Class 12th

Let C₁ be the curve obtained by the solution of differential equation 2xy dy/dx = y² - x², x > 0. Let the curve C₂ be the solution of 2xy/(x²-y²) = dy/dx. If both the curves pass through (1, 1), then the area enclosed by the curves C₁ and C₂ is equal to :

Let f:S → S where S = (0,∞) be a twice differentiable function such that f(x+1) = xf(x). If g: S → R be defined as g(x) = logₑf(x), then the value of |g''(5)-g''(1)| is equal to :

Let P(x) = x² + bx + c be a quadratic polynomial with real coefficients such that ∫₀¹ P(x)dx = 1 and P(x) leaves remainder 5 when it is divided by (x - 2). Then the value of 9(b + c) is equal to:

If y = y(x) is the solution of the differential equation dy/dx + (tanx)y = sinx, 0 ≤ x ≤ π/3, with y(0) = 0, then y(π/4) equal to:

Let A denote the event that a 6-digit integer formed by 0,1,2,3,4,5,6 without repetitions, is divisible by 3. Then probability of event A is equal to :

Let A = {2,3,4,5,.,30} and '~' be an equivalence relation on A*A, defined by (a,b) ~ (c,d), if and only if ad = bc. Then the number of ordered pairs which satisfy this equivalence relation with ordered pair (4, 3) is equal to :

When light of wavelength 248 nm falls on a metal of threshold energy 3.0eV, the de-Broglie wavelength of emitted electrons is__________ Angstrom.

The decomposition of formic acid on gold surface follows first order kinetics. If the rate constant at 300 K is 1.0x10^-3  /s and the activation energy E_a = 11.488 kJ/mol, the rate constant at 200 K is ___________ x10-5  /s. (Round off to the nearest Integer).

[Given R = 8.314 J/molK]

The equivalents of ethylene diamine required to replace the neutral ligands from the coordination sphere of the trans-complex of CoCl3.4NH3 is _________.

(Round off to the Nearest Integer).

Let α ∈ R be such that the function f(x) = { [cos⁻¹(1-{x}²)sin⁻¹(1-{x})] / ({x}-{x}³), x≠0; α, x=0 } is continuous at x=0, where {x} = x-[x], [x] is the greatest integer less than or equal to x. Then :