Commerce (English Medium) Class 12 - CBSE Question Bank Solutions

Which of the following differential equation has y = x as one of its particular solution?

A. `(d^2y)/(dx^2) - x^2 (dy)/(dx) + xy = x`

B. `(d^2y)/(dx^2) + x dy/dx + xy = x`

C. `(d^2y)/(dx^2) - x^2 dy/dx + xy = 0`

D. `(d^2y)/(dx^2) + x dy/dx + xy = 0`

Form the differential equation representing the family of curves given by (x – a)² + 2y² = a², where a is an arbitrary constant.

Form the differential equation of the family of circles in the first quadrant which touch the coordinate axes.

Find the slope of the tangent to the curve y = 3x⁴ − 4x at x = 4.

Find the slope of the tangent to the curve y = (x -1)/(x - 2), x != 2 at x = 10.

Find the slope of the tangent to curve y = x³ − x + 1 at the point whose x-coordinate is 2.

Find the slope of the tangent to the curve y = x³ − 3x + 2 at the point whose x-coordinate is 3.

Find the slope of the normal to the curve x = acos³θ, y = asin³θ at `theta = pi/4`

Find a point on the curve y = (x − 2)² at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4).

Find the equation of all lines having slope −1 that are tangents to the curve  `y = 1/(x -1), x != 1`

Find the equation of all lines having slope 2 which are tangents to the curve `y =   1/(x- 3), x != 3`

Find the equations of all lines having slope 0 which are tangent to the curve  y =   `1/(x^2-2x + 3)`

Find points on the curve `x^2/9 + "y"^2/16 = 1` at which the tangent is parallel to x-axis.

Find the equations of the tangent and normal to the given curves at the indicated points:

y = x⁴ − 6x³ + 13x² − 10x + 5 at (0, 5)

Find the equations of the tangent and normal to the given curves at the indicated points:

y = x⁴ − 6x³ + 13x² − 10x + 5 at (1, 3)

Find the equations of the tangent and normal to the given curves at the indicated points:

y = x³ at (1, 1)

Find the equations of the tangent and normal to the given curves at the indicated points:

y = x² at (0, 0)

Find the equations of the tangent and normal to the given curves at the indicated points:

x = cos t, y = sin t at  t = `pi/4`