Question 7

The length *x* of a rectangle is decreasing at the rate of 5 cm/minute and the width *y* is increasing at the rate of 4 cm/minute. When *x* = 8 cm and *y* = 6 cm, find the rates of change of (a) the perimeter, and (b) the area of the rectangle.

Answer

Since the length (*x)* is decreasing at the rate of 5 cm/minute and the width (*y)* is increasing at the rate of 4 cm/minute, we have:

\begin{align} \frac{dx}{dt} = -5 \;cm/min\; and \; \frac{dy}{dt}= 4 \;cm/min\end{align}

(a) The perimeter (*P)* of a rectangle is given by,

*P* = 2(*x + y*)

\begin{align} \therefore\frac{dp}{dt} = 2\left(\frac{dx}{dt} + \frac{dy}{dt}\right)= 2(-5+4)=-2\;cm/min\end{align}

Hence, the perimeter is decreasing at the rate of 2 cm/min.

(b) The area (*A)* of a rectangle is given by,

*A* = *x**⋅** y*

\begin{align} \therefore\frac{dA}{dt} = \frac{dx}{dt}.y + x.\frac{dy}{dt}=-5y + 4x \end{align}

When *x* = 8 cm and *y* = 6 cm,

\begin{align} \frac{dA}{dt} = (-5 \times 6 + 4 \times 8)\; cm^2/min = 2\; cm^2/min\end{align}

Hence, the area of the rectangle is increasing at the rate of 2 cm^{2}/min.

- 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:- Find the principal value of \begin{align} cosec^{-1}\left({2}\right)\end{align}
- Q:-
Let

*f*: N → N be defined by

State whether the function*f*is bijective. Justify your answer. - 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 - 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:- Find the value of if \begin{align} sin^{-1} x= y,\end{align}

then

\begin{align} (A) 0 ≤ y ≤ x \;\;\;\;\; (B) -\frac{\pi}{2} ≤ y ≤ \frac{\pi}{2} \end{align}

\begin{align} (C)\; 0 < y < \pi \;\;\;\;\; (D) -\frac{\pi}{2} < y < \frac{\pi}{2} \end{align} - Q:-
A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the later is 10 cm.

- Q:-
Let

*f*: R → R be defined as f(x) = 3x. Choose the correct answer.(A)

*f*is one-one onto(B)

*f*is many-one onto(C)

*f*is one-one but not onto(D)

*f*is neither one-one nor onto. - Q:-
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?

- Q:-
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/s^{2} - Q:- sin 2x – 4e
^{3x}

- NCERT Chapter