Answers posted by meena.p

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

answered Feb 22, 2013

Given the relation $R=\{(a,b):a=b-2, \; b>6 \}$ in a set N, we can arrive at the right option by ...

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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.

answered Feb 22, 2013

Toolbox:A relation R in a set A is a equivalance relation if it is symmetric, reflexive and transiti...

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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 $y = 2x + 4.$

answered Feb 22, 2013

Toolbox:A relation R in a set A is a equivalance relation if it is symmetric, reflexive and transiti...

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Show that the relation $R$ defined in the set $A$ of all polygons as $R = \{(P_1, P_2) :\: P_1\ and \;P_2$ have same number of sides$\}$, is an equivalence relation. What is the set of all elements in $A$ related to the right angle triangle $T$ with sides $3, 4$ and $5$?

answered Feb 22, 2013

Toolbox:A relation R in a set A is a equivalance relation if it is symmetric, reflexive and transiti...

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Show that the relation $R$ defined in the set $A$ of all triangles as $R = \{(T_1, T_2) : T_1\,is\, similar\, to\, T_2\}$, is equivalence relation. Consider three right angle triangles $(T_1 \;$ with sides $3, 4, 5, \; T_2$ with sides $5, 12, 13$, and $T_3$, with sides $6, 8, 10$) Which triangles among $T_1\, T_2\, and\, T_3$ are related?

answered Feb 22, 2013

Toolbox:A relation R in a set A is a equivalance relation if it is symmetric, reflexive and transiti...

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The relation $R$ in the set $A$ of points in a plane given by $R = \{(P, Q)$ : distance of the point $P$ from the origin is same as the distance of the point $Q$ from the origin $\}$, is

answered Feb 22, 2013

Toolbox:R is an equivalance relation if it is reflexive, symmetric and transitive.A relation R in a ...

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Show that the relation $R$ in the set $A= \{1,2,3,4,5\} $given by $R = \{(a, b) : |a – b| \,is\, even \} $, is an equivalence relation. Show that all the elements of $\{1, 3, 5\} $ are related to each other and all the elements of $\{2, 4\}$ are related to each other. But no element of $ \{1, 3, 5 \} $ is related to any element of $\{2, 4\}$.

answered Feb 21, 2013

Toolbox:A relation R in a set A is an equivalence relation if R is reflexive, symmetric and transiti...

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Show that the relation $R$ in the set $A$ of all the books in a library of a college, given by ( $R = \{ (x, y) : x\;$ and $\; y\;$ have same number of pages$\}$ ) is an equivalence relation.

answered Feb 21, 2013

Toolbox:A relation R in a set A in an equivalance relation. if R is reflexive, symmetric and transit...

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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.

answered Feb 21, 2013

Toolbox:A relation R in a set A is called reflexive. if $(a,a) \in R\;for\; all\; a\in A$A relation ...

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Show that the relation $R$ in $R$ defined as $R = {(a, b) : a \leq b} $, is reflexive and  transitive but not symmetric.

answered Feb 20, 2013

Toolbox: A relation R in a set A is called reflexive. if $(a,a) \in R\;for\; all\; a\in A$A relat...

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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.

answered Feb 20, 2013

Toolbox: A relation R in a set A is called reflexive. if $(a,a) \in R\;for\; all\; a\in A$A relat...

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Show that the relation $R$ in the set $R$ of real numbers, defined as $(R) =\{(a, b: (a \leq b^2 )\}$ is neither reflexive nor symmetric nor transitive.

answered Feb 20, 2013

Toolbox: A relation R in a set A is called reflexive. if $(a,a) \in R\;for\; all\; a\in A$A relat...

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Evaluate the definite integrals: $\int\limits_1^4 [\; |x-1|+|x-2|+|x-3|\;]dx$

answered Feb 19, 2013

Toolbox: $\int |x+a| dx$ where $-(x+a) \geq 0\; for\; -a \leq x \leq 0\qquad (x+a) \leq 0\...

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Evaluate the definite integrals\[\int\limits_0^\frac{\pi}{2}\sin 2x\tan^{-1}(sin x)\;dx\]

answered Feb 19, 2013

Toolbox: (i)$\int \limits_a^b f(x)dx=F(b)-F(a)$ (ii) In a integral function, $\int f...

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Prove the following\[\int\limits_0^1x\;e^x\;dx=1\]

answered Feb 19, 2013

Toolbox: (i)$\int \limits_a^bf(x)dx=F(b)-F(a)$ (ii)$ \int udv=uv-\int vdu$ (ii...

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Evaluate the definite integrals\[\int\limits_0^\frac{\pi}{4}\frac{\sin x+\cos x}{9+16\sin 2x}\;dx\]

answered Feb 19, 2013

Toolbox: (i)Let $f(x)=t,\;then \;f'(x)=dt \;Therefore\; \int f(x)dx=\int t.dt$ (ii)$...

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Evaluate the definite integrals\[\int\limits_0^1\frac{dx}{\sqrt{1+x}-\sqrt x}\]

answered Feb 19, 2013

Toolbox: (i)$\int \limits_a^bf(x)dx=F(b)-F(a)+c$ $\int x^n dx=[\frac{x^{n+1}}{n+1}]+...

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Evaluate the definite integrals\[\int\limits_\frac{\pi}{6}^\frac{\pi}{3}\frac{\sin x+\cos x}{\sqrt{\sin2x}}dx\]

answered Feb 19, 2013

Toolbox: (i)when f(x) is an integral function is substitute on t,then $f'(x)dx=dt.Hence\;\int f(x...

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Evaluate the definite integrals\[\int\limits_0^\frac{\pi}{2}\frac{\cos^2xdx}{\cos^2x+4\sin^2x}\]

answered Feb 19, 2013

Toolbox: (i)when f(x) is an integral function is substitute on t,then $f'(x)dx=dt.Hence\;\...

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Evaluate the definite integrals\[\int\limits_0^\frac{\pi}{4}\frac{\sin x\cos x}{\cos^4x+\sin^4x}dx\]

answered Feb 19, 2013

Toolbox: (i)f(x) is an integral function and we substitute f(x) for t, then $f'(x)dx=dt,$ ...

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Evaluate the definite integrals\[\int\limits_\frac{\pi}{2}^\pi e^x\bigg(\frac{1-\sin x}{1-\cos x}\bigg)dx\]

answered Feb 18, 2013

Toolbox: (i)$\int e^x[f(x)+f'(x)]dx=e^x f(x)$ (ii)$\sin^2x+\cos^2x=1$ (iii)$\s...

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Integrate the function\[\int\frac{\sqrt{x^2+1}[log(x^2+1)-2logx]}{x^4}\]

answered Feb 18, 2013

Toolbox: (i)If an integral function $f(x)=t,$ then $f'(x)dx=dt$ hence the integral functio...

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Integrate the function\[\int\tan^{-1}\sqrt{\frac{1-x}{1+x}}\]

answered Feb 18, 2013

Toolbox: (i)If an integral function $f(x)=t,$ then $f'(x)dx=dt$ hence the integral functio...

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Integrate the function\[\int\sqrt{\frac{1-\sqrt x}{1+\sqrt x}}\]

answered Feb 18, 2013

Toolbox: (i)$\sin 2\theta=2\sin \theta \cos \theta$ (ii)$\int \sin x dx= -\cos x+c$ ...

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Integrate the function\[\int f'(ax+b)[f(ax+b)]^n\]

answered Feb 17, 2013

Toolbox: If an integral function $f(x)=t,$ then $f'(x)dx=dt$ then $\int f(x)dx=\int tdt$ ...

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Integrate the function $\int\cos^3x\;e^{log\sin x}dx$

answered Feb 17, 2013

Toolbox: (i)$ e^{logx}=x$ (ii)$\int \frac{x^n}{1}=\frac{x^{n+1}}{n+1}$ (iii) I...

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Integrate the function\[\int\frac{1}{(x^2+1)(x^2+4)}\]

answered Feb 17, 2013

Toolbox: (i)An rational function is of the form. $\frac{1}{(x^2+a)(x^2+b)}$ can be written...

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Integrate the function $\begin{align*}\int\frac{x^3}{\sqrt{1-x^8}} \end{align*}$

answered Feb 17, 2013

Toolbox: (i) If in a function $\int f(x)dx,\;let\;f(x)=t,\;then\;f'(x)=dt,\;then \;\int f(...

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Integrate the function\[\int\frac{1}{\cos(x+a)\cos(x+b)}\]

answered Feb 17, 2013

Toolbox: (i)$\sin (A+B)=\sin A\cos B+\cos A\sin B$ (ii)$\cos (A+B)=\cos A\cos B-\sin...

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Integrate the function\[\int\frac{\sin^8x-\cos^8x}{1-2\sin^2x\cos^2x}\]

answered Feb 16, 2013

Toolbox: (i)$\sin ax=-\frac{1}{a}\cos ax+c$ (ii)$a^2-b^2=(a+b)(a-b)$ (iii)$\si...

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Integrate the function $\begin{align*}\int\frac{\cos x}{\sqrt{4-\sin^2x}} \end{align*}$

answered Feb 16, 2013

Toolbox: (i) if given $I=\int f(x)dx,$ let $f(x)=t,$ then $f'(x)dx=dt$,hence $\int f(x)dx=...

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Integrate the function $\begin{align*}\frac{e^{5logx}-e^{4logx}}{e^{3logx}-e^{2logx}}\end{align*}$

answered Feb 15, 2013

Toolbox: (i)$e^{log x}=x$ (ii)$alogx=logx^a$ (iii)$\int x^ndx=\frac{x^{n+1}}{n+1}+c$ Give...

0 votes

Integrate the function\[\frac{5x}{(x+1)(x^2+9)}\]

answered Feb 15, 2013

Toolbox: If the function is of a rational form $\frac{1}{(x+a)(x^2+b)}$ then it can be res...

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Integrate the function\[\frac{1}{x^\frac{1}{2}+x^\frac{1}{3}}\qquad[Hint:\frac{1}{x^\frac{1}{2}+x^\frac{1}{3}}=\frac{1}{x^\frac{1}{3}\bigg(1+x^\frac{1}{6}\bigg)},put\;x=t^6]\]

answered Feb 15, 2013

Toolbox: (i)If f(x)=t, then f'(x)=dt.then if $I=\int f(x)dx,$it can be written as $\int t....

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Integrate the function\[\int\frac{1}{x^2(x^4+1)^\frac{3}{4}}\]

answered Feb 15, 2013

Toolbox: (i)If f(x)=t, then f'(x)=dt.Henc $int f(x)dx=\int t.dt$ (ii)$\int x^n dx=\f...

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Integrate the function\[\frac{1}{x\sqrt{ax-x^2}}[Hint:Put\;x=\frac{a}{t}]\]

answered Feb 15, 2013

Toolbox: (i)If f(x)=t, then f'(x)=dt.Henc $int f(x)dx=\int t.dt$ (ii)$\int x^n dx=\f...

0 votes

Integrate the function\[\int\frac{1}{x-x^3}\]

answered Feb 15, 2013

Toolbox: (i)If the integral function is of the form $\int\frac{dx}{(x-a)(x-b)}$ then the f...

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Integrate the function\[\frac{1}{\sqrt{x+a}+\sqrt{x+b}}\]

answered Feb 15, 2013

Toolbox: $=\int \sqrt{x+a}=\frac{(x+a)^{3/2}}{3/2}=\frac{2}{3}(x+a)^{3/2}$ Given:$...

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Choose the correct answer in the value of $\displaystyle\int\limits_\frac{\Large -\pi}{\Large 2}^\frac{\Large \pi}{\Large 2}(x^3+x\cos x+\tan^5x+1)dx$ is

answered Feb 14, 2013

Toolbox: (i)$\int \limits_a^bf(x)dx=F(b)-F(a)$ (ii)If $f(-x)=-f(x)$ it is an odd fun...

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Choose the correct answer in the value of $\int\limits_0^\frac{\pi}{2}log\bigg(\frac{\large 4+3\sin x}{\large 4+3\cos x}\bigg)dx$ is

answered Feb 14, 2013

Toolbox: (i)$\int \limits_a^bf(x)dx=F(b)-F(a)$ (ii)$\int \limits_0^a f(x)dx=\int \limits_0^a f...

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Show that $\int\limits_0^af(x)g(x)dx=2\int\limits_0^af(x)dx,$ if $f$ and $g$ are defined as $f(x) = f(a - x)$ and $g(x) + g(a - x)=4$

answered Feb 14, 2013

Toolbox: (i)$\int \limits_a^bf(x)dx=F(b)-F(a)$ (ii)$\int \limits_0^a f(x)dx=\int \li...

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By using the properties of definite integrals,evaluate the integral\[\int\limits_0^\pi\; log(1+\cos x)\;dx\]

answered Feb 14, 2013

Toolbox: (i)$\int f(x)dx=F(b)-F(a)$ (ii)$\int \limits_0^a f(x)dx=\int \limits_0^a f(...

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By using the properties of definite integrals,evaluate the integral $\begin{align*}\int\limits_0^4\mid x-1\mid dx \end{align*}$

answered Feb 14, 2013

Toolbox: (i)$\int\limits_a^b f(x)dx=F(b)-F(a)$ (ii)$\int\limits_0^a f(x)dx=\int \lim...

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By using the properties of definite integrals,evaluate the integral $\begin{align*}\int\limits_0^a\frac{\sqrt x}{\sqrt x+\sqrt{a-x}}dx \end{align*}$

answered Feb 14, 2013

Toolbox: (i)$\int\limits_a^b f(x)dx=F(b)-F(a)$ $\int\limits_0^a f(x)dx=\int \limits_...

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By using the properties of definite integrals,evaluate the integral $\begin{align*}\int\limits_0^\frac{\Large \pi}{\Large 2}\frac{\sin x-\cos x}{1+\sin x\cos x}\;dx \end{align*}$

answered Feb 14, 2013

Toolbox: (i)$\int\limits_a^b f(x)dx=F(b)-F(a)$ $\int\limits_0^a f(x)dx=\int \limits_...

0 votes

By using the properties of definite integrals,evaluate the integral $\begin{align*}\int\limits_0^{2\pi}\cos^5x\;dx \end{align*}$

answered Feb 14, 2013

Toolbox: (i)$\int\limits_a^b f(x)dx=F(b)-F(a)$ $\int \limits_0^{2a}f(x)dx=2\int f(x)...

0 votes

By using the properties of definite integrals,evaluate the integral $\begin{align*}\int\limits_\frac{\Large -\pi}{\Large 2}^\frac{\Large \pi}{\Large 2}\sin^7\;x\;dx \end{align*}$

answered Feb 14, 2013

Toolbox: (i)$\int\limits_a^b f(x)dx=F(b)-F(a)$ If $ f(-x)=-f(x)$ then the function i...

0 votes

By using the properties of definite integrals,evaluate the integral\[\int\limits_0^\pi\frac{x\;dx}{1+\sin x}\]

answered Feb 14, 2013

Toolbox: (i)$\int\limits_a^b f(x)dx=F(b)-F(a)$ (ii)$\int \limits_0^af(x)dx=\int \lim...

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By using the properties of definite integrals,evaluate the integral $\begin{align*}\int\limits_\frac{\Large -\pi}{\Large 2}^\frac{\Large \pi}{\Large 2}\sin^2\;x\;dx \end{align*}$

answered Feb 14, 2013

Toolbox: (i)$\int\limits_a^b f(x)dx=F(b)-F(a)$ (ii)$ If \;f(-x)=-f(x)$ then it is an...

0 votes

By using the properties of definite integrals,evaluate the integral $\begin{align*}\int\limits_0^2x\sqrt{2-x}dx \end{align*}$

answered Feb 14, 2013

Toolbox: (i)$\int \limits_a^b f(x)dx=F(b)-F(a)$ (ii)$\int \limits_0^af(x)dx=\int \li...

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