In Figure, identify the following vectors.
(i) Coinitial (ii) Equal (iii) Collinear but not equal
\begin{align} (i) \;Vectors\; \overrightarrow{a}\; and\; \overrightarrow{d}\; are \;coinitial\; because\; they\; have\; the\; same \;initial \;point. \end{align}
\begin{align}(ii)\; Vectors\;\overrightarrow{b} \;and\;\overrightarrow{d}\; are\; equal\; because\; they\; have\; the\; same \;magnitude \;and\; direction. \end{align}
\begin{align}(iii)\; Vectors\;\overrightarrow{a} \;and\; \overrightarrow{c} \;are\; collinear\; but\; not\; equal\;. This\; is\; because\; although\; they\; are \;parallel,\; their\; directions\; are\; not \;the\; same.\end{align}
Answer the following as true or false.
\begin{align}(i) \overrightarrow{a}\; and\; \overrightarrow{-a}\; are\; collinear.\end{align}
(ii) Two collinear vectors are always equal in magnitude.
(iii) Two vectors having same magnitude are collinear.
(iv) Two collinear vectors having the same magnitude are equal.
Classify the following measures as scalars and vectors.
(i) 10 kg (ii) 2 metres north-west (iii) 40°
(iv) 40 watt (v) 10–19 coulomb (vi) 20 m/s2
Classify the following as scalar and vector quantities.
(i) time period (ii) distance (iii) force
(iv) velocity (v) work done
Represent graphically a displacement of 40 km, 30° east of north.
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}
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).
Consider f : R+ → [– 5, ∞) given by f(x) = 9x2 + 6x – 5. Show that f is invertible
with .
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 =ex
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