Class 12 Mathematics - Chapter Inverse Trigonometric Functions NCERT Solutions | Find the principal value of \begin{align

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.

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).

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

Consider f : R₊ → [– 5, ∞) given by f(x) = 9x² + 6x – 5. Show that f is invertible
with .

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 f : N → N be defined by

 State whether the function f is bijective. Justify your answer.

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.,)

Classify the following as scalar and vector quantities.

(i) time period (ii) distance (iii) force

(iv) velocity (v) work done

y = Ax : xy^' = y (x ≠ 0)

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

 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.