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

Solve the following :

Find the vector equation of the plane which is at a distance of 5 units from the origin and which is normal to the vector `2hat"i" + hat"j" + 2hat"k"`.

Find the vector equation of the plane passing through the points A(1, -2, 1), B(2, -1, -3) and C(0, 1, 5).

Solve the following :

Find the cartesian equation of the plane passing through A(1,-2, 3) and direction ratios of whose normal are 0, 2, 0.

Solve the following :

Find the cartesian equation of the plane passing through A(7, 8, 6) and parallel to the plane `bar"r".(6hat"i" + 8hat"j" + 7hat"k")` = 0.

The foot of the perpendicular drawn from the origin to a plane is M(1, 2, 0). Find the vector equation of the plane.

Solve the following :

A plane makes non zero intercepts a, b, c on the coordinate axes. Show that the vector equation of the plane is `bar"r".(bchat"i" + cahat"j" + abhat"k")` = abc.

Solve the following :

Find the vector equation of the plane passing through the point A(– 2, 3, 5) and parallel to the vectors `4hat"i" + 3hat"k" and hat"i" + hat"j"`.

Solve the following :

Find the cartesian equation of the plane `bar"r" = lambda(hat"i" + hat"j" - hat"k") + mu(hat"i" + 2hat"j" + 3hat"k")`.

Solve the following :

Find the cartesian equations of the planes which pass through A(1, 2, 3), B(3, 2, 1) and make equal intercepts on the coordinate axes.

Solve the following :

Find the vector equation of the plane which makes equal non zero intercepts on the coordinate axes and passes through (1, 1, 1).

Solve the following :

Find the vector equation of the plane passing through the origin and containing the line `bar"r" = (hat"i" + 4hat"j" + hat"k") + lambda(hat"i" + 2hat"j" + hat"k")`.

Solve the following :

Find the vector equation of the plane which bisects the segment joining A(2, 3, 6) and B(4, 3, –2) at right angle.

Solve the following :

Show that the lines x = y, z = 0 and x + y = 0, z = 0 intersect each other. Find the vector equation of the plane determined by them.

Find the equation of the tangent to the curve at the point on it.

y = x² + 2e^x + 2 at (0, 4)

Find the equations of tangents and normals to the following curves at the indicated points on them : x³ + y³ – 9xy = 0 at (2, 4)

Find the equations of tangents and normals to the following curves at the indicated points on them:

`x^2 - sqrt(3)xy + 2y^2 = 5  at  (sqrt(3), 2)`

Find the equations of tangents and normals to the following curves at the indicated points on them : 2xy + π sin y = `2pi  "at" (1, pi/2)`

Find the equations of tangents and normals to the following curves at the indicated points on them : x sin 2y = y cos 2x at `(pi/4, pi/2)`

Find the equations of tangents and normals to the following curve at the indicated points on them:

x = sin θ and y = cos 2θ at θ = `pi/(6)`

Find the equations of tangents and normals to the following curves at the indicated points on them : `x = sqrt(t), y = t  - (1)/sqrt(t)` at = 4.