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If P represents radiation pressure, c represents speed of light and Q represents radiation energy striking a unit area per second, the nonzero integers x, y and z such that P^{x}Q^{y}c^{z} is dimensionless, are.
The position x of a particle varies with time (t) as x = A t^{2} – B t^{3}. The acceleration at time t of the particle will be equal to zero. What is the value of t?
Given that x = A t^{2} – B t^{3}
∴ velocity dx/dy = 2A t  3 Bt ^{2}
and acceleration = = 2 A  6Bt
For acceleration to be zero 2A – 6Bt = 0.
Two projectiles A and B are thrown with the same speed but angles are 40º and 50º with the horizontal. Then
lesser is the value of θ, lesser is sinq and hence lesser will be the time taken. Hence A will fall earlier.
A body is travelling in a circle at a constant speed. It
Body moves with constant speed it means that tangential acceleration a_{T} = 0 & only centripetal acceleration a_{C} exists whose direction is always towards the centre or inward (along the radius of the circle).
Two blocks are connected over a massless pulley as shown in fig. The mass of block A is 10 kg and the coefficient of kinetic friction is 0.2. Block A slides down the incline at constant speed. The mass of block B in kg is:
Considering the equilibrium of A, we get 10a = 10g sin 30º – T – mN
where N = 10g cos 30°
but a = 0, T = m_{B}g
⇒ m_{B} = 3.268 ≈ 3.3 kg
A spring is compressed between two toy carts of mass m_{1} and m_{2}. When the toy carts are released, the springs exert equal and opposite average forces for the same time on each toy cart. If v_{1} and v_{2} are the velocities of the toy carts and there is no friction between the toy carts and the ground, then :
Applyin g law of conservation of lin ear momentum m_{1}v_{1 }+ m_{2}v_{2 }= 0,
The potential energy for a force field is given by U (x, y) = cos (x + y). The force acting on a particle at position given by coordinates (0, π/4) is –
A long string is stretched by 2 cm and the potential energy is V. If the spring is stretched by 10 cm, its potential energy will be
The ratio of the accelerations for a solid sphere (mass ‘m’ and radius ‘R’) rolling down an incline of angle ‘θ’ without slipping and slipping down the incline without rolling is:
For solid sphere rolling without slipping on inclined plane, acceleration
For solid sphere slipping on inclined plane without rolling, acceleration
a_{2} = g sin θ
Therefore required ratio =
A system consists of three particles, each of mass m and located at (1, 1), (2, 2) and (3, 3). The coordinates of the centre of mass are
The coordinates of C.M of three particle are
so coordinates of C.M. of three particle are (2, 2) V = 300 m/s
Suppose the gravitational force varies inversely as the nth power of distance. Then the time period of a planet in circular orbit of radius ‘R’ around the sun will be proportional to
F = KR^{ n} = MRω^{2} ⇒ ω^{2} = KR^{ (n +1)}
[where K' = K^{1/2}, a constant]
Two planets A and B have the same material density. If the radius of A is twice that of B, then the ratio of the escape velocity v_{A}/v_{B} is
as v_{e} ∝ R for same density,
The upper end of a wire of diameter 12mm and length 1m is clamped and its other end is twisted through an angle of 30°. The angle of shear is
A spherical ball is dropped in a long column of a viscous liquid. The speed (v) of the ball as a function of time (t) may be best represented by
Two mercury drops (each of radius r) merge to form a bigger drop. The surface energy of the bigger drop, if T is the surface tension, is
Surface energy of bigger drop,
E = 4πR^{2}T = 4 x 2^{2/3 }πr^{2}T = 2^{8/3} πr^{2}T
Two circular plates of radius 5 cm each, have a 0.01 mm thick water film between them. Then what will be the force required to separate these plate (S.T. of water = 73 dyne/cm) ?
= 36.5 π ≈ 115 newton
One kilogram of ice at 0°C is mixed with one kilogram of water at 80°C. The final temperature of the mixture is (Take specific heat of water = 4200 kJ/kg°C, Latent heat of ice = 336 kJ/kg)
In the equation PV^{γ} = constant, the value of γ is unity. Then the process is
PV = constant represents isothermal process.
An ideal refrigerator has a freezer at a temperature of 13ºC. The coefficient of performance of the engine is 5. The temperature of the air (to which heat is rejected) is
T_{2} = 273 – 13 = 260,
or T_{1} – 260 = 52; T_{1} = 312 K,
T_{2} = 312 – 273 = 39°C
3 moles of an ideal gas at a temperature of 27°C are mixed with 2 moles of an ideal gas at a temperature 227°C, determine the equilibrium temperature of the mixture, assuming no loss of energy.
Energy possessed by the ideal gas at 27°C is
Energy possessed by the ideal gas at 227°C is
If T be the equilibrium temperature, of the mixture, then its energy will be
Since, energy remains conserved,
E_{m} = E_{1 }+ E_{2}
or T = 380 K or 107°C
A simple pendulum has time period 't'. Its time period in a lift which is moving upwards with acceleration 3 ms^{–2} is
A wave y = a sin (ωt – kx) on a string meets with another wave producing a node at x = 0. Then the equation of the unknown wave is
Equation of a wave
y_{1} = a sin (ωt – kx) ....(i)
Let equations of another wave may be,
y_{2} = a sin (ωt + kx) ....(ii)
y_{3} = –a sin (ωt + kx) ....(iii)
If Eq. (i) propagate with Eq. (ii), we get y = 2a cos kx sin ωt
If Eq. (i), propagate with Eq. (iii), we get y = –2a sin kx cos ωt
At x = 0, y = 0, wave produce node
So, Eq.(iii) is the equation of unknown wave
A source of sound produces waves of wavelength 60 cm when it is stationary. If the speed of sound in air is 320 m s^{1} and source moves with speed 20 m s^{1}, the wavelength of sound in the forward direction will be nearest to
A charge +q is at a distance L/2 above a square of side L. Then what is the flux linked with the surface?
The given square of side L may be considered as one of the faces of a cube with edge L. Then given charge q will be considered to be placed at the centre of the cube. Then according to Gauss's theorem, the magnitude of the electric flux through the faces (six) of the cube is given by
φ = q/ε_{0}
Hence, electric flux through one face of the cube for the given square will be
Two metallic spheres of radii 1 cm and 3 cm are given charges of 1 × 10^{2} C and 5 × 10^{2 }C, respectively. If these are connected by a conducting wire, the final charge on the bigger sphere is
In a region, the potential is represented by V(x, y, z) = 6x  8xy  8y + 6yz, where V is in volts and x, y, z are in metres. The electric force experienced by a charge of 2 coulomb situated at point (1, 1, 1) is :
The power dissipated in the circuit shown in the figure is 30 Watts. The value of R is
...(i)
Substituting the values in equation (i)
Which of the following quantities do not change when a resistor connected to a battery is heated due to the current?
Only number of free electrons is constant, other factors are temperature dependent.
The magnetic field at the origin due to the current flowing in the wire is –
The back emf induced in a coil, when current changes from 1 ampere to zero in one millisecond, is 4 volts, the self inductance of the coil is
Two solenoids of same crosssectional area have their lengths and number of turns in ratio of 1 : 2.
The ratio of selfinductance of two solenoids is
From
we get,
An alternating voltage V = V_{0} sin ωt is applied across a circuit. As a result, a current I = I_{0} sin (ωt – π/2) flows in it. The power consumed per cycle is
The phase angle between voltage V and current I is π/2.
A resistance R and inductance L and a capacitor C all are connected in series with an AC supply. The resistance of R is 16 ohm and for a given frequency, the inductive reactance of L is 24 ohm and capacitive reactance of C is 12 ohm. If the current in the circuit is 5 amp., find the potential difference across R, L and C.
V_{R} = iR = 5 × 16 = 80 Volt
V_{L} = i × (ωL) = 5 × 24 = 120 Volt
V_{C} = i × (l /ωC) = 5 × 12 = 60 Volt
The diameter of the objective of a telescope is a, its magnifying power is m and wavelength of light is λ, The resolving power of the telescope is:
The photoelectric threshold of metal is 2000Å. The energy of the electrons ejected from the surface by ultraviolet light of wavelength 1500Å is
A material particle with a rest mass m_{0} is moving with a velocity of light c. Then the wavelength of the de Broglie wave associated with it is :
Hydrogen atom in ground state is excited by a monochromatic radiation of λ = 975 Å. Number of spectral lines in the resulting spectrum emitted will be
Number of spectral lines =
Pluton ium 239 is pr ocessed by br eeder mechanism to be used as nuclear feul.
A transistor has a base current of 1 mA and emitter current 90 mA. The collector current will
I_{C} = I_{E} –I_{B} = 90 – 1 = 89 m A
A d.c. battery of V volt is connected to a series combination of a resistor R and an ideal diode D as shown in the figure below. The potential difference across R will be
In forward biasing, the diode conducts. For ideal junction diode, the forward resistance is zero; therefore, entire applied voltage occurs across external resistance R i.e., there occurs no potential drop, so potential across R is V in forward biased.
We know that,
Molecular weight of compound or molecules = 2 × V. D.
Vapour density (V. D.) of ozone molecules
Hence, V. D. of O_{3} is 24.
In redox reaction 1 geq of reducing agent requires P gmeq. of oxidising agent. The value of P is
In redox reaction,
g equivalent of reducing agent = g. equivalent of oxidising agent
Hence 1g equ. of reducing agent = P g equ. of oxidising agent.
Chloride ion and potassium ion are isoelectronic. Then :
Which of the following pairs has both members from the same period of periodic table :
_{11}Na ⇒ 2, 8, 1; _{17}Cl ⇒ 2, 8, 7
These have same number of shells. Hence, they are the elements of the same period.
In the periodic table, with the increase in atomic number, the metallic character of an element
Which of the following statements is/are true?
1. PH_{5} and BiCl_{5} do not exist.
2. pπ – dπ bond is present in SO_{2}.
3. I_{3}^{+} has bent geometry.
4. SeF_{4} and CH_{4} have same shape.
All the three statements are correct.
When the temperature is raised, the viscosity of liquid decreases this is because
At a constant volume the specific heat of a gas is 0.075 and its molecular weight is 40. The gas is:
⇒ C_{V} = q × m
= 0.075 × 40
= 3.0 cal
C_{P} – C_{v} = R
C_{P} = C_{V} + R
= 3 + 2 = 5
Monoatomic gas.
Which of these is least likely to act as Lewis base?
BF_{3} is Lewis acid (e^{–} pair acceptor)
The K_{sp} of CuS, Ag_{2}S and HgS are 10^{–31},10^{–44} and 10^{–54} respectively. The solubility of these sulphides are in the order :
For CuS, solubility = (10^{–31})^{1/2};
for HgS =
In which of the following reactions, H_{2}O_{2} is acting as a reducing agent?
SO_{2} changes to H_{2}SO_{4} (O.N. changes from +4 to +6 oxidation)
2KI → I_{2} (O.S. changes from –1 to 0 oxidation)
PbS → PbSO_{4} (O.S. changes from –2 to +6 oxidation)
Ag_{2}O → 2Ag (O.S. changes from +1 to 0 reduction)
Sodium peroxide in contact with moist air turns white due to formation of :
Na_{2}O_{2} + H_{2}O → 2NaOH + 1/2 O_{2}
Graphite and boron nitride have similar structure.
The number of geometrical isomers of CH_{3} – CH = CH – CH = CH – CH = CHCl is
The given structure has three double bonds whose each carbon atom is differently substituted hence number of geometrical isomers will be 2^{n }= 2^{3}’ = 8, where n is the number of double bonds whose each carbon atom is differently substituted.
According to IUPAC system, the correct name of the compound having the formula
The given reaction is Diel’s Alder reaction.
The unsaturated hydrocarbon which on ozonolysis gives one mole each of formaldehyde, acetaldehyde and methyl glyoxal (CH_{3}COCHO) is
Minamata is caused by Hg poisoning.
Phosphate pollution is caused by sewage and agricultural fertilizers.
Eutrophication causes reduction in dissolved oxygen.
Coolant used in car radiator is aqueous solution of ethylene glycol. In order to prevent the solution from freezing at – 0.3°C. How much ethylene glycol must be added to 5 kg of water?
(K_{f} = 1.86 K Kg mol^{–1})
ΔT_{f} = 0.3 ° C
∴ W_{B} = 50 g
The amount used should be more than 50 g.
Which of the following will form the cathode with respect to iron anode in an electrolyte cell?
In electrolytic cell the cathode is of higher reduction potential.
The activation energy for a simple chemical reaction A → B is E_{a} in forward direction. The activation energy for reverse reaction
Since the nature of reaction (i.e. exothermic or endothermic) is not given, E_{a} for reverse reaction can be more or less.
The following data are for the decomposition of ammonium nitrite in aqueous solution:
The order of reaction is:
NH_{4} NO_{2} → N_{2} + 2H_{2}O
Volume of N_{2} formed in successive five minutes are 2.75 cc, 2.40 cc and 2.25 cc which is in decreasing order. So rate of reaction is dependent on concentration of NH_{4} NO_{2}. As decrease is not very fast so it will be first order reaction.
Which liberates ammonia when treated with water?
All nitrides react with H_{2}O to give NH_{3} and CaCN_{2} also react with H_{2}O
CaNCN + 3H_{2}O → CaCO_{3} + NH_{3}
The correct order of reactivity of halogens with alkalies is
Reactivity follows the order F > Cl > Br > I
In the manufacture of iron from haematite, limestone is added to act as:
Limestone (CaCO_{3}) is mixed with Fe_{2}O_{3} and it acts as flux to form slag (CaSiO_{3}).
Which of the following has square planar geometry?
[PtCl_{4}]^{2–} has square planar geometry. Pt : 5d^{9}6s^{1}
Two electrons are removed from 5d shell and 6s shell. So, hybridisation takes place is dsp^{2} i.e. square planar geometry.
In which of the following conversions, phosphorus pentachloride is used as the reagent?
When ethyl alcohol is treated with PCl 5 , then ethyl chloride is formed.
CH_{3}CH_{2}  OH + PCl_{5} CH_{3}CH_{2}  Cl + HCl + POCl_{3}
Match List I (Reaction) with List II (Reagent) and select the correct answer using the codes given below the lists:
Which of the following will not form a yellow precipitate on heating with an alkaline solution of iodine?
CH_{3}OH does not have – CH(OH)CH_{3} group hence it will not form yellow precipitate with an alkaline solution of iodine (haloform reaction).
Formic acid and acetic acid can be distinguished by:
Formic acid(HCOOH) has aldehydic group.
When ethylamine react with sodium metal, the gas evolved is
When ethylamine is heated with sodium metal, then hydrogen gas is evolved.
2C_{2}H_{5}NH_{2} + 2Na → 2C_{2}H_{5}NHNa^{+} + H_{2} ↑
The secondary structure of a protein refers to
The secondary structure of a protein refers to the shape in which a long peptide chain can exist. There are two different conformations of the peptide linkage present in protein are αhelix and βconformation.
The αhelix always has a right handed arrangement.
In βconformation all peptide chains are streched out to nearly maximum extension and then laid side by side held together by intermolecular hydrogen bonds. The structure resembles the pleated folds of drapery and therefore is known as βpleated shelt.
When H_{2}S gas is passed through the HCl containing aqueous solution of CuCl_{2}, HgCl_{2}, BiCl_{3} and CoCl_{2} it does not precipitate out
CoS is not precipitated in acidic medium.
Which one of the following statements is correct?
Ammonia can dissolve ppt. of AgCl only due to formation of complex as given below:
AgCl + 2NH_{3} → [Ag (NH_{3})_{2}]Cl
Three separate samples of a solution of a single salt gave these results. One formed a white precipitate with excess ammonia solution, one formed a white precipitate with dil. HCl solution and one formed a black precipitate with H_{2}S. The salt could be
Pb(NO_{3})_{2} + 2NH_{4}OH → Pb(OH)_{2} ↓ + 2NH_{4}NO_{3} (white ppt)
Pb(NO_{3})_{2 }+ 2HCl → PbCl_{2} ↓ + 2HNO_{3} (white ppt)
Pb(NO_{3})_{2} + H_{2}S → PbS ↓ + 2HNO_{3} (black)
Experiment to study kinetics of the dissociation of hydrogen peroxide must be performed by group of two or three so that–
Directions: In the following questions, choose the alternative which can most appropriately replace the group of words italicised in the sentence.
Despite being in the career of singing for the last 10 yr, he has not been able to earn fame on account of his practice of borrowing ideas and words from others and using them as his own.
Directions: In the following questions, choose the alternative which can most appropriately replace the group of words italicised in the sentence.
Every person is not allowed to enter the place where public, government or historical records are kept
Directions: In the following questions, choose the alternative which can most appropriately replace the group of words italicised in the sentence.
The advertisement assured the public that the medicine would give back to the users, their youthful vigour and appearance.
Directions: Choose the alternative which is most similar in meaning to the word given in capital letters.
PARAMOUR
Directions: Choose the alternative which is most similar in meaning to the word given in capital letters.
REFECTORY
Directions: Choose the alternative which is most similar in meaning to the word given in capital letters.
ASSENT
Directions: Choose the alternative which best expresses the meaning of the given idiom/ phrase.
To show one’s teeth
Directions: Choose the alternative which best expresses the meaning of the given idiom/ phrase.
To pour oil in troubled water
Directions: Rearrange the following six sentences (A), (B), (C), (D), (E), (F) in the proper sequence to form a meaningful paragraph; then answer the questions given below them.
(A) This is being done to ensure that when the cess is removed or merged ,the flexibility to impose higher rate on luxury goods is not taken away ,a senior finance ministry official told ET.
(B) This is only an enabling provision and the highest rate levied on goods will still be 28% (14% central GST and 14% state GST).The demerit and luxury goods will attract higher 28% rate plus cess.
(C) India has decided to peg the peak goods and services tax (GST) rate at 40% in the legislation instead of 28%, giving it the flexibility to raise rates without having to reach out to parliament.
(D) "Some members of the council felt such an enabling provision was needed." the official privy to the development said.
(E) The GST council has proposed to peg the peak tax rate at 40% (20% central GST and 20% state GST) in the model GST law to preclude the requirement of approaching parliament or state assemblies for any change in future.
(F) This enabling provision will also allow the government to remove the cess at some stage and instead have a higher GST rate only, which will make for a neater GST.
Q. Which sentence should come second in the paragraph?
Directions: Rearrange the following six sentences (A), (B), (C), (D), (E), (F) in the proper sequence to form a meaningful paragraph; then answer the questions given below them.
(A) This is being done to ensure that when the cess is removed or merged ,the flexibility to impose higher rate on luxury goods is not taken away ,a senior finance ministry official told ET.
(B) This is only an enabling provision and the highest rate levied on goods will still be 28% (14% central GST and 14% state GST).The demerit and luxury goods will attract higher 28% rate plus cess.
(C) India has decided to peg the peak goods and services tax (GST) rate at 40% in the legislation instead of 28%, giving it the flexibility to raise rates without having to reach out to parliament.
(D) "Some members of the council felt such an enabling provision was needed." the official privy to the development said.
(E) The GST council has proposed to peg the peak tax rate at 40% (20% central GST and 20% state GST) in the model GST law to preclude the requirement of approaching parliament or state assemblies for any change in future.
(F) This enabling provision will also allow the government to remove the cess at some stage and instead have a higher GST rate only, which will make for a neater GST.
Q. Which sentence shouldcome before the last?
Directions: Rearrange the following six sentences (A), (B), (C), (D), (E), (F) in the proper sequence to form a meaningful paragraph; then answer the questions given below them.
(A) This is being done to ensure that when the cess is removed or merged ,the flexibility to impose higher rate on luxury goods is not taken away ,a senior finance ministry official told ET.
(B) This is only an enabling provision and the highest rate levied on goods will still be 28% (14% central GST and 14% state GST).The demerit and luxury goods will attract higher 28% rate plus cess.
(C) India has decided to peg the peak goods and services tax (GST) rate at 40% in the legislation instead of 28%, giving it the flexibility to raise rates without having to reach out to parliament.
(D) "Some members of the council felt such an enabling provision was needed." the official privy to the development said.
(E) The GST council has proposed to peg the peak tax rate at 40% (20% central GST and 20% state GST) in the model GST law to preclude the requirement of approaching parliament or state assemblies for any change in future.
(F) This enabling provision will also allow the government to remove the cess at some stage and instead have a higher GST rate only, which will make for a neater GST.
Q. Which sentence will come complete the passage?
Directions: Rearrange the following six sentences (A), (B), (C), (D), (E), (F) in the proper sequence to form a meaningful paragraph; then answer the questions given below them.
(A) This is being done to ensure that when the cess is removed or merged ,the flexibility to impose higher rate on luxury goods is not taken away ,a senior finance ministry official told ET.
(B) This is only an enabling provision and the highest rate levied on goods will still be 28% (14% central GST and 14% state GST).The demerit and luxury goods will attract higher 28% rate plus cess.
(C) India has decided to peg the peak goods and services tax (GST) rate at 40% in the legislation instead of 28%, giving it the flexibility to raise rates without having to reach out to parliament.
(D) "Some members of the council felt such an enabling provision was needed." the official privy to the development said.
(E) The GST council has proposed to peg the peak tax rate at 40% (20% central GST and 20% state GST) in the model GST law to preclude the requirement of approaching parliament or state assemblies for any change in future.
(F) This enabling provision will also allow the government to remove the cess at some stage and instead have a higher GST rate only, which will make for a neater GST.
Q. Which sentence will be third after the rearrangement?
Directions: Rearrange the following six sentences (A), (B), (C), (D), (E), (F) in the proper sequence to form a meaningful paragraph; then answer the questions given below them.
(A) This is being done to ensure that when the cess is removed or merged ,the flexibility to impose higher rate on luxury goods is not taken away ,a senior finance ministry official told ET.
(B) This is only an enabling provision and the highest rate levied on goods will still be 28% (14% central GST and 14% state GST).The demerit and luxury goods will attract higher 28% rate plus cess.
(C) India has decided to peg the peak goods and services tax (GST) rate at 40% in the legislation instead of 28%, giving it the flexibility to raise rates without having to reach out to parliament.
(D) "Some members of the council felt such an enabling provision was needed." the official privy to the development said.
(E) The GST council has proposed to peg the peak tax rate at 40% (20% central GST and 20% state GST) in the model GST law to preclude the requirement of approaching parliament or state assemblies for any change in future.
(F) This enabling provision will also allow the government to remove the cess at some stage and instead have a higher GST rate only, which will make for a neater GST.
Q. Which sentence will start the passage?
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.
A novel of real_____________ must invent its own language, and this one does,
Ambitiona cherished desire Abstrusehard to understand intricate complex
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.
Information technology, and the hardware and software______________with the IT industry.
Amalgamationjoining of two organization to form one large organization Regulated means controlled.
Directions: In the following question, select the related letter/word/number from the given alternatives.
EFJK : LMQR :: LMQR : ?
Directions: In the following question, select the related letter/word/number from the given alternatives.
Mahatma Gandhi : Porbandar : : Pt. Jawaharlal Nehru : ?
As, the birth place of Mahatma Gandhi was Porbandar. Similarly, the birth place of Pt. Jawaharlal Nehru was Allahabad.
Directions: In the question below is given a statement followed by three assumptions numbered I, II and III. You have to consider the statement and the following assumptions, decide which of the assumptions is implicit in the statement and choose your answer accordingly.
Statement: The education of a student at collegiate level, not taking into account maintenance expenses, costs four hundred rupees a year. Collegiate education is thus drawing heavily upon the national resources of an impoverished community. So college education should be restricted to a brilliant few.
Assumptions:
I. Our resources are very limited.
II. Only a few students should be admitted to the colleges.
The use of the words 'impoverished community' in the statement makes I implicit while the phrase 'college education should be restricted to a brilliant few' makes II implicit.
In a code language, if BANGED is coded as JJKQCC, then the word STRAY will be coded as
Directions: In the following question, a series is given with one terms missing. Choose the correct alternative from the given ones that will complete the series:
2, 3, 7, 22, 155, ?
Which one of the following diagram represents the correct relationship among Colour, Black and White.
Find out the alternative figure which contains figure (X) as its part.
A piece of paper is folded and cut. From the figures given, indicate how it will appear when opened?
Identify the figure that completes the pattern.
Find out which of the figures (a), (b), (c) and (d) can be formed from the pieces given in figure (X).
Let A = {x : x ∈ R x < 1};
B = {x : x ∈ R, x  1 ≥ 1} and A ∪ B = R D, then the set D is
A = [ x : x ∈ R, 1 < x < 1]
B = [x : x ∈ R : x 1 ≤ 1 or x 1 ≥ 1]
= [x: x ∈ R : x ≤ 0 or x ≥ 2]
∴ A ∪ B = R  D,
where D = [ x : x ∈ R,1 ≤ x < 2]
If 12 cot^{2}θ – 31 coses θ + 32 = 0, then the value of sin θ is
12 cot^{2}θ  31cosec θ + 32 = 0
⇒ 12(cosec^{2}θ  1)  31cosec θ + 32 = 0
⇒ 12cosec^{2}θ  31cosec θ + 20 = 0
⇒ 12cosec^{2}θ  16 cosec θ  15 cosecθ + 20 = 0
⇒ (4cosec θ  5)(3cosec θ  4) = 0
⇒
tan 20° + tan 40° + √3 tan 20° tan 40° is equal to
√3 = tan 60° = tan (40° + 20°)
∴ √3 – √3 tan 40° tan 20°
= tan 40° + tan 20°
Hence tan 40° + tan 20° + √3 tan 40° tan20°
= √3
The roots of the equation x^{2} – 2 √2x + 1 = 0 are
The discriminant of the equation
( 2√2)^{2} – 4 (1) (1) = 8 – 4 = 4 > 0 and a perfect square, so roots are real and different but we can't say that roots are rational because coefficients are not rational therefore.
this is irrational
∴ the roots are real and different.
In a polygon no three diagonals are concurrent. If the total number of points of intersection of diagonals interior to the polygon be 70 then the number of diagonals of the polygon is
A combination of four vertices is equivalent to one interior poin of intersection of diagonals.
∴ No. of interior points of intersection
⇒ ^{n}C_{4} = 70
⇒ n (n – 1) (n – 2) (n – 3) = 5. 6. 7. 8
∴ n = 8
So, number of diagonals = ^{8}C_{2} – 8 = 20
With 17 consonants and 5 vowels the number of words of four letters that can be formed having two different vowels in the middle and one consonant, repeated or different at each end is
The two letters, th e first and the last of the four lettered word can be chosen in (17)^{2} ways, as repetition is allowed for consonants. The two vowels in the middle are distinct so that the number of ways of filling up the two places is ^{5}P_{2} = 20.
The no. of different words = (17)^{2 }. 20 = 5780.
The coefficient of x^{32} in the expansion of is:
We know by Binomial expansion, that (x + a)^{n }
Given expansion is
On comparing we get n = 15, x = x^{4},
⇒ x^{60 –7r} = x^{32} ⇒ 60 – 7r = 32
⇒ 7r = 28 ⇒ r = 4
So, 5th term, contains x^{32}
= ^{15}C_{4}X^{32}.
Thus, coefficient of x^{32} = ^{15}C_{4}.
If m arithmetic means are inserted between 1 and 31 so that the ratio of the 7^{th} and (m  1)^{th} means is 5 : 9, then find the value of m.
Let the means be x_{1}, x_{2},.....x_{m} so that 1, x_{1}, x_{2},....x_{m}, 31 is an A.P. of (m + 2) terms.
Now, 31 = T_{m+2 }= a + (m + 1)d = 1 + (m + 1)d
⇒ 9a + 63d = 5a + (5m – 5)d
The reflection of the point (4, –13) in the line 5x + y + 6 = 0 , is
Let Q(a, b) be the reflection of P(4, – 13) in the line 5x + y + 6 = 0
Then the midpoint lies on
5x + y + 6 = 0
⇒ 5a + b + 19 = 0 ...(i)
Also PQ is perpendicular to 5x + y + 6 = 0
Therefore
⇒ a – 5b – 69 = 0 ..(ii)
Solving (i) and (ii), we get a = – 1, b = – 14
If the equations of the pairs of opposite sides of a parallelogram are x^{2} – 5x + 6 = 0 and y^{2} – 6y + 5 = 0, then equations of its diagonals are
Equations of the sides of the parallelogram are
(x – 3) (x – 2) = 0 and (y – 5) (y – 1) = 0
i.e. x = 3, x = 2; y = 5, y = 1
Hence its vertices are : A (2 ,1 ); B (3, 1); C (3, 5); D (2, 5)
Equation of the diagonal AC is
y – 1 = 4/1 (x – 2) ⇒ y = 4x – 7
Equation of the diagonal BD is
If the line 2x – 1 = 0 is the directrix of the parabola y^{2} – kx + 6 = 0, then one of the values of k is
Given eq^{n} of parabola is y^{2} – kx + 6 = 0
The line ax + by = 1 cuts ellipse cx^{2} + dy^{2} = 1 only once if
Clearly ax + by = 1
Find the equation of chord of the circle x^{2} + y^{2} = 8x bisected at the point (4, 3)
T = S_{1} ⇒ x (4) + y (3) – 4 (x + 4) = 16 + 9 – 32
⇒ 3y – 9 = 0 ⇒ y = 3
Mean of 25 observations was found to be 78.4. But later on it was found that 96 was misread 69. The correct mean is
Mean
∑x = 25 × 78.4 = 1960
But this ∑x is incorrect as 96 was misread as 69.
∴ correct ∑x = 1960 + (96 – 69) = 1987
∴ correct mean = 1987/25 = 79.48
If the mean, mode and S.D. of a fr equency distribution are 41, 45 and 8 respectively, then its Pearson’s coefficient of skewness is
A black die and a white die are rolled. Find the probability that the number shown by the black die will be more than twice that shown by the white die.
The number of favourable cases are shown below:
There are 6 favourable cases in which the number on black die is more than twice the number on the white die.
∴ m = 6
n = Total number of cases = 6 × 6
(∴ with each die there are six possibilities)
∴ Probability
Let E = {1, 2, 3, 4} and F = {1, 2}. Then the number of onto functions from E to F is
If set A has m elements and set B has n elements then number of onto functions from A to B is
Here E = {1, 2, 3, 4}, F = {1, 2}
m = 4, n = 2
∴ No. of onto functions from E to F
Let cos^{–1}x = y
If and A + B – D = 0 (zero matrix), then D matrix will be 
= 1 (3 × 9 – 6 (–7)) –2 (–4 × 9 –2 × 6) +3 ((–4) (–7) – 3 × 2)
= (27 + 42) –2 (–36 – 12)+3 (28 – 6) = 231
If a circular plate is heated uniformly, its area expands 3c times as fast as its radius, then the value of c when the radius is 6 units, is
Let A sq. units in the area measure when the radius is r units.
their A = πr^{2}
Differentiate both side w.r.t ‘t’
We have,
From eqn (i), we get
The function f(x) = tan x – 4x is strictly decreasing on
f(x) = tanx – 4x ⇒ f'(x) = sec^{2}x – 4
When
Therefore, 1 < sec^{2}x < 4
⇒ – 3 < (sec^{2}x – 4) < 0
Thus, for
Hence, f is strictly decreasing on
The slope of the tangent to the hyperbola 2x^{2 }– 3y^{2} = 6 at (3, 2) is
Differentiating the given equation of the curve
4x – 6y. (dy/dx) = 0 ∴ dy/dx = 2x / 3y
If , where m ∈ N, then I_{10} + 10 I_{9} is equal to 
The area of the region bounded by the curve y = x  x , xaxis and the ordinates x = 1, x = –1 is given by:
The area of the region bounded by the curve y = f (x) and the ordinates x = a, x = b is given by
According to the question,
Required area
= area of region OAB + area of region OCD
= 2 × Area of region OAB
What is the solution of dy/dx + 2y = 1 satisfying y(0) = 0 ?
The solution of differential equation – y = 3 represents a family of
We have,
integrating, 2 ln (y + 3) = ln x + ln c = ln cx
⇒ ln (y + 3)^{2} = ln cx ⇒ (y + 3)^{2} = cx
which is a family of parabolas.
Which one of the following is the unit vector perpendicular to both and
With respect to a rectangular cartesian coordinate system, three vectors are expressed as:
and where are unit vectors, along the X, Y and Zaxis respectively. The unit vector along the direction of sum of these vector is –
If the middle points of sides BC, CA & AB of triangle ABC are respectively D, E, F then position vector of centre of triangle DEF, when position vector of A, B, C are respectively
The position vector of points D, E, F are respectively
So, position vector of centre of ΔDEF
The perpendicular distance of P(1, 2, 3) from the line is
The point A (6, 7, 7) is on the line . Let the perpendicular from P meet the line in L. Then AP^{2} = (6  1) 2 + (7  2) 2 + (7  3) 2 = 66
∴ ⊥ distance d of P from the line is given by
d^{2} = AP^{2 } AL^{2} = 66  17 = 49 so that d = 7
The equation of the plane containing the line is a (x – x_{1}) + b (y – y_{1}) + c (z – z_{1}) = 0, then
If the given plane contains the given line then the normal to the plane, must be perpendicular to the line and the condition for the same is aℓ + bm + cn = 0.
If mean of a poisson distribution of a random variable X is 2, then the value of P(X > 1.5) is
If P(A ∪ B) = 2/3 , P(A ∪ B) = 1/6 and P(A) = 1/3 then
Now, P(AB) = P(A) P(B), A and B are independent events.
If a flagstaff of 6 metres high placed on the top of a tower throws a shadow of 2√3 metres along the ground, then the angle (in degrees) that the sun makes with the ground is
Accordingly,
A wholesale merchant wants to start the business of cereal with ₹ 24000. Wheat is ₹ 400 per quintal and rice is ₹ 600 per quintal. He has capacity to store 200 quintal cereal. He earns the profit ₹ 25 per quintal on wheat and ₹ 40 per quintal on rice. If he store x quintal rice and y quintal wheat, then for maximum profit the objective function is
For maximum profit, z = 40x + 25y.
The minimum value of for positive real number x, y, z is
By A.M. ≥ G.M.
Let and f(0) = 12. If f is continuous at x = 0, then the value of a is equal to
Which of the following functions is differentiable at x = 0?
x is nondifferentiable function at x = 0 as L.H.D = –1 and R.H. D = 1
But cos h is differentiable
∴ Any combination of two such functions will be nondifferentiable. Hence option (a) and (b) are ruled out.
Now, consider sin x + x
Hence, sin x  x is diffble at x = 0.
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