HSC Arts (English Medium) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions

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.

Find the point on the curve y = `sqrt(x - 3)` where the tangent is perpendicular to the line 6x + 3y – 5 = 0.

Find the points on the curve y = x³ – 2x² – x where the tangents are parllel to 3x – y + 1 = 0.

Find the equation of the tangents to the curve x² + y² – 2x – 4y + 1 =0 which a parallel to the X-axis.

Find the equations of the normals to the curve 3x² – y² = 8, which are parallel to the line x + 3y = 4.

If the line y = 4x – 5 touches the curves y² = ax³ + b at the point (2, 3), find a and b.

A particle moves along the curve 6y = x³ + 2. Find the points on the curve at which y-coordinate is changing 8 times as fast as the x-coordinate.