Question 14

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?

Answer

The volume of a cone (*V)* with radius (*r*) and height (*h)* is given by,

\begin{align}V=\frac{1}{3}\pi r^2h\end{align}

It is given that,

\begin{align}h=\frac{1}{6} r\Rightarrow r =6h\end{align}

\begin{align}\therefore V=\frac{1}{3}\pi (6h)^2.h = 12\pi h^3\end{align}

The rate of change of volume with respect to time (*t)* is given by,

\begin{align} \frac{dV}{dt}=12 \pi \frac{d}{dh}(h^3).\frac{dh}{dt}[By\; Chain\; Rule]\end{align}

\begin{align}=12 \pi (3h^2).\frac{dh}{dt}\end{align}

\begin{align}=36 \pi h^2.\frac{dh}{dt}\end{align}

It is also given that

\begin{align}\frac{dV}{dt}=12\;cm^3/s \end{align}

Therefore, when *h* = 4 cm, we have:

\begin{align}12=36\pi (4)^2.\frac{dh}{dt}\end{align}

\begin{align}\Rightarrow \frac{dh}{dt}=\frac{12}{36\pi (16)}=\frac{1}{48\pi}\end{align}

Hence, when the height of the sand cone is 4 cm, its height is increasing at the rate of

\begin{align}\frac{1}{48\pi}.\end{align}

- 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:-
If

*f(x)*=_{}, show that*fof*(x) = x, for all x ≠ 2/3. What is the inverse of*f*? - 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:-
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.,) - Q:-
A particle moves along the curve 6y = x

^{3}+ 2. Find the points on the curve at which the*y*-coordinate is changing 8 times as fast as the*x*-coordinate. - Q:- Evaluate the determinants

(i) \(\begin{vmatrix}3 & -1 & -2\\0 & 1 & 2\\0 & 0 & 4\end{vmatrix}\) (iii) \(\begin{vmatrix}3 & -4 & 5\\1 & 1 & -2\\2 & 3 & 1\end{vmatrix}\)

(ii) \(\begin{vmatrix}0 & 1 & 2\\-1 & 0 & -3\\-2 & 3 & 0\end{vmatrix}\)(iv) \(\begin{vmatrix}2 & -1 & -2\\0 & 2 & -1\\3 & -5 & 0\end{vmatrix}\) - Q:- Find the principal value of \begin{align} cos^{-1}\left(-\frac{1}{\sqrt2}\right)\end{align}
- Q:-
Consider

*f*: R → R given by*f(x)*= 4x + 3. Show that*f*is invertible. Find the inverse of*f*. - Q:- If a matrix has 24 elements, what are the possible order it can have? What, if it has 13 elements?
- Q:- Integrals (ax + b)
^{2} - Q:-
The degree of the differential equation

\begin{align}\left(\frac{d^2y}{dx^2}\right)^3\;+ \left(\frac{dy}{dx}\right)^2+\;sin\left(\frac{dy}{dx}\right)\;+ 1=\;0\end{align}

is

**(A)**3**(B)**2**(C)**1**(D)**not defined

- NCERT Chapter