectum
If α and β are roots of the equation such that then p belongs to the set :
Given quadratic eqn. is
Now, given
Given the system of straight lines a(2x + y – 3) + b(3x + 2y – 5) = 0, the line of the system situated farthest from the point (4, –3) has the equation
The given system of lines passes through the point of intersection of the straight lines 2x + y – 3 = 0 and 3x + 2y – 5 = 0 [L_{1} + λL_{2} = 0 form], which is (1, 1).
The required line will also pass through this point. Further, the line will be farthest from point (4, –3) if it is in direction perpendicular to line joining (1, 1) and (4, –3).
∴ The equation of the required line is
One mapping is selected at random from all mappings of the set S = {1, 2, 3, ......n} into itself. The probability that it is oneone is 3/32. Then the value of n is
The integer just greater than (3 + √5)^{2n} is divisible by (n ∈ N)
Let [R] + 1 = I ( ∵ [ . ] greatest integer function)
⇒ R + G = I ( ∵ 0 < G < 1)
seeing the option put n = 1
I = 28 is divisible by 4 i.e 2^{n+1}
For f (x) to be defined, we must have
Form (2) and (3), we get the domain of
The marks obtained by 60 students in a certain test are given below:
Median of the above data is
We construct the followin g table taking assumed mean a = 55 (step deviation method).
Here n = 60 ⇒ n/2 = 30 therefore, 60–70 is
the median class
Using the formula :
If A, B, C are the angles of a triangle and e^{iA} , e^{iB} , e^{iC} ar e in A.P. Then the trian gle must be
Squaring and adding we get
cos( A  C) = 1 ⇒ A  C = 0
∴ A = C, From (i) and (ii) cos B = cos A and sin B = sin A
So, A = B.
⇒ A = B = C
An observer on the top of a tree, finds the angle of depression of a car moving towards the tree to be 30°. After 3 minutes this angle becomes 60°.After how much more time, the car will reach the tree?
It will travel distance h cot 60° in
After striking the floor a certain ball rebounds 4/5th of its height from which it has fallen. Thetotal distance that the ball travels before coming to rest if it is gently released from a height of 120m is
Clearly, the total distance described
Except in the first fall the same ball will travel twice in each step the same distance one upward and second downward travel.
∴ Distance travelled
An equilateral triangle is inscribed in the circle x^{2} + y^{2} = a^{2} with one of the vertices at (a, 0). What is the equation of the side opposite to this vertex?
Since the equilateral triangle is inscribed in the circle with centre at the origin, centroid lies on the origin.
So, other vertices of triangle have coordinates,
∴ Equation of line BC is :
The function f(x) = x –  x – x2 , –1≤ x ≤ 1 is continuous on the interval
we have,
∴ Continuity is to be checked at x = 0 and x = 1. At x = 0
and f(0) = 0
Since LHL = RHL = f(0),
∴ f(x) is continuous at x = 0.
At x = 1
∴ f(x) is contin uous at x = 1
Hence f(x) is con tinuous for all x ∈ [1, 1]
which is true.
Let for n = m ≥ 2, P(m) is true.
Hence, for n ≥ 2, P(n) is true.
If a system of equation
– ax + y + z = 0
x – by + z = 0 x + y – cz = 0 (a, b, c≠ –1)
has a nonzero solution then
for nonzero
solution
⇒ abc – a – b – c – 2 = 0
⇒ abc = a + b + c + 2
If f (x) = x^{x}, then f (x) is increasing in interval :
A function f(x) is said to be increasing function in
Differentiate equation (i)
f '(x) = x^{x} (1 + log x)
Put f '(x) = 0
0 = x^{x} (1 + log x)
∴ f (x) is increasing in interval
⇒ x^{2}y – (5y + 1)x + 9y = 0 for real x, Discriminant = b^{2} – 4ac ≥ 0
(5y + 1)^{2} – 36y^{2} ≥ 0
⇒ (5y + 1 – 6y) (5y + 1 + 6y) ≥ 0
⇒ (– y + 1) (11y + 1) ≥ 0
The value of
a_{i} > 0, i = 1, 2, ...... n, is
The value of cot^{–1} 7 + cot^{–1} 8 + cot^{–1} 18 is
We have cot^{1} 7 + cot^{1} 8+ cot^{1} 18
A random variable X has the probability distribution
For the events E = {X is a prime number} and F = {X < 4} then P(E ∪ F) is
P(E) = P(2 or 3 or 5 or 7)
= 0.23 + 0.12 + 0.20 + 0.07 = 0.62
P( F ) =P(1 or2 or3)
= 0.15 + 0.23 + 0.12 = 0.50
P(E Ç F) = P(2 or 3)
= 0.23 + 0.12 = 0.35
∴ P(EUF ) = P(E) + P(F )  P(E∩ F )
= 0.62 + 0.50  0.35 = 0.77
The number of roots of equation cos x + cos 2x + cos 3x = 0 is (0 ≤ x ≤ 2 π)
We have cos x + cos 2 x + cos 3x = 0
or (cos 3x + cos x ) + cos 2 x = 0
or 2 cos 2x. cos x + cos 2x = 0
or cos 2x (2 cos x + 1) = 0
We have either cos 2x = 0 or 2 cos x + 1 = 0
Hence the required general solution are
The area under the curve y = cos x – sin x, and above xaxis is :
y =  cos x – sin x 
Required area
we have
Since Lf'(0) = Rf' (0), therefore f(x) is differentiable at x = 0
Since differentiability
⇒ continutity, therefore f(x) is continuous at x = 0.
The maximum value of z = 3x + 2y subject to x + 2y ≥ 2, x + 2y ≤ 8, x, y ≥ 0 is :
and x, y ≥ 0
For equation (1)
and for equation (2)
Given : z = 3x + 2y
At point (2, 0); z = 3 x 2 + 0 = 6
At point (0, 1); z = 3 x 0 + 2 x 1 = 2
At point (8, 0); z = 3 x 8 + 2 x 0 = 24
At point (0, 4); z = 3 x 0 +2 x 4 = 8
∴maximum value of z is 24 at point (8, 0).
A cylindircal gas container is closed at the top and open at the bottom. if the iron plate of the top is 5/4 time as thick as the plate forming thecylindrical sides. The ratio of the radius to the height of the cylinder using minimum material for the same capacity is
V = πr^{2}h = constant. If k be the thickness of the sides then that of the top will be (5/4)k.
∴ S = (2πrh)k + (πr^{2}). (5/4)k
(‘S’ is vol. of material used)
When r^{3} = 4V/5π or 5πr^{3} = 4πr^{2}h.
∴ r/h = 5/4
Let A, B, C be finite sets. Suppose that n (A) = 10, n (B) = 15, n (C) = 20, n(A ∩ B) = 8 and n(B ∩ C) = 9. Then the possible value of n (A ∪ B ∪ C) is
We have
n (A ∪ B ∪ C) = n (A) + n (B) + n (C) – n (A ∩ B) – n(B∩C) – n (C ∩ A) + n (A∩B ∩ C)
= 10 +15 + 20 – 8 – 9 – n (C ∩ A) + n (A ∩ B ∩C)
= 28 – {n(C ∩ A) – n (A ∩ B ∩ C)} ...(i)
Since n (C ∩ A) ≥ n (A ∩ B ∩ C)
We have
n (C ∩ A) – n (A ∩ B ∩ C) ≥ 0 ...(ii)
From (i) and (ii)
n (A ∪ B ∪ C) ≤ 28 ...(iii)
Now, n(A ∪ B) = n (A) +n (B) – n (A Ç B) = 10 + 15 – 8 = 17
and n (B ∪ C) = n (B) + n (C) – n (B Ç C) = 15 + 20 – 9 = 26
Since, n (A ∪ B ∪ C) ≥ n (A∪C) and n (A∪B∪C) ≥ n (B∪C),
we have n (A∪B∪C) ≥ 17 and n (A∪B∪C) ≥ 26
Hence n (A∪B∪C) ≥ 26 ...(iv)
From (iii) and (iv)
we obtain 26 ≤ n (A∪B∪C)≤ 28
Also n (A∪B∪C) is a positive integer
∴ n(A∪B∪C) = 26 or 27 or 28
If where z = 1 + 2i , then f(z) is equal to :
Put log x = t in f (x)
Now, put t = tan θ , we get
= cos–1 [cos 2 θ] =2θ = 2 tan^{–1} t = 2
tan^{–1} (log x)
Diff. both side w.r.t 'x', we get
Now,
Statement 1: A five digit number divisible by 3 is to be formed using the digits 0, 1, 2, 3, 4 and 5 with repetition. The total number formed are 216.
Statement 2 : If sum of digits of any number is divisible by 3 then the number must be divisible by 3.
Number form by using 1, 2, 3, 4, 5 = 5! = 120 Number formed by using 0, 1, 2, 4, 5
Total number formed, divisible by 3 (taking numbers without repetition) = 216
Statement 1 is false and statement 2 is true.
The equation of one of the common tangents to the parabola y^{2} = 8x and x^{2} + y^{2}  12x + 4 = 0 is
Any tangent to parabola y2 = 8x is
It touches the circle x^{2} + y^{2}  12x + 4 = 0 , if the length of perpendicular from the centre (6, 0) is equal to radius
⇒ (3m^{2}^{ }+ 1)^{2} = 8(m^{4} + m^{2})
⇒ m^{4} – 2m^{2} + 1 = 0
⇒ m = ±1
Hence, the required tangents are y = x + 2 and y = –x – 2.
Let be noncoplanar unit vectors equally inclined to one another at an acute angle θ . Then in terms of θ is equal to
are unit vectors and equally inclined to each other at an acute angle θ.
∴ ABC is an equilateral triangle and
∴ Area of Δ ABC
If G is the centroid of the D ABC, then
2^{1/4}. 2^{2/8}. 2^{3/16}. 2^{4/32}......∞ is equal to
The given product
Apply R_{1} → R_{1} – R_{3} and R_{2} → R_{2} – R_{3}, we get
[Expansion along first row]
⇒ xyr + xzq  xzy + yzp  zyx = 0
An urn contains five balls. Two balls are drawn and found to be white. The probability that all the balls are white is
Let A_{i} ( i = 2, 3, 4, 5) be the event that urn contains 2, 3, 4, 5 white balls and let B be the event that two white balls have been drawn then we have to find P (A_{5}/B).
Since the four events A_{2}, A_{3}, A_{4} and A_{5} are equally likely we have P (A_{2}) = P (A_{3}) = P(A_{4}) = P(A_{5}) = 1/4
P(B/A_{2}) is probability of event that the urn contains 2 white balls and both have been drawn.
By Baye’s theorem,
The ratio in which the join of ( 2, 1, 5) and (3, 4, 3) is divided by the plane (x + y – z) = 1/2 is:
As given plane divides the linejoining the points A (2, 1, 5) and B (3, 4, 3) at a point C in the ratio k : 1.
Then coordinates of C
Point C lies on the plane,
⇒ Coordinates of C must satisfy the equation of plane.
then
Adding (i) and (ii), we get
The dot product of a vector with the vectors are 0, 5 and 8 respectively. The vector is
Let the required vector be
Subtracting (ii) from (i),we have
– 2y – z = –5 Þ 2y + z = 5 ..(iv)
Multiply (ii) by 2 and subtracting (iii) from it, we obtain
5y – 8z = 2 ...(v)
Multiply (iv) by 8 and adding (v) to it, we have 21y = 42
⇒ y = 2 ...(v)
Substituting y = 2 in (iv),
we get 2 × 2 + z = 5
⇒ z = 5 – 4 = 1
Substituting these values in (i),we get x + 2 – 3 = 0
⇒ x = 3 – 2 = 1
Hence, the required vector is
The angle between the lines whose intercepts on the axes are a, –b and b, –a respectively , is
Equation of lines are and
If the line through the points A (k, 1, –1) and B (2k, 0, 2) is perpendicular to the line through the points B and C (2 + 2k , k, 1), then what is the value of k?
Given points are A (k, 1, –1), B (2k, 0, 2) and C (2 + 2k, k, 1)
Let r_{1} = length of line
and r_{2} = length of line BC
Now, let ℓ_{1} , m_{1},n_{1} be directioncosines of line AB and ℓ_{2}, m_{2}, n_{2} be the direction cosines of BC.
Since AB is perpendicular to BC
⇒ 2k k  3=0
⇒ k= 3 For k = 3,
AB is perpendicular to BC.
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