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A passenger in a open car travelling at 30 m/s throws a ball out over the bonnet. Relative to the car the initial velocity of the ball is 20 m/s at 60° to the horizontal. The angle of projection of the ball with respect to the horizontal road will be
A particle is moving in a straight line with initial velocity and uniform acceleration a. If the sum of the distance travelled in tth and (t + 1)th seconds is 100 cm, then its velocity after t seconds, in cm/s, is
The two vectors are drawn from a common point and , then angle between is –
(1) 90° if C^{2} = A^{2} + B^{2}
(2) greater than 90° if C^{2} < A^{2} + B^{2}
(3) greater than 90° if C^{2} > A^{2} + B2
(4) less than 90° if C^{2} > A^{2} + B^{2}
Correct options are –
If then find the dimensions of q. Where T is the time period of bar of mass M, length L and Young modulus Y.
, writing dimensions of both the sides, we get
or q = [L^{4}]
An object experiences a net force and accelerates from rest to its final position in 16s. How long would the object take to reach the same final position from rest if the object's mass was four times larger ?
When the mass increases by a factor of 4 the acceleration must decrease by a factor of four if the same force is applied. The question asks about position so we need to relate acceleration and time to position. We can do this by the equation : x_{f }– x_{i} = v_{xi} t + ½ a_{x} t^{2} We want the change in position to stay the same. The initial velocity is zero so in order for the change in position to remain constant the term (1/2) at^{2} must remain the same. If the acceleration is reduced by a factor of 4 you can see that the time must be increased by a factor of 2 in order for the term to remain the same.
Three blocks of masses m_{1}, m_{2} and m_{3} are connected by massless strings, as shown, on a frictionless table. They are pulled with a force T_{3} = 40 N. If m_{1} = 10 kg, m_{2} = 6 kg and m_{3} = 4kg, the tension T_{2} will be
For equilibrium of all 3 masses,
For equilibrium of m_{1} & m_{2}
T_{2} = (m_{1} + m_{2}) × a or, 2
Given m_{1} = 10 kg, m_{2} = 6 kg, m_{3} = 4 kg, T_{3} = 40 N
A massless platform is kept on a light elastic spring as shown in fig. When a sand particle of mass 0.1 kg is dropped on the pan from a height of 0.24 m, the particle strikes the pan and spring is compressed by 0.01 m. From what height should the particle be dropped to cause a compression of 0.04 m.
A constant torque of 31.4 Nm is exerted on a pivoted wheel. If angular acceleration of wheel is 4 p rad/s^{2}, then the moment of inertia of the wheel is
A man of mass m starts falling towards a planet of mass M and radius R. As he reaches near to the surface, he realizes that he will pass through a small hole in the planet. As he enters the hole, he sees that the planet is really made of two pieces a spherical shell of negligible thickness of mass 2M/3 and a point mass M/3 at the centre.
Change in the force of gravity experienced by the man is
Change in force of gravity
(only due to mass M/3 due to shell gravitational field is zero (inside the shell))
= 2GMm/3R^{2}
Geostationary satellites are also called synchronous satellite. They always remain about the same path on equater, i.e., it has a period of exactly one day (86400 sec) So orbit radius comes out to be 42400 km, which is nearly equal to the circumference of earth. So height of Geostationary satellite from the earth surface is 42,400 – 6400 = 36,000 km.
Two wires are made of the same material and have the same volume. However wire 1 has crosssectional area A and wire 2 has crosssectional area 3A. If the length of wire 1 increases by Dx on applying force F, how much force is needed to stretch wire 2 by the same amount?
As shown in the figure, the wires will have the same Young’s modulus (same material) and the length of the wire of area of crosssection 3A will be l/3 (same volume as wire 1).
⇒ F' = 9F
An iron rod of length 2m and crosssectional area of 50 mm2 stretched by 0.5 mm, when a mass of 250 kg is hung from its lower end. Young’s modulus of iron rod is
Viscosity is the property of a liquid due to which it :
The radiation emitted by a perfectly black body is proportional to
A copper sphere cools from 62°C to 50°C in 10 minutes and to 42°C in the next 10 minutes.Calculate the temperature of the surroundings.
By Newton's law of cooling,
A sphere cools from 62°C to 50°C in 10 min.
Now, sphere cools from 50°C to 42°C in next 10 min.
Dividing eqn. (2) by (3) we get,
Hence θ_{0} = 26°C
An air bubble of volume v0 is released by a fish at a depth h in a lake. The bubble rises to the surface. Assume constant temperature and standard atmospheric pressure above the lake.
The volume of the bubble just before touching the surface will be (density) of water is ρ
As the bubble rises the pr essur e gets reduced for constant temperature, if P is the standard atmospheric pressure, then (P + ρgh ) V_{0 }= PV
or,
The molecules of a given mass of gas have a root mean square velocity of 200m s^{–1} at 27°C and 1.0 × 105 N m^{–2} pressure. When the temperature is 127°C and the pressure 0.5 × 105 Nm^{–2}, the root mean square velocity in ms^{–1}, is
Which of the following expressions corresponds to simple harmonic motion along a straight line, where x is the displacement and a, b, c are positive constants?
In linear S.H.M., the restoring force acting on particle should always be proportional to the displacement of the particle and directed towards the equilibrium position. i.e., F ∝ x
or F = –bx where b is a positive constant.
A mass m is suspended from a spring of force constant k and just touches another identical spring fixed to the floor as shown in the figure. The time period of small oscillations is
When the spring undergoes displacement in the downward direction it completes one half oscillation while it completes another half oscillation in the upward direction. The total time period is:
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 = V2l, where n_{0} is the fundamental frequency of open pipe.
As freq. of 1st overtone of open pipe = freq. of 1st overtone of closed pipe.
= 41.25 cm
In an uniformly charged sphere of total charge Q and radius R, the electric field E is plotted as function of distance from the centre. The graph which would correspond to the above will be
Electric field inside the uniformly charged sphere varies linearly, while outside the sphere, it varies as inverse square of distance,which is correctly represented in option (c).
A charge Q_{1} exerts some force on a second charge Q_{2}. If a 3rd charge Q_{3 }is brought near, then the force of Q_{1} exerted on Q_{2} –
A hollow metal sphere of radius 5 cm is charged such that the potential on its surface is 10 V. The potential at a distance of 2 cm from the centre of the sphere is
Potential at any point inside the sphere = potential at the surface of the sphere = 10V.
If the potential of a capacitor having capacity 6 mF is increased from 10 V to 20 V, then increase in its energy will be
Capacitan ce of capacitor (C) = 6 m F = 6 × 10^{–6} F; Initial potential (V_{1}) = 10 V and final potential (V_{2}) = 20 V.
The increase in energy (ΔU)
= 1/2 x (6x10^{6}) x [(20)^{2}  (10)^{2}]
= (3x10^{6}) x 300 = 9 x 10^{4} J.
Calculate the effective resistance between A and B in following network.
Equivalent resistance = (5 + 10 + 15)   (10 + 20 + 30)
So,
A steady current is set up in a cubic network composed of wires of equal resistance and length d as shown in figure. What is the magnetic field at the centre P due to the cubic network
By symmetry, the magnetic field at the centre P is zero.
If M is magnetic moment and B is the magnetic field, then the torque is given by
Torque,
A metal rod of length 1 m is rotated about one of its ends in a plane right angles to a field of inductance 2.5 × 10–3 Wb/m². If it makes 1800 revolutions/min. Calculate induced e.m.f. between its ends.
Given: l = 1m, B = 5 × 10^{–3} Wb/m²
f = 1800/60 = 30 rotations/sec In one rotation, the moving rod of the metal traces a circle of radius r = l
∴ Area swept in one rotation = πr^{2}
= Bfπr^{2} = (5 × 10^{–3}) × 3.14 × 30 × 1 = 0.471 V
Which one of the following curves represents the variation of impedance (Z) with frequency f in series LCR circuit?
An electromagnetic wave passes through space and its equation is given by E = E_{0} sin (ωt – kx) where E is electric field. Energy density of electromagnetic wave in space is
Energy density
A thin convergent glass lens (μg = 1.5) has a power of + 5.0 D. When this lens is immersed in a liquid of refractive index m, it acts as a divergent lens of focal length 100 cm. The value of m must be
A vessel of depth 2d cm. is half filled with a liquid of refractive index µ1 and the upper half with a liquid of refractive index µ_{2}. The apparent depth of the vessel seen perpendicularly is –
If the distance between the first maxima and fifth minima of a double slit pattern is 7mm and the slits are separated by 0.15 mm with the screen 50 cm. from the slits, then the wavelength of the light used is :
There are three and a half fringes from first maxima to fifth minima as shown.
If the energy of a photon is 10 eV, then its momentum is
Momentum of a photon
= 5.33 × 10^{–27} kg ms^{–1}
The energies of energy levels A, B and C for a given atom are in the sequence E_{A} < E_{B} < E_{C}. If the radiations of wavelengths λ_{1}, λ_{2} and λ_{3} are emitted due to the atomic transitions C to B, B to A and C to A respectively then which of the following relations is correct ?
The output of an OR gate is connected to both the inputs of a NAND gate. The combination will serve as a:
= NOR gate When both inputs of NAND gate are connected, it behaves as NOT gate OR + NOT = NOR.
In a semiconductor diode, the barrier potential offers opposition to
An electron, in a hydrogenlike atom, is in an excited state. It has a total energy of –3.4 eV. The kinetic energy and the deBroglie wavelength of the electron are respectively
E_{n} = – 3.4 eV
The kinetic energy is eq
∴ K.E. = + 3.4 eV
The de Broglie wavelength of electron
= 0.66 × 10^{–9} m
Light of wavelength 180 nm ejects photoelectron from a plate of a metal whose work function is 2 eV. If a uniform magnetic field of 5 × 10^{–5} T is applied parallel to plate, what would be the radius of the path followed by electrons ejected normally from the plate with maximum energy ?
If v_{max }is the speed of the fastest electron emitted from the metal surface, then
∴ v = 1.31 × 10^{6 }m/s
The radius of the electron is given by
The product of atomic weight and specific heat of any element is a constant, approximately 6.4.This is known as :
According to Dulong and Pettit’s law Atomic weight × Specific heat = 6.4 (approx) This law is applicable only to solid elements but it fails to explain very high specific heat of diamond.
1.520 g of hydroxide of a metal on ignition gave 0.995g of oxide. The equivalent weight of metal is:
Let E be the equivalent weight of the metal
[17 is equivalent weight of OH and 8 is equivalent weight of oxygen]
⇒ 0.995 E + 17 × 0.995 = E × 1.52 + 8 × 1.52
⇒ 0.525 E = 16. 915 – 12.16 = 4.755
Effective nuclear char ge (i,e. Z/e ratio) decreases from F^{–} to N^{3–} hence the radii follows the order: F^{–} < O^{2–} < N^{3–}. Z/e for F^{–} = 9/10 = 0.9, for O^{2}– = 8/10 = .8, for N^{3 –} = 0.7
Beryllium and aluminium exhibit many properties which are similar. But, the two elements differ in
The valency of beryllium is +2 while that of aluminium is +3
Among Al_{2}O_{3}, SiO_{2}, P_{2}O_{3} and SO_{2} the correct order of acid strength is:
A σ bonded molecule MX_{3} is Tshaped. The number of non bonded pair of electrons is
For Tshape geometry the molecule must have 3 bonded pair and 2 lone pair of electrons.
The correct bond order in the following species is:
ion^{  }Total number of electrons (16 – 1) = 15.
E.C. :
(Super oxide ion): Total number of electrons (16 + 1) = 17 .
E.C. :
ion: Total number of electrons = (16 – 2) = 14
E.C:
What is the free energy change, DG , when 1.0 mole of water at 100º C and 1 atm pressure is converted into steam at 100°C and 1 atm. pressure?
Condition of equilibrium, hence ΔG = 0.
H_{2}S gas wh en passed thr ough a solution of cations containing HCl precipitates the cations of second group of qualitative analysis but not those belonging to the fourth group. It is because
IV^{th} group needs higher S^{2} ion concentration. In presence of HCl, the dissociation of H_{2}S decreases hence produces less amount of sulphide ions due to common ion effect, thus HCl decreases the solubility of H_{2}S which is sufficient to precipitate II^{nd} group radicals.
The pH of a solution is increased from 3 to 6; its H^{+} ion concentration will be
pH = 3. ∴ [H^{+}] = 10^{–3}; pH = 6,
∴ [H^{+}] = 10^{–6}. Hence [H^{+}] is reduced by 10^{–3} times
A gas X at 1 atm is bubbled through a solution containing a mixture of 1 M Y^{–} and 1 M Z^{–} at 25°C. If the reduction potential is Z > Y > X, then
The more the reduction potential, the more the oxidising power.
When a crystal of caustic soda is exposed to air, a liquid layer is deposited because :
It is hygroscopic and deliquescent and hence absorbs moisture and CO_{2} to form Na_{2}CO_{3}
2NaOH + CO_{2} → Na_{2}CO_{3} + H_{2}O
Which of the following compound is not chiral?
None of the carbon atoms in
DCH_{2}CH_{2}CH_{2}Cl is chiral i.e., each carbon atom is achiral (symmetric).
C_{6}H_{5}C ≡ N and C_{6}H_{5}N C exhibit which type of isomerism?
Nucleophilicity increases with the decrease in electronegativity of the central atom.
Since electronegativity follows the order: F > O > N > C; nucleophilicity of the concerned group will follow the reverse order i.e.,
CH_{3}^{} > NH_{2}^{} > OH^{} > F^{}
In the anion HCOO^{– }the two carbonoxygen bonds are found to be of equal length. What is the reason for it ?
What will be the product in the following reaction?
The fraction of total volume occupied by the atoms present in a simple cube is
Number of atoms per unit cell = 1 Atoms touch each other along edges.
Hence r = a/2
(r = radius of atom and a = edge length)
Therefore % fraction =
1.00 g of a nonelectrolyte solute (molar mass 250 g mol^{–1}) was dissolved in 51.2 g of benzene.If the freezing point depression constant, Kf of benzene is 5.12 K kg mol^{–1}, the freezing point of benzene will be lowered by
The number of coulombs required for the deposition of 108 g of silver is
Amt. deposited =
During the kinetic study of the reaction, 2A + B → C + D, following results were obtained:
Based on the above data which one of the following is correct?
In case of (II ) and (I II) Keeping concentration of [A] constant, when the concentration of [B] is doubled, the rate quadruples. Hence it is second order with respect to B. In case of I & IV Keeping the concentration of [B] constant, when the concentration of [A] is increased four times, rate also increases four times. Hence, the order with respect to A is one. Hence Rate = k [A] [B]^{2}
Position of nonpolar and polar part in micelle is
For adsorption of a gas on a solid, the plot of log x/m vs log P is linear with slope equal to (n being whole number)
According to Freundlich adsorption
isotherm, at intermediate pressure, extent of adsorption
plot of log x/m vs log P is linear with slope = 1/n
Calcination is used in metallurgy for removal of?
Calcination is used for removal of volatile impurities and decompose carbonates.
Red P does not react with NaOH to give PH_{3}.
Which of the following halides is not oxidized by MnO_{2}
F_{2} is strongest oxidising agent. F^{–} is not oxidised by MnO_{2}
Which of the following exhibit only + 3 oxidation state?
Ac (89) = [Rn] [6d^{1}] [7s^{2}]
Which of the following pairs has the same size?
Due to lanthanide contraction, the size of Zr and Hf (atom and ions) become nearly similar.
Which of the following is not considered as an organometallic compound?
The structural formula of cisplatin is
Since no carbon is involved it is not a organometallic compound.
A more basic ligand forms stable bond with metal ion, Cl^{–} is most basic amongst all.
A is an optically inactive alkyl chloride which on reaction with aqueous KOH gives B. B on heating with Cu at 300°C gives an alkene C, what are A and C
Williamson syn thesis is on e of the best methods for the preparation of symmetrical and unsymmetrical ethers. In this method, an alkyl halide is allowed to react with sodium alkoxide.
Which of the following esters cannot undergo Claisen self condensation ?
Claisen conden sation is given by esters having two ahydrogen atoms.
Schotten Baumann reaction is a reaction of phenols with
The reagent (s) which can be used to distinguish acetophenone from benzophenone is (are)
I_{2} and NaOH react with acetophenone (C_{6}H_{5}COCH_{3}) to give yellow ppt. of CHI_{3} but benzophenone (C_{6}H_{5}COC_{6}H_{5}) does not give haloform test.
When aniline is treated with nitrous acid in the presence of HCl, then benzene diazonium chloride is obtained.
Th e structural feature wh ich distinguish es proline from natural aamino acids?
Proline contains imino (secondary amino),group.
Which of the following cann ot give iodometric titration?
There is no r eaction between I^{–} and Fe^{3+.}
Acetaldehyde and acetone can be distinguished by :
Acetaldehyde is easily oxidised to acetic acid by a mild oxidising agent like Fehling solution. Acetone is not easily oxidised.
Both acetone and acetaldehyde give iodoform test. Other two conditions are not relevant to aldehydes and ketones.
DIRECTIONS: In the following questions, two sentences are given. There may be an error in the sentence(s). Mark as your correct answer.
I. Although he was innocent, baseless accusations were leveled at him.
II. Despite of repeated representations from the people, the authorities have failed to take any action.
Sentence I, accusations were leveled against him not at him. Sentence II, despite is not followed by of.
DIRECTIONS: In the following questions, two sentences are given. There may be an error in the sentence(s). Mark as your correct answer.
I. I deem it as a privilege to address the gathering.
II. Perfection can be achieved with practice.
Sentence I: I deem it a privilege not as a privilege.
Sentence II: …… achieved through practice not with practice.
DIRECTIONS: For each of the following questions, select the option which is CLOSEST in meaning to the capitalized word.
TURBULENCE
Commotion means an disorderly outburst or tumult. Its most close to tubulence which means unstable flow of a liquid or gas.
Turbulence also refers to a state of disturbance.
DIRECTIONS: For each of the following questions, select the option which is CLOSEST in meaning to the capitalized word.
DEFER
Other synonyms are prorogue, put off, set back, shelve.
DIRECTIONS: For each of the following questions, select the option which is CLOSEST in meaning to the capitalized word.
ADAGE
An adage is a proverb or byword.
DIRECTIONS: Choose the word, which is most OPPOSITE in meaning as the word given in bold.
FRAGRANCE
DIRECTIONS: Choose the word, which is most OPPOSITE in meaning as the word given in bold.
PECULIAR
DIRECTIONS: Choose the word, which is most OPPOSITE in meaning as the word given in bold.
ETERNAL
DIRECTIONS: Pick out the most effective word from the given words to fill in the blanks to make the sentence meaningfully complete in the contest of the sentence.
______ to popular belief that red meat makes human aggressive, scientist have found that it actually has a calming effect.
Stickingextending out above a surface or boundary contraryvery opposite in nature or character or purpose.
DIRECTIONS: Pick out the most effective word from the given words to fill in the blanks to make the sentence meaningfully complete in the contest of the sentence.
From its ______ opening sequence, it is clear that we are in the grip of a delicious new voice, a voice of breathtaking.
Evocativerecreate strong feelings, memory etc.
Mesmerizingattract strongly.
DIRECTIONS: In the following passages, the first and the last parts of the sentence are numbered 1 and 6. The rest of the sentence is split into four parts and named, P, Q, R and S. These four parts are not given in their proper order. Read the parts and find out which of the four combinations is correct.
Then find the correct answer.
1. making ourselves
P. our language
Q. part of growing into
R. Masters of
S. is an important
6. full manhood or womanhood
DIRECTIONS: In the following passages, the first and the last parts of the sentence are numbered 1 and 6. The rest of the sentence is split into four parts and named, P, Q, R and S. These four parts are not given in their proper order. Read the parts and find out which of the four combinations is correct.
Then find the correct answer.
1. The very first battle they fought
P. and they had to fall back
Q. cross the border
R. was lost
S. letting the enemy
6. an enter the country
DIRECTIONS: In the following passages, the first and the last parts of the sentence are numbered 1 and 6. The rest of the sentence is split into four parts and named, P, Q, R and S. These four parts are not given in their proper order. Read the parts and find out which of the four combinations is correct.
Then find the correct answer.
1. A nation
P. the material assets it possesses
Q. is not made by
R. and collective determination
S. but by the will
6. of the people
DIRECTIONS: In the following passages, the first and the last parts of the sentence are numbered 1 and 6. The rest of the sentence is split into four parts and named, P, Q, R and S. These four parts are not given in their proper order. Read the parts and find out which of the four combinations is correct.
Then find the correct answer.
1. When the Governor
P. the bell had rung
Q. justice should be immediately
R. he ordered that
S. found out why
6. done to the horse
DIRECTIONS: In the following passages, the first and the last parts of the sentence are numbered 1 and 6. The rest of the sentence is split into four parts and named, P, Q, R and S. These four parts are not given in their proper order. Read the parts and find out which of the four combinations is correct.
Then find the correct answer.
1. When you ponder over
P. that the only hope
Q. you will realize
R. of world peace lies
S. the question deeply
6. in the United Nations
DIRECTIONS: In the following question, a series is given with one term missing.
Choose the correct alternative from the given ones that will complete the series:
One of the, numbers does not fit into the series. Find the wrong number.
15, 20, 45, 145, 565, 2830
The number should be 140.
× 1 + 5, × 2 + 5, × 3 + 5……
DIRECTIONS: In the following question, a series is given with one term missing.
Choose the correct alternative from the given ones that will complete the series:
VWX, BCD, HIJ, ?
The pattern is:
In a code lan guage, if TARGET is coded as 201187520, then the word WILLIUM will be coded as
Sanjay is taller than Suresh but sh orter than Rakesh. Rakesh is taller than Harish but shorter than Binit. Who among is the tallest?
From the question we get, Rakesh > Sanjay > Suresh Binit > Rakesh > Harish So, Binit is the tallest among them.
In a row of 62 persons. Rahul is 36th from left side of the row and Nitesh is 29th form the right side of the row. Find out the number of persons sitting between them?
No. of Persons between Rahul and Nitesh = (36 + 29)  622 = 65  622 = 1
Select the combination of number so that the letters arranged will from a meaningful word.
Which of the given Venn diagrams out of (a), (b), (c) or (d) correctly represents the relationship among the following classes?
Rose, Flower, Lotus
A piece of paper is folded and a cut is made as shown below. From the given responses indicate how it will appear when opened?
Which answer figure will complete the question figure?
Question figure
If f(x) is a function that is odd and even simultaneously, then f(3)  f(2) is equal to
f (x) = 0 x ∈ R ⇒ f (3) – f (2) = 0
If, tan A = 1/2 and tan B = 1/3, then find the valueof A + B
∴ A + B = 45° = π/4
We shall first consider values of θ between 0 and 2π sin θ =
or sin (2π – π/6)
∴ θ = 7π/6 ; 11π/6
tan θ = 1 /3 = tan (π/6) or tan (π + π/6)
∴ θ = π/6, 7π/6
The value of θ which satisfies both the equations is 7π/6
Hence the general value of θ is 2nπ + 7π/6 where n ∈ I.
For n ∈ N, x^{n+1} + (x + 1)^{2n–1} is divisible by
For n = 1, we have;
x^{n+1} + (x + 1)^{2n–1} = x^{2} + (x + 1) = x^{2} + x + 1,
which is divisible by x^{2} + x + 1
For n = 2, we have; x^{n+1} + (x + 1)^{2n–1}
= x^{3} + (x + 1)^{3} = (2x + 1) (x2 + x + 1),
which is divisible by x^{2} + x + 1.
If a, b are the roots of the equation ax^{2} + bx + c = 0, then the roots of the equation ax^{2} + bx (x + 1) + c (x + 1)^{2 }= 0 are
Putting z = a + 2i in the given equation and comparing imaginary parts, we get a^{2} + 4 = a^{2}, which is not possible.
If a > 0, a ∈ R, z = a + 2i and z z – az + 1 = 0 then
Putting z = a + 2i in the given equation and comparing imaginary parts, we get a^{2} + 4 = a^{2}, which is not possible.
Which of the following is not a vertex of the positive region bounded by the inqualities 2x + 3y ≤ 6, 5x + 3y ≤ 15 and x, y ≥ 0
Here (0, 2), (0, 0) and (3, 0) all are vertices of feasible region.
If ^{20}C_{r} = ^{20}C_{r – 10} then ^{18}C_{r} is equal to
^{20}C_{r }= ^{20}C_{r–10} ⇒ r + (r – 10) = 20 ⇒ r = 15
∴ ^{18}C_{r} = ^{18}C_{15} = ^{18}C_{3} = 18.17.16/1.2.3 = 816
The term independent of x in the expansion of , is a times the corresponding binomial coefficient. Then a is
is independent of x provided r = 12 and then a = 1.
In the binomial (2^{1/3 }+ 3^{1/3})n, if the ratio of the seventh term from the beginning of the expansion to the seventh term from its end is 1/6, then n equal to
T_{r + 1} = ^{n}C_{r} a^{n  r} . b^{r} where a = 2^{1/3} and
b = 3^{1/3}
T_{7} from beginning = ^{n}C_{6} a^{n  6} b^{6} and
T_{7} from end = ^{n}C_{6} b^{n  6} a^{6}
⇒ n  12 =  3 ⇒ n = 9
If p^{th},q^{th} and r^{th} terms of H.P. are u,v,w respectively, then find the value of the expression (q  r) vw + (r  p) wu + (p  q) uv.
Let H.P. be
If the sum of the first 2n terms of 2, 5, 8, ....... is equal to the sum of the first n terms of 57, 59, 61......., then n is equal to
Given,
2n/2 {2.2 + (2n  1)3} = n/2 {2.57 + (n  1)2}
or 2 (6n + 1) = 112 + 2n or 10n = 110, ∴ n = 11
The distance of the point (–1, 1) from the line 12(x + 6) = 5 (y – 2) is
The given line is 12 (x + 6) = 5(y – 2)
⇒ 12x + 72 = 5y – 10 or 12x – 5y + 72 + 10 = 0
⇒ 12x – 5y + 82 = 0 The perpendicular distance from (x_{1}, y_{1}) to the line ax + by + c = 0 is
The point (x_{1}, y_{1}) is (–1, 1), therefore, perpendicular distance from (–1, 1) to the line 12x – 5y + 82 = 0 is
The family of straight lines (2a + 3b) x + (a – b) y + 2a – 4b = 0 is concurrent at the point
Rewriting the equation
(2x + y + 2) a + (3x – y – 4)b = 0 and for all a, b the straight lines pass through the intersection of 2x + y + 2 = 0 and 3x – y – 4 = 0
i.e., the point
The length of the latusrectum of the parabola whose focus is and directrix is is
According to the figure, the length of latus rectum is
The equation of the ellipse with focus at (±5, 0) and x =36/5 as one directrix is
We have ae = 5 [Since focus is (±ae, 0)] and a/e = 36/5
On solving we get a = 6
Thus, the required equation of the ellipse is
For what value of k the circles x^{2} + y^{2} + 5x + 3y + 7 = 0 and x^{2} + y^{2} – 8x + 6y + k = 0 cuts orthogonally
Let the two circles be 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 where g_{1} = 5/2, f_{1} = 3/2, c_{1} =7, g_{2} =–4, f_{2} =3 and c_{2} = k
If the two circles intersects orthogonally, then
⇒ 11 = 7 + k ⇒ k = – 18
If the lines 3x – 4y + 4 = 0 and 6x – 8y – 7 = 0 are tangents to a circle, then the radius of the circle is
The diameter of the circle is perpendicular distance between the parallel lines (tangents) 3x – 4y + 4 = 0 and 3x – 4y – 7/2 = 0 and so it is equal to Hence radius is 3/4.
Negation of “Paris in France and London is in England” is
Let p : Paris is in France, q : London is in England
∴ i.e., Paris is not in France or London is not in England..
First ten odd numbers are 1, 3, 5, 7, 11, 13, 15, 17, 19 respectively. So A.M.
If A and B are mutually exclusive events and if P(B) = 1/3, P(A ∪ B) = 13/21 then P(A) is equal to
For mutually exclusive events
P(A ∪ B) = P(A) + P(B) ⇒ P(A) = 2/7
A die is loaded such that th e pr obability of throwing the number i is proportional to its reciprocal. The probability that 3 appears in a single throw is
Clearly, f –1 (x) is defined for 1 + x ≠ 0.
Hence, domain of f^{ –1} (x) is R – {– 1}
The given expression = 0
The matrix A^{2} + 4A – 5I, where I is identity matrix and equals:
A^{2} + 4A – 5 I = A × A + 4A – 5I
Con tinuous as well as differ entiable, so f '(1) = 0
The function f(x) = sin x – kx – c, where k and c are constants, decreases always when
Let f (x) = sin x – kx – c where k and c are constants. f'(x) = cos x – k
∴ f decreases if cos x ≤ k
Thus, f (x) = sin x – kx – c decrease always when k ≥ 1.
The minimum value of f (x) = sin^{4} x + cos^{4} x in the interval is
Let y = sin^{4}x + cos^{4}x
dy/dx = 4 sin^{3} x cos x 4 cos3 x ( sin x)
= 4 sin xcox (sin^{2} x  cos^{2}x)
= (2 sin 2x) (– cos 2x) = – sin 4x
∴ dy/dx = 0 ⇒ sin 4x = 0 ⇒ 4x = 0, π, 2π, 3π
The curve y –e^{xy}+ x = 0 has a vertical tangent at
y – e^{xy} + x = 0
for the vertical tangents
1 – x (x + y) = 0 i.e.,
∴ x = 1 and y = 0
The function f(x) = 2x^{3} – 3x^{2} – 12x + 4, has
f(x) = 2x^{3} – 3x^{2} – 12x + 4
⇒ f '(x) = 6x^{2 }– 6x – 12 = 6(x^{2} – x – 2)
= 6(x – 2) (x + 1)
For maxima and minima f '(x) = 0
∴ 6(x – 2)(x + 1) = 0 Þ x = 2, – 1
Now, f ''(x) = 12x6
At x = 2; f ''(x) = 24  6 = 18> 0
∴ x = 2 , local min. point
At x = – 1; f ''(x) = 12 ( 1)  6 = 18<0
∴ x = –1 local max. point
Given integral
We know that sinx is a periodic function of π
What is the area bounded by y = tan x , y = 0 and x = π/4?
Required area = tan x dx
It is a differential equation of degree 2.
Two vectors are such that The angle between the two vectors will be–
Squaring both the sides, we get
or dot product is zero) . Therefore angle between is 90°
Gives the line L: and theplane π : x – 2y – z = 0. Of the following assertions, the only one that is always true is
Since 3(1) + 2(– 2) + (–1) (–1) = 3 – 4 + 1 = 0
∴ Given line is ^ to the normal to the planei.e., given line is parallel to the given plane.
Also, (1, –1, 3) lies on the plane x – 2y – z = 0 if 1 – 2 (–1) – 3 = 0 i.e., 1 + 2 – 3 = 0 which is true ∴ L lies in plane π.
A ladder rests against a wall so th at its top touches the roof of the house. If the ladder makes an angle of 60° with the horizontal and height of the house be 63 meters, then the length of the ladder in meters is
Length of ladder =
In an equilateral triangle, the in radius, circumradius and one of the exradii are in the ratio
For the constraints of a L.P. Problem given by x_{1} + 2x_{2} ≤ 2000, x_{1} + x_{2} ≤ 1500 and x_{2} ≤ 600 and x_{1}, x_{2} ≥ 0, which one of the following points does not lie in the positive bounded region
Clearly point (2000, 0) is outside.
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