PUC Science 2nd PUC Class 12 - Karnataka Board PUC Question Bank Solutions for Mathematics

Solve the following initial value problem:
\[\left\{ x \sin^2 \left( \frac{y}{x} \right) - y \right\}dx + x dy = 0, y\left( 1 \right) = \frac{\pi}{4}\]

Solve the following initial value problem:
\[x\frac{dy}{dx} - y + x \sin\left( \frac{y}{x} \right) = 0, y\left( 2 \right) = x\]

Find the particular solution of the differential equation x cos\[\left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x\], given that when x = 1, \[y = \frac{\pi}{4}\]

Find the particular solution of the differential equation \[\left( x - y \right)\frac{dy}{dx} = x + 2y\], given that when x = 1, y = 0.

Show that the family of curves for which \[\frac{dy}{dx} = \frac{x^2 + y^2}{2xy}\], is given by \[x^2 - y^2 = Cx\]

A homogeneous differential equation of the form \[\frac{dx}{dy} = h\left( \frac{x}{y} \right)\] can be solved by making the substitution

Which of the following is a homogeneous differential equation?

Verify Rolle's theorem for the following function on the indicated interval f(x) = x² − 8x + 12 on [2, 6] ?

Verify Rolle's theorem for the following function on the indicated interval f(x) = x² − 4x + 3 on [1, 3] ?

Verify Rolle's theorem for the following function on the indicated interval f (x) = (x − 1) (x − 2)² on [1, 2] ?

Verify Rolle's theorem for the following function on the indicated interval  f (x) = x(x − 1)² on [0, 1] ?

Verify Rolle's theorem for the following function on the indicated interval  f (x) = (x² − 1) (x − 2) on [−1, 2] ?

Verify Rolle's theorem for the following function on the indicated interval   f (x) = x(x − 4)² on the interval [0, 4] ?

Verify Rolle's theorem for the following function on the indicated interval  f(x) = x(x −2)² on the interval [0, 2] ?

Verify Rolle's theorem for the following function on the indicated interval f (x) = x² + 5x + 6 on the interval [−3, −2]  ?

Verify Rolle's theorem for each of the following function on the indicated interval f (x) = cos 2 (x − π/4) on [0, π/2] ?

Verify Rolle's theorem for the following function on the indicated interval  f(x) = sin 2x on [0, π/2] ?

Verify Rolle's theorem for the following function on the indicated interval f(x) = cos 2x on [−π/4, π/4] ?

Verify Rolle's theorem for the following function on the indicated interval f(x) = e^x sin x on [0, π] ?

Verify Rolle's theorem for the following function on the indicated interval f(x) = e^x cos x on [−π/2, π/2] ?