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}

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

Show that f : [–1, 1] → R, given by is one-one. Find the inverse of the function f : [–1, 1] → Range f.

(Hint: For y ∈ Range f, y =, for some x in [ - 1, 1], i.e.,)

 Let f : X → Y be an invertible function. Show that f has unique inverse.

(Hint: suppose g1 and g2 are two inverses of f. Then for all y ∈ Y, fog1(y) = 1Y(y) = fog2(y). Use one-one ness of f). 

 Consider f : R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f. 

A balloon, which always remains spherical, has a variable diameter

\begin{align} \frac{3}{2}(2x+1)\end{align}

Find the rate of change of its volume with respect to x.

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

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

The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x = 8 cm and y = 6 cm, find the rates of change of (a) the perimeter, and (b) the area of the rectangle.

In Figure, identify the following vectors.

(i) Coinitial (ii) Equal (iii) Collinear but not equal