\begin{align} \int \left(\frac {2-3sin x}{cos^2 x}\right) . dx\end{align}
\begin{align} =\int \left(\frac {2}{cos^2 x} - \frac{3sinx}{cos^2x}\right) . dx\end{align}
\begin{align} =2\int sec^2 x .dx - 3 \int tan x . sec x. dx\end{align}
\begin{align} =2 tan x - 3 sec x + C\end{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).
The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of increase of its circumference?
If a line has the direction ratios −18, 12, −4, then what are its direction cosines?
Let A and B be sets. Show that f : A × B → B × A such that f(a, b) = (b, a) is bijective function.
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.
In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
(i) f : R → R defined by f(x) = 3 – 4x
(ii) f : R → R defined by f(x) = 1 + x2
Let f: X → Y be an invertible function. Show that the inverse of f –1 is f, i.e., (f–1)–1 = f.
\begin{align} y= \sqrt{1+x^2} : y^{'}=\frac{xy}{1+x^2}\end{align}