HSC Arts (English Medium) इयत्ता १२ वी - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

If f(x) = x³ + x − 2, find (f⁻¹)'(0).

Using derivative, prove that: tan ^–1x + cot^–1x = `pi/(2)`

Using derivative, prove that: sec^–1x + cosec^–1x = `pi/(2)`    ...[for |x| ≥ 1]

Choose the correct option from the given alternatives :

If g is the inverse of function f and f'(x) = `(1)/(1 + x)`, then the value of g'(x) is equal to :

If y = f(x) is a differentiable function of x, then show that `(d^2x)/(dy^2) = -(dy/dx)^-3.("d^2y)/(dx^2)`.

Find the length of the perpendicular (2, –3, 1) to the line `(x + 1)/(2) = (y - 3)/(3) = (z + 1)/(-1)`.

Find the co-ordinates of the foot of the perpendicular drawn from the point `2hati - hatj + 5hatk` to the line `barr = (11hati - 2hatj - 8hatk) + λ(10hati - 4hatj - 11hatk).` Also find the length of the perpendicular.

Find the perpendicular distance of the point (1, 0, 0) from the line `(x - 1)/(2) = (y + 1)/(-3) = (z + 10)/(8)` Also find the co-ordinates of the foot of the perpendicular.

A(1, 0, 4), B(0, -11, 13), C(2, -3, 1) are three points and D is the foot of the perpendicular from A to BC. Find the co-ordinates of D.

If the lines `(x - 1)/2 = (y + 1)/3 = (z - 1)/4 and (x - 3)/1 = (y - k)/2 = z/1` intersect each other, then find k.

Find the vector equation of a plane which is at 42 unit distance from the origin and which is normal to the vector `2hati + hatj - 2hatk`.

Find the perpendicular distance of the origin from the plane 6x – 2y + 3z – 7 = 0.

Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 2x + 6y – 3z = 63.

Reduce the equation `bar"r".(3hat"i" + 4hat"j" + 12hat"k")` to normal form and hence find
(i) the length of the perpendicular from the origin to the plane
(ii) direction cosines of the normal.

Find the vector equation of the plane passing through the point having position vector `hati + hatj + hatk` and perpendicular to the vector `4hati + 5hatj + 6hatk`.

Show that the line `bar"r" = (2hat"j" - 3hat"k") + lambda(hat"i" + 2hat"j" + 3hat"k") and bar"r" = (2hat"i" + 6hat"j" + 3hat"k") + mu(2hat"i" + 3hat"j" + 4hat"k")` are coplanar. Find the equation of the plane determined by them.

Find the co-ordinates of the foot of the perpendicular drawn from the point (0, 2, 3) to the line `(x + 3)/(5) = (y - 1)/(2) = (z + 4)/(3)`.

Choose correct alternatives:

If the line `x/(3) = y/(4)` = z is perpendicular to the line `(x - 1)/k = (y + 2)/(3) = (z - 3)/(k - 1)`, then the value of k is ______.

Choose correct alternatives :

The length of the perpendicular from (1, 6,3) to the line `x/(1) = (y - 1)/(2) =(z - 2)/(3)`

Choose correct alternatives :

The lines `x/(1) = y/(2) = z/(3) and (x - 1)/(-2) = (y - 2)/(-4) = (z - 3)/(6)` are