Maths

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 :

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 :

Given that the inverse trigonometric functions take principal values only. Then, the number of real values of x which satisfy sin⁻¹(3x/5) + sin⁻¹(4x/5) = sin⁻¹x is equal to :

Consider a rectangle ABCD having 5,7,6,9 points in the interior of the segments AB, CD, BC, DA respectively. Let α be the number of triangles having these points from different sides as vertices and β be the number of quadrilaterals having these points from different sides as vertices. Then (β-α) is equal to :

Let f be a real valued function, defined on R - {-1, 1} and given by f(x) = 3logₑ|(x-1)/(x+1)| - 2/(x-1). Then in which of the following intervals, function f(x) is increasing?

If the foot of the perpendicular from point (4, 3, 8) on the line L₁ : (x-a)/l = (y-2)/3 = (z-b)/4, l ≠ 0 is (3, 5, 7) then the shortest distance between the line L₁ and line L₂ : (x-2)/3 = (y-4)/4 = (z-5)/5 is equal to:

If (x,y,z) be an arbitrary point lying on a plane P which passes through the points (42, 0, 0), (0, 42, 0) and (0, 0, 42), then the value of the expression
3 + (x-11)/((y-19)²(z-12)²) + (y-9)/((x-11)²(z-12)²) + (z-12)/((x-11)²(y-19)²) - (x+y+z)/(14(x-11)(y-19)(z-12))
is equal to :

The maximum value of f(x) =

| sin²x 1+cos²x cos2x |
| 1+sin²x cos²x cos2x |, x ∈ R is:
| sin²x cos²x sin2x |

Kindly considwer the following