Class 12 Mathematics - Chapter Probability NCERT Solutions | Given that E and F are events such that

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

Determine order and degree(if defined) of differential equation \begin{align} \frac{d^4y}{dx^4}\;+\;\sin(y^m)\;=0\end{align}

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.

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.,)

Maximise Z = 3x + 4y

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

Show that the Signum Function f : R → R, given by

is neither one-one nor onto.

 Consider f : R₊ → [4, ∞) given by f(x) = x² + 4. Show that f is invertible with the inverse f^–1 of f given by , where R₊ is the set of all non-negative real numbers.

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

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

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

Letbe a function defined as. The inverse of f is map g: Range

(A)  

(B) 

(C)  

(D)