Recent questions tagged bookproblem

Show that the function given by $f(x) = e^{2x}$ is strictly increasing on $R$.

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Show that the function given by $f (x) = 3x + 17$ is strictly increasing on $R$.

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Choose the correct answer in the total revenue in Rupees received from the sale of $x$ units of a product is given by $Rx = 3x^2 +36x + 5$. The marginal revenue, when $x = 15$ is

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Choose the correct answer in the rate of change of the area of a circle with respect to its radius $r$ at $r = 6$ cm is

asked Nov 23, 2012 by thanvigandhi_1

1 answer

The total revenue in Rupees received from the sale of $x$ units of a product is given by $ R(x) = 13x^2 + 26x + 15.$ Find the marginal revenue when $x = 7$.

asked Nov 23, 2012 by thanvigandhi_1

1 answer

The total cost $C(x)$ in Rupees associated with the production of $x$ units of an item is given by $ C(x) = 0.007x^3 - 0.003x^2 + 15x + 4000.$ Find the marginal cost when 17 units are produced.

asked Nov 23, 2012 by thanvigandhi_1

1 answer

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?

asked Nov 23, 2012 by thanvigandhi_1

1 answer

A balloon, which always remains spherical, has a variable diameter $\large\frac{3}{2}$$(2x+1)$. Find the rate of change of its volume with respect to $x.$

asked Nov 23, 2012 by thanvigandhi_1

1 answer

The radius of an air bubble is increasing at the rate of $ \large\frac{1}{2}$ cm/s. At what rate is the volume of the bubble increasing when the radius is $1\; cm$?

asked Nov 23, 2012 by thanvigandhi_1

1 answer

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.

asked Nov 23, 2012 by thanvigandhi_1

1 answer

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 2cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall ?

asked Nov 23, 2012 by thanvigandhi_1

1 answer

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 $10\; cm$.

asked Nov 22, 2012 by thanvigandhi_1

1 answer

A balloon, which always remains spherical on inflation, is being inflated by pumping in $900$ cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is $15\; cm$.

asked Nov 22, 2012 by thanvigandhi_1

1 answer

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

asked Nov 22, 2012 by thanvigandhi_1

1 answer

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

asked Nov 22, 2012 by thanvigandhi_1

1 answer

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?

asked Nov 22, 2012 by thanvigandhi_1

1 answer

An edge of a variable cube is increasing at the rate of $3\; cm/s$. How fast is the volume of the cube increasing when the edge is $10\; cm$ long?

asked Nov 22, 2012 by thanvigandhi_1

1 answer

The radius of a circle is increasing uniformly at the rate of $3\; cm/s.$ Find the rate at which the area of the circle is increasing when the radius is $10\; cm$.

asked Nov 22, 2012 by thanvigandhi_1

1 answer

A manufacturer produces three products $ x, y, z $ which he sells in two markets. Annual sales are indicated below: \[ \begin{array} { c c } \textbf{Market} & \textbf{Products} \\ I & 10,000 \quad 2,000 \quad 18,000 \\ II & 6,000 \quad 20,000 \quad 8,000 \end{array} \]
If unit sale prices of x, y and z are Rs 2.50, Rs 1.50 and Rs 1.00, respectively, find the total revenue in each market

asked Nov 22, 2012 by pady_1

1 answer

Define a binary operation $\ast$ on the set $\{0, 1, 2, 3, 4, 5\}$ as \[ a \ast b = \left\{ \begin{array} {1 1} a+b, & \quad \text{ if a$+$b $<$ 6} \\ a+b-6, & \quad \text{ if a+b $\geq$ 6} \\ \end{array} \right. \] Show that zero is the identity for this operation and each element $a\neq0$ of the set is invertible with $6-a$ being the inverse of $a$.

asked Nov 22, 2012 by vaishali.a

1 answer

Find the matrix $ X $ so that $ X \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} \: = \: \begin{bmatrix} -7 & -8 & -9 \\ 2 & 4 & 6 \end{bmatrix} $

asked Nov 22, 2012 by pady_1

1 answer

Given a non-empty set $ X,$ let $\ast :\; P(X)\; \times\; P(X) \to P(X) $ be defined as $A \ast B = \; ( A-B)\; \cup \; (B-A),\; \forall A, B \in \; P(X).$. Show that the empty set $\emptyset $ is the identity for the operation $\ast$ and all the elemnets $A$ of $ P(X) $ are invertible with $ A^{-1} \;= A$.

asked Nov 22, 2012 by vaishali.a

1 answer

Consider the binary operation $\ast :\; R \times R \rightarrow R$ and $o :\; R \times R \rightarrow R$ defined as $a \ast b = | a \text{-b}|$ and $\;a\;o\;b=a, \forall a,\;b \in R.$ Show that $\ast$ is commutative but not associative, $o$ is associative but not commutative. Further, show that $\forall\; a,\; b,\; c \in R,\; a\; \ast\; (b\; o\; c) = (a \ast b) \;o\; (a \ast c)$. [If it is so, we say that the operation $\ast$ distributes over $o$]. Does $o$ distribute over? Justify your answer.

asked Nov 22, 2012 by vaishali.a

1 answer

If $A$ and $B$ are square matrices of the same order such that $AB = BA $, then prove by induction that $AB^n = B^nA $. Further, prove that $ (AB)^n = A^nB^n \: $ for all $ n ∈ N. $

asked Nov 22, 2012 by pady_1

1 answer

Find the number of all onto functions from the set $\{1, 2, 3, ... , n\}$ to itself.

asked Nov 22, 2012 by vaishali.a

1 answer

Given a non-empty set $X$, consider the binary operation $\ast : P(X) × P(X) \to P(X)$ given by $A \ast B=A \cap B\; \forall A, $ $ B \;in\; P(X),$ where $P(X)$ is the power set of $X$. Show that $X$ is the identity element for this operation and $X$ is the only invertible element in $P(X)$ with respect to the operation $\ast$.

asked Nov 22, 2012 by vaishali.a

1 answer

If the matrix A is both symmetric and skew symmetric, then

asked Nov 22, 2012 by pady_1

1 answer

If $A $ is a square matrix, such that $ A^2 = A $ , then $ (1 + A )^3 - 7A $ is equal to:

asked Nov 22, 2012 by pady_1

1 answer

Let $S=\{a,b,c\}\;$ and$ \;T = \{1,2,3\}$. Find the inverse of the following function $F$ from $S$ to $T$, if it exists - \[\;\; F=\{(a,3), (b,2), (c,1)\}\]

asked Nov 22, 2012 by vaishali.a

1 answer

Given a non empty set $X$, consider $P(X)$ which is the set of all subsets of $X$. Define the relation $R$ in $P(X)$ as follows: For subsets $A,\; B$ in $ P(X),\; ARB$ if and only if $ A \subset B $ Is $R$ an equivalence relation on $P(X)$?

asked Nov 22, 2012 by vaishali.a

1 answer

Give examples of two functions $f: N \to N$ and $g: N \to N$ such that $g\;o\;f $ is onto but $f$ is not onto.

asked Nov 21, 2012 by vaishali.a

1 answer

Give examples of two functions $f : N \to Z $ and $g: Z \to Z$ such that $g\;o\;f$ is injective but $g$ is not injective.

asked Nov 21, 2012 by vaishali.a

1 answer

Show that the function $f : R \to R$ given by $f (x) = x^3$ is injective.

asked Nov 21, 2012 by vaishali.a

1 answer

Show that the function f:R $\rightarrow \{ x \in$ R:-1$<$x$<$1 $\}$ defined by $f(x) = \frac {x} { 1+|\;x\;|}, x \in R$ is one-one and onto function.

asked Nov 21, 2012 by vaishali.a

1 answer

If $(f:R \to R)$ is defined by $f(x) = x^2$ - $3x+2$. Find $f(f(x))$:

asked Nov 21, 2012 by vaishali.a

1 answer

Let $f:W \to W$ be defined as $f(n)=n$ - $1$, if $n\;is\;odd\;and\; f(n)=n+1,\;if\;n\;is\; even.$ Show that $f$ is invertible. Find the inverse of $f$. Here, $W$ is the set of all whole numbers.

asked Nov 21, 2012 by vaishali.a

1 answer

Let $f:R \to R$ be defined as $f(x)=10x+7.$Find the function $g:R \to R$ such that $g\;o\;f = f\;o\;g = I_R.$

asked Nov 21, 2012 by vaishali.a

1 answer

Consider a binary operation $\ast$ on $N$ defined as $a \ast b = a^3 + b^3$. Choose the correct answer:

asked Nov 21, 2012 by vaishali.a

1 answer

If $( x-a)^2 + (y-b)^2 = c^2$, for some $ c > 0$, prove that $\Large\frac{\begin{bmatrix} 1 + \left(\frac{dy}{dx}\right)^2 \\[0.3em] \end{bmatrix}^{\frac{\Large 3}{\Large 2}}}{\Large\frac{d^2y}{dx^2}}$is a constant independent of $a$ and $b$.

asked Nov 21, 2012 by thanvigandhi_1

1 answer

If $ x \sqrt {1+y} + y \sqrt{1+x} = 0$, for, $-1 < x < 1$, prove that $\large \frac{dy}{dx} = -\frac{1}{(1+x)^2} $

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Let $(A=N \times N \,and\, * )$ be the binary operation on $(A)$ defined by $( (a, b) * (c, d) = (a + c, b + d))$. Show that * is commutative and associative. Find the identity element for * on $( A )$, if any.

asked Nov 21, 2012 by vaishali.a

1 answer

Find if the given operation has identity: $\;\; a \ast b = a^2 + b^2$

asked Nov 21, 2012 by vaishali.a

2 answers

Let $\ast$ be a binary operation on the set $Q$ of rational numbers as follows: $\;\; a \ast b = a-b$. Find which of the binary operations are commutative and which are associative.

asked Nov 21, 2012 by vaishali.a

1 answer

Let $\ast$ be the binary operation on $N$ defined by $a \ast b=H.C.F. $ of a and b. Is $\ast$ commutative? Is $\ast$ associative? Does there exist identity for this binary operation on $N$?

asked Nov 21, 2012 by vaishali.a

1 answer

Differentiate w.r.t. x the function $cot^{-1} \begin{bmatrix} \frac{{\sqrt {1+sin \: x}} + {\sqrt{1-sin \: x}}} {{\sqrt {1+sin \: x}} - {\sqrt{1-sin \: x}}} \\[0.3em] \end{bmatrix}$, where $0 < x < \frac{\pi}{2}$

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Is $\ast$ defined on the set $\{1,2,3,4,5\}$ by $a\ast b=L.C.M$.of $a$ and $b$ a binary operation?

asked Nov 21, 2012 by vaishali.a

1 answer

Find $\large \frac{dy}{dx} $, if $ y = \sin^{-1}x + \sin^{-1} \: \sqrt {1-x^2} , -1 \leq x \leq 1 $

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Find $ \large\frac{dy}{dx} $, if $ y = 12 ( 1 - \cos t ) , x = 10 ( t - \sin t ) , -\large\frac{\pi}{2}\normalsize < t <\large \frac{\pi}{2} $

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Let $\ast$ be the binary operation on $N$ given by $a\ast b=L.C.M\,.of\,a\,and\,b$. Find $\begin{array}{1 1}(i)\;\; 5 \ast7,\; 20 \ast 16 & (ii)\;\; Is\; \ast \;commutative?\\(iii)\;\; Is\; \ast \;associative? & (iv)\;\; Find\, the\,identity \, of \ast \, in N\\(v)\;\; Which \, elements\, of \, N\, are\, invertible\, for\, the\, operaation\,\ast ? & \;\end{array}$

asked Nov 21, 2012 by vaishali.a

2 answers

Let $\ast '$ be the binary operation on the set $\{1, 2, 3, 4, 5\}$ defined by $a \ast ' b = H.C.F$ of a and b. Is the operation same as the operation $\ast$ defined in the table below? $\begin{matrix} *&1&2&3&4&5 \\ 1&1&1&1&1&1 \\ 2&1&2&1&2&1 \\ 3&1&1&3&1&1 \\ 4&1&2&1&4&1 \\ 5&1&1&1&1&5 \end{matrix}$

asked Nov 21, 2012 by vaishali.a

1 answer

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