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For all real x, the minimum value of 1  x + x^{2}/1 + x + x^{2} is
AOB is the positive quadrant of the ellipse x^{2}/a^{2} + y^{2}/b^{2} = 1, where OA = a, OB = b. The area between the arc AB and the chord AB of the ellipse is
Required area
If z₁ and z₂ are two nonzero complex numberssuch that z₁ + z₂ =z₁ + z₂,then arg z₁ arg z₂ is equal to
where [x] denotes the greatest integer less than or equal to x, is continuous at x = 2, then the value of a is equal to
The differential equation representing the family of curves y^{2 }= 2c(x+√c), where c is a positive perimeter, is of
The differential equation  dy/dx  +  y  + 3 = 0 admits
If x dy = y(dx + y dy), y > 0 and y (1) = 1, then y (3) is equal to
If the function f(x) = cos^{2} x + cos^{2}(π/3 + x)  cos x.cos(π/3 + x) is constant, then the value of this constant is
d/dx(x^{2} + cos x)^{4}
= 4.(x^{2} + cos x)^{3} x d/dx (x^{2} + cos x)
= 4(x^{2} + cos x)^{3} x (2x  sin x)
if f : R→R be such that f(1) = 4 and f^{'}(1) = 12 then
Which of the following is the converse of the statement: "If x > 4, then x + 2 > 5"?
To form the CONVERSE of a statement, the "If" and the "then" sections of the statement switch places.
If A and B are two matrices such that AB = B and BA = A, then A^{2}+ B^{2} =
In how many ways can 5 prizes be given to 4 students if any student can receive more than one prize?
Required ways = 4 x 4 x 4 x 4 x 4 = 4^{5}
The determinant of skewsymmetric matrix of odd order is always zero
Therefore the inverse of skewsymmetric matrix does not exist
If A and B are two independent events in a given sample space and the probability that both A and B occur is 0.16 while the probability that neither occurs is 0.36. The probabilities P(A) and P(B) are respectively
From a well shuffled pack of playing cards, two cards drawn one by one with replacement. The probability that both are aces is
For the first card drawn , probability of ace is 4/52 .
Then for the second card, after the first ace is drawn, there are only 3 aces left out of 51 cards,
So probability is 3/51 .
The probability of performing two independent activities such as this is actually the product of the probabilities:
( 4/52 ) x ( 3/51 ).
This reduces to ( 1/13 ) x ( 1/17 ) = 1 221
In shuffling a pack of cards 3 are accedentally dropped, then the chance that missing card should be of different suit is
We have to find out the probability of 3 cards of different suits out of four suits
The ways of choosing three suits out of four = ^{4}C_{3}
Since, each suit has 13 cards, the ways of choosing 1 card = ^{13}C_{1} = 13
If b,c and sinB are given such that ∠B is acute and b<c sinB, then
When two sides and angle opposite to one of then be given. In this case, the triangle is not always uniquely determined. It is quite possible to have no triangle, one triangle and two triangles with this type of data.
So, it is called an ambiguous case.
Let b, c and B are given parts.
When B is acute and b < c sin B, then there is no triangle
By Cosine Law
b^{2} = c^{2} + a^{2}  2ac cos B
⇒ a^{2}  2ac cos B = b^{2}  c^{2}
⇒ a^{2}  2ac cos B + c^{2} cos^{2} B
= b^{2}  c^{2} + c^{2} cos^{2} B
⇒ (a  c cos B)^{2} = b^{2}  c^{2} (1  cos^{2} B)
= b^{2}  c^{2} sin^{2} B
This equation helps us to determine a, when b, c and B are being given. We observe that:
When b < c sin B, then a is imaginary and so there is no solution.
The number of terms of the A.P. 3, 7, 11, 15,...to be taken so that the sum is 406 is
The mean of 5 observations is 4.4 and their variance is 8.24. If three observations are 1, 2 and 6, the other two observations are
The mode of the following items is 0,1,6,7,2,3,7,6,6,2,6,0,5,6,0, is
p and q are the roots of x^{2} + ax + b = 0, then equation whose roots are p^{2}q, q^{2}p is
p + q =  a, p.q = b
Now, p^{2} q + q^{2} p = pq(p + q)
= b(a) =  ab
p^{2}q . q^{2}p = (pq)^{3} = b^{3}
The required equation is
x^{2}  (p^{2}q + q^{2}p) x+ (p^{2}q . q^{2}p) = 0
x^{2} + abx + b^{3} = 0
If the roots of the equation x^{2}+2mx+m^{2}2m+6=0 are equal, then the value of m is
If the pair of lines ax^{2}+ 2hxy + by^{2} + 2gx + 2fy + c = 0 intersect on the yaxis then
If a=3, b=4, c=5 and a+b+c=0, the angle between a and b is
The point on the axis of y which is equidistant from (1, 2) and (3, 4) is
A parallel plate condenser with oil between the plates (dielectric constant of oil K = 2) has a capacitance C. If the oil is removed, then capacitance of the capacitors becomes
On removing the oil, the capacitance of the capacitor will become.
C ′ = C/K = C/2
A steady current of 1.5 A flows through a copper voltameter for 10 min. If the electrochemical equivalent of copper is 30 x 10^{5} g C^{1} , the mass of copper deposited on the electrode will be
The coefficient of mutual inductance, when magnetic flux changes by 2 X 10^{2} Wb and current changes by 0.01 A is
n small drops of same size are charged to V volt each. They coalesce to form a bigger drop. The potential of the bigger drop is
If an unchanged capacitor is charged by connecting it to a battery, then the amount of energy lost as heat is
A charged particle of mass 5 x 10^{6} Kg is held stationary in space by placing it in an electric field of strength 10^{6} N C^{1} directed vertically downwards. The charge on the particle is g = 10 m s^{2}
A cyclotron oscillator frequency is 10 MHz. If radius of its dees 60 cm, then magnetic field for accelerating the protons is (Given e = 1.6 x 10^{19}C, mp = 1.67 x 10^{27} kg)
v = 10 MHz = 10 x 10^{6} Hz = 10^{7} Hz
R = rm = 60 cm = 0.6 m
e = 1.6 x 10^{19} C
m = 1.67 x 10^{27} kg
resonance v = Be/2πm
or B = 2πmv/e
A ray of light is incident at 60^{o} on a prism of refracting angle 30^{o} . The emergent ray is at an angle 30^{o} with the incident ray. The value of refractive index of the prism is
Angle of prism = 30^{o} = A
Angle of incidence = i_{1} = 60^{o}
Angle of emergence = i_{2} = ?
Angle of devation = D = 30^{o}
∴ i_{2} = A + D  i_{2} = 30 + 30  60 = 0^{o}
⇒ r_{2} = 0^{o}
∴ r_{1} = Ar_{2} = 30  0 = 30^{o}
∴ μ = sin i 1 sin r 1 = sin 60^{o} /sin 30^{o} = √ 3
A ray of light falls on a transparent glass slab of refractive index 1.62. If the reflected ray and the refracted ray are mutually perpendicular, the angle of incidence is
For diffraction of light, the condition for nth secondary minimum is that path difference is equal to [Here a is the width of slit, n is the order of diffraction, λ is the wavelength of light and θ n gives the direction of nth secondary minimum ]
A compound A has a molecular formula C₂Cl₃OH. It reduces Fehling solution and on oxidation produces a monocarboxylic acid B. A can also be obtained by the action of Cl₂ on Ethanol. A is
Which of the following gives iodoform on heating with a solution of l_{2} containing Na_{2}CO_{3} ?
Propyne on hydrolysis in presence of HCl and HgSO₄ gives
An alkane C_{7}H_{16} is produced by the reaction of Lithium di (3pentyl) cuperate with ethyl bromide. The structural formula of product is
The ratio of the molar amounts of H₂S needed to precipitate the metal ions from 20 mL each of 1M Cd(NO₃)₂ and 0.5 M CuSO₄ is
As Cd^{2+} and Cu^{2+}, both are dipositive ions but Cu^{2+} has one half of the concentration compared to that of the Cd^{2+}. Thus, the molar amount of the H_{2}S used to precipitate the metals ions Cd^{2+} and Cu^{2+} is 2 : 1.
What is the main reason for the fact that carboxylic acids can undergo ionization?
When propanamide reacts with Br₂ and NaOH then which of the following compounds is formed ?
A reaction reaches a state of chemical equilibrium only when
Which of the following favours the backward reaction in a chemical equilibrium?
The minimum energy required by molecules to react is called
The synthetic product which closely resembles with the natural musk in odour is
The effective atomic number of Cr (at. no. 24) in [(Cr(NH₃)₆] Cl₃ is
When dilute H_{2}SO_{4} is electrolysed at room temperature between Pt electrodes. The substance liberated at anode is
Which of the following will exhibits geometrical isomerism ?
How many litres of O₂ at STP are needed to complete the combustion of 39 g of liquid benzene as per equation given below ?
2C₆H₆(l) + 15O₂(g) →12CO₂(g) + 6H₂O(g)
The correct order of the mobility of the alkali metal ions in aqueous solution is
tertButyl methyl ether on heating with HI of one molar concentration gives
The standard E^{0}_{red.} values of A,B,C are 0.68V, 2.54 V,0.50V respectively.The order of their reducing power is
If molecular weight of KMnO₄ is M, then its equivalent weight in acidic medium would be
P_{4}O_{10} has short and long PO bonds. The number of short PO bonds in this compound is
P_{4}O_{10} has four P = O bonds which are shorter than PO single bonds.
Each P atom has three PO single bonds and one P = O bond, i.e.,
a total of four PO linkages
Which of the following oxides of Nitrogen is anhydride of HNO_{3}
Fill in the blank with appropriate word.
Manuring in agricultural fields increases........
Out of the given alternatives, choose the one which can be substituted for the given capitalised word.
Being an introvert, he will eat his HEART OUT.
Improve the sentence by choosing best alternative for capitalised part of the sentence.
No one could explain how a calm and balanced person like him could PENETRATE such a mindless act on his friends.
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