Class 12 Mathematics - Chapter Differential Equations NCERT Solutions | Determine order and degree(if defined) o

Determine order and degree(if defined) of differential equation y' + 5y = 0

Determine order and degree(if defined) of differential equation

\begin{align}\left(\frac{d^2y}{dx^2}\right)^2\;+\;cos\left(\frac{dy}{dx}\right)\;=\;0\end{align}

Determine order and degree(if defined) of differential equation (y^m)² + (yⁿ)³ + (y')⁴ + y⁵ =0

Determine order and degree(if defined) of differential equation yⁿ + 2y^' + siny = 0

The order of the differential equation

\begin{align}2x^2\frac{d^2y}{dx^2}\;- \;3\frac{dy}{dx}\;+ y=\;0\end{align}

is (A) 2 (B) 1 (C) 0 (D) not defined

Determine order and degree(if defined) of differential y^' + y =e^x

y = Ax : xy^' = y (x ≠ 0)

Determine order and degree(if defined) of differential equation y^m + 2yⁿ + y' =0

\begin{align} y = xsinx:xy{'}=y +x\sqrt{x^2 -y^2}(x\neq0\; and\; x>y\; or\; x<-y)\end{align}

y = x² + 2x + C : y^' - 2x - 2 = 0   

Prove that the function f(x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5.

Represent graphically a displacement of 40 km, 30° east of north.

If a line makes angles 90°, 135°, 45° with x, y and z-axes respectively, find its direction cosines.

Maximise Z = 3x + 4y

Subject to the constraints:x + y ≤ 4, x ≥ 0, y ≥ 0

Find the area of the region bounded by the curve y² = x and the lines x = 1, x = 4 and the x-axis.

Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E ∩ F) = 0.2, find P (E|F) and P(F|E).

If a line has the direction ratios −18, 12, −4, then what are its direction cosines?

Let f, g and h be functions from R to R. Show that

(f + g)oh = foh + goh

(f . g)oh = (foh) . (goh) 

Check the injectivity and surjectivity of the following functions:

(i) f : N → N given by f(x) = x²

(ii) f : Z → Z given by f(x) = x²

(iii) f : R → R given by f(x) = x²

(iv) f : N → N given by f(x) = x³

(v) f : Z → Z given by f(x) = x³ 

 Let A and B be sets. Show that f : A × B → B × A such that f(a, b) = (b, a) is bijective function.

The rate of change of the area of a circle with respect to its radius r at r = 6 cm is

(A) 10π (B) 12π (C) 8π (D) 11π

 Prove that the Greatest Integer Function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.