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The corner points of the feasible region determined by the following system of linear inequalities: 2x + y ≤ 10, x + 3y ≤ 15, x, y ≥ 0 are (0, 0), (5, 0), (3, 4) and (0, 5). Let Z = px + qy, where p, q > 0. The condition on p and q so that the maximum of Z occurs at both (3, 4) and (0, 5) is (A) p = q (B) p = 2q (C) p = 3q (D) q = 3p?

asked Nov 16, 2012 by fingeazy

1 answer

Find $\large \frac {dy}{dx} $ in the following: $y = \sin^{-1} \big( 2x \sqrt{1 -x^2} \big), -\;{ \frac{1}{\sqrt2} }\;< x < \;{\frac{1}{\sqrt2} }\\ $

asked Nov 16, 2012 by thanvigandhi_1

1 answer

A merchant plans to sell two types of personal computers, a desktop model and a portable model that will cost Rs 25000 and Rs 40000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant would stock to get maximum profit if he does not want to invest more than Rs 70 lakhs and if his profit on the desktop model is Rs 4500 and on portable model is Rs 5000.

asked Nov 16, 2012 by fingeazy

1 answer

A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of Rs17.50 per package on nuts and Rs 7.00 per package on bolts. How many packages of each should be produced each day so as to maximise his profit, if he operates his machines for at the most 12 hours a day?

asked Nov 16, 2012 by fingeazy

1 answer

One kind of cake requires 200g of flour and 25g of fat, and another kind of cake requires 100g of flour and 50g of fat. Find the maximum number of cakes which can be made from 5kg of flour and 1 kg of fat assuming that there is no shortage of the other ingredients used in making the cakes.

asked Nov 16, 2012 by fingeazy

1 answer

Find \( \large\frac {dy}{dx} \) in the following: $ y = cos^{-1} \left(\large\frac{2x}{1 + x^2}\right), -1 < x < 1 $

asked Nov 15, 2012 by thanvigandhi_1

1 answer

Find $ \frac {dy}{dx} $ in the following:$ y = sin^{-1} \left(\frac{1 - x^2}{1 + x^2}\right), 0 < x < 1 $

asked Nov 15, 2012 by thanvigandhi_1

1 answer

Find $ \frac {dy}{dx}$ in the following: $y = tan^{-1} \left(\frac{3x - x^3}{1 - 3x^2}\right), - \; { \frac{1}{\sqrt3} } < x < \;{\frac{1}{\sqrt3} }\\ $

asked Nov 15, 2012 by thanvigandhi_1

1 answer

Find $ \frac {dy}{dx} $ in the following: $ y = sin^{-1} \left(\frac{2x}{1 + x^2}\right) $

asked Nov 15, 2012 by thanvigandhi_1

1 answer

Consider\(f:\{1,2,3\} \to\{a,b,c\}\) given by \(f(1)=a, \,f(2)=b\) and \(f(3)=c\). Find \(f^{-1} \) and show that \((f^{-1})^{-1} = f\).

asked Nov 15, 2012 by vaishali.a

1 answer

Let \(f : X \to Y\) be an invertible function. Show that \(f\) has unique inverse.

asked Nov 15, 2012 by vaishali.a

1 answer

Find $ \frac {dy}{dx} $ in the following: $ \sin^2x +\cos^2y = 1 $

asked Nov 11, 2012 by thanvigandhi_1

1 answer

Find $ \frac {dy}{dx} $ in the following: $ x^3 +x^2y + xy^2 +y^3 = 81 $

asked Nov 11, 2012 by thanvigandhi_1

1 answer

Find $ \frac {dy}{dx} $ in the following: $ x^2 +xy + y^2 = 100 $

asked Nov 11, 2012 by thanvigandhi_1

1 answer

Find $ \frac {dy}{dx} $ in the following: $ xy +y^2 = tan \: x + y $

asked Nov 11, 2012 by thanvigandhi_1

1 answer

Find $\frac {dy}{dx}$ in the following: $ax +by^2 = cos \: y$

asked Nov 11, 2012 by thanvigandhi_1

1 answer

Find $\frac {dy}{dx}$ in the following: $2x +3y = sin \: y$

asked Nov 11, 2012 by thanvigandhi_1

1 answer

Find $\frac {dy}{dx}$ in the following: $2x +3y = sin \: x$

asked Nov 11, 2012 by thanvigandhi_1

1 answer

Find the values of $k$ so that the function $f$ is continuous at the indicated point in $ f(x) = \left\{ \begin{array} {1 1} \large\frac{k\cos x}{\pi - 2x}\normalsize,& \quad { if \; x \neq \large\frac{\pi}{2} }\\ \normalsize 3 ,& \quad { if \; x = \large\frac{\pi}{2} }\\ \end{array} \right. \qquad at \; x = \; \large\frac{\pi}{2} $

asked Nov 11, 2012 by thanvigandhi_1

1 answer

Find the values of $k$ so that the function $f$ is continuous at the indicated point in $ f(x) = \left\{ \begin{array} {1 1} kx+1 ,& \quad\text{ if $ x $ \(\leq 5\)}\\ 3x-5,& \quad \text{if $x$ > 5}\\ \end{array} \right.\qquad at \; x = 5 $

asked Nov 11, 2012 by thanvigandhi_1

1 answer

Find the values of $k$ so that the function $f$ is continuous at the indicated point in $f(x) = \left\{ \begin{array} {1 1} kx+1 ,& \quad\text{ if $ x $ \(\leq \pi\)}\\ \cos x,& \quad \text{if $x$ > \(\pi\)}\\ \end{array} \right.$ at $x = \pi$

asked Nov 11, 2012 by thanvigandhi_1

1 answer

Find the values of $k$ so that the function $f$ is continuous at the indicated point in $ f(x) = \left\{ \begin{array} {1 1} kx^2 ,& \quad\text{ if $ x $ \(\leq 2\)}\\ 3,& \quad \text{if $x$ > 2}\\ \end{array} \right. \qquad at \;x = 2 $

asked Nov 11, 2012 by thanvigandhi_1

1 answer

Examine the continuity of $f$, where $f$ is defined by $ f(x) = \left\{ \begin{array} {1 1} \sin x - \cos x,& \quad\text{ if \(x\) \(\neq\) 0 }\\ -1 ,& \quad \text{if $x$ =0}\\ \end{array} \right. $

asked Nov 10, 2012 by thanvigandhi_1

1 answer

Determine if $f$ defined by $f(x) = \left\{ \begin{array} {1 1} x^2sin \large\frac{ 1 }{x},\normalsize & \quad\text{ if x \(\neq\) 0 }\\ 0 ,& \quad \text{if $x$ =0}\\ \end{array} \right. $ is a continuous function?

asked Nov 10, 2012 by thanvigandhi_1

1 answer

Find all points of discontinuity of \(f\), where $ f(x) = \left\{ \begin{array} {1 1} \large\frac {sinx} {x},\normalsize & \quad\text{ if \(x\) < 0 }\\ x + 1 ,& \quad \text{if $x$ \( \geq 0\)}\\ \end{array} \right. $

asked Nov 10, 2012 by thanvigandhi_1

1 answer

For what value of \( \lambda \) is the function defined by \(f\) defined by $ f(x) = \left\{ \begin{array} {1 1} \lambda(x^2 - 2x) ,& \quad\text{ if $ x $ \(\leq 0\)}\\ 4x + 1,& \quad \text{if $x$ > 0}\\ \end{array} \right. $ Continuous at \(x = 0\)?

asked Nov 10, 2012 by thanvigandhi_1

1 answer

Find the relationship between \(a\) and \(b\) so that the function \(f\) defined by $ f(x) = \left\{ \begin{array} {1 1} ax + 1 ,& \quad\text{ if $ x $ \(\leq 3\)}\\ bx + 3,& \quad \text{if $x$ > 3}\\ \end{array} \right. $ is continuous at \(x = 3\).

asked Nov 10, 2012 by thanvigandhi_1

1 answer

State with reason whether following functions have inverse: (iii) \(h: \{2,3,4,5,\} \to \{7,9,11,13\}\) with \(h=\{(2,7),(3,9),(4,11),(5,13) \} \)

asked Nov 9, 2012 by vaishali.a

1 answer

Find \( gof\) and \(fog\), if (i) \( f(x) = |\;x\;| \, and \, g(x) = |\;5x-2\;| \)

asked Nov 9, 2012 by vaishali.a

1 answer

Let \(f,\, g\, and\, h\) be functions from \(R\, to\, R.\) Show that \[ (f+g) oh = foh + goh\]

asked Nov 9, 2012 by vaishali.a

1 answer

Let $f:\{1,3,4\} \to \{1,2,5\} $ and $g:\{1,2,5\}\to\{1,3\}$ be given by $f = \{(1, 2), (3, 5), (4, 1)\}$ and $ g = \{(1, 3), (2, 3), (5, 1)\}$. Write down $gof$.

asked Nov 9, 2012 by vaishali.a

1 answer

Let $f : R \to R$ be defined as $f (x) = 3x$. Choose the correct answer:

asked Nov 9, 2012 by vaishali.a

1 answer

Let $f: R \to R $, be defined as $f(x) = x^4$. Choose the correct answer.

asked Nov 9, 2012 by vaishali.a

1 answer

Discuss the continuity of the function $f$, where $f$ is defined by \[ f(x) = \left\{ \begin{array} {1 1} 3 ,& \quad\text{ if $ 0 $ \(\leq x\) \(\leq1\)}\\ 4,& \quad\text{if $ 1 $ < \(x\) < 3 }\\ 5 ,& \quad\text{ if $ 3 $ \(\leq x\) \(\leq10\)}\\ \end{array} \right. \]

asked Nov 9, 2012 by thanvigandhi_1

1 answer

Find all points of discontinuity of $f$, where $f$ is defined by $ f(x) = \left\{ \begin{array} {1 1} x^3-3 ,& \quad\text{ if $ x $ \(\leq 2\)}\\ x^2+1,& \quad \text{if $x$ > 2}\\ \end{array} \right. $

asked Nov 9, 2012 by thanvigandhi_1

1 answer

Find all points of discontinuity of $f$, where $f$ is defined by $f(x) = \left\{ \begin{array} {1 1} x + 1,& \quad\text{ if $ x $ \(\geq 1\)}\\ x^2+1,& \quad \text{if $x$ < 1}\\ \end{array} \right. $

asked Nov 9, 2012 by thanvigandhi_1

1 answer

Find all points of discontinuity of $f$, where $f$ is defined by $ f(x) = \left\{ \begin{array} {1 1} \large \frac{ |\;x\;| }x,& \quad\text{ if x \(\neq\) 0 }\\ 0 ,& \quad \text{if $x$ =0}\\ \end{array} \right. $

asked Nov 9, 2012 by thanvigandhi_1

1 answer

Prove that the greatest integer function defined by $ f(x) = [x], 0<x<3$ is not differentiable at $x = 1 $ and $ x = 2 $

asked Nov 8, 2012 by thanvigandhi_1

1 answer

Differentiate the functions with respect to x in $\;\cos(\sqrt {x} \;) $

asked Nov 8, 2012 by thanvigandhi_1

1 answer

Differentiate the functions with respect to x: $\;2\sqrt {cot (x^2)}$

asked Nov 8, 2012 by thanvigandhi_1

1 answer

In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.(i) \(f : R\; \to R\; defined \; by \; f(x)\; =\; 3-4x\)

asked Nov 8, 2012 by vaishali.a

1 answer

Differentiate the functions with respect to $x: \large \frac{ sin\; (ax + b) } { cos \;( cx + d)} $

asked Nov 8, 2012 by thanvigandhi_1

1 answer

Let \(A=\{1,2,3\},\; B = \{4,5,6,7\}\) and let \(f = \{(1,4),(2,5),(3,6)\}\)be a function from \(A\) to \(B\). Show that \(f\) is one-one.

asked Nov 8, 2012 by vaishali.a

1 answer

Show that the Signum Function \(f : R \to R\), given by \[ f(x) = \left\{ \begin{array}{l l} 1, & if \; x \; > 0 \\ 0, & if\; x\;= 0 \text{ is neither one-one nor onto } \\ -1, & if\; x \;< 0 \end{array} \right. \]

asked Nov 8, 2012 by vaishali.a

1 answer

Prove that the Greatest Integer Function \(f : R \to R\), given by \(f (x) = [x]\), is neither one-one nor onto, where \([x]\) denotes the greatest integer less than or equal to \(x\).

asked Nov 8, 2012 by vaishali.a

1 answer

Check the injectivity and surjectivity of the function: $f : N\to N\; given\; by\; f(x)\; = x^2 $

asked Nov 8, 2012 by vaishali.a

1 answer

Show that the function \(f : R_* \rightarrow R_* \) defined by \( f(x) = (\frac{1} {x})\) is one-one and onto, where \(R_* \) is the set of all non-zero real numbers. Is the result true, if the domain \(R_* \) is replaced by \(N\) with co-domain being same as \(R_*\)?

asked Nov 8, 2012 by vaishali.a

1 answer

Let $R$ be the relation in the set $N$ given by $R=\{(a,b): a=b-2, b>6\}$. Choose the correct answer:

asked Nov 8, 2012 by vaishali.a

1 answer

Let $R$ be the relation in the set $\{1, 2, 3, 4\}$ given by $R = \{(1, 2), (2, 2), (1, 1), (4,4), (1, 3), (3, 3), (3, 2)\}$. Choose the correct answer.

asked Nov 8, 2012 by vaishali.a

1 answer

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 \(y = 2x + 4.\)

asked Nov 8, 2012 by vaishali.a

1 answer

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