In a mercury thermometer the ice point (lower fixed point) is marked as 10^{o} and the steam point (upper fixed point) is marked as 130^{o}. At 40^{o}C temperature, what will this thermometer read?
The magnetic flux linked with a coil satisfies the relation = 4t^{2 }+ 6t + 9 Wb, where t is the time in second. The e.m.f. induced in the coil at t = 2 second is
Water is flowing through a very narrow tube. The velocity of water below which the flow remains a streamline flow is known as
If the velocity of light in vacuum is 3 x 10^{8} ms^{–1}, the time taken (in nanosecond) to travel through a glass of thickness 10 cm and refractive index 1.5 is
A charge +q is placed at the orgin O of X – Y axes as shown in the figure. The work done in taking a charge Q from A to B along the straight line AB is
What current will flow through 2 k resistor in the circuit shown in the figure?
I_{1}= current through 2 kΩ
In a region, the intensity of an electric field is given by in NC^{–1}. The electric flux through a surface in the region is
A train approaching a railway platform with a speed of 20 ms^{–1 }starts blowing the whistle. Speed of sound in air is 340 ms^{–1}. If the frequency of the emitted sound from the whistle is 640 Hz, the frequency of sound to a person standing on the platform will appear to be
A straight wire of length 2 m carries a current of 10 A. If this wire is placed in a uniform magnetic field of 0.15 T making an angle of 45^{o} with the magnetic field, the applied force on the wire will be
What is the phase difference between two simple harmonic motions represented by and x_{2} = A cos (ωt)?
Heat is produced at a rate given by H in a resistor when it is connected across a supply of voltage V. If now the resistance of the resistor is doubled and the supply voltage is made V/3 then the rate of production of heat in the resistor will be
Two elements A and B with atomic numbers Z_{A} and Z_{B} are used to produce characteristic x–rays with frequencies A and B respectively. If Z_{A} : Z_{B}= 1 : 2, then v_{A} : v_{B} will be
√v = a(z  b) , Ignoring screening effect (i.e. b=0)
The de Broglie wavelength of an electron moving with a velocity c/2 (c = velocity of light in vacuum) is equal to the wavelength of a photon. The ratio of the kinetic energies of electron and photon is
Two infinite parallel metal planes, contain electric charges with charge densities + and – respectively and they are separated by a small distance in air. If the permittivity of air is 0 then the magnitude of the field between the two planes with its direction will be
A box of mass 2 kg is placed on the roof of a car. The box would remain stationary until the car attains a maximum acceleration. Coefficient of static friction between the box and the roof of the car is 0.2 and g = 10 ms^{–2.}This maximum acceleration of the car, for the box to remain stationary, is
a_{max} = μg =0.2x 10 = 2 m/s^{2}
The dimension of angular momentum is
[L]= [m r v] = [M^{1}L^{2}T^{–1}]
have scalar magnitudes of 5, 4 , 3 units respectivley then the angle between A and C is
A particle is travelling along a straight line OX. The distance x (in meters) of the particle from O at a time t is given by x = 37 + 27 t – t^{3} where t is time in seconds. The distance of the particle from O when it comes to rest is
t = 3 sec. x = 37 + 27 x 3 – 3^{3}, = 37 + 81 – 27 = 91 m
A particle is projected from the ground with kinetic energy E at an angle of 60^{0} with the horizontal. Its kinetic energy at the highest point of its motion will be
A bullet on penetrating 30 cm into its target loses its velocity by 50%. What additional distance will it penetrate into the target before it comes to rest?
From (1) and (2) s = 10 m
When a spring is stretched by 10 cm, the potential energy stored is E. When the spring is stretched by 10 cm more, the potential energy stored in the spring becomes
Average distance of the earth from the Sun is L_{1.} If one year of the Earth = D days, one year of another planet whose average distance from the Sun is L_{2} will be
A spherical ball A of mass 4 kg, moving along a straight line strikes another spherical ball B of mass 1 kg at rest.
After the collision, A and B move with velocities v_{1} ms^{–1} and v_{2} ms^{–1 }respectively making angles of 30^{o} and 60^{o }with respect to the original direction of motion of A. The ratio v_{1}/v_{2} will be
Apply conservation of momentum along Normal
The decimal number equivalent ot a binary number 1011001 is
(1011001)_{2}= 1 x 2^{6} + 0 x 2^{5 }+1x 2^{4} + 1x 2^{3} + 0x 2^{2} + 0x 2^{1} + 1x 2^{0} = 64 + 16 + 8 + 1 =89
The frequency of the first overtone of a closed pipe of length l_{1 }is equal to that of the first overtone of an open pipe of length l_{2}. The ratio of their lenghts (l_{1} : l_{2}) is
The I – V characteristics of a metal wire at two different temperatures (T_{1} and T_{2}) are given in the adjoining figure. Here, we can conclude that
1/V = 1/R
slope of curve, R ∞T greater the slope smaller will be temperature.
T_{1} < T_{2}
In a slide calipers, (m + 1) number of vernier divisions is equal to m number of smallest main scale divisions. If d unit is the magnitude of the smallest main scale division, then the magnitude of the vernier constant is
From the top of a tower, 80 m high from the ground, a stone is thrown in the horizontal direction with a velocity of 8 ms^{–1}. The stone reaches the ground after a time ‘t’ and falls at a distance of ‘d’ from the foot of the tower. Assuming g = 10 ms^{–2}, the time t and distance d are given respectively by
time of flight (t) =
A Wheatstone bridge has the resistances 10 Ω , 10 Ω , 10Ω and 30 Ω in its four arms. What resistance joined in parallel to the 30 resistance will bring it to the balanced condition?
for a balanced bridge. R = 10 Ω
An electric bulb marked as 50 W–200 V is connected across a 100 V supply. The present power of the bulb is
A magnetic needle is placed in a uniform magnetic field and is aligned with the field. The needle is now rotated by an angle of 600 and the work done is W. The torque on the magnetic needle at this poition is
W = MB(1  cos ) = MB(1  cos 60^{o} ) = MB/2
In the adjoining figure the potential difference between X and Y is 60 V. The potential difference between the points M and N will be
A body when fully immersed in a liquid of specific gravity 1.2 weighs 44 gwt. The same body when fully immersed in water weighs 50 gwt. The mass of the body is
mg – σvg = 44, σ = density of liquid,
mg –1.2
ρvg = 44 ..............(1),
ρ = density of water
mg – ρvg = 50.............(2),
(1) – 1.2 x(2)
M = 80 g
When a certain metal surface is illuminated with light of frequency ν , the stopping potential for photoelectric current is ν_{o} When the same surface is illuminated by light of frequency v/2 the stopping potential is v_{0}/4 The threshold frequency for photoelectric emission is
solving equation (1) and (2) ,v/3
Three blocks of mass 4 kg, 2 kg, 1 kg respectively are in contact on a frictionless table as shown in the figure. If a force of 14 N is applied on the 4 kg block, the contact force between the 4 kg block, the contact force between the 4 kg and the 2 kg block will be
Let L be the length and d be the diameter of cross section of a wire. Wires of the same material with different L and d are subjected to the same tension along the length of the wire. In which of the following cases, the extension of wire will be the maximum?
An object placed in front of a concave mirror at a distance of x cm from the pole gives a 3 times magnified real image.If it is moved to a distance of (x + 5) cm, the magnification of the image becomes 2. The focal length of the mirror is
so f = 30 cm
22320 cal of heat is supplied to 100 g of ice at 0^{o}C If the latent heat of fusion of ice is 80 cal g^{–1 }and latent heat of vaporization of water is 540 cal g^{–1,} the final amount of water thus obtained and its temperature respectively are
total energy required to melt and boil at 100^{o}C is
(Q)min = 8000 + 10000 = 18000 cal,
(Qrequired)min < Qgiven for amount of vapour 22320=18000+ m x 540
= m = 8 gm. temperature = 100^{o}C and water remaining = 92 gm.
A progressive wave moving along x–axis is represented by The wavelength (λ) at which themaximum particle velocity is 3 times the wave velocity is
Two radioactive substances A and B have decay constant 5λ and λ respectively. At t = 0, they have the same number of nuclei the ratio of number of nuclei of A to that of B will be (1/e)^{2} after a time interval of
Li occupies higher position in the electrochemical series of metals as compared to Cu since
_{11}Na^{24} is radioactive and it decays to
The paramagnetic behavior of B_{2} is due to the presence of
A 100 ml 0.1 (M) solution of ammonium acetate is diluted by adding 100 ml of water. The pH of the resulting solution will be (pK_{a} of acetic acid is nearly equal to pK_{b} of NH_{4}OH)
Hydrolysis of a salt of weak acid and weak base
pH = 7 + ½ (pK_{a}– pK_{b}) = 7 [ pK_{a} = pK_{b} ]
In 2butene, which one of the following statements is true ?
The well known compounds, (+) lactic acid and (–) lactic acid have the same molecular formula, C_{3}H_{6}O_{3}. The correct relationship between them is
They are enantiomers and correct relationship between them is optical isomerism.
The stability of Me_{2}C=CH_{2} is more than that of MeCH_{2}CH = CH_{2} due to
Me_{2}C = CH_{2} is more stable than MeCH_{2}CH = CH_{2 }due to hyperconjugative effect.
Which one of the following characteristics belongs to an electrophile ?
Which one of the following methods is used to prepare Me_{3}COEt with a good yield?
Williamson’s synthesis (SN_{2}) given by 1° alkyl halide.
58.5 gm of NaCl and 180 gm of glucose were separately dissolved in 1000 ml of water. Identify the correct statement regarding the elevation of boiling point (b.p.) of the resulting solutions.
For NaCl (i = 2) for glucose (i = 1), Hence NaCl solution will show higher elevation of bp (Colligative property) as both are one Molar Solution.
Equal weights of CH_{4} and H_{2} are mixed in an empty container at 25^{o}C. The fraction of the total pressure exerted by H_{2 }is
Let equal weights be w g.
Partial pressure = mole fraction × Total pressure
P_{H}_{2} = 8/9 x Total pressure
Which of the following will show a negative deviation from Raoult’s law?
Acetone – chloroform due to formation of intermolecular Hydrogen bonding
In a reversible chemical reaction at equilibrium, if the concentration of any one of the reactants is doubled, then the equilibrium constant will
Equilibrium constant does not depend on conc. It is only a function of temperature
Identify the correct statement from the following in a chemical reaction.
Spontaneity of a reaction is decided by a combined effect of suitable change in values of enthalpy and entropy. [ ΔG = ΔH – TΔS]
Which one of the following is wrong about molecularity of a reaction?
Molecularity can never be a fraction.
Which of the following does not represent the mathematical expression for the Heisenberg uncertainty principle?
The stable bivalency of Pb and trivalency of Bi is
Lower oxidation state becomes more stable for group 14 and group 15 elements as we move down the group due to inert pair effect. Hence Pb^{+2 }and Bi^{+3} are more stable.
The equivalent weight of K_{2}Cr_{2}O_{7 }in acidic medium is expressed in terms of its molecular weight (M) as
Number of electrons gained by one ion = 6
Eq.wt = M/6
Which of the following is correct ?
Ca^{+2}, S^{–2}, Cl^{– }are isoelectronic (consist of 18e^{–}) and for isoelectronic species, ionic radii ∞e/z
CO is practically nonpolar since
The number of acidic protons in H_{3}PO_{3 }are
Number of OH group is 2 hence dibasic in nature.
When H_{2}O_{2} is shaken with an acidified solution of K_{2}Cr_{2}O_{7} in presence of ether, the ethereal layer turns blue due to the formation of
The state of hybridization of the central atom and the number of lone pairs over the central atom in POCl_{3} are
Number of lone pairs on P atom = 0
Upon treatment with l_{2} and aqueous NaOH, which of the following compounds will form iodoform?
Iodoform reaction is given by all alcohols having
Upon treatment with Al(OEt)_{3} followed by usual reactions (work up), CH_{3}CHO will produce
This reaction is known as Tischenko reaction
FriedelCraft’s reaction using MeCl and anhydrous AlCl_{3} will take place most efficiently with
As – CH_{3} group in toluene activates the benzene ring for electrophilic substitution reaction.
Which one of the following properties is exhibited by phenol?
Phenol is less acidic than H_{2}CO_{3.} So it cannot liberate CO_{2} from NaHCO_{3} (aq).
The basicity of aniline is weaker in comparison to that of methyl amine due to
Under identical conditions, the SN_{1} reaction will occur most efficiently with
Reactivity order for SN_{1} reaction in case of alkyl halide is 3°> 2°> 1°
Identify the method by which Me_{3}CCO_{2}H can be prepared
20 ml 0.1 (N) acetic acid is mixed with 10 ml 0.1 (N) solution of NaOH. The pH of the resulting solution is (pK_{a} of acetic acid is 4.74)
CH_{3}COOH + NaOH ⇒ CH_{3}COONa + H_{2}O
In the brown ring complex [Fe(H_{2}O)_{5}(NO)]SO_{4}, nitric oxide behaves as
‘NO’ ligand when attached to Fe behaves as positively charged ligand.
The most contributing tautomeric enol form of MeCOCH_{2}CO_{2}Et is
By passing excess Cl_{2}(g) in boiling toluene, which one of the following compounds is exclusively formed?
An equimolar mixture of toluene and chlorobenzene is treated with a mixture of conc. H_{2}SO_{4} and conc. HNO_{3}.Indicate the correct statement from the following
– CH_{3} is activating group due to +I effect and hyperconjugation.
But for – Cl, – I > + R effect.
∴CH_{3} is weakly activating and – Cl is weakly deactivating
Among the following carbocations :
Ph_{2}C*CH_{2}Me(I),
PhCH_{2}CH_{2}CH*Ph (II),
Ph_{2}CHCH*Me (III)
and Ph_{2}C(Me)CH_{2}^{+ }(IV),
the order of stability is
Which of the following is correct?
Evaparation increases randomness as liquid water is converted to water vapours.
On passing ‘C’ Ampere of current for time ‘t’ sec through 1 litre of 2 (M) CuSO_{4} solution (atomic weight of Cu = 63.5), the amount ‘m’ of Cu (in gm) deposited on cathode will be
1000 ml 2(M) CuSO_{4} ≡ 2(M) CuSO_{4} solution contains 2 moles Cu^{+2}
Cu^{2 }+ 2 →Cu
2F 63.5g
q = c × t coulomb
or, 2 × 96500C
If the 1st ionization energy of H atom is 13.6 eV, then the 2nd ionization energy of He atom is
The weight of oxalic acid that will be required to prepare a 1000 ml (N/20) solution is
The maximum value of z when the complex number z satisfies the condition z + Z/2 = 2 is
where x and y are real, then the ordered pair (x, y) is
There are 100 students in a class. In an examination, 50 of them failed in Mathematics, 45 failed in Physics, 40 failed in Biology and 32 failed in exactly two of the three subjects. Only one student passed in all the subjects. Then the number of students failing in all the three subjects
n(MUPUB) = n(M) + n(P) + n(B) n(M ∩ P)  n(PMB)  n(B ∩ P) + n(M ∩ P ∩ B)
given n(M) = 50 , n (P) = 45, n(B) = 40;
n(M∩ P) + n(P∩ B) + n(B∩ M)  3(M∩ P ∩ B) = 32
99 = 50 + 45 + 40 – (32 + 3 n (M ∩ P∩ B) )+ n(M ∩ P∩ B) ;
2 n(M ∩ P ∩ B) = 36 – 32;
n(M∩ P ∩ B) = 2
A vehicle registration number consists of 2 letters of English alphabets followed by 4 digits, where the first digit is not zero. Then the total number of vehicles with distinct registration numbers is
Total No. of ways = 26^{2} x 9 x 10^{3}
The number of words that can be written using all the letters of the word ‘IRRATIONAL’ is
IRRATIONAL. There are 2 I, 2 R, 2 A, one T, one O, One N, One L. Total Number of words
Four speakers will address a meeting where speaker Q will always speak after speaker P. Then the number of ways in which the order of speakers can be prepared is
The number of ways in which speaker P and Q can deliver their speach is ^{4}C_{2}.Total number of ways = ^{4}C_{2} x 2! = 12
The number of diagonals in a regular polygon of 100 sides is
Number of Diagonals in n sided Polygon is equal to ^{n}C_{2} – n. Total number of Diagonals = ^{100}C_{2} – 100=4850
Let the coefficients of powers of x in the 2nd, 3rd and 4th terms in the expansion of (1+x)^{n}, where n is a positive integer, be in arithmetic progression. Then the sum of the coefficients of odd powers of x in the expansion is
Coefficient of 2nd, 3rd and 4th terms are ^{n}C_{1}, ^{n}C_{2}, ^{n}C_{3}. It is given 2.^{n}C_{2 }= ^{n}C_{2 }+ ^{n}C_{3} = n = 7.
Sum of Coefficient of odd power of x = 2^{6}
Let f(x) = ax^{2} + bx + c, g(x) = px^{2} + qx + r, such that f(1) = g(1), f(2) = g(2) and f(3) – g(3) = 2. Then f(4) – g(4) is
Let a – p = w, b – q = y and c – r = z
f(1) = g(1) w + y + z = 0;
f(2) = g(2) 4w + 2y + z = 0;
f(3) = g(3) + 2 9w + 2y + z = 2w = 1,
y = –3, z = 2
f(4) – g(4) = 16 w + 4y + z = 6
The sum 1x 1! + 2x 2! + .... + 50x 50! equals
Six numbers are in A.P. such that their sum is 3. The first term is 4 times the third term. Then the fifth term is
Let α, α  5d, α  3d, α  d, + d, + 3d, α + 5d are six numbers in A.P.
The sum of the infinite series
The equations x^{2} +x +a=0 and x^{2} +ax +1=0 have a common real root
x^{2} +x +a=0 .....(A) and x^{2} +ax +1=0 ...(B)
A  B
(1–a)x + (a–1) = 0; (1–a) (x–1) = 0 if a = 1, x = 1, so, only sol is a = 2
If 64, 27, 36 are the Pth , Qth and Rth terms of a G.P., then P + 2Q is equal to
t_{P} = 64,
t_{R} = 36,
a . r^{P–1} = 64.............(1);
t_{Q} = 27, a . r^{Q–1} = 27.............(2)
a . r^{R–1} = 36.............(3)
(2)^{2}x (1)/(3)^{3} ; 2 Q + P = 3R
If (α+√β) and (α√β) are the roots of the equation x^{2} + px + q = 0 where α,β p and q are real, then the roots of the equation (p^{2}– 4q) (p^{2}x^{2} + 4 px) – 16 q = 0 are
2α = p, α^{2}  β = q,
so the given equation is 4β(4α^{2}x^{2}  8αx)  16α^{2} 16+b = 0
The number of solutions of the equation log_{2}(x^{2} + 2x –1) = 1 is
x^{2 }+ 2x –1 = 2
x^{2} + 2x – 3=0
(x + 3) (x – 1) = 0;
x = 1, –3
The sum of the series
then the value of the determinant of Q is equal to
The remainder obtained when 1!+2!+...+95! is divided by 15 is
Starting from 5! all the other numbers are divisible by 15 since 15 will be a factor So 1! + 2! + 3! + 4! = 1 + 2 + 6 + 24 = 33 So the remainder on dividing 33 by 15 is 3.
If P,Q,R are angles of triangle PQR, then the value of
On expanding the determinant,
Δ = – 1 + cos^{2}p + cos^{2}Q + cos^{2}R + 2cosP cosQ cosR
= – 1 cos^{2} P + cos^{2}Q + cos^{2}Q + cos^{2}R + (cos(P+Q) + cos(P – Q)) cosR
now P + Q = π – R
So Δ = – 1+ cos^{2}P + cos^{2}Q + cos2R – cos2R – cos(P – Q) cos(P + Q)
= – 1 + cos^{2}P + cos^{2}Q – cos.^{2}P + sin^{2}Q = – 1 + 1 = 0
The number of real values of for which the system of equations
x + 3y+5z = αx
5x+y+3z =α y
3x+5y+z = αz
The total number of injections (oneone into mappings) from {a_{1}, a_{2}, a_{3 }, a_{4}} to {b_{1},b_{2} ,b_{3} ,b_{4} ,b_{5} ,b_{6} , b_{7} is}
So total = 7 × 6 × 5 × 4 = 840 oneone into functions
Two decks of playing cards are well schuffled and 26 cards are randomly distributed to a player. Then the probability that the player gets all distinct cards is
Total no. of way of distribution = ^{104} C_{26} Favourable no. of ways of ditribution =
(selecting any one decki) × (selecting 26 cards out of it) = 2 x ^{52 }C_{26}
An urn contains 8 red and 5 white balls. Three balls are drawn at random. Then the probability that balls of both colours are drawn is
Total no. of selection = ^{13 }C_{3}
Two coins are avilable, one fair and the other twoheaded. Choose a coin and toss it once; assume that the unbiased coin is chosen with probalility 3/4. Given that the outcome is head, the probability that the twoheaded coin waschosen is
B is the event of selecting a biased coin
UB is the event of selecting an unbiased coin
Let be the set of real numbers and the funtions f: R → R and g: R → R be defined by f(x)=x^{2} + 2x3 and g(x) = x+1. Then the value of x for which f(g(x)) = g(f(x)) is
fog(x) = (x + 1)^{2 }+ 2(x + 1) – 3 = x^{2} + 4x
= x^{2} + 4x
gof(x) = x^{2} + 2x – 3 + 1 = x^{2} + 2x – 2
fog(x) = gof(x)
x^{2} + 4x = x^{2} + 2x – 2 x = – 1
If a,b,c are in arithmetic progression, then the roots of the equation ax^{2}  2bx + c = 0 are
a – 2b + c = 0
So x = 1 is a root
Now product of roots = c/a
So the other root is = c/a
If sin^{1}x + sin^{1}y + sin^{1}z = 3π/2, then the value of
sin^{1}x + sin^{1}y + sin^{1}z = π/2
x = y = z = 1
Let p,q,r be the sides opposite to the angles P,Q,R respectively in a triangle PQR, if r^{2}sin P sin Q = pq, then the triangle is
Let p,q,r be the sides opposite to the angles P,Q,R respectively in a triangle PQR. Then
Let P (2, 3) , Q (2,1) be the vertices of the triangle PQR. If the centroid of PQR lies on the line 2x+3y = 1, then the locus of R is
Let R(h, k)
If f is a realvalued differentiable function such that f(x) f’(x) < 0 for all real x, then
f(x) f (x) < 0 ....(i)
f(x) < 0 ; f (x) > 0 or f(x) > 0 ; f (x) < 0 ............... (ii)
So possible graphs of f(x) are
f(x) is decreasing function in both cases.
Rolle’s theorem is applicable in the interval [2,2] for the function.
For f(x) = 4x^{4} f(–2) = f(2) & f'(0) exist
The solution of
y(0) = 1,
y(1) = 2e^{1/5} is
Let y = e^{mx }is a trial solution
dy/dx = m
25m^{2}  10m  1 = 0
m = 1/5(equal roots)
The differential solution y = (1 + x)e^{x/5} (A & B are two arbitrary constant)
Initially y (0) = 1 ( A + 0) e^{o}
1 ⇒ A = 1
Let P be the midpoint of a chord joining the vertex of the parabola y^{2} = 8x to another point on it. Then the locus of P is
Let P(h, k) be the midpoint of chord AB through vertex (0, 0) of parabola y^{2} = 8x
Coordinate of B= (2h  0, 2k  0) = (2h, 2k )
B (2h, 2k) lies on y^{2} = 8x
⇒ 4k^{2} = 8 × 2h
⇒ k^{2} = 4h
The line x = 2y intersects the ellipse x^{2}/4 + y^{2} = 1 at the points P and Q. The equation of the circle with PQ as diameter is
Solving the given equations then common points are
Eqn. of Circle PQ as
diameter
The eccentric angle in the first quadrant of a point on the ellipse x^{2} /10 + y^{2}/8 = 1 at a distance 3 units from the centre of the ellipse is
The transverse axis of a hyperbola is along the xaxis and its length is 2a. The vertex of the hyperbola bisects the line segment joining the centre and the focus. The equation of the hyperbola is
A point moves in such a way that the difference of its distance from two points (8, 0) and (–8, 0) always remains 4.Then the locus of the point is
Let P(h, k) be the moving point
Conjugate equation of (i)
The number of integer values of m, for which the xcoordinate of the point of intersection of the lines 3x + 4y = 9 and y = mx + 1 is also an integer, is
From given equation x = 5/3+4m, x is an integer
3 + 4m = ± 5
or ± 1 m = – 2
or m = – 1
If a straight line passes through the point (α,β ) and the portion of the line intercepted between the axes is divided equally at that point, then x/α + y/β
Let, the equation of line be x/a + y/b = 1 , passes through P(α,β )
(α,β ),is the mid point of the portion intercepted between the axes
The equation y^{2} + 4x + 4y + k = 0 represents a parabola whose latus rectum is
y^{2} + 4x + 4y + k = 0
⇒ (y + 2)^{2} = 4 (–x + 1 – k/4)
Latus rectum = 4
If the circles x^{2} + y^{2} + 2x + 2ky + 6 = 0 and x^{2} + y^{2} + 2ky + k = 0 intersect orthogonally, then k is equal to
2g_{1}g_{2 }+ 2f_{1}f_{2 }
= c_{1} + c_{2}
⇒ 2k_{2} – k – 6 = 0
⇒ k = 2, –3/2
If four distinct points (2k, 3k), (2, 0), (0, 3) (0, 0) lie on a circle, then
Equation of circle x(x – 2) + y (y – 3) = 0
(2k, 3k) will satisfy
So, K = 1
The line joining A (b cosα , b sinα ) and B (a cosβ , a sinβ ), where a ≠ b , is produced to the point M(x, y) so that AM : MB = b : a.
Let the foci of the ellipse x^{2}/9+ y^{2} = 1 subtend a right angle at a point P. Then the locus of P is
P(h, k) be a point on the ellipse,, e = 2√2/3
= h^{2}+ k^{2} = 8
The general solution of the differential equation
Put x + y = θ
The value of the integral
the value of the integral
I = π/4
The integrating factor of the differential equation
Number of solutions of the equation tan x + sec x = 2 cos x, x ∈ [0, π] is
tanx + secx = 2cosx
2sin^{2}x + sinx – 1 = 0
sinx = 1/2
The value of the integral
Maximum value of function f(x) = x/8 + 2/x on the interval [1, 6] is
The value of the integral
The points representing the complex number z for which
Let a, b, c, p,q, r be positive real numbers such that a, b, c are in G.P. and a^{p} = b^{q} = c^{r}. Then
b^{2} = ac ; 2logb = loga + log c
a^{p }= b^{q}
loga/logb = q/p....(ii)
Let S_{k} be the sum of an infinite G.P. series whose first term is k and common ratio is k/k+1.Then the value of
The quadratic equation 2x^{2} – (a^{3}+ 8a – 1)x + a^{2} – 4a = 0 possesses roots of opposite sign. Then
If log_{e}(x^{2 }– 16) log_{e}(4x – 11), then
x^{2 }– 16 > 0
The coefficient of x10 in the expansion of 1 + (1 + x) + ......... + (1 + x)^{20} is
1 + (1 + x) + ......... + (1 + x)^{20}
The coefficient of x10 in the expansion of 1 + (1 + x) + ......... + (1 + x)^{20} is
1 + (1 + x) + ......... + (1 + x)^{20}
The system of linear equations
λx + y + z = 3
x – y – 2z = 6
– x + y + z = μ has
if λ = – 1
then Δ = 0 so equation has infinite number of solution for λ = – 1 and μ = 3
Let A and B be two events with P(A^{C}) = 0.3, P(B) = 0.4 and P(A ∩ B^{C}) = 0.5. Then P(B/A U B^{C}) is equal to
P(A ∩ B^{C}) = P(A)  P(A ∩ B) ; P(A ∩ B) = 0.2
Let p, q, r be the altitudes of a triangle with area S and perimeter 2t. Then the value of 1/p + 1/q +1/r is
Let C_{1} and C_{2} denote the centres of the circles x^{2} + y^{2} = 4 and (x – 2)^{2} + y^{2 }= 1 respectively and let P and Q be their points of intersection. Then the areas of triangles C_{1}PQ and C_{2}PQ are in the ratio
Equation of chord S_{2} – S_{1 }= 0 ; x_{2} + y_{2} – 4x + 3 = 0
x = 7/4
C_{1}T = Perpendicular distance of point (0, 0) from x = 7/4 is 7/4
C_{2}T = Perpendicular distance of point (0, 0) from x = 7/4 is 1/4
A straight line through the point of intersection of the lines x + 2y = 4 and 2x + y = 4 meets the coordinate axes at A and B. The locus of the midpoint of AB is
Point of intersection of given lines x = 4/3, y = 4/3
Let P and Q be the points on the parabola y^{2} = 4x so that the line segment PQ subtends right angle at the vertex.If PQ intersects the axis of the parabola at R, then the distance of the vertex from R is
PQ subtants right angle at vertex.
so t_{1},t_{2} = 4
equation of PQ is
it intersect the x axis so, put y = 0 we get x = 4
The incentre of an equilateral triangle is (1, 1) and the equation of one side is 3x + 4y + 3 = 0. Then the equation of the circumcircle of the triangle is
For equalateral triangle R = 2r = 2 × 2 = 4 equation of circumcircle (x – 1)^{2} + (y – 1)^{2} = 42 ; x^{2} + y^{2} – 2x – 2y – 14 = 0
The area of the region bounded by the curves y = x^{3} , y = 1/x, x = 2 is
Let y be the solution of the differential equation satisfying y(1) = 1. Then y satisfies
The area of the region, bounded by the curves y = sin^{ –1}x +x (1 – x) and y = sin^{ –1}x – x(1 – x) in the first quadrant, is
The value of the integral
Let [x] denote the greatest integer less than or equal to x, then the value of the integral
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