Let \( f(x) = \log x \) and \[ g(x) = \frac{x^4 - 2x^3 + 3x^2 - 2x + 2}{2x^2 - 2x + 1} \] Then the domain of \( f \circ g \) is:
Step 1: Understanding domain constraints. The function \( f(x) = \log x \) requires \( x>0 \), so we must ensure \( g(x)>0 \) for \( f(g(x)) \) to be defined.
Step 2: Finding domain of \( g(x) \). Given: \[ g(x) = \frac{x^4 - 2x^3 + 3x^2 - 2x + 2}{2x^2 - 2x + 1} \] The denominator is a quadratic equation: \[ 2x^2 - 2x + 1 \] Since the discriminant is negative, it is always positive.
Step 3: Solving for \( g(x)>0 \). Setting the numerator \( x^4 - 2x^3 + 3x^2 - 2x + 2>0 \), we find that \( x<0 \) satisfies this condition. Hence, \( g(x) \) is always positive. Thus, \( g(x)>0 \) for all \( x \), meaning the domain of \( f \circ g \) is \( \mathbb{R} \).
Match List - I with List - II:
List - I:
(A) Electric field inside (distance \( r > 0 \) from center) of a uniformly charged spherical shell with surface charge density \( \sigma \), and radius \( R \).
(B) Electric field at distance \( r > 0 \) from a uniformly charged infinite plane sheet with surface charge density \( \sigma \).
(C) Electric field outside (distance \( r > 0 \) from center) of a uniformly charged spherical shell with surface charge density \( \sigma \), and radius \( R \).
(D) Electric field between two oppositely charged infinite plane parallel sheets with uniform surface charge density \( \sigma \).
List - II:
(I) \( \frac{\sigma}{\epsilon_0} \)
(II) \( \frac{\sigma}{2\epsilon_0} \)
(III) 0
(IV) \( \frac{\sigma}{\epsilon_0 r^2} \) Choose the correct answer from the options given below:
Consider the following statements:
A. Surface tension arises due to extra energy of the molecules at the interior as compared to the molecules at the surface of a liquid.
B. As the temperature of liquid rises, the coefficient of viscosity increases.
C. As the temperature of gas increases, the coefficient of viscosity increases.
D. The onset of turbulence is determined by Reynolds number.
E. In a steady flow, two streamlines never intersect.
Choose the correct answer from the options given below: