Find the area of the region bounded by the curve y2 = x and the lines x = 1, x = 4 and the x-axis.
The area of the region bounded by the curve, y2 = x, the lines, x = 1 and x = 4, and the x-axis is the area ABCD.
\begin{align}Area \;of\; ABCD = \int_{1}^{4} y.dx \end{align}
\begin{align} = \int_{1}^{4} \sqrt{x}.dx \end{align}
\begin{align} =\left[\frac{x^\frac{3}{2}}{\frac{3}{2}}\right]_1^4 \end{align}
\begin{align} =\frac{2}{3}\left[(4)^\frac{3}{2} - (1)^{\frac{3}{2}}\right] \end{align}
\begin{align} =\frac{2}{3}\left[8 -1\right] \end{align}
\begin{align} =\frac{14}{3}\; Units \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
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).
A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall?
Maximise Z = 3x + 4y
Subject to the constraints:x + y ≤ 4, x ≥ 0, y ≥ 0
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
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.,
)
Find the direction cosines of a line which makes equal angles with the coordinate axes.
If f: R → R be given by f(x) = , then fof(x) is
(A)
(B) x3
(C) x
(D) (3 – x3).
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