Question 6

The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of increase of its circumference?

Answer

The circumference of a circle (*C*) with radius (*r) *is given by

*C* = 2π*r.*

Therefore, the rate of change of circumference (C) with respect to time (*t)* is given by,

\begin{align} \frac{dC}{dt}=\frac{dC}{dr}.\frac{dr}{dt}\;\;\; [By\; Chain\; Rule]\end{align}

\begin{align} =\frac{d}{dr}(2\pi r).\frac{dr}{dt}\end{align}

\begin{align} =2\pi.\frac{dr}{dt}\end{align}

It is given that

\begin{align} \frac{dr}{dt}=0.7\; cm/s\end{align}

Hence, the rate of increase of the circumference 2π(0.7)=1.4π cm/s

- Q:- Given an example of a relation. Which is

(i) Symmetric but neither reflexive nor transitive.

(ii) Transitive but neither reflexive nor symmetric.

(iii) Reflexive and symmetric but not transitive.

(iv) Reflexive and transitive but not symmetric.

(v) Symmetric and transitive but not reflexive. - Q:- Determine whether each of the following relations are reflexive, symmetric and transitive:

(i) Relation R in the set A = {1, 2, 3,13, 14} defined as

R = {(x, y): 3x − y = 0}

(ii) Relation R in the set N of natural numbers defined as

R = {(x, y): y = x + 5 and x < 4}

(iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as

R = {(x, y): y is divisible by x}

(iv) Relation R in the set Z of all integers defined as

R = {(x, y): x − y is as integer}

(v) Relation R in the set A of human beings in a town at a particular time given by

(a) R = {(x, y): x and y work at the same place}

(b) R = {(x, y): x and y live in the same locality}

(c) R = {(x, y): x is exactly 7 cm taller than y}

(d) R = {(x, y): x is wife of y}

(e) R = {(x, y): x is father of y} - Q:- Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.
- Q:- Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as

R = {(a, b): b = a + 1} is reflexive, symmetric or transitive. - Q:- Show that the relation R in the set R of real numbers, defined as R = {(a, b): a ≤ b
^{2}} is neither reflexive nor symmetric nor transitive. - Q:-
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 + x^{2 } - Q:- Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1, L2): L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.
- Q:-
Show that the Modulus Function

*f*: R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative. - Q:-
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. - Q:- Show that the relation R in R defined as R = {(a, b): a ≤ b}, is reflexive and transitive but not symmetric.

- Q:-
The total revenue in Rupees received from the sale of

*x*units of a product is given byR (x) = 3x

^{2}+ 36x + 5. The marginal revenue, when x = 15 is(A) 116 (B) 96 (C) 90 (D) 126

- Q:-
Consider

*f*: R_{+}→ [– 5, ∞) given by*f(x)*= 9x^{2}+ 6x – 5. Show that*f*is invertible

with**.** - Q:- Find the principal value of \begin{align} cos^{-1}\left(\frac{1}{2}\right) + 2sin^{-1}\left(\frac{1}{2}\right)\end{align}
- Q:-
A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/s. At the instant when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing?

- Q:-
If

*f(x)*=_{}, show that*fof*(x) = x, for all x ≠ 2/3. What is the inverse of*f*? - Q:- Given an example of a relation. Which is

(i) Symmetric but neither reflexive nor transitive.

(ii) Transitive but neither reflexive nor symmetric.

(iii) Reflexive and symmetric but not transitive.

(iv) Reflexive and transitive but not symmetric.

(v) Symmetric and transitive but not reflexive. - Q:-
A balloon, which always remains spherical, has a variable diameter

\begin{align} \frac{3}{2}(2x+1)\end{align}

Find the rate of change of its volume with respect to

*x*. - Q:-
Determine order and degree(if defined) of differential equation y

^{m}+ 2y^{n}+ y' =0 - Q:-
Let

*f*: X → Y be an invertible function. Show that the inverse of*f*is f, i.e.,^{–1}*(f*=^{–1})^{–1}*f*. - Q:-
Sand is pouring from a pipe at the rate of 12 cm

^{3}/s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?

- NCERT Chapter