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The perimeter of a sector is p. the area of the sector is maximum when its radius is
Let z,w be complex numbers such that z̅ + iw ̅ = 0 and arg zw = π.Then arg z equals
The area, in square unit, bounded by the curves y = x^{3}, y = x^{2} and the ordinates x = 1, x = 2 is
If (a+ib)(c+id)(e+if)(g+ih) = A+iB, then (a^{2}+b^{2})(c^{2}+d^{2})(c^{2}+f^{2})(g^{2}+h^{2}) =
Put sin^{2}x = t
Then sin2x dx = d t ; x = 0, π/2 ⇒ t = 0,1
The solution of the equation (xy^{2}x)dx=(yx^{2}y)dy is
Multiply R_{1}, R_{2}, R_{3} by a, b, c respectively and divide by abc
Take abc common from C_{3} so that C_{1}, C_{3} are identical
Two functions f : R → R , g : R → R are defined as follows :
What is the value of the integral ∫ e^{x} [ f'x + f x ] dx ?
True statement for
lim _{x → 0} 1 + x − 1 − x^{2} + 3x − 2 − 3 x is
The statement "If x is divisible by 8, then it is divisible by 6" is false if x equals
A sentence in "If....then..." form is called an implication. The only time when an implication is false is when the "If" part of the sentence is true and the "then" part of the sentence is false. The number 32 makes the first part of the statement true and the second part of the statement false.
The system of equation
2x + y = 5
x  3y = 1
3x + 4y = k
is consistent, when k is
Five digit number divisible by 3 is formed using the digits 0, 1 , 2, 3, 4 and 5 without repetition. Total number of such numbers is
A number is divisible by 3 if and only if the sum of its digits are divisible by 3
Notice that 1 + 2 + 3 + 4 + 5 = 15, which is divisible by 3
The only other way we can have a sum of 5 digits divisible by 3 is to replace the 3 by the 0 making the sum 3 less:
1 + 2 + 0 + 4 + 5 = 12, which is divisible by 3
No other choice of 5 digits can have a sum divisible by 3, because there is no other way to make the sum 12 or 15, and we certainly can't have a sum of 9 or 18
So the number of 5digit numbers that can be formed from the digits {1,2,3,4,5} is
Number of ways = 1 x 2 x 3 x 4 x 5 = 120
And the number of 5digit numbers that can be formed from the digits {1, 2, 0, 4, 5} is figured this way
Number of ways = 1 x 2 x 3 x 4 x 4 = 96
Total Number of ways = 120 + 96 = 216
If A and B are two events such that P(A ∪ B) = 0 .65, P(A ∩ B) = 0.15 then
You are given a box with 20 cards in it. 10 of these cards have the letter 1 printed on them. The other ten have the letter T printed on them. If you pick up 3 cards at random and keep them in the same order, the probability of making the word IIT is
If to form a ΔABC it is given that a=5, b=7, sinA=( 3/4 ), then possible triangles are
Given a = 5, b = 7 and sin A = 3/4
∴ b sin A = 7 x ( 3/4 ) = 21/4 > a (= 5),
Thus is this case no triangle in possible
If the A.M. and G.M, of the roots of a quadratic equation in x are p and q respectively, then its equation is
If v is the variance and σ is the standard deviation, then
Coefficients of correlation between observations (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) is
The condition that one root of the equation (1+m^{2}) x^{2} + 2 cmx + (c^{2}  a^{2}) = 0 is reciprocal of the other is____
Let the roots of the equation
(1+m^{2}) x^{2} + 2 cmx + (c^{2}  a^{2}) = 0
are α and 1/α .
If  x + 1  −  x  + 3  x − 1  x − 2  = x + 2 , then solution for − 1 ≤ x ≤ 0 is ___
When − 1 ≤ x < 0
(x + 1) − (− x) + 3 (− x + 1)( − x + 2) = x + 2
⇒ 3 x^{2} − 8 x + 5 = 0
⇒ 3 x^{2} − 3 x − 5 x + 5 = 0
⇒ 3 x − 5 x − 1 = 0
⇒ x = 1 , x = 5 ∕ 3
− 1 ≤ x < 0
Hence, given equation has no solution in − 1 ≤ x < 0. .
If cos α, cos β, cos γ are the d.c. of a line, then value of sinα^{2 }+ sin^{2}β + sin^{2}γ is
Let p be a variable point on the ellipse [(x^{2}/a^{2}) + (y^{2}/b^{2})] = 1 with foci F_{1} and F_{2}. If A is area of triangle PF_{1}F_{2}, then the maximum value of A is
The centre of a circle passing through the points (0, 0), (1, 0) and touching the circle x^{2} + y^{2} = 9 is
If a,b,c are three noncoplanar vectors, then [a+b,b+c,c+a]=
In a circuit containing an inductance of zero resistance, the emf of the applied A.C. voltage leads the current by
In the photoelectric effect is that the stopping potential is a function of the frequency of the incident light. As shown in Figure, there is even a cutoff frequency ν_{0} below which there is no photoelectric effect occurs, regardless of the intensity of the light.
A plot of stopping potential vs. frequency of incident light (Cathode: Sodium ; Cutoff frequency νo =4.39x 10^{14} Hz
Hence option (A) is correct.
Which of the following metal thermionically emit an electron at a relatively lowest temperature among them?
A metal foil of negligible thickness is introduced between two plates of a capacitor of capacitance C at the centre. The new capacitance of the capacitor will be
A steady current of 1.5A flows through a copper voltameter for 10 min. If the electrochemical equivalent of copper is 30 x 10^{5} gC^{1}, the mass of copper deposited on the electrode will be
If sky wave with a frequency of 50 MHz is incident on Dregion at an angle of 30^{0} then angle of refraction is
In a metre bridge, the balancing length from the left end (standard resistance of one ohm is in the right gap) is found to be 20 cm. The value of the unknown resistance is
In a Wheatstone's bridge, all the four arms have equal resistance R. If resistance of the galvanometer arm is also R, then equivalent resistance of the combination is
Four two ohm resistors are connecte together along the edges of a square. A 10V battery of negligible internal resistance is connected across a pair of the diagonally opposite corners of the square. The power dissipated in the circuit is
Particle nature and wave nature of electromagnetic waves and electrons be shown by
R, L and C represent the physical quantities resistance, inductance and capacitance respectively. Which one of the following combinations has dimension of frequency?
A metal disc of radius 100 cm is rotated at a constant angular speed of 60 rad/s in a plane at right angles to an external field of magnetic induction 0.05 Wb/m^{2}. The emf induced between the centre and a point on the rim will be
If the electric flux entering and leaving an enclosed surface respectively is φ_{1} and φ_{2} , the electric charge inside the surface will be
A hollow metal sphere of radious 10 cm is charged such that the potential on its surface is 80V. The potential at the centre of the sphere is
Equal charges are given to two spheres of different radii. The potential will be
Two equal and unlike charges places 3 cm apart in air, attract each other with a force of 40 N. The magnitude of each charge in micro coulomb is
A 2 μ F condenser is charged to 500V and then its plates are joined through of consistance. The heat produced in the resistance in joules is
The figure shows a charge q placed inside a cavity in an uncharged conductor. Now if an external electric field is switched on:
A straight wire carrying current I is turned into a circular loop. If the magnitude of magnetic moment associated with it in M.K.S. unit is M, the length of wire will be
Which of the following is more effective in inducing nuclear fission
A radioactive element has a half life of 15 years. The fraction that will decay in 30 years is
The photoelectric threshold of a certain metal is 30 Å. If the radiation of 2000 Å is incident on the metal
Light of wavelength λ and photon energy 2 eV falling on a metal surface produces photoelectrons of maximum velocity ν. If λ is decreased by 25% and the maximum velocity is doubled, the work function of metal (in eV) is
A glass slab is placed in the path of convergent light. The point of convergence of light
The image formed by an objective of a compound microscope is
The element that can be used as acceptor impurity to dope silicon is
In a commonbase amplifier, the phase difference between the input signal voltage and output voltage is
A commonemitter amplifier is shown in the figure (A).
In this Circuit , the output signal is an amplified version of the input signal and is 180 degrees out of phase with the input signal.
Common emitter Amplifier
Common Base Amplifier
Figure (B) is a commonbase amplifier. In this circuit the output signal is an amplified version of the input signal and is in phase with the input signal. In both of these circuits, the output signal is controlled by the basetoemitter bias. As this bias changes (because of the input signal) the current through the transistor changes.
A halfwave rectifier is being used to rectify an alternating voltage of frequency 50 Hz. The number of pulses of rectified current obtained in one second is
The difference in the variation of resistance with temperature in a metal and a semiconductor arises essentially due to the difference in the
Which of the following property of light waves is not observed in sound waves?
If the phase differenc between light waves from two slits of Young's double slit experiment is π radian, then the central fringe will be
Bubble's of colourless soap solution appear coloured in sun light. This is due to
The reagent required to convert propene to 1Propanol is
In nucleophilic substitution reactions, the reactivity of carbonyl compounds follows the order
The most reactive compound towards formation of cyanohydrin on treatment with KCN followed by acidification is
In which of the following reactions, both oxidised and reduced forms of the same compound are obtained?
Wurtz reaction involves the interaction of alkyl halides in dry ether with
Ozonolysis of C₇H₁₄ gave 2methyl3pentanone toluene. The alkene is
In the following reaction, which choice has value twice that of the equivalent mass of the oxidising agent SO₂ + H₂O → 3S + 2H₂O
The volume in litres of CO_{2} liberated at STP when 10 grams of 90% pure limestone is heated completely is
Limestone (CaCO_{3}) has a molar mass of = 100 g/mol
So, 10 gm of 100% pure CaCO_{3} would = 10 gm of CaCO_{3}
90% pure CaCO_{3} means, each 10 gm of sample will have, = ( 90/100 ) x 10 = 9 gm CaCO_{3}
So, number of moles of CaCO_{3} = given mass/ molar mass = 9/100 = 0.09
Now, we have the reaction,
CaCO_{3} → CaO + CO_{2}
When 1 mole of CaCO_{3} yields 1 moles of CO_{2}
0.09 moles of CaCO_{3} will yield 0.09 moles of CO_{2}
1 mole of CO_{2} = 22.4 liters (at STP)
So, total volume = 22.4 x 0.09 = 2.016 liters
Which of the following biomolecule is insoluble in water?
Pick out the one which does not belong to family of enzymes.
Acetyl chloride can not be obtained by treating acetic acid with
A colourless liquid at room temperature reacts with soda lime to form sodium salt of carboxylic acid and ammonia gas.The liquid is
The two opposite processes, forward reaction and reverse reaction, keep on occurring in it all the time but at exactly equal rates (or equal speeds)
A chemical reaction does not stop after reaching the equilibrium state, only the rate of the forward and the reverse reactions become equal at the equilibrium state
Latent heat of vaporisation of a liquid at 500 K and 1 atm pressure is 10.0 kcal/mol. What will be the change in internal energy (ΔE) of 3 mol of liquid at same temperature?
Vaporization of 3 moles of H_{2}O vapors is:
3H_{2}O(l) → 3H_{2}O(g)
Δn = 3  0 = 3
Therefore,
ΔU = ΔH  ΔnRT
= (3 x10)  3 (0.002) (500) = 27
change in internal energy is:
ΔU = 27 Kcal
The minimum energy required by molecules to react is called
Given that
2C s + 2O_{2 } g → 2CO_{2} g , Δ H = − 787 k J
H_{2} g + 12O_{2} g → H_{2}O l , Δ H = − 286 k J
C_{2}H_{2} g + 2 1/2 O_{2} g → 2CO_{2} + H_{2}O l , Δ H = − 1301 k J
Heat of formation of acetylene is
The primary and secondary valencies of chromium in the complex ion, dichlorodioxalatochromium (III) are respectively
In which of the following complex the oxidation number of the central metal atom is zero?
In the diazotisation of aniline with sodium nitrite and hydrochloric acid, the excess of hydrochloric acid is used primarily to
Which of the following is produced by reaction of RCN in sodium and alcohol ?
The pH of aqueous KCl solution is 7.0. This solution was electrolysed for a few seconds using Pt electrodes. Which of the following is correct?
What is the standard cell potential for the cell? Zn ; Zn^{2+}(1M) ∥ Cu^{2+}(1M) ; Cu (E^{0} for Zn^{2+}∕Zn = −0.76 ; E^{0}for Cu^{2}⁺∕Cu = +0.34)
When diethyl ether is treated with excess of Cl₂ in the presence of sunlight, then the product formed is
When the densities of the ore particles and gangue particles are different, then the ore is concentrated by
If the pressure and absolute temperatre of 2 litres of CO₂ are doubles, the volume of CO₂ would become
When iron or zinc is added to CuSO₄ solution, copper is precipitated. It is due to
When concentrated H₂SO₄ is heated with dry KNO₃, brown fumes evolve. These fumes are due to
The ratio of cationic radius to anionic radius in an ionic crystal is greater than 0.732. Its coordination number is
The separation of colloidal particles from those of molecular dimensions is known as
Fill in the blank with appropriate word.
Soft minded individuals are........to embrace all kinds of superstitions.
Improve the sentence by choosing best alternative for capitalised part of the sentence.
Poor Tom LAID in the shade of a tree before he could walk further.
Fill in the blanks with the most suitable idiom/phrase.
Let us now _____ and work for the advancement of the country .
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