Maths

Let a complex number z, |z| ≥ 1, satisfy log?/√² (|z|+11)/(|z|-1)² ≤ 2. Then, the largest value of |z| is equal to…

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 :

The least value of |z| where z is complex number which satisfies the inequality exp( ((|z|+3)(|z|-1))/(|z|+1) logₑ2 ) ≥ log_√2 |5√7 + 9i|, i = √-1, is equal to :

Let C be the locus of the mirror image of a point on the parabola y² = 4x with respect to the line y = x. Then the equation of tangent to C at P(2, 1) is :

Let the lengths of intercepts on x-axis and y-axis made by the circle x² + y² + ax + 2ay + c = 0, (a < 0) be 22 and 25, respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line x + 2y = 0, 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 :