Answers posted by meena.p

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Show that the differential equation $[xsin^2\big(\frac{y}{x}\big)-y]dx+xdy=0.$ is homogeneous.Find the particular solution of this differential equation ,given that $y=\frac{\pi}{4}$ when x=1.

answered Mar 24, 2013

$[x \sin ^2(\frac{y}{x})-y]dx+xdy=0$$xdy=-(x \sin ^2(y/x)-y)dx$=>$\large\frac{dy}{dx}=-\frac{(x \...

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If $x^y=e^{x-y}$,prove that $\Large \frac{dy}{dx}=\frac{log x}{(1+log x)^2}$

answered Mar 23, 2013

$x^y=e^{x-y}$Take log on both sides$y\; log x=(x-y) log\; e \qquad (But\; log\;e=1)$=>$y\; log\; ...

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Evaluate :$\int\limits_0^{\pi}\large\frac{xsin x}{1+cos^2x}dx.$

answered Mar 22, 2013

$I=\large\int \limits_0^{\pi} \frac{x \sin x}{1+\cos ^2 x}$----(1) using the properties of inte...

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Evaluate :$\int \large \frac{dx}{x(x^3+8)}$

answered Mar 22, 2013

$I=\int \large\frac{dx}{x(x^3+8)}$Mutiply and divide by $x^2$$I=\large\frac {x^2dx}{x^3(x^3+8)}$Let ...

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A and B are two points with position vectors $2\overrightarrow{a}-3\overrightarrow{b}$ and $6\overrightarrow{b}-\overrightarrow{a}$ respectively.Write the position vector of a point P which divides the line segment AB internally in the ratio 1 : 2.

answered Mar 22, 2013

$\overrightarrow{OA}=2\overrightarrow{a}-3\overrightarrow{b}$ $\overrightarrow{OB}=6\overri...

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If Matrix A = $\begin{bmatrix} 2 &-2 \\ -2 & 2 \end{bmatrix}$, and $A^2=pA$, then write the value of $p$.

answered Mar 22, 2013

$A=\begin{bmatrix} 2 & -2 \\ -2 & 2 \end{bmatrix}$$A^2=\begin{bmatrix} 4 & -4 \\ -4 &...

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Find the area of the region bounded by the parabola $y=x^2$ and y=| x |.

answered Mar 21, 2013

Given $y=x^2\; and\; y=x^2$ $x^2=y$ represents a parabola with vertex (0,0),open upwards, in th...

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Find the equations of tangents to the curve $3x^2-y^2=8$,which pass through the point$\big(\frac{4}{3},0\big)$. (Note: This question has been split into 2 questions)

answered Mar 21, 2013

Given $y=x^2\; and\; y=x^2$ $x^2=y$ represents a parabola with vertex (0,0),open upwards, in th...

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The probabilities of two students A and B coming to the school in time are $\frac{3}{7}$ and $\frac{5}{7}$ respectively.Assuming that the events,'A coming in time' and 'B coming in time' are independent ,find the probability of only one of them coming to the school in time.Write at least one advantage of coming to school in time.

answered Mar 21, 2013

Let $P(A)=\large \frac{3}{7}$ (ie) Probability of A coming in time. Therefore $P(A^1)=1-\la...

-1 votes

Find the vector equation of the plane through the points (2,1,-1) and (-1,3,4) and perpendicular to the plane x-2y+4z=10.

answered Mar 21, 2013

$\textbf{Step 1}$:The required plane passes through: $P(2,1,-1)$ and $Q(-1,3,4)$.Then we can write t...

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Show that the lines $\begin{array}{1 1}\overrightarrow{r}=3\hat{i}+2\hat{j}-4\hat{k}+\lambda(\hat{i}+2\hat{j}+2\hat{k});\\\overrightarrow{r}=5\hat{i}-2\hat{j}+\mu(3\hat{i}+2\hat{j}+6\hat{k})\end{array} $ are intersecting. Hence find their point of intersection.

answered Mar 21, 2013

If the lines are intersecting then$(a_2^2-a_1).(b_1 \times b_2)=0$$a_1=3\overrightarrow{i}+2\overrig...

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If $\overrightarrow{a}=\hat{i}-\hat{j}+7\hat{k}$ and $\overrightarrow{b}=5\hat{i}-\hat{j}+\lambda\hat{k}$,then find the value of $\lambda$,so that $\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{a}-\overrightarrow{b}$ are perpendicular vectors.

answered Mar 21, 2013

Toolbox:Since $\overrightarrow{a}+\overrightarrow{b}\; and\; \overrightarrow{a}-\overrightarrow{b}$ ...

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The management committee of a residential colony decided to award some of its members (say x) for honesty,some (say y) for helping others and some others (say z) for supervising the workers to keep the colony neat and clean . The sum of all the awardees is 12. Three times the sum of awardees for cooperation and supervision added in two times the number of awardees for honesty is 33. If the sum of the number of awardees for honesty and supervision is twice the number of awardees for helping others, using matrix method, find the number of awardees of each category. Apart from these values, namely, honesty, cooperation and supervision, suggest one more value which the management of the colony must include for awards.

answered Mar 21, 2013

$\textbf{Step 1}$: Given that $x$ = # of members awarded for honesty, $y$ = # of members awarde...

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In a hockey match,both teams A and B scored same number of goals up to the end of the game,so to decide the winner ,the referee asked both the captains to throw a die alternately and decided that the team,whose captain gets a six first,will be declared the winner.If the captain of team A was asked to start,find their respective probabilities of winning the match and state whether the decision of the referee was fair or not.

answered Mar 21, 2013

Let the probability that A gets a six be $\frac{1}{6}$$\Rightarrow$ $p=\frac{1}{6}$ and hence $q=1-\...

0 votes

Find the vector equation of the line passing through the point (1,2,3) and parallel to the planes $\overrightarrow{r}.(\hat{i}-\hat{j}+2\hat{k})=5$ and $\overrightarrow{r}.(3\hat{i}+\hat{j}+\hat{k})=6.$

answered Mar 21, 2013

The required line planes through the point having its position vector $\overrightarrow{a}=\overright...

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Find the equation of the plane passing through the line of intersection of the planes $\overrightarrow{r}.(\hat{i}+3\hat{j})-6=0$ and $\overrightarrow{r}.(3\hat{i}-\hat{j}-4\hat{k})=0$,whose perpendicular distance from origin is unity.

answered Mar 21, 2013

Let the plane$\overrightarrow{r}.\overrightarrow{n_1}=d_1$ be $\overrightarrow{r}.(\overrightarrow{i...

0 votes

Find the particular solution of the differential equation $(\tan^{-1}y-x)dy=(1+y^2)dx$,given that when x=0,y=0.

answered Mar 21, 2013

Toolbox: A first order differential equation is an equation of the form $\frac{dy}{dx}+Px=...

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In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.(ii) $f : R\; \to R\; defined \; by \; f(x)\; =\; 1+x^2$

answered Mar 18, 2013

Toolbox: A function $f: X \rightarrow Y$ where for every $x1, x2 \in X, f(x1) = f(x2) \Rig...

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Evaluate:\[\int \limits_0^{\pi /2} \frac{1}{\cos (x-\pi/3) \cos (x-\pi/6)}\]

answered Mar 15, 2013

Toolbox: (i)$\int \limits_a^b f(x)dx=F(b)-F(a)$ $ \sin (A - B)=\sin A \cos B-\cos A ...

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Evaluate:\[\int_{-1}^1 log \bigg(\large\frac{4-x}{4+x}\bigg)dx\]

answered Mar 15, 2013

Toolbox: (i)$\int \limits_a^b f(x)dx=F(b)-F(a)$ (ii)$ \int _{-a}^a f(x) dx =0 $ if f...

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Evaluate:\[\int\limits_0^{16} \frac{x^{1/4}}{1+x^{1/2}}dx\]

answered Mar 15, 2013

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

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Evaluate:\[\int\limits_0^{\pi/2} \frac {\sqrt {\tan x}}{1+\sqrt {\tan x}}dx\]

answered Mar 15, 2013

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

0 votes

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

answered Mar 15, 2013

Toolbox: (i)$\int \limits_a^b f(x)dx=F(b)-F(a)$ (ii) $ 1-\cos 2x =2 \sin ^2 x$ ...

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Evaluate:\[\int_{-\pi}^\pi (\sin ^{-1}x+x^{295})\]

answered Mar 14, 2013

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

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Evaluate:\[\int \limits _0 ^ \pi \Large\frac{e^{\cos x}}{[e^{\cos x}+e^{-\cos x}]}dx\]

answered Mar 14, 2013

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

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Evaluate:\[\int \limits _{\pi/6}^{\pi/3} \frac{dx}{1+\sqrt {\tan x}}\]

answered Mar 14, 2013

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

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Choose the correct answers in the value of $\int\limits_0^1\tan^{-1}\bigg(\frac{\large 2x-1}{\large 1+x-x^2}\bigg)dx\;is$\[(A)\;1\qquad(B)\;0\qquad(C)\;-1\qquad(D)\;\frac{\pi}{4}\]

answered Mar 14, 2013

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

0 votes

Prove the following\[\int\limits_0^1\sin^{-1}x\;dx=\frac{\pi}{2}-1\]

answered Mar 14, 2013

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

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Prove the following\[\int\limits_0^\frac{\pi}{4}2\tan^3x\;dx=1-log2\]

answered Mar 14, 2013

Toolbox: (i)$\int \limits _a^b f(x)dx=F(b)-F(a)$ (ii) $ \tan ^2 x=\sec ^2 x-1$ ...

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

answered Mar 14, 2013

Toolbox: A function $f: A \rightarrow B$ where for every $x1, x2 \in X, f(x1) = f(x2) \Rightarrow...

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Choose the correct answers in $\int\frac{\large \cos 2x}{\large (\sin x+\cos x)^2}dx$ is equal to

answered Mar 13, 2013

Toolbox: (i)In a function f(x) is substituted by t, then $f'(x)dx=dt$ then $\int f(x)dx=\i...

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Choose the correct answers in $\int\frac{dx}{e^x+e^{-x}}$ is equal to

answered Mar 13, 2013

Toolbox: (i)In a function $ \int f(x)dx$ if f(x)=t, then $f'(x)dx=dt$ Therefore $\int f(x)...

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Prove the following\[\int\limits_0^\frac{\pi}{2}\sin^3x\;dx=\frac{2}{3}\]

answered Mar 13, 2013

Toolbox: $\int \limits _a^b f(x)dx=F(b)-F(a)$ (ii) $ \sin ^2 x=1-\cos ^2 x$ (i...

0 votes

Find $ gof$ and $fog$, if (ii) $ f(x) = 8x^3$ and $ g(x) = x ^ \frac { 1} {3} $

answered Mar 7, 2013

Toolbox:$(gof)(x) =g(f(x))$$(fog)(x)=f(g(x))$ $f(x)=8x^3 \;and\;g(x)=x^{1/3}$ $(gof) (x)=g(f(x...

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

answered Mar 7, 2013

Toolbox:Given two functions $f:A \to B $ and $g:B \to C$, then composition of $f$ and $g$, $gof:A \...

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Prove the following\[\int\limits_1^3\frac{dx}{x^2(x+1)}=\frac{2}{3}+log\frac{2}{3}\]

answered Mar 7, 2013

Toolbox: If the given integral function in a rational function of the form $\large\frac{A}...

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Choose the correct answers if $f(a+b-x) = f(x)$ , then $\int\limits_a^bx\;f(x)dx$ is equal to

answered Mar 6, 2013

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

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Prove the following\[\int\limits_{-1}^1x^{17}\cos^4x\;dx=0\]

answered Mar 6, 2013

Toolbox: (i) If $f(-x)=f(x)$ then it is an odd function (ii) $\int \limits_{-a}^a f(...

0 votes

True or False: A binary operation on a set has always the identity element.

answered Mar 6, 2013

Toolbox: An identify element 'e' is defined binary operation * meet A if $ e \in A$ and $a...

0 votes

True or False: Every function is invertible.

answered Mar 6, 2013

Toolbox: A function f is invertible if it is one-one and onto A function $f:N \to ...

0 votes

True or False: The composition of functions is associative

answered Mar 6, 2013

Toolbox: Let f(x) g(x) and h(x) be the three functions Then composition of functions...

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True or False: The composition of functions is commutative.

answered Mar 6, 2013

Toolbox: The composition of function fog is commutative if $fog=gof$ Let $f(x)=1+x...

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True or False: The relation R on the set A={1,2,3} defined as R={(1,1),(1,2),(2,1),(3,3)} is reflexive,symmetric and transitive.

answered Mar 6, 2013

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

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True or False: Let A={0,1} and N be the set of natural numbers.Then the mapping $f:N \rightarrow A$ defined by f(2n-1)=0,f(2n)=1,$\quad n\in N$,is onto.

answered Mar 6, 2013

Toolbox: A function $f: N \to A \qquad A=\{0,1\}$ is onto if the exists every element y an...

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True or False: An integer m is said to be related to another integer n if m is a integral multiple of n.This relation in Z is reflexive,symmetric and transitive.

answered Mar 6, 2013

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

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True or False: Every relation which is symmetric and transitive is also reflexive.

answered Mar 6, 2013

Toolbox: 1.For a given function R in A. A relation R in a set A is called $\mathbf{ ...

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True or False: Let $f:R \rightarrow R$ be the function defined by f(x)=sin (3x+2)$\quad x\in R$.Then f is invertible.

answered Mar 6, 2013

Toolbox: 1.A function f is invertible if f is one -one (ie) $ f(x)=f(y)=>x=y \qquad x \...

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True or False: Let R={(3,1),(1,3),(3,3)} be a relation defined on the set A={1,2,3}.Then R is symmetric,transitive but not reflexive.

answered Mar 6, 2013

Toolbox: 1. A relation R defined on A is reflexive if $(a,a) \in R \qquad a \in A$ 2...

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If $f(x)=(4-(x-7)^3\},then\;f^{-1}(x)$=____________.

answered Mar 6, 2013

Toolbox: we define $f^{-1}(x)$ as a function . such that $(fof^{-1})(x)=x$ ie $g=f^{...

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Let $f:R \rightarrow R$ be defined by $f(x)=\Large {\frac{x}{\sqrt{1+x^2}}}$.Then ( f o f o f )(x)=____________.

answered Mar 6, 2013

Toolbox: $(fofof)x=f(f(f(x)))$ $f:R \to R$ $f(x)=\frac {x}{\sqrt {1...

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