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

Minimize z = x + 2y,

Subject to x + 2y ≥ 50, 2x – y ≤ 0, 2x + y ≤ 100, x ≥ 0, y ≥ 0.

Find the coordinates of the point of intersection of the pair of lines 6x² + 5xy – 4y² + 7x + 13y – 3 = 0.

Find the equation of the plane containing the lines `(x - 1)/2 = (y + 1)/-1 = z/3` and `x/2 = (y - 2)/-1 = (z + 1)/3`.

Find the point of intersection of the lines given by x² + 3xy + 2y² + x – y – 6 = 0

Find the values of p and q if the equation px² – 6xy + y² + 18x – qy + 8 = 0 represents a pair of parallel lines.

If Rolle's theorem holds for the function f(x) = x³ + bx² + ax + 5, x ∈ [1, 3] with c = `2 + 1/sqrt(3)`, find the values of a and b.

Reduce the equation `barr*(3hati - 4hatj + 12hatk)` = 3 to the normal form and hence find the length of perpendicular from the origin to the plane.

The slope of tangent at any point on the curve is 3. lf the curve passes through (1, 1), then the equation of curve is ______.

Draw the rough graph and shade the feasible region for the inequalities x + y ≥ 2, 2x + y ≤ 8, x ≥ 0, y ≥ 0.

Find the equation of plane which is at a distance of 4 units from the origin and which is normal to the vector `2hati - 2hatj + hatk`.

Find `dy/dx`, if y = `sec^-1((1 + x^2)/(1 - x^2))`.

If x – y ≥ 8, x ≥ 3, y ≥ 3, x ≥ 0, y ≥ 0 then find the coordinates of the corner points of the feasible region.

Solve:

`xsinx dy/dx + (xcosx + sinx)y` = sin x

The coordinates of the foot of the perpendicular from the point P(1, 0, 0) in the line `(x - 1)/2 = (y + 1)/-3 = (z + 10)/8` are ______.

If y = `cos^-1 sqrt((1 + x^2)/2`, then `dy/dx` = ______.

If y = `sin^-1((2tanx)/(1 + tan^2x))`, find `dy/dx`.

Find the vector equation of the line passing through the point (–2, 1, 4) and perpendicular to the plane `barr*(4hati - 5hatj + 7hatk)` = 15

Find the equation of the plane which contains the line of intersection of the planes x + 2y + 4z = 4 and 2x – 3y – z = 9 and which is perpendicular to the plane 4x – 3y + 5z = 10.

Find the point of intersection of the line `(x + 1)/2 = (y - 1)/3 = (z - 2)/1` with the plane x + 2y – z = 6.

Evaluate:

`int x/((x + 2)(x - 1)^2)dx`