Which of the following is true?

The value of \(\int_0^1[\log x-\log (1-x)] d x\) is

Integrating factor of the differential equation \(x \frac{d y}{d x}-y=3 x^2\) is:

If \(x=a \sec ^3 \theta, y=a \tan ^3 \theta\), then \(\frac{d y}{d x}\) at \(\theta=\frac{\pi}{3}\) is

A random variable X has the following probability distribution :

x	1	2	3	4	5
\(P ( X = x )\)	k	k	3k	\(2 k^2\)	\(4 k^2\)

Based on the above information, the value of ' \(K\) ' is :

If A is a skew-symmetric matrix of order \(n \times n\) and \(a_{i j}\) is the \((i, j)^{\text {th }}\) elements, then :

Corner points of the feasible region for LPP are: \((4,0),(0,5),(7,0),(5,2)\) and \((0,3)\). Let \(Z=3 x+6 y\) be the objective function. Then, the value of (Max. \(Z-\) Min. \(Z\) ) i :

Which cell is rechargeable?

Read the passage carefully and answer the Questions
Molar conductivity \(\left(\Lambda_m\right)\) of a solution at a given concentration (c) is the conductance of volume, V of solution, containing one mole of electrolyte kept between the two electrodes with area of cross-section A and at a distance of unit length. It increases with the decrease in concentration and when the concentration approaches zero, the molar conductivity is called limiting molar conductivity \(\left(\Lambda_m^0\right)\). For a strong electrolyte, \(\Lambda_m\) increases linearly with dilution and is given by \(\Lambda_m=\Lambda_m^0-A c^{1 / 2}\). The value of the constant A for a given solvent depends on the type of electrolyte along with temperature. According to Kohlrausch law, the value of \(\left(\Lambda_m^0\right)\) for an electrolyte is A\(\Lambda_m^0=\nu_{+} \lambda_{+}^0+\nu_{-} \lambda_{-}^0\), where \(\nu_{+}\)and \(\nu_{-}\) are the number of cations and anions, respectively, per molecule of the electrolyte and \(\lambda_{+}^0\) and \(\lambda_{-}^0\) are limiting molar conductivities of cation and anion, respectively. Kohlrausch law
finds many applications, like determining the solubility of a sparingly soluble salt, determining the degree of dissociation\(\left(\Lambda_m / \Lambda_m^0\right)\), and the dissociation constant of a weak electrolyte.
If the equivalent conductivity of 1 M Phthalic acid is \(12.8 S cm ^2 eq ^{-1}\) and the limiting equivalent conductivities of the
hydrogen and Phthalate ions are 42 and \(288.42 S cm ^2 eq ^{-1}\), respectively, the degree of dissociation of Phthalic acid will be :

Which of the following is the wrong statement? Coulomb's law describes the electric force that

Match List - I with List - II. Match the integrating factors :

List - I (Differential Equation)	List - II (Integrating factor)
(A) \(\frac{d y}{d x}+3 y=e^{-2 x}\)	(I) \(\frac{1}{x}\)
(B) \(x \frac{d y}{d x}+y=3 x^2\)	(II) \(e^{-x}\)
(C) \(x \frac{d y}{d x}-y=3 x^2\)	(III) \(x\)
(D) \(\frac{d y}{d x}-y=x\)	(IV) \(e^{3 x}\)

Choose the correct answer from the options given below :

Two concentric circular loops of radius 5 cm and 15 cm are placed co-axially. Find e.m.f induced in the inner loop, if the current through the outer loop is changed at a rate of 10 A/ms. Assume the magnetic field on the inner loop to be uniform.

Which one of the followings is responsible for producing the large holes in "Swiss cheese"?