Arts (English Medium) Class 12 - CBSE Question Bank Solutions for Mathematics

Find the area bounded by the parabola y² = 4x and the line y = 2x − 4 By using vertical strips.

In Figure ABCD is a regular hexagon, which vectors are:
(i) Collinear
(ii) Equal
(iii) Coinitial
(iv) Collinear but not equal.

Show that y = ae^2x + be^−x is a solution of the differential equation \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} - 2y = 0\]

y² dx + (x² − xy + y²) dy = 0

Verify that the function y = e^−3x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + \frac{dy}{dx} - 6y = 0.\]

In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-

y = e^x + 1            y'' − y' = 0

In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-

`y=sqrt(a^2-x^2)`              `x+y(dy/dx)=0`

Form the differential equation representing the family of curves y = a sin (x + b), where a, b are arbitrary constant.

Form the differential equation representing the family of parabolas having vertex at origin and axis along positive direction of x-axis.

Form the differential equation of the family of circles having centre on y-axis and radius 3 unit.

Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.

Show that the function f given by:

`f(x)={((e^(1/x)-1)/(e^(1/x)+1),"if",x,!=,0),(-1,"if",x,=,0):}"`

is discontinuous at x = 0.

Solve for x `tan^-1((1 - x)/(1 + x)) = 1/2 tan^-1x, x > 0`

The domain of the function y = sin^–1 (– x²) is ______.

The domain of y = cos^–1(x² – 4) is ______.

The domain of the function defined by f(x) = sin^–1x + cosx is ______.

The equation tan^–1x – cot^–1x = `(1/sqrt(3))` has ______.

Prove that `cot(pi/4 - 2cot^-1 3)` = 7

Show that `2tan^-1 (-3) = (-pi)/2 + tan^-1 ((-4)/3)`

If 2 tan^–1(cos θ) = tan^–1(2 cosec θ), then show that θ = π 4, where n is any integer.