NDA || 25 day challenge || Math part II

The differential equation of the system of circles touching the y-axis at the origin is

Let  .Then the number of points in , at which  assumes its local maximum value, is

Find the area of the common region between

Consider the following statements :

1) The function f(x) = [x], where [.] is the greatest integer function defined on R, is continuous at all points except at x = 0.

2) The function f(x) = sin |x| is continuous for all x ∈ R.

Which of the above statements is/are correct?

Find the value of  :

 is equal to:

is equal to:

If , then what is

equal to?

If , then f(x) f(y) f(z) is equal to

The equation of the plane passing through the intersection of the planes 2x + y + 2z = 9, 4x – 5y – 4z = 1 and the point (3, 2, 1) is

If the lines 3y + 4x = 1, y = x + 5 and 5y + bx = 3 are concurrent, then what is the value of b?

 are tangents to a parabola . If  are the lengths of perpendicular from  on any tangent to the parabola, then

The foci of a hyperbola are  and . it touches the y-axis. The length of its transverse axis is

What is the value of for which the vectors  are  perpendicular?

A circle is drawn on the chord of a circle x² + y² = a² as diameter. The chord lies on the line x + y = a . What is the equation of the circle?